The description and use of the trianguler quadrant being a particular and general instrument, useful at land or sea, both for observation and operation : more universally useful, portable and convenient, than any other yet discovered, with its uses in arithmetick, geometry, superficial and solid, astronomy, dyalling, three wayes, gaging, navigation, in a method not before used / by John Brown, philomath.
         Brown, John, philomath.
      
       
         
           1671
        
      
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         A29761
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         61037
         
           
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             The description and use of the trianguler quadrant being a particular and general instrument, useful at land or sea, both for observation and operation : more universally useful, portable and convenient, than any other yet discovered, with its uses in arithmetick, geometry, superficial and solid, astronomy, dyalling, three wayes, gaging, navigation, in a method not before used / by John Brown, philomath.
             Brown, John, philomath.
          
           [16], 483, [13] p., [19] p. of plates : ill.
           
             Printed by John Darby, for John Wingfield, and are to be sold at his house ... and by John Brown ... and by John Sellers ...,
             London :
             1671.
          
           
             Reproduction of original in Huntington Library.
             Does not include 2d pt., "Horologiographia, or, The art of dyalling ..." Wing B5042.
             Errata: p. [16]
          
        
      
    
     
       
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         eng
      
       
         
           Quadrant -- Early works to 1800.
           Dialing -- Early works to 1800.
        
      
    
     
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           THE
           Description
           and
           Use
           OF
           THE
           TRIANGULER-QUADRANT
           :
           BEING
           A
           Particular
           and
           General
           Instrument
           ,
           useful
           at
           Land
           or
           Sea
           ;
           both
           for
           Observation
           and
           Operation
           .
           More
           Universally
           useful
           ,
           Portable
           and
           Convenient
           ,
           than
           any
           other
           yet
           discovered
           .
           With
           its
           Uses
           in
           
             
               
                 Arithmetick
                 ▪
              
            
             
               Geometry
               ,
               Superficial
               and
               Solid
               .
            
             
               Astronomy
               .
            
             
               Dyalling
               ,
               Three
               wayes
               .
            
             
               Gaging
               .
            
             
               Navigation
               .
            
          
           In
           a
           Method
           not
           before
           used
           .
        
         
           By
           
             Iohn
             Brown
          
           ,
           Philomath
           .
        
         
           London
           ,
           Printed
           by
           
             Iohn
             Darby
          
           ,
           for
           
             Iohn
             Wingfield
          
           ,
           and
           are
           to
           be
           sold
           at
           his
           house
           in
           Crutched-Fryers
           ;
           and
           by
           
             Iohn
             Brown
          
           at
           the
           Sphear
           and
           Sun-Dial
           in
           the
           Minories
           ;
           and
           by
           
             Iohn
             Sellers
          
           at
           the
           Hermitage-stairs
           in
           Wapping
           .
           1671.
           
        
      
       
         
         
         
           To
           the
           Reader
           .
        
         
           
             FRiendly
             Reader
          
           ,
           Thou
           hast
           once
           more
           presented
           to
           thy
           view
           ,
           a
           further
           Improvement
           and
           use
           of
           the
           Sector
           ,
           under
           the
           name
           of
           the
           
             Trianguler
             Quadrant
          
           ,
           so
           called
           from
           the
           shape
           thereof
           .
        
         
           In
           the
           year
           1660
           ,
           it
           was
           my
           lot
           ,
           first
           ,
           to
           apply
           and
           improve
           this
           former
           Contrivance
           of
           Mr.
           
             Samuel
             Foster
          
           on
           a
           Quadrant
           ,
           to
           a
           joynt
           Rule
           or
           Sector
           ;
           and
           did
           ,
           in
           1661
           ,
           publish
           my
           present
           Thoughts
           thereof
           ,
           in
           a
           small
           Discourse
           ,
           under
           the
           name
           of
           the
           
             Ioynt
             Rule
          
           .
        
         
           Since
           then
           ,
           through
           my
           perswasions
           ,
           and
           assistance
           ,
           another
           Piece
           was
           published
           1667
           ,
           by
           
             I.
             T.
          
           under
           the
           name
           of
           the
           
             Semi-Circle
             on
             a
             Sector
          
           :
           But
           neither
           of
           these
           ,
           that
           is
           to
           say
           ,
           neither
           my
           own
           nor
           his
           ,
           spoke
           what
           
           I
           would
           have
           it
           speak
           ;
           neither
           have
           I
           hopes
           ever
           to
           produce
           a
           Discourse
           either
           for
           method
           or
           matter
           ,
           worthy
           or
           becoming
           so
           excellent
           ,
           universal
           ,
           and
           useful
           an
           Instrument
           ,
           for
           the
           most
           Mathematical
           Occasions
           ,
           being
           for
           acurateness
           ,
           conveniency
           ,
           cheapness
           ,
           and
           universality
           ,
           before
           all
           others
           .
           For
           ,
        
         
           1.
           
           If
           it
           is
           made
           of
           Wood
           ,
           if
           the
           Wood
           keep
           but
           streight
           ,
           it
           is
           as
           true
           to
           be
           made
           use
           of
           as
           of
           Metal
           .
        
         
           2.
           
           It
           may
           be
           made
           of
           any
           Radius
           or
           bigness
           ,
           and
           yet
           in
           little
           Room
           in
           comparison
           of
           other
           Quadrants
           .
        
         
           3.
           
           More
           convenient
           to
           use
           whe●
           large
           ,
           than
           other
           Quadrants
           .
        
         
           4.
           
           As
           to
           the
           Projection
           for
           〈◊〉
           and
           Azimuth
           ,
           particularly
           using
           on●ly
           two
           Lines
           of
           Natural
           Sines
           ,
           th●
           Thred
           and
           Compasses
           for
           those
           tw●
           difficult
           (
           and
           many
           more
           easie
           )
           Pro●positions
           .
        
         
           5.
           
           The
           neat
           Conveniency
           of
           greater
           and
           a
           less
           Radius
           ,
           doubl●
           
           treble
           ,
           or
           quadruple
           one
           to
           another
           .
        
         
           6.
           
           The
           convenient
           Contrivance
           that
           happens
           to
           it
           ,
           of
           three
           Instruments
           in
           one
           ,
           viz.
           A
           
             Sector
             ,
             Quadrant
          
           ,
           and
           Gunter's
           Rule
           ;
           all
           three
           conveniently
           in
           one
           .
        
         
           The
           consideration
           of
           these
           things
           ,
           and
           the
           love
           and
           willingness
           I
           alwayes
           had
           ,
           to
           the
           communicating
           of
           them
           to
           others
           ,
           hath
           put
           me
           on
           this
           hard
           task
           of
           writing
           this
           Collection
           of
           the
           use
           thereof
           .
        
         
           Wherein
           I
           do
           most
           heartily
           beg
           thy
           Pardon
           and
           Acceptance
           ,
           to
           accept
           in
           good
           part
           the
           willing
           endeavours
           of
           my
           poor
           Ability
           ,
           which
           I
           doubt
           not
           but
           to
           have
           from
           most
           that
           know
           me
           ;
           For
           ,
           first
           ,
           my
           insufficiency
           ●n
           the
           Tongues
           ,
           Arts
           ,
           and
           Sciences
           :
           Secondly
           ,
           my
           Meanness
           and
           Poverty
           in
           the
           World
           ,
           for
           these
           Imployments
           ,
           which
           take
           up
           so
           much
           of
           a
           mans
           time
           ,
           and
           ability
           ,
           to
           perform
           them
           to
           purpose
           ,
           my
           plead
           my
           excuse
           ;
           for
           first
           ,
           here
           is
           the
           Product
           
           of
           more
           than
           Two
           years
           Improvement
           of
           more
           than
           vacant
           Hours
           ;
           with
           the
           great
           disadvantage
           of
           taking
           three
           Weeks
           at
           times
           ,
           to
           do
           that
           which
           three
           Dayes
           together
           might
           have
           as
           well
           ,
           if
           not
           much
           better
           ,
           performed
           ;
           And
           at
           last
           ,
           to
           cal●
           the
           Assistance
           of
           two
           others
           ,
           to
           un●dertake
           the
           Charge
           thereof
           ,
           to
           mid●wife
           it
           into
           the
           World.
           
        
         
           Thus
           ,
           as
           Widows
           Mites
           are
           accepted
           ,
           which
           are
           offered
           in
           sinceri●ty
           ;
           so
           I
           hope
           will
           mine
           ,
           though
           at●tended
           with
           much
           disorder
           ,
           as
           〈◊〉
           Method
           ;
           more
           uncouthness
           ,
           as
           〈◊〉
           Stile
           and
           Matter
           ;
           What
           it
           is
           ,
           it
           〈◊〉
           as
           at
           first
           Composing
           ,
           for
           I
           could
           ne●ver
           get
           Time
           nor
           Liberty
           ,
           from
           〈◊〉
           daily
           Trade
           and
           Calling
           ,
           to
           tran●scribe
           it
           twice
           .
        
         
           Yet
           was
           it
           not
           done
           at
           any
           tim●
           carelesly
           ,
           but
           with
           good
           will
           and
           free
           intent
           of
           plainness
           and
           useful●ness
           for
           the
           publick
           good
           of
           others
           as
           well
           as
           my
           own
           recreation
           〈◊〉
           delight
           .
        
         
         
           The
           Gunters
           Rule
           ,
           the
           Quadrant
           ,
           and
           Sector
           ,
           I
           need
           not
           commend
           ,
           they
           are
           so
           well
           known
           already
           ;
           but
           this
           I
           will
           add
           ,
           a
           better
           Contrivance
           and
           more
           general
           hath
           not
           yet
           to
           my
           knowledge
           been
           produced
           ;
           nor
           a
           Discourse
           where
           the
           use
           of
           all
           the
           Three
           together
           hath
           so
           been
           handled
           ,
           nor
           many
           more
           Examples
           ,
           though
           Mr.
           Windgate
           and
           Mr.
           Patridge
           have
           done
           sufficiently
           for
           the
           Gunters-Lines
           ,
           and
           Mr.
           Gunter
           for
           the
           Sector
           ,
           and
           Mr.
           Collins
           with
           the
           Quadrant
           ,
           and
           all
           of
           them
           distinctly
           far
           beyond
           this
           ;
           yet
           this
           Discourse
           of
           all
           the
           Three
           together
           ,
           may
           give
           content
           to
           some
           others
           ,
           as
           well
           as
           to
           me
           .
        
         
           The
           Discourse
           of
           Dialling
           ,
           is
           gathered
           from
           Mr.
           Wells
           ;
           and
           yet
           those
           that
           shall
           read
           Mr.
           Wells
           ,
           and
           this
           ,
           may
           often-times
           think
           otherwise
           ;
           for
           I
           assure
           you
           ,
           I
           saw
           not
           one
           leaf
           of
           his
           Book
           all
           the
           while
           it
           was
           doing
           ;
           but
           ,
           I
           hope
           ,
           it
           may
           please
           
           in
           moderate
           sort
           ,
           and
           ordinary
           capacity
           ,
           both
           for
           plainness
           ,
           convenience
           ,
           and
           variety
           .
        
         
           The
           cutting
           of
           the
           Regular
           Bodies
           ,
           I
           learned
           from
           Mr.
           
             Iohn
             Leake
          
           ,
           and
           the
           way
           is
           ready
           ,
           convenient
           ,
           and
           exact
           ,
           and
           worthy
           of
           remembrance
           .
        
         
           The
           Theorems
           ,
           from
           Mr.
           
             Thomas
             Diggs
          
           ,
           as
           in
           its
           due
           place
           ,
           is
           observed
           .
        
         
           The
           way
           of
           Measuring
           Superficies
           and
           Solids
           ,
           from
           Mr.
           Gunter
           :
           and
           my
           constant
           Experience
           in
           those
           Imployments
           ;
           and
           the
           Learner
           may
           here
           be
           supplied
           with
           what
           is
           often
           complained
           on
           ,
           viz.
           the
           Interpretation
           of
           Hard-words
           ,
           as
           much
           as
           I
           could
           call
           to
           mind
           ,
           or
           think
           to
           be
           convenient
           for
           that
           purpose
           .
        
         
           In
           the
           15th
           Chapter
           ,
           I
           have
           gathered
           many
           Cannons
           from
           Mr.
           Collins
           his
           Workes
           ,
           and
           applied
           them
           to
           the
           
             Trianguler
             Quadrant
          
           ;
           and
           been
           more
           large
           than
           needs
           in
           
           some
           places
           ,
           yet
           I
           hope
           to
           the
           content
           of
           some
           inquiring
           Persons
           .
        
         
           The
           business
           of
           Navigation
           ,
           I
           fear
           ,
           may
           prove
           most
           defective
           ;
           for
           my
           part
           ,
           I
           never
           yet
           saw
           Graves-end
           ,
           much
           less
           the
           Streights
           of
           Gibralter
           ;
           but
           for
           Observation
           and
           Operation
           ,
           the
           Instrument
           will
           do
           as
           well
           as
           any
           ,
           if
           well
           made
           and
           applied
           .
        
         
           So
           for
           the
           present
           ,
           I
           rest
           and
           remain
           ,
           ready
           to
           serve
           you
           in
           ,
           and
           supply
           defects
           by
           well
           making
           of
           these
           Instruments
           ,
           at
           the
           Sphear
           and
           Sun-Dial
           in
           the
           
             Great
             Minories
          
           .
        
         
           
             Iohn
             Browne
             .
          
        
      
       
         
         
           The
           Argument
           of
           the
           Book
           ,
           and
           the
           Authors
           Apologie
           .
        
         
           AT
           length
           my
           pains
           hath
           brought
           to
           pass
        
         
           the
           things
           I
           long
           intended
           ,
        
         
           And
           doubt
           not
           but
           in
           every
           place
           ,
        
         
           hereafter
           't
           may
           be
           mended
           .
        
         
           To
           me
           it
           hath
           been
           of
           great
           use
           ,
        
         
           to
           others
           more
           likewise
           ;
        
         
           Therefore
           let
           no
           man
           it
           abuse
           ,
        
         
           before
           he
           doth
           advise
           .
        
         
           One
           Part
           thereof
           hath
           had
           renown
           ,
        
         
           with
           Artists
           far
           and
           near
           :
        
         
           The
           other
           Part
           I
           strive
           to
           crown
           ,
        
         
           with
           use
           and
           plainness
           here
           .
        
         
           Although
           my
           Parts
           and
           Time
           be
           small
           ,
        
         
           to
           hold
           forth
           Arts
           aright
           ;
        
         
           Yet
           have
           I
           plainly
           set
           forth
           all
           ,
        
         
           seemed
           useful
           in
           my
           sight
           .
        
         
           And
           though
           I
           have
           not
           seen
           so
           far
           ,
        
         
           as
           some
           perhaps
           might
           see
           ;
        
         
           I
           doubt
           not
           but
           that
           some
           there
           are
           ,
        
         
           will
           pleased
           with
           it
           be
           .
        
         
           For
           first
           the
           Tyroes
           young
           may
           find
           ,
        
         
           some
           terms
           to
           be
           explained
           ;
        
         
         
           Which
           when
           well
           fixed
           in
           his
           mind
           ,
        
         
           time
           quickly
           will
           be
           gained
           .
        
         
           In
           the
           next
           place
           Mechanicks
           mean
           ,
        
         
           that
           have
           small
           time
           to
           spare
           ;
        
         
           But
           yet
           may
           have
           a
           Love
           extream
           ,
        
         
           to
           Mathematicks
           fair
           .
        
         
           And
           others
           that
           of
           wordly
           Means
           ,
        
         
           have
           little
           to
           afford
           ,
        
         
           For
           various
           Mathematick
           Theams
           ,
        
         
           this
           having
           they
           are
           stor'd
           ;
        
         
           As
           first
           with
           
             Gunters
             Sector
          
           ,
           and
           ,
        
         
           his
           Quadrant
           eke
           also
           ;
        
         
           By
           Foster
           altred
           after
           ,
           and
           ,
        
         
           with
           
             Gunters
             Rule
          
           and
           Bow.
        
         
           The
           
             Traviss
             Quadrant
          
           and
           Cross-staves
           ,
        
         
           the
           
             Davis
             Quadrant
          
           too
           ;
        
         
           Their
           uses
           all
           to
           more
           than
           halfs
           ,
        
         
           this
           Instrument
           will
           do
           :
        
         
           With
           this
           advantage
           more
           beside
           ,
        
         
           of
           lying
           in
           less
           room
           ,
        
         
           A
           fault
           that
           Saylors
           must
           abide
           ,
        
         
           when
           they
           on
           Ship-board
           come
           .
        
         
           In
           the
           next
           place
           ,
           the
           Rudiments
        
         
           of
           Geometry
           exact
           .
        
         
           The
           right
           Sines
           &
           ●heir
           complements
           ,
        
         
           and
           how
           they
           lie
           compact
           ,
        
         
         
           Within
           a
           Circle
           ,
           and
           the
           rest
           ,
        
         
           the
           Chords
           and
           
             versed
             Sines
          
        
         
           About
           a
           Circle
           are
           exprest
           ,
        
         
           the
           
             Tangents
             ,
             Secants
             ,
             Lines
          
           .
        
         
           And
           how
           their
           use
           and
           place
           is
           seen
           ,
        
         
           in
           Round
           and
           
             Plain
             Triangles
          
           ;
        
         
           Which
           serve
           to
           deck
           Urania
           Queen
           ,
        
         
           as
           Iewels
           ,
           Beads
           ,
           and
           Spangles
           .
        
         
           In
           the
           next
           place
           Arithmetick
           ,
        
         
           by
           Numbers
           and
           by
           Lines
           ;
        
         
           In
           wayes
           that
           won't
           be
           far
           to
           seek
           ,
        
         
           by
           them
           that
           use
           their
           times
           ;
        
         
           Because
           the
           Precepts
           are
           explain'd
           ,
        
         
           by
           things
           of
           frequent
           use
           ,
        
         
           That
           for
           the
           most
           part
           are
           contain'd
           ,
        
         
           in
           City
           ,
           Town
           ,
           or
           House
           ;
        
         
           As
           Land
           and
           Timber
           ,
           Boards
           &
           Stones
           ,
        
         
           Roofs
           ,
           Chimneys
           ,
           Walls
           and
           Floor
           ,
        
         
           Computed
           and
           reduc●d
           at
           once
           ,
        
         
           in
           Thickness
           ,
           Less
           or
           More
           .
        
         
           The
           cutting
           Platoe's
           Bodies
           five
           ,
        
         
           which
           are
           not
           yet
           made
           six
           ;
        
         
           And
           them
           the
           best
           way
           to
           contrive
           ,
        
         
           and
           Dials
           on
           them
           fix
           :
        
         
           Their
           Measure
           and
           their
           Magnitude
           ,
        
         
           in
           Circle
           circumscribed
           ;
        
         
         
           Whose
           Properties
           by
           old
           Euclide
           ,
        
         
           and
           Diggs
           ,
           have
           been
           described
           .
        
         
           Then
           also
           in
           Astronomy
           ,
        
         
           are
           many
           Propositions
           ,
        
         
           Which
           fitly
           to
           th'
           Rule
           I
           apply
           ,
        
         
           avoiding
           repetitions
           .
        
         
           And
           after
           ,
           in
           the
           pleasant
           Art
           ,
        
         
           of
           Shadows
           ,
           I
           do
           wander
           ,
        
         
           To
           draw
           Hour-lines
           in
           every
           part
           ,
        
         
           both
           
             upright
             ,
             over
          
           ,
           and
           under
           :
        
         
           And
           all
           the
           usual
           Ornaments
           ,
        
         
           that
           on
           Sun-Dials
           be
           ,
        
         
           Which
           are
           describ'd
           to
           the
           intent
           ,
        
         
           Sol's
           travels
           for
           to
           see
           ;
        
         
           As
           first
           ,
           his
           Place
           and
           Altitude
           ,
        
         
           his
           Azimuth
           likewise
           ;
        
         
           His
           
             Right
             Ascention
             ,
             Amplitude
          
           ,
        
         
           and
           how
           soon
           he
           doth
           Rise
           .
        
         
           The
           same
           also
           to
           Moon
           and
           Stars
           ,
        
         
           is
           moderately
           appli'd
           ;
        
         
           Whereby
           the
           time
           of
           Night
           appears
           ,
        
         
           the
           
             Moons
             Age
          
           ,
           and
           the
           Tide
           .
        
         
           Then
           Heights
           and
           Distances
           to
           take
           ,
        
         
           at
           one
           ,
           or
           at
           two
           Stations
           ,
        
         
           Performed
           by
           those
           wayes
           that
           make
           ,
        
         
           the
           fewest
           Operations
           .
        
         
         
           And
           also
           ready
           Rules
           to
           use
           ,
        
         
           the
           
             Logarithmal
             Table
          
           ;
        
         
           Which
           may
           prove
           ready
           Hints
           to
           these
           ,
        
         
           that
           are
           in
           those
           most
           able
           :
        
         
           And
           many
           other
           useful
           Thing
           ,
        
         
           is
           scattered
           here
           and
           there
           ,
        
         
           Which
           formerly
           by
           Me
           hath
           been
           ,
        
         
           accounted
           very
           rare
           .
        
         
           And
           lastly
           ,
           for
           the
           Saylors
           sake
           ,
        
         
           I
           have
           spent
           many
           an
           Hour
           ,
        
         
           Th'
           Trianguler-Quadrant
           for
           to
           make
           ,
        
         
           more
           useful
           than
           all
           other
           :
        
         
           Sea
           -
           Instruments
           that
           they
           do
           use
           ,
        
         
           at
           Sea
           for
           Observation
           ;
        
         
           And
           sure
           I
           am
           ,
           it
           won't
           abuse
        
         
           them
           in
           their
           Operation
           ;
        
         
           As
           in
           the
           following
           Discourse
           ,
        
         
           to
           them
           that
           willing
           be
           ,
        
         
           It
           will
           appear
           with
           easie
           force
           ,
        
         
           if
           they
           have
           eyes
           to
           see
           :
        
         
           The
           Method
           and
           the
           Manner
           us'd
           ,
        
         
           as
           neer
           as
           I
           was
           able
           ,
        
         
           To
           follow
           the
           old
           Wayes
           still
           us'd
           ,
        
         
           and
           counted
           warrantable
           .
        
         
           And
           in
           this
           ,
           having
           done
           my
           best
           ,
        
         
           〈…〉
           up
           my
           male
           ;
        
         
         
           Ascribing
           to
           my self
           the
           least
           ,
        
         
           would
           have
           the
           Truth
           prevail
           ;
        
         
           And
           give
           the
           honour
           and
           the
           praise
           ,
        
         
           to
           him
           that
           hath
           us
           made
           ,
        
         
           Of
           willing
           minds
           his
           Fame
           to
           raise
           ,
        
         
           by
           his
           assisting
           aid
           .
        
         
           To
           whom
           be
           honour
           now
           and
           eke
           ,
        
         
           henceforth
           for
           evermore
           ,
        
         
           Ascribed
           by
           all
           them
           that
           seek
        
         
           the
           Truth
           for
           to
           adore
           .
        
         
           
             J.
             B.
             
          
        
      
       
         
         
           ERRATA
           .
        
         
           PAge
           28.
           line
           8.
           for
           Rombords
           ,
           read
           Romboides
           .
           P.
           73.
           l.
           last
           .
           f.
           337
           ,
           r.
           247.
           p.
           75.
           l.
           1.
           f.
           7.
           r.
           8.
           p
           87.
           l.
           14.
           r.
           
             multiplied
             by
          
           .
           p.
           89.
           l.
           14.
           f.
           5
           371616.
           r.
           538.1616
           ▪
           &
           l.
           21.
           f.
           537
           ▪
           r.
           538.
           p.
           90.
           l.
           4.
           f.
           537
           ,
           r.
           538.
           &
           l.
           5.
           add
           ,
           
             being
             better
             done
             with
             a
             parallel
             answer
             .
          
           p.
           100.
           l.
           2
           add
           ,
           
             the
             Thred
          
           .
           p.
           128.
           l
           ,
           2.
           dele
           10
           min.
           p.
           133.
           l.
           6.
           f.
           60
           ,
           r.
           16.
           p.
           143.
           l.
           10
           ,
           11.
           f.
           from
           12
           to
           7
           ,
           r.
           from
           7
           to
           12.
           p.
           146.
           l.
           22.
           f
           12
           Section
           ,
           r.
           13
           Section
           .
           p.
           158
           l.
           last
           ,
           dele
           and.
           p.
           160.
           l.
           11.
           f.
           72
           ,
           r.
           720
           ,
           also
           in
           line
           15
           &
           23.
           p.
           164.
           l.
           19.
           f.
           Diameter
           ,
           r.
           Area
           .
           p.
           165.
           l.
           last
           ,
           add
           ,
           to
           707.
           p.
           184.
           l.
           10
           ▪
           f.
           foot
           ,
           r.
           brick
           .
           &
           l.
           20.
           f.
           ½
           ,
           r.
           1
           ½
           .
           p.
           187.
           l.
           17.
           f.
           Ceiling
           ,
           r.
           Tileing
           p.
           201.
           l.
           11.
           f.
           52
           Links
           .
           r.
           55
           Links
           .
           &
           l.
           12.
           f.
           48
           Acres
           ,
           r.
           4
           Acres
           ,
           3
           
             Roods
             ,
             &
          
           8000
           Links
           .
           p
           102.
           l.
           5.
           f.
           21
           Acres
           42
           Links
           ,
           r.
           2
           Acres
           ,
           0
           
             Roods
             ,
             but
          
           14760
           Links
           ;
           read
           so
           likewise
           in
           l.
           11.
           of
           the
           same
           page
           .
           p.
           204.
           l.
           1.
           f.
           16
           ½
           r.
           18
           ½
           .
           p.
           205.
           l.
           8.
           f.
           55
           ,
           r.
           50.
           &
           r.
           50
           f.
           55
           in
           l.
           21
           &
           22.
           p.
           206.
           l.
           19.
           f
           4-50
           ,
           r.
           4-50000
           .
           &
           l.
           21.
           f.
           1
           Chain
           25
           ,
           r.
           11
           Chains
           23.
           p.
           229.
           l.
           16.
           f.
           8-10
           th
           ,
           r.
           8-100
           .
           p.
           231.
           l.
           15.
           f
           of
           ,
           r.
           at
           .
           p.
           234.
           l.
           22.
           f.
           1
           
             of
             a
             foot
          
           ,
           r.
           1.10
           
             th
             of
             a
             foot
          
           .
           p.
           236
           ,
           the
           3
           lines
           over
           134-5
           ,
           are
           to
           come
           in
           after
           134-5
           .
           Also
           ,
           the
           two
           lines
           over
           3-545
           ,
           should
           come
           in
           after
           3-545
           .
           p.
           257.
           l.
           13.
           f.
           ●496
           ,
           r.
           249-6
           .
           p.
           370.
           l.
           3.
           f.
           sine
           r.
           Co-sine
           .
           p.
           383.
           l.
           22.
           add
           ,
           
             by
             the
             general
             Scale
          
           .
           p.
           384.
           l.
           14.
           f.
           =
           S.
           ☉
           .
           r.
           =
           Co-sine
           .
           p.
           414.
           l.
           11.
           f.
           or
           r.
           on
           .
           p.
           420.
           l.
           22.
           f.
           71
           r.
           31.
           p.
           429.
           l.
           15.
           f.
           Declination
           ,
           r.
           
             Suns
             Right
             Ascention
          
           .
        
      
    
     
       
         
         
           The
           Description
           ,
           and
           some
           Uses
           of
           the
           
             Triangular
             Quadrant
          
           ,
           or
           the
           Sector
           made
           a
           Quadrant
           ;
           being
           an
           excellent
           Instrument
           for
           Observations
           and
           Operations
           at
           Land
           or
           Sea
           ,
           performing
           all
           the
           Uses
           of
           the
           Fore-staff
           ,
           Davis-Quadrant
           ,
           Gunter's-Bow
           ,
           Gunter's-Cross-staff
           ,
           Gunter's-Quadrant
           and
           Sector
           ,
           with
           far
           more
           conveniency
           and
           as
           much
           exactness
           as
           any
           ,
           or
           all
           of
           them
           will
           do
           .
        
         
           
             The
             Description
             thereof
             .
          
           
             1.
             
             FIrst
             ,
             it
             is
             a
             joynted
             Rule
             (
             or
             Sector
             )
             made
             to
             what
             Length
             or
             Radius
             you
             please
             ,
             (
             as
             to
             6
             ,
             9
             ,
             12
             ,
             18
             ,
             24
             ,
             30
             ,
             or
             36
             inches
             Length
             ,
             when
             it
             is
             folded
             or
             shut
             together
             ;
             the
             shorter
             of
             which
             Lengths
             is
             big
             enough
             for
             Land
             uses
             ,
             or
             Paper
             draughts
             ;
             the
             four
             last
             for
             Sea
             uses
             ,
             or
             Observations
             .
             )
             To
             which
             is
             
             added
             ,
             a
             third
             Piece
             of
             the
             same
             length
             of
             the
             Sector
             ,
             with
             a
             Tennon
             at
             each
             end
             ,
             to
             fit
             into
             two
             Mortice-holes
             at
             the
             two
             ends
             of
             the
             inside
             of
             the
             Sector
             ,
             to
             make
             it
             an
             Aequilateral
             Triangle
             ;
             from
             which
             shape
             ,
             and
             its
             use
             ,
             it
             is
             properly
             called
             a
             
               Triangular
               Quadrant
            
             .
          
           
             2.
             
             Secondly
             ,
             as
             to
             the
             Lines
             graduated
             thereon
             ,
             they
             may
             be
             more
             or
             less
             ,
             as
             your
             use
             of
             them
             ,
             and
             as
             the
             cost
             you
             will
             bestow
             ,
             shall
             please
             to
             command
             :
             But
             to
             make
             it
             compleat
             for
             the
             promised
             Premises
             ,
             these
             that
             follow
             are
             necessary
             to
             be
             inscribed
             thereon
             ,
             as
             in
             the
             Figure
             thereof
             .
          
           
             And
             first
             you
             are
             in
             order
             hereunto
             to
             consider
             ,
             The
             outer-edges
             of
             the
             (
             Sector
             or
             )
             Instrument
             ,
             the
             inner-edges
             ,
             the
             Quadrantal-side
             ,
             the
             Sector-side
             ,
             and
             the
             third
             or
             loose-piece
             ,
             also
             the
             fixed
             or
             Head-leg
             ,
             the
             moving-leg
             ,
             the
             head
             ,
             and
             the
             end
             of
             each
             leg
             ,
             also
             the
             head
             and
             leg
             center
             ;
             of
             which
             more
             in
             its
             proper
             place
             .
          
           
             1.
             
             And
             first
             ,
             on
             the
             outer-edge
             is
             placed
             the
             Lines
             of
             Artificial
             Numbers
             ,
             Tangents
             ,
             Sines
             ,
             and
             versed
             Sines
             ,
             to
             as
             large
             a
             Radius
             as
             the
             Instrument
             will
             bear
             .
          
           
             2.
             
             Secondly
             ,
             on
             the
             in-side
             or
             edge
             ,
             on
             short
             Rules
             is
             placed
             inches
             ,
             foot
             measure
             ,
             the
             line
             of
             112
             ,
             or
             such-like
             .
             But
             on
             larger
             Instruments
             ,
             a
             Meridian
             line
             to
             one
             inch
             ,
             
             or
             half
             an
             inch
             (
             more
             or
             less
             )
             for
             one
             degree
             of
             the
             Aequinoctial
             ,
             for
             the
             drawing
             of
             Charts
             ,
             according
             to
             Mercator
             ,
             or
             any
             other
             more
             useful
             Line
             you
             shall
             appoint
             for
             your
             particular
             purpose
             .
          
           
             3.
             
             Thirdly
             ,
             in
             decribing
             the
             Lines
             on
             the
             two
             sides
             ;
             first
             I
             shall
             speak
             to
             the
             Sector-side
             ,
             where
             the
             middle
             Lines
             all
             meet
             at
             the
             Center
             at
             the
             head
             where
             the
             Joynt
             is
             :
             the
             order
             of
             which
             (
             went
             the
             head
             or
             joynted
             end
             lyeth
             toward
             your
             left
             hand
             ,
             the
             Sector
             being
             shut
             ,
             and
             the
             Sector-side
             upermost
             )
             is
             thus
             :
          
           
             1.
             
             The
             first
             pair
             of
             Lines
             ,
             and
             lying
             next
             to
             you
             ,
             is
             the
             Line
             of
             Sines
             ,
             and
             Line
             of
             Lines
             ,
             noted
             at
             the
             end
             with
             S
             ,
             and
             L
             :
             for
             Sines
             and
             Lines
             ,
             the
             middle
             Line
             between
             them
             that
             runs
             up
             to
             the
             Center
             ,
             and
             wherein
             the
             Brass
             center
             pricks
             be
             ,
             is
             common
             both
             to
             the
             Sines
             and
             Lines
             in
             all
             Parallel
             uses
             ,
             or
             entrances
             .
          
           
             2.
             
             The
             Line
             next
             these
             ,
             and
             counting
             from
             you
             ,
             is
             ,
             the
             Line
             of
             Secants
             beginning
             at
             the
             middle
             of
             the
             Rule
             ,
             and
             proceeding
             to
             60
             at
             the
             end
             ,
             and
             noted
             also
             with
             Se
             for
             Secants
             ,
             one
             of
             which
             marginal
             Lines
             continued
             ,
             would
             run
             to
             the
             center
             as
             the
             other
             did
             .
          
           
             3.
             
             The
             next
             Lines
             forward
             ,
             and
             next
             the
             inner-edge
             on
             the
             moving-leg
             are
             the
             Lines
             of
             
             Tangents
             ;
             the
             first
             of
             which
             ,
             and
             next
             to
             you
             ,
             is
             the
             Tangent
             of
             45
             ,
             being
             the
             largest
             Radius
             ,
             (
             as
             to
             the
             length
             of
             the
             Rule
             :
             )
             the
             other
             is
             another
             Tangent
             to
             one
             fourth
             part
             of
             the
             length
             of
             the
             other
             ,
             and
             proceeds
             to
             76
             degrees
             ,
             a
             little
             beyond
             the
             other
             45
             :
             the
             middle
             Line
             of
             these
             also
             is
             common
             to
             both
             ,
             in
             which
             the
             Center
             pricks
             must
             be
             .
             At
             the
             end
             of
             these
             Lines
             is
             usually
             set
             T.
             T.
             for
             Tangents
             .
          
           
             4.
             
             On
             the
             other
             Leg
             of
             the
             Sector
             ,
             are
             the
             same
             Lines
             again
             ,
             in
             the
             same
             order
             counting
             from
             you
             ;
             wherein
             you
             may
             note
             ,
             That
             as
             the
             Lines
             of
             Sines
             and
             Lines
             on
             one
             Leg
             ,
             are
             next
             the
             outward-edge
             ;
             on
             the
             other
             Leg
             ,
             they
             are
             next
             the
             inward-edge
             :
             so
             that
             at
             every
             ,
             or
             any
             Angle
             whatsoever
             the
             Sector
             stands
             at
             ,
             you
             have
             Lines
             ,
             Sines
             ,
             and
             Tangents
             to
             the
             same
             Radius
             :
             and
             the
             Secants
             to
             just
             half
             the
             Radius
             ,
             and
             consequently
             to
             the
             same
             Radius
             by
             turning
             the
             Compasses
             twice
             ;
             Also
             any
             Tangent
             to
             the
             greater
             Radius
             above
             45
             ,
             and
             under
             76
             ,
             by
             turning
             the
             Compasses
             four
             times
             ,
             as
             afterwards
             will
             more
             appear
             :
             Which
             contrivance
             is
             of
             excellent
             convenience
             to
             avoid
             trouble
             ,
             and
             save
             time
             ;
             and
             happily
             made
             use
             of
             ,
             in
             this
             contrary
             manner
             to
             the
             former
             wayes
             of
             ordering
             them
             .
          
           
             5.
             
             Fifthly
             ,
             without
             or
             beyond
             ,
             yet
             next
             
             to
             the
             greater
             Line
             of
             Tangents
             on
             the
             head-leg
             ,
             is
             placed
             the
             first
             45
             degrees
             of
             the
             lesser
             Tangents
             ,
             which
             begin
             from
             the
             Center
             at
             45
             degrees
             ,
             because
             of
             the
             straitness
             of
             the
             room
             next
             the
             Center
             ,
             where
             they
             meet
             in
             a
             Point
             :
             yet
             this
             is
             almost
             of
             as
             good
             use
             ,
             as
             if
             it
             had
             gone
             quite
             to
             the
             Center
             ,
             by
             taking
             any
             parallel
             Tangent
             from
             the
             middle
             or
             common
             Line
             on
             the
             great
             Tangents
             ,
             right
             against
             the
             requisite
             Number
             counted
             on
             the
             small
             Tangent
             under
             45.
             
          
           
             6.
             
             Sixthly
             ,
             next
             to
             this
             will
             not
             be
             amiss
             to
             adde
             a
             Line
             of
             Sines
             ,
             to
             the
             same
             Radius
             of
             the
             small
             Tangent
             last
             mentioned
             ,
             and
             figured
             both
             wayes
             for
             Sine
             and
             co-Sine
             ,
             or
             sometimes
             versed
             Sines
             .
          
           
             7.
             
             Seventhly
             ,
             next
             to
             this
             a
             Line
             of
             Equal
             Parts
             ,
             and
             Chords
             ,
             and
             the
             Secants
             in
             a
             pricked
             line
             beyond
             the
             little
             Tangent
             of
             45
             ,
             all
             to
             one
             Radius
             :
             To
             which
             (
             if
             you
             please
             )
             may
             be
             added
             ,
             Mr.
             Fosters
             Line
             Soll
             ,
             and
             his
             Line
             of
             Latitudes
             ;
             but
             these
             at
             pleasure
             .
          
           
             8.
             
             Eighthly
             ,
             on
             the
             outermost-part
             of
             both
             Legs
             next
             the
             out-side
             ,
             in
             Rules
             of
             half
             an
             inch
             thick
             and
             under
             ,
             is
             set
             the
             Line
             of
             Artificial
             versed
             Sines
             ,
             laid
             next
             to
             the
             Line
             of
             Artificial
             Sines
             ,
             on
             the
             outer-edge
             ;
             but
             if
             the
             Rule
             be
             thick
             enough
             to
             bear
             four
             
             Lines
             ,
             then
             in
             this
             place
             may
             be
             set
             the
             Meridian
             Line
             ,
             according
             to
             Mr.
             Gunter
             ,
             counting
             the
             Line
             of
             Lines
             as
             a
             Scale
             of
             Equal
             Parts
             .
             Thus
             much
             as
             for
             the
             Sector-side
             of
             the
             Instrument
             .
          
           
             4.
             
             Fourthly
             ,
             The
             last
             side
             to
             be
             described
             is
             the
             Quadrantal-side
             of
             the
             Instrument
             ,
             wherein
             it
             chiefly
             is
             new
             .
             Therefore
             I
             shall
             be
             as
             plain
             as
             I
             can
             herein
             .
          
           
             To
             that
             purpose
             I
             shall
             in
             the
             description
             thereof
             imagine
             the
             loose
             piece
             ,
             (
             or
             third
             piece
             )
             to
             be
             put
             into
             the
             two
             Mortise-holes
             ,
             which
             position
             makes
             it
             in
             form
             of
             an
             Aequilateral
             Triangle
             ,
             according
             to
             the
             Figure
             annexed
             ,
             noted
             with
             ABCD
             ;
             where
             in
             AB
             is
             for
             brevity
             and
             plainness
             sake
             called
             the
             Moveable-leg
             ,
             DB
             the
             Head
             or
             Fixed-leg
             ,
             DA
             the
             loose-piece
             ,
             B
             the
             Head
             ,
             A
             and
             D
             the
             ends
             ,
             C
             the
             Leg-center
             ,
             at
             the
             beginning
             of
             the
             general
             Scale
             ;
             the
             center
             at
             B
             the
             head-center
             ,
             used
             only
             in
             large
             Instruments
             ,
             and
             when
             you
             please
             on
             any
             oother
             .
          
           
             
               For
               the
               Lines
               graduated
               on
               this
               side
               .
            
             
               First
               ,
               On
               the
               outer-edge
               of
               the
               moveable-Leg
               ,
               and
               loose-piece
               ,
               is
               graduated
               ,
               the
               180
               degrees
               of
               a
               semi-circle
               ,
               C
               being
               the
               center
               thereof
               .
            
             
             
               And
               these
               degrees
               are
               numbred
               from
               060
               on
               the
               loose-piece
               toward
               both
               ends
               ,
               with
               10
               ,
               20
               ,
               30
               ,
               40
               ,
               &c.
               and
               about
               on
               the
               moveable-leg
               ,
               with
               20
               ,
               30
               ,
               40
               ,
               50
               ,
               60
               ,
               70
               ,
               80
               ,
               and
               90
               at
               the
               head
               :
               Also
               it
               is
               numbred
               from
               600
               on
               the
               moveable-leg
               ,
               with
               10
               ,
               20
               ,
               toward
               the
               head
               ;
               and
               the
               other
               way
               ,
               with
               10
               ,
               20
               ,
               30
               ,
               40
               ,
               50
               ,
               60
               on
               the
               loose-piece
               ;
               and
               sometimes
               also
               from
               the
               Head
               along
               the
               Moveable-leg
               ,
               with
               10
               ,
               20
               ,
               30
               ,
               &c.
               to
               ●0
               on
               the
               loose-piece
               ;
               and
               the
               like
               also
               from
               the
               end
               of
               the
               Head-leg
               ,
               and
               sometimes
               from
               60
               on
               the
               loose-piece
               both
               wayes
               ,
               as
               your
               use
               and
               occasion
               shall
               require
               .
            
             
               Secondly
               ,
               On
               the
               Quadrantal-side
               of
               the
               loose-piece
               ,
               but
               next
               the
               inward-edge
               is
               graduated
               60
               degrees
               ,
               or
               the
               Tangent
               of
               twice
               30
               degrees
               ,
               whose
               center
               ,
               is
               the
               center-hole
               or
               Pin
               at
               B
               ,
               on
               the
               Head
               or
               Joynt
               of
               the
               Sector
               .
            
             
               Which
               degrees
               are
               numbred
               three
               wayes
               ,
               viz.
               First
               from
               D
               to
               A
               for
               forward
               Observations
               ;
               and
               from
               the
               middle
               at
               30
               to
               A
               the
               end
               of
               the
               Moving-leg
               ,
               with
               10
               ,
               20
               ,
               30
               ;
               and
               again
               ,
               from
               D
               the
               end
               of
               the
               Head-leg
               to
               A
               ,
               with
               40
               ,
               50
               ,
               60
               ,
               70
               ,
               80
               ,
               90
               ,
               for
               Observations
               with
               Thred
               and
               Plummet
               .
            
             
             
               Thirdly
               ,
               Next
               to
               these
               degrees
               on
               the
               Moving-leg
               ,
               is
               the
               Line
               of
               the
               Suns
               right
               Ascention
               ,
               numbred
               from
               600
               on
               the
               degrees
               ,
               with
               1
               ,
               2
               ,
               3
               ,
               4
               ,
               5
               ,
               6
               ,
               toward
               the
               Head
               ,
               and
               then
               back
               again
               with
               7
               ,
               8
               ,
               9
               ,
               10
               ,
               11
               ,
               12
               ,
               &c.
               1
               ,
               2
               ,
               3
               ,
               4
               ,
               5
               ,
               on
               the
               other
               side
               of
               the
               Line
               ,
               as
               the
               Figure
               annexed
               sheweth
               :
               The
               divisions
               on
               this
               Line
               is
               (
               for
               the
               most
               part
               )
               whole
               degrees
               ,
               or
               every
               four
               minutes
               of
               time
               .
            
             
               Fourthly
               ,
               Next
               above
               this
               is
               the
               Line
               of
               the
               Suns
               place
               in
               the
               Zodiack
               ,
               noted
               with
               ♈
               ♉
               ♊
               ♋
               toward
               the
               Head
               ;
               then
               back
               again
               with
               ♌
               ♍
               ♎
               over
               600
               in
               the
               degrees
               ,
               and
               12
               and
               24
               in
               the
               Line
               of
               the
               Suns
               right
               Ascentions
               ▪
               then
               toward
               the
               end
               ,
               with
               ♏
               ♐
               ♑
               ;
               then
               back
               again
               with
               ♒
               and
               ♓
               ,
               being
               the
               Characters
               of
               the
               12
               Signes
               of
               the
               Zodiack
               ,
               wherein
               you
               have
               exprest
               every
               whole
               degree
               ,
               as
               the
               number
               of
               them
               do
               shew
               ,
               there
               being
               30
               degrees
               in
               one
               Sign
               .
            
             
               Fiftly
               ,
               Next
               above
               this
               is
               a
               Kalender
               of
               Months
               and
               Dayes
               ;
               every
               single
               Day
               being
               exprest
               ,
               and
               three
               or
               more
               Letters
               ,
               of
               the
               name
               of
               every
               Month
               being
               set
               in
               the
               Month
               ,
               and
               also
               at
               the
               beginning
               of
               each
               Month
               ,
               and
               every
               10th
               day
               noted
               with
               a
               Prick
               on
               the
               top
               of
               the
               Line
               representing
               it
               ,
               as
               is
               usual
               in
               such
               work
               .
            
             
             
               Sixtly
               ,
               Next
               over
               the
               Months
               ,
               is
               the
               Line
               to
               find
               the
               Hour
               and
               Azimuth
               in
               a
               particular
               Latitude
               .
               Put
               alwayes
               on
               smaller
               Instruments
               (
               and
               very
               rarely
               on
               large
               Triangular
               Quadrants
               for
               Sea
               Observations
               )
               the
               lowest
               Margent
               whereof
               ,
               and
               next
               the
               Months
               ,
               is
               numbred
               from
               the
               end
               toward
               the
               Head
               ,
               with
               10
               ,
               20
               ,
               30
               ,
               40
               ,
               50
               ,
               60
               ,
               70
               ,
               80
               ,
               90
               ,
               100
               ,
               110
               ,
               120
               ,
               130
               ,
               near
               the
               Head
               Center
               .
               For
               the
               Semi-diurnal
               Ark
               of
               the
               Suns
               Azimuth
               ,
               and
               in
               the
               Margent
               next
               above
               this
               ,
               with
               4
               ,
               5
               ,
               6
               ,
               7
               ,
               8
               ,
               9
               ,
               10
               ,
               11
               ,
               12
               ,
               near
               the
               end
               ,
               for
               the
               Morning
               hours
               ;
               then
               the
               other
               way
               ,
               viz.
               toward
               the
               Head
               on
               the
               other-side
               the
               Hour
               Line
               ,
               with
               1
               ,
               2
               ,
               3
               ,
               4
               ,
               5
               ,
               6
               ,
               7
               ,
               8
               ,
               for
               the
               Afternoon
               hours
               .
            
             
               Seventhly
               ,
               On
               the
               same
               Quadrantal-side
               ,
               and
               Moveable-leg
               on
               the
               spare
               places
               ,
               beyond
               the
               Months
               toward
               the
               end
               ,
               is
               set
               an
               Almanack
               ;
               and
               the
               Names
               of
               12
               or
               more
               Stars
               ,
               to
               find
               the
               hour
               of
               the
               Night
               ;
               which
               12
               Stars
               are
               noted
               with
               1
               ,
               2
               ,
               3
               ,
               4
               ,
               5
               ,
               6
               ,
               7
               ,
               8
               ,
               9
               ,
               10
               ,
               11
               ,
               12.
               among
               the
               degrees
               in
               small
               Figures
               ,
               as
               in
               the
               Figure
               .
            
             
               Eightly
               ,
               Next
               of
               all
               to
               the
               in-side
               ,
               is
               the
               Line
               of
               Natural
               versed
               Sines
               drawn
               to
               the
               Center
               ,
               with
               his
               correspondent
               Line
               on
               the
               other
               ,
               or
               Head-leg
               .
               Exprest
               sometimes
               in
               a
               pricked
               Line
               ,
               for
               want
               of
               room
               .
            
             
               Ninthly
               ,
               On
               the
               Head-leg
               ,
               and
               next
               to
               
               the
               versed
               Sines
               last
               mentioned
               ,
               is
               first
               the
               Line
               of
               Equal
               Parts
               ,
               or
               Line
               of
               Lines
               :
               and
               on
               the
               same
               common
               Line
               wherein
               is
               the
               Center
               ,
               is
               the
               Line
               of
               Natural
               Sines
               ,
               whose
               length
               is
               equal
               to
               the
               measure
               from
               the
               center
               at
               C
               to
               600
               on
               the
               moveable-leg
               ;
               so
               that
               the
               Line
               of
               degrees
               is
               a
               Tangent
               ,
               and
               the
               measure
               from
               C
               to
               any
               Tangent
               ,
               a
               Secant
               ,
               to
               the
               same
               Radius
               of
               the
               Natural
               Lines
               of
               Sines
               ,
               and
               Lines
               :
               Also
               beyond
               the
               Center
               C
               on
               the
               same
               common
               middle
               Line
               is
               another
               smaller
               Line
               of
               Natural
               Sines
               ,
               whose
               length
               is
               equal
               to
               the
               measure
               from
               C
               to
               60
               on
               the
               loose-piece
               ;
               then
               if
               you
               count
               from
               the
               Center
               pin
               at
               60
               ,
               on
               the
               loose-piece
               ,
               toward
               the
               end
               of
               the
               movable-leg
               ,
               they
               shall
               be
               Tangents
               to
               the
               same
               Radius
               ,
               and
               the
               measure
               from
               the
               Center
               C
               to
               those
               Tangents
               ,
               shall
               be
               Secants
               to
               the
               same
               Radius
               ,
               which
               may
               be
               well
               to
               be
               ordered
               ,
               to
               a
               third
               ,
               or
               fourth
               part
               of
               the
               former
               ,
               from
               the
               Center
               downwards
               :
               These
               two
               Lines
               of
               Sines
               are
               best
               figur'd
               with
               their
               Sines
               ;
               and
               Cosines
               ,
               the
               other
               way
               with
               a
               smaller
               figure
               ,
               and
               the
               Line
               of
               Lines
               from
               the
               Center
               downward
               from
               1
               to
               10
               where
               90
               is
               ,
               which
               Lines
               of
               Sines
               may
               be
               called
               a
               general
               Scale
               for
               all
               Latitudes
               .
            
             
               Tenthly
               ,
               Next
               to
               this
               toward
               the
               outer-edge
               is
               another
               Line
               of
               Natural
               Sines
               ,
               fitted
               
               to
               the
               particular
               Line
               of
               Hour
               and
               Azimuths
               ,
               for
               one
               particular
               Latitude
               ,
               noted
               ,
               
                 Pert.
                 Scale
                 of
                 Altitudes
                 ;
                 or
                 Sines
                 .
              
            
             
               Eleventhly
               ,
               Next
               to
               this
               is
               the
               Line
               of
               29
               ½
               ,
               for
               so
               many
               dayes
               of
               the
               Moon
               's
               age
               ,
               in
               short
               Rules
               of
               the
               whole
               length
               ,
               but
               in
               longer
               not
               ;
               being
               easily
               known
               by
               the
               single
               strokes
               ,
               and
               Figures
               annexed
               to
               those
               strokes
               .
            
             
               Twelfthly
               ,
               Next
               the
               outer-edge
               is
               a
               Line
               of
               24
               hours
               ,
               360
               degrees
               ,
               or
               12
               Signs
               ,
               or
               in
               most
               Rules
               inches
               also
               ,
               used
               together
               with
               the
               former
               Line
               of
               29
               ½
               ,
               and
               as
               a
               Theory
               of
               the
               Sun
               and
               Moon
               ,
               and
               ready
               way
               of
               finding
               the
               Hour
               by
               the
               Moon
               or
               fixed
               Stars
               .
            
             
               Thirteenthly
               ,
               To
               this
               Instrument
               also
               belongs
               a
               Thred
               and
               Plummet
               ,
               and
               Sights
               ,
               as
               to
               other
               Quadrants
               ;
               and
               a
               pair
               of
               Compasses
               as
               to
               other
               Sectors
               ;
               a
               Staff
               and
               Ball
               socket
               also
               ,
               if
               you
               will
               be
               curious
               and
               accurate
               .
            
             
               And
               for
               large
               Instruments
               for
               Sea
               ,
               a
               Square
               and
               an
               Index
               ,
               which
               makes
               it
               a
               perfect
               sinical
               Quadrant
               ,
               and
               two
               sliding
               sights
               also
               ,
               which
               makes
               it
               a
               fore
               and
               back-staff
               ,
               and
               bow
               ,
               as
               will
               appear
               more
               at
               large
               afterward
               .
            
          
        
         
           
           
             Some
             Uses
             of
             the
             Trianguler
             Quadrant
             ,
             for
             Land
             and
             Sea
             Observations
             and
             Operations
             .
          
           
             CHAP.
             I.
             Numeration
             on
             the
             Lines
             graduated
             on
             the
             Instrument
             .
          
           
             IN
             the
             first
             place
             it
             will
             not
             be
             amiss
             to
             hint
             a
             few
             words
             ,
             as
             to
             the
             reading
             the
             Lines
             ,
             or
             (
             more
             properly
             )
             Numeration
             on
             the
             Lines
             ;
             wherein
             take
             notice
             ,
             That
             all
             Lines
             of
             Equal
             Parts
             ,
             or
             Lines
             applicable
             to
             Arithmetick
             ,
             as
             the
             Line
             of
             Lines
             ,
             the
             Line
             of
             Numbers
             ,
             the
             Line
             of
             Foot-measure
             ,
             and
             the
             like
             ;
             wherein
             Fractions
             of
             Numbers
             are
             requisite
             :
             they
             are
             most
             commonly
             accounted
             in
             a
             Decimal
             way
             ,
             and
             as
             much
             as
             may
             be
             ,
             the
             small
             divisions
             are
             numbred
             ,
             and
             counted
             accordingly
             .
          
           
           
             But
             in
             the
             Lines
             of
             Sines
             ,
             Tangents
             ,
             Secants
             ,
             and
             Chords
             ;
             being
             Lines
             belonging
             properly
             to
             a
             Circle
             :
             in
             regard
             that
             the
             Sexagenary
             Fraction
             is
             still
             in
             use
             ,
             the
             intermediate
             Divisions
             are
             ,
             as
             much
             as
             may
             be
             ,
             fitted
             to
             that
             way
             of
             account
             ,
             viz.
             by
             whole
             degrees
             ,
             where
             they
             come
             close
             together
             ,
             (
             or
             the
             Line
             of
             no
             great
             use
             .
             )
             And
             if
             more
             room
             is
             ,
             to
             half
             degrees
             or
             30
             minuts
             ,
             and
             sometimes
             to
             quarters
             of
             degrees
             or
             15
             minuts
             ;
             but
             toward
             the
             beginning
             of
             the
             Line
             of
             Natural
             Sines
             ,
             or
             the
             end
             of
             the
             Natural
             Tangents
             and
             Secants
             :
             where
             the
             degrees
             are
             largest
             ,
             they
             are
             divided
             to
             every
             10th
             minute
             in
             all
             large
             Rules
             ,
             as
             by
             considering
             and
             accounting
             you
             may
             plainly
             perceive
             .
          
           
             
               Take
               two
               or
               three
               Examples
               of
               each
               kind
               .
            
          
           
             1.
             
             First
             ,
             On
             the
             Line
             of
             Lines
             ,
             to
             find
             the
             Point
             that
             represents
             15.
             
             In
             the
             doing
             of
             this
             ,
             or
             any
             the
             like
             ,
             you
             must
             consider
             your
             whole
             Scale
             ,
             Radius
             ,
             or
             length
             of
             the
             Line
             ,
             may
             be
             accounted
             as
             1
             ,
             as
             10
             ,
             as
             100
             ,
             as
             1000
             ,
             or
             as
             10000
             ;
             and
             no
             further
             can
             be
             applicable
             to
             any
             ordinary
             Instrument
             .
          
           
             Wherein
             observe
             ,
             That
             if
             the
             whole
             Line
             be
             one
             ,
             then
             the
             long
             stroke
             by
             every
             Figure
             doth
             represent
             one
             tenth
             of
             that
             Integer
             :
             and
             the
             next
             shorter
             without
             Figures
             ,
             are
             
             hundredth
             parts
             of
             that
             one
             Integer
             ;
             and
             a
             1000th
             part
             is
             estimated
             in
             smaller
             Instruments
             ,
             and
             sometimes
             exprest
             in
             larger
             :
             But
             the
             hundredth
             thousand
             part
             is
             alwayes
             to
             be
             estimated
             by
             the
             eye
             in
             all
             Instruments
             whatsoever
             .
          
           
             2.
             
             But
             if
             the
             whole
             Line
             of
             Lines
             shall
             represent
             10
             ,
             as
             it
             usually
             doth
             ,
             and
             as
             it
             is
             figured
             ,
             then
             the
             long
             stroke
             at
             every
             Figure
             is
             1
             ,
             and
             the
             next
             longer
             are
             tenths
             ,
             and
             the
             shortest
             are
             hundred
             parts
             ,
             and
             the
             thousand
             parts
             as
             near
             as
             can
             be
             estimated
             .
          
           
             3.
             
             But
             if
             the
             whole
             Line
             represents
             a
             hundred
             ,
             as
             here
             in
             our
             present
             Example
             ,
             then
             the
             long
             stroke
             by
             every
             Figure
             represents
             10
             ,
             and
             every
             shorter
             stroke
             is
             one
             ,
             and
             the
             shortest
             strokes
             are
             tenths
             ,
             and
             the
             hundredth
             parts
             as
             much
             as
             can
             be
             estimated
             .
          
           
             4.
             
             But
             if
             the
             whole
             Line
             shall
             represent
             a
             1000
             ,
             then
             the
             long
             stroke
             by
             the
             Figure
             shall
             represent
             a
             hundred
             ,
             and
             every
             shorter
             10
             ,
             and
             every
             of
             the
             shortest
             strokes
             is
             one
             Integer
             ,
             and
             a
             10th
             part
             as
             near
             as
             can
             be
             estimated
             .
          
           
             5.
             
             But
             lastly
             ,
             if
             the
             whole
             Line
             represent
             10000
             ,
             then
             every
             long
             stroke
             is
             100
             ,
             and
             every
             shortest
             cut
             is
             ten
             ,
             and
             every
             single
             Integer
             is
             as
             near
             as
             can
             be
             estimated
             by
             any
             ordinary
             Instrument
             .
          
           
           
             Now
             our
             present
             Example
             will
             properly
             come
             under
             the
             third
             Rule
             ,
             by
             conceiving
             the
             whole
             Line
             to
             represent
             100
             ;
             then
             the
             first
             long
             stroke
             by
             1
             is
             10
             ,
             then
             the
             next
             shorter
             is
             for
             11
             ,
             the
             next
             12
             ,
             &c.
             to
             15
             ;
             which
             is
             cut
             up
             a
             little
             above
             the
             Line
             ,
             for
             the
             more
             ready
             reckoning
             without
             telling
             the
             parts
             :
             which
             15
             is
             the
             Point
             required
             to
             be
             found
             .
          
           
             
               Example
               the
               second
               ,
               to
               find
               out
               1550
               on
               the
               Line
               .
            
          
           
             This
             will
             come
             under
             the
             Notion
             of
             the
             5th
             Rule
             ,
             wherein
             the
             whole
             Line
             is
             conceived
             to
             represent
             10000
             ;
             then
             the
             first
             1
             is
             for
             the
             1
             thousand
             ,
             then
             the
             fifth
             longer
             stroke
             next
             is
             for
             the
             500
             ;
             and
             lastly
             ,
             the
             middle
             between
             the
             500
             stroke
             and
             the
             6000
             stroke
             is
             for
             the
             50
             ,
             being
             a
             little
             beyond
             the
             Point
             for
             15
             in
             the
             first
             Example
             .
          
           
             
               A
               third
               Example
               of
               5025.
            
             
          
           
             This
             third
             Example
             may
             su●fice
             for
             this
             work
             ,
             being
             so
             plain
             after
             a
             little
             due
             consideration
             :
             For
             first
             ,
             the
             whole
             Line
             is
             conceived
             to
             represent
             10000
             ,
             then
             the
             long
             stroke
             by
             5
             is
             for
             5000
             ,
             then
             there
             is
             no
             hundreds
             ,
             therefore
             the
             Point
             required
             must
             be
             short
             of
             the
             next
             longer
             stroke
             ,
             which
             signifies
             hundreds
             ,
             and
             being
             it
             is
             just
             25
             ,
             which
             is
             ¼
             of
             an
             hundred
             ,
             the
             true
             Point
             readily
             sheweth
             it self
             ;
             If
             you
             require
             
             a
             more
             plainer
             and
             larger
             wording
             o●
             this
             matter
             ,
             I
             refer
             you
             to
             the
             third
             Chapte●
             of
             Mr.
             Windgates
             Rule
             of
             Proportion
             ;
             〈◊〉
             the
             first
             Chapter
             of
             the
             
               Carpenters
               Rule
            
             ,
             b●
             
               I.
               Brown.
            
             
          
           
             Lastly
             ,
             In
             nameing
             of
             any
             Point
             found
             out
             on
             the
             Line
             ,
             great
             care
             and
             respect
             mus●
             be
             had
             as
             to
             the
             true
             value
             of
             the
             Number
             according
             to
             the
             rate
             of
             the
             question
             propounded
             :
             for
             the
             same
             Point
             that
             represents
             15
             ,
             doth
             represent
             150
             ;
             and
             also
             1500
             ,
             or
             15000
             ,
             (
             increasing
             above
             the
             bounds
             before
             mentioned
             )
             also
             it
             signifies
             one
             and
             a
             half
             ,
             or
             15
             of
             one
             hundred
             ,
             which
             is
             usually
             exprest
             thus
             in
             a
             Decimal
             Fraction
             15
             /
             100
             ,
             or
             more
             readily
             ,
             0.15
             .
          
           
             Also
             if
             it
             should
             be
             a
             Number
             with
             a
             digit
             ,
             two
             ciphers
             and
             another
             digit
             ,
             as
             2.005
             ,
             this
             Number
             would
             be
             found
             close
             to
             the
             long
             stroke
             ,
             by
             the
             figure
             2
             :
             and
             may
             represent
             either
             two
             thousand
             ,
             and
             5
             of
             1000
             more
             ;
             or
             20
             and
             5
             of
             a
             hundred
             ;
             or
             2
             hundred
             and
             5
             of
             another
             10
             more
             ,
             or
             plainly
             as
             it
             is
             set
             down
             ,
             two
             thousand
             ,
             no
             hundred
             ,
             but
             five
             :
             Thus
             you
             see
             the
             manner
             of
             expressing
             whole
             Numbers
             ,
             or
             whole
             Numbers
             and
             Decimal
             Fractions
             ,
             which
             on
             the
             Lines
             is
             one
             and
             the
             same
             thing
             ;
             and
             thus
             all
             Decimal
             Scales
             are
             to
             be
             accounted
             ,
             and
             in
             the
             same
             manner
             is
             
             the
             Line
             of
             Numbers
             to
             be
             read
             ,
             as
             you
             may
             see
             more
             at
             large
             in
             the
             two
             Books
             before
             mentioned
             .
          
           
             
               SECT
               .
               II.
               
            
             
               But
               for
               Numerateon
               on
               all
               Circular
               Lines
               ,
               it
               is
               much
               easier
               :
               For
               first
               ,
               very
               few
               Instruments
               ,
               unless
               at
               one
               part
               of
               the
               Line
               ,
               can
               express
               nearer
               than
               minutes
               of
               a
               degree
               .
            
             
               Secondly
               ,
               The
               whole
               Radius
               or
               Line
               of
               Lines
               is
               but
               90
               degrees
               ,
               or
               but
               45
               of
               the
               Tangents
               ,
               or
               60
               of
               the
               Chords
               or
               Secants
               :
               So
               that
               in
               Instruments
               of
               12
               or
               18
               inches
               Radius
               ,
               you
               may
               express
               very
               well
               every
               tenth
               minute
               ,
               to
               60
               on
               the
               Line
               of
               Sines
               :
               and
               every
               half
               degree
               to
               75
               ,
               and
               whole
               degrees
               to
               90.
               
               And
               on
               the
               Tangents
               or
               Chords
               ,
               every
               10th
               minute
               quite
               through
               :
               and
               the
               Secants
               as
               the
               Sines
               .
            
             
               So
               that
               any
               degree
               or
               minute
               being
               named
               ,
               t●
               find
               the
               same
               on
               the
               respective
               Line
               ,
               count
               thus
               ;
            
             
               First
               ,
               every
               10th
               degree
               is
               noted
               with
               a
               long
               stroke
               ,
               and
               figures
               set
               thereunto
               .
               Secondly
               ,
               every
               whole
               degree
               is
               cut
               between
               two
               ,
               or
               three
               Lines
               ,
               and
               sometimes
               with
               a
               Point
               or
               Mark
               on
               the
               end
               of
               the
               stroke
               ;
               and
               every
               5th
               degree
               cut
               up
               higher
               than
               the
               rest
               ,
               and
               sometimes
               with
               three
               Points
               ,
               on
               the
               end
               
               of
               the
               Line
               ,
               or
               some
               other
               convenient
               distinction
               ,
               for
               readiness
               sake
               :
               and
               every
               10th
               ,
               15th
               ,
               or
               30th
               minute
               ,
               is
               cut
               only
               between
               two
               Lines
               and
               no
               more
               ;
               as
               will
               appear
               very
               plain
               with
               a
               little
               practice
               .
            
             
               
                 Example
                 ,
                 to
                 find
                 the
                 Sine
                 of
                 the
                 Latitude
                 being
                 at
                 London
                 ,
                 51
                 degrees
                 ,
                 32
                 minutes
                 .
              
            
             
               1.
               
               First
               ,
               look
               on
               any
               Line
               of
               Sines
               ,
               on
               the
               Quadrantal
               ,
               or
               Sector
               side
               ,
               according
               a●
               you
               have
               occasion
               ,
               till
               you
               see
               50
               ,
               which
               i●
               50
               degrees
               ;
               then
               one
               degree
               forward
               ,
               toward
               60
               is
               51
               degrees
               ,
               then
               count
               thre●
               10ths
               of
               minutes
               more
               for
               30
               minutes
               ,
               and
               then
               for
               the
               odde
               two
               minutes
               ,
               estimate
               on●
               fifth
               part
               of
               the
               next
               10
               minutes
               forwarder
               and
               that
               is
               the
               precise
               Point
               for
               the
               Sine
               〈◊〉
               51
               degrees
               32
               minutes
               ,
               the
               latitu●e
               of
               London
               ,
               where
               sometimes
               is
               set
               a
               Bra●s
               Center-Pin
               .
            
             
               
                 Example
                 the
                 second
              
               .
            
             
               2.
               
               To
               find
               the
               Cosine
               of
               the
               Latitude
               there
               are
               two
               wayes
               to
               count
               the
               Comple●ment
               of
               any
               Ark
               or
               Angle
               .
            
             
               First
               ,
               by
               substracting
               the
               Ark
               or
               Angle
               out
               of
               90
               by
               the
               Pen
               ,
               and
               count
               the
               residu●
               from
               the
               beginning
               of
               the
               Line
               of
               Sines
               ,
               and
               
               that
               shall
               be
               the
               Sine
               Complement
               of
               the
               Latitude
               required
               .
            
             
               Example
               .
            
             
               51
               32
               taken
               from
               90
               ,
               the
               remainder
               is
               38
               28
               ,
               now
               if
               you
               count
               so
               much
               from
               the
               beginning
               of
               the
               Line
               of
               Sines
               ,
               according
               to
               the
               last
               Rule
               that
               shall
               be
               the
               Point
               for
               the
               Sine
               of
               38
               28
               ,
               the
               Complement
               of
               51
               32
               ,
               or
               the
               Sine
               Complement
               of
               the
               Latitude
               .
            
             
               Or
               Secondly
               ,
               If
               you
               count
               51
               32
               from
               90
               ,
               calling
               80
               ,
               10
               ;
               and
               70
               ,
               20
               ;
               and
               60
               ,
               30
               ;
               50
               ,
               40
               ;
               40
               ,
               50
               ,
               &c.
               wheresoever
               the
               Number
               whose
               Complement
               you
               would
               have
               shall
               end
               ,
               that
               is
               the
               Sine
               Complement
               required
               ,
               which
               will
               be
               at
               38
               28
               ,
               from
               the
               Center
               or
               beginning
               ,
               for
               the
               Co-sine
               of
               51
               32
               ;
               The
               like
               work
               serves
               for
               any
               other
               Number
               ,
               or
               on
               any
               other
               Line
               ,
               as
               on
               the
               Degrees
               ,
               Tangents
               ,
               or
               Secants
               ,
               Natural
               or
               Artificial
               ,
               as
               by
               practice
               will
               more
               plainly
               appear
               ,
               to
               the
               willing
               Practitioner
               .
            
          
           
             
               SECT
               .
               III.
               
            
             
               To
               find
               the
               versed
               Sine
               of
               an
               Ark
               or
               Angle
               ,
               or
               the
               Sine
               of
               an
               Ark
               or
               Angle
               above
               90
               degrees
               ,
               or
               the
               Chord
               above
               180
               degrees
               ,
               observe
               these
               Rules
               .
            
             
             
               1.
               
               First
               ,
               a
               right
               Sine
               ,
               is
               the
               measure
               on
               the
               Line
               of
               Sines
               ,
               from
               the
               center
               or
               beginning
               of
               that
               Line
               ,
               to
               the
               Point
               that
               doth
               represent
               the
               Ark
               or
               Angle
               required
               .
            
             
               2.
               
               The
               right
               Sine
               of
               an
               Ark
               or
               Angle
               above
               90
               degrees
               ,
               is
               equal
               ,
               to
               the
               right
               Sine
               of
               the
               Complement
               thereof
               to
               180
               degrees
               ,
               being
               readily
               accounted
               ,
               thus
               ;
               Count
               the
               excess
               above
               90
               backwards
               ,
               from
               90
               toward
               the
               Center
               ;
               then
               the
               measure
               or
               distance
               from
               the
               end
               of
               the
               account
               to
               the
               Center
               ,
               is
               the
               Sine
               of
               the
               Ark
               above
               90
               required
               :
               Example
               .
               Let
               the
               Sine
               of
               130
               be
               required
               ,
               first
               ,
               if
               you
               take
               130
               from
               180
               ,
               the
               remainder
               is
               50
               ;
               then
               I
               say
               that
               the
               right
               Sine
               of
               50
               ,
               is
               also
               right
               Sine
               of
               130
               ;
               for
               if
               you
               count
               backwards
               from
               90
               ,
               calling
               80
               ,
               100
               ;
               and
               70
               ,
               110
               ;
               and
               60
               ,
               120
               ;
               and
               50
               ,
               130
               ;
               the
               measure
               from
               thence
               to
               00
               ,
               or
               the
               Center
               ,
               is
               the
               right
               Sine
               of
               130
               degrees
               .
            
             
               3.
               
               The
               versed
               Sine
               of
               an
               Ark
               or
               Angle
               ,
               is
               the
               measure
               on
               the
               Line
               of
               Sines
               from
               90
               toward
               the
               Center
               ,
               counted
               backwards
               ,
               as
               the
               small
               figures
               for
               Complements
               shew
               ,
               counting
               90
               for
               00
               ,
               and
               the
               Center
               for
               90
               ,
               (
               as
               the
               Azimuth
               Line
               is
               figured
               )
               opening
               the
               Line
               of
               Sines
               to
               a
               strait
               Line
               ,
               and
               then
               counting
               beyond
               90
               for
               the
               versed
               Sines
               above
               90
               ,
               as
               on
               the
               versed
               Sines
               is
               plainly
               
               seen
               in
               the
               figure
               of
               the
               Rule
               .
            
             
               4.
               
               For
               Chords
               of
               any
               Ark
               or
               Angle
               ,
               do
               thus
               :
            
             
               Halve
               the
               Ark
               or
               Angle
               required
               ,
               and
               take
               the
               right
               Sine
               thereof
               ,
               and
               that
               shall
               be
               the
               Chord
               thereof
               .
            
             
               Example
               .
            
             
               I
               would
               have
               the
               Chord
               of
               40
               ,
               the
               half
               of
               40
               is
               20
               ;
               then
               I
               say
               the
               right
               Sine
               of
               20
               is
               the
               Chord
               of
               40
               ,
               to
               that
               Radius
               that
               is
               equal
               to
               the
               right
               Sine
               of
               30
               degrees
               ,
               at
               the
               Radius
               the
               Rule
               stands
               at
               .
            
             
               5.
               
               To
               find
               a
               Chord
               to
               an
               Ark
               or
               Angle
               above
               180
               degrees
               ,
               you
               must
               count
               as
               you
               did
               the
               right
               Sines
               ;
               for
               note
               ,
               the
               Chord
               of
               180
               is
               equal
               to
               the
               right
               Sine
               of
               90
               doubled
               ,
               which
               is
               the
               full
               Diameter
               of
               a
               Circle
               :
               and
               a
               longer
               right
               Line
               than
               the
               Diameter
               cannot
               be
               taken
               in
               a
               Circle
               ;
               therefore
               it
               must
               needs
               follow
               that
               Chords
               of
               above
               180
               ,
               are
               shorter
               than
               the
               Diameter
               which
               is
               the
               biggest
               Chord
               ;
               therefore
               the
               Chord
               of
               260
               ,
               is
               equal
               to
               the
               Chord
               of
               100
               degrees
               ,
               or
               right
               Sine
               of
               50
               ,
               the
               Sine
               of
               30
               being
               Radius
               .
            
             
               6.
               
               In
               using
               the
               Artificial
               Sines
               and
               Tangents
               ,
               or
               Secants
               ;
               if
               you
               are
               to
               use
               a
               Sine
               above
               90
               ,
               then
               count
               80
               for
               100
               ,
               70
               for
               110
               ,
               60
               for
               120
               ,
               &c.
               
               But
               for
               Secants
               ,
               
               then
               count
               after
               the
               manner
               of
               versed
               Sines
               :
               Thus
               the
               Secant
               of
               60
               is
               as
               far
               beyond
               90
               ,
               as
               it
               is
               from
               30
               to
               90
               ;
               so
               that
               when
               you
               have
               occasion
               to
               use
               an
               Artificial
               Secant
               ,
               which
               is
               not
               often
               ,
               Then
               set
               the
               end
               of
               the
               Rule
               against
               a
               Table
               ,
               and
               counting
               backwards
               from
               90
               to
               the
               number
               of
               the
               Secant
               required
               ,
               turn
               that
               distance
               beyond
               90
               on
               the
               Board
               or
               Table
               ,
               and
               that
               shall
               be
               your
               Secant
               required
               ,
               as
               will
               be
               afterward
               hinted
               ,
               as
               they
               come
               in
               use
               .
            
          
        
         
           
             CHAP.
             II.
             A
             brief
             Description
             of
             the
             Lines
             of
             a
             Circle
             ,
             and
             the
             Explanation
             of
             some
             termes
             used
             in
             the
             following
             Discourse
             .
          
           
             FOR
             the
             better
             understanding
             of
             the
             following
             discourse
             ,
             it
             is
             needful
             to
             understand
             these
             Elements
             or
             Principles
             ,
             as
             the
             Letters
             are
             necessary
             to
             be
             known
             before
             reading
             .
          
           
             1.
             
             A
             Circle
             is
             a
             figure
             enclosed
             in
             one
             circular
             Line
             ,
             called
             the
             Circumference
             ;
             in
             the
             middle
             whereof
             is
             a
             Point
             called
             the
             Center
             :
             
             From
             which
             Point
             all
             right
             Lines
             drawn
             to
             the
             Circumference
             ,
             are
             equal
             one
             to
             another
             ;
             as
             in
             the
             Circle
             ABCD
             ,
             E
             is
             the
             Center
             ,
             ABCD
             the
             Circumference
             ,
             the
             Lines
             EA
             ,
             EB
             ,
             EC
             ,
             ED
             are
             equal
             .
          
           
             2.
             
             Any
             right
             Line
             crossing
             the
             Circumference
             ,
             and
             passing
             through
             the
             Center
             of
             a
             Circle
             ,
             is
             called
             the
             Diameter
             ;
             and
             it
             divides
             the
             whole
             Circle
             into
             two
             equal
             parts
             ,
             called
             Semi-circles
             (
             or
             half-Circles
             .
             )
             And
             the
             half
             of
             that
             Line
             is
             called
             the
             Semi-diameter
             or
             Radius
             to
             that
             Circle
             .
             As
             the
             Line
             AC
             is
             the
             Diameter
             ,
             and
             EC
             the
             half-Diameter
             or
             Radius
             .
          
           
             3.
             
             Any
             other
             Right-line
             crossing
             the
             Circumference
             is
             called
             a
             Chord
             ,
             or
             Subtence
             ,
             as
             the
             Line
             FG
             ,
             which
             divides
             the
             Circle
             into
             two
             unequal
             parts
             :
             And
             note
             ,
             that
             this
             Subtence
             belongs
             both
             to
             the
             lesser
             ,
             and
             also
             to
             the
             greater
             part
             of
             the
             Circumference
             ;
             that
             is
             to
             say
             ,
             the
             Chord
             of
             90
             deg
             .
             is
             also
             the
             Chord
             of
             270
             deg
             .
             so
             that
             FG
             is
             Chord
             to
             the
             Ark
             FBG
             90
             deg
             .
             and
             also
             to
             the
             Ark
             FDG
             being
             270
             deg
             .
             much
             more
             than
             half
             the
             Circle
             .
          
           
             4.
             
             Half
             the
             Chord
             of
             any
             Ark
             ,
             is
             the
             right
             Sine
             of
             half
             that
             Ark
             :
             thus
             the
             right
             Line
             HG
             ,
             the
             half
             of
             FG
             ,
             is
             the
             right
             Sine
             of
             the
             Ark
             BG
             the
             half
             of
             FBG
             .
          
           
             5.
             
             The
             Sine
             Complement
             or
             Cosine
             of
             
             any
             Ark
             is
             the
             nearest
             distance
             from
             the
             Circumference
             to
             the
             Diameter
             :
             Perpendiculer
             to
             that
             Diameter
             from
             whence
             you
             counted
             the
             degrees
             and
             minutes
             of
             the
             Ark
             or
             Angle
             .
             As
             thus
             ,
             GI
             is
             the
             Cosine
             of
             the
             Ark
             BG
             ;
             and
             the
             Right
             Sine
             of
             any
             Ark
             is
             the
             nearest
             distance
             from
             the
             Circumference
             to
             the
             Diameter
             you
             counted
             the
             degrees
             from
             ,
             as
             GH
             is
             the
             Right
             Sine
             of
             BG
             .
          
           
             6.
             
             The
             versed
             Sine
             of
             any
             Ark
             or
             Angle
             ,
             is
             the
             Segment
             of
             the
             Diameter
             between
             the
             right
             Sine
             of
             the
             same
             Ark
             and
             the
             Circumference
             .
          
           
             Thus
             HB
             is
             the
             versed
             Sine
             of
             the
             Ark
             BG
             ,
             and
             HD
             the
             versed
             Sine
             of
             GD
             .
             So
             also
             is
             GH
             the
             right
             Sine
             of
             the
             Ark
             GCD
             or
             the
             Angle
             GED
             45
             degrees
             above
             90
             ,
             viz.
             135
             degrees
             .
          
           
             7.
             
             A
             Tangent
             is
             a
             right
             Line
             drawn
             perpendiculer
             to
             the
             Diameter
             ,
             beginning
             at
             one
             extreme
             of
             the
             given-Ark
             ,
             and
             terminated
             by
             a
             right
             Line
             drawn
             from
             the
             Center
             to
             the
             other
             extreme
             ,
             of
             the
             given-Ark
             in
             the
             Circumference
             ,
             till
             it
             inter-sect
             the
             perpendiculer
             ;
             Thus
             CK
             is
             the
             Tangent
             of
             the
             Ark
             CG
             ,
             or
             the
             Angle
             CEG
             ,
             45
             degrees
             .
          
           
             8.
             
             A
             Secant
             is
             a
             right
             Line
             drawn
             from
             the
             Center
             thorow
             one
             extreme
             of
             the
             given-Ark
             ,
             
             till
             it
             meet
             with
             the
             Tangent
             rais'd
             perpendicularly
             from
             the
             Diameter
             ,
             drawn
             to
             the
             other
             Extreme
             of
             the
             said
             Ark
             ;
             Thus
             the
             Line
             EK
             is
             the
             Secant
             of
             the
             Ark
             CG
             ,
             or
             the
             Angle
             GEC
             .
          
           
             9.
             
             Note
             ,
             as
             in
             a
             (
             Natural
             )
             Sine
             ,
             the
             nearest
             distance
             from
             the
             Ark
             to
             one
             Diameter
             ,
             from
             whence
             you
             counted
             the
             degrees
             o●
             the
             Ark
             or
             Angle
             ,
             was
             the
             Right
             Sine
             ;
             and
             the
             nearest
             distance
             from
             the
             same
             Point
             to
             the
             other
             perpendiculer
             Diameter
             ,
             is
             the
             Cosine
             of
             that
             Ark
             or
             Angle
             .
          
           
             So
             likewise
             the
             nearest
             distance
             from
             the
             Point
             where
             the
             Tangent
             and
             Secant
             meets
             ,
             to
             one
             of
             the
             Diameters
             aforesaid
             ,
             is
             the
             Tangent
             of
             the
             Ark
             or
             Angle
             ;
             so
             the
             nearest
             distance
             from
             the
             meeting
             Point
             of
             the
             same
             Secant-line
             is
             the
             other
             Tangent-line
             to
             the
             other
             Diameter
             abovesaid
             ,
             is
             the
             Co-Tangent
             of
             the
             Ark
             or
             Angle
             abovesaid
             .
          
           
             Thus
             the
             Right-line
             KC
             is
             the
             Tangent
             of
             45
             ,
             and
             the
             Right
             Line
             KB
             the
             Co-Tangent
             of
             45
             ;
             Also
             the
             Line
             LC
             is
             the
             Tangent
             of
             53
             ,
             30
             ;
             and
             the
             Line
             MB
             is
             the
             co-Tangent
             thereof
             ,
             viz.
             the
             Tangent
             of
             36
             ,
             30.
             
          
           
             Also
             the
             nearest
             distance
             from
             L
             to
             EB
             ,
             is
             the
             Tangent
             of
             36
             ,
             30
             ,
             to
             the
             Radius
             LC
             .
          
           
             10.
             
             Every
             Circle
             is
             supposed
             to
             be
             divided
             
             into
             360
             degrees
             ;
             the
             Semi-circle
             into
             180
             ,
             the
             Quadrant
             or
             Quarter
             into
             90.
             
          
           
             11.
             
             Every
             Degree
             is
             supposed
             to
             be
             divided
             into
             60
             minutes
             ,
             and
             every
             minute
             into
             60
             Seconds
             ,
             and
             every
             Second
             into
             60
             Thirds
             ,
             &c.
             
          
           
             12.
             
             A
             Radius
             ,
             or
             Semidiameter
             ,
             is
             in
             our
             Instrumental
             Practice
             ,
             supposed
             to
             be
             divided
             into
             10000
             parts
             ,
             and
             every
             Chord
             ,
             Sine
             ,
             Tangent
             ,
             or
             Secant
             ,
             is
             to
             be
             divided
             by
             the
             Parts
             of
             the
             same
             Radius
             ,
             or
             Radius
             and
             Parts
             more
             .
          
           
             13.
             
             An
             Angle
             is
             the
             meeting
             of
             two
             Right
             Lines
             ,
             as
             GE
             ,
             and
             EC
             ,
             meeting
             at
             E
             ,
             do
             constitute
             the
             Angle
             GEC
             ,
             called
             a
             Right-lined
             Angle
             ;
             or
             when
             two
             Circles
             cross
             one
             another
             ,
             it
             is
             called
             a
             Spherical
             Angle
             ,
             the
             Anguler
             Point
             being
             noted
             alwayes
             by
             the
             middle
             Letter
             of
             three
             that
             shew
             the
             Triangle
             .
          
           
             14.
             
             A
             
               Plain
               Triangle
            
             is
             the
             meeting
             of
             three
             Right
             Lines
             crossing
             one
             another
             ;
             and
             a
             
               Spherical
               Triangle
            
             is
             constituted
             by
             the
             crossing
             of
             three
             Circles
             ,
             as
             in
             the
             two
             Figures
             noted
             II
             and
             III
             ,
             you
             may
             plainly
             see
             .
          
           
             15.
             
             All
             Angles
             ,
             Plain
             and
             Spherical
             ,
             are
             either
             Acute
             ,
             Right
             ,
             or
             Obtuce
             .
          
           
           
             17.
             
             A
             
               Right
               Angle
            
             is
             alwa●●●
             just
             90
             degrees
             ,
             as
             you
             may
             〈◊〉
             in
             the
             Figures
             II
             ,
             and
             III
             ,
             by
             the
             Angles
             at
             A
             in
             both
             of
             them
             .
          
           
             18.
             
             An
             
               Obtuce
               Angle
            
             is
             alwayes
             more
             than
             90
             degrees
             ,
             as
             the
             Angles
             at
             D
             in
             both
             Figures
             shew
             .
          
           
             19.
             
             A
             
               Parallel
               Line
            
             ,
             is
             any
             Line
             drawn
             by
             another
             Line
             in
             such
             a
             way
             ,
             that
             though
             it
             were
             infinitely
             produced
             ,
             yet
             they
             would
             never
             meet
             or
             cross
             one
             another
             ,
             as
             the
             Lines
             AB
             ,
             CD
             .
          
           
             20.
             
             A
             
               Perpendiculer
               Line
            
             ,
             is
             when
             one
             Line
             so
             falleth
             on
             another
             Line
             ,
             that
             the
             Angles
             on
             each
             side
             are
             equal
             ,
             as
             CA
             falls
             of
             the
             Line
             BA
             ,
             Figure
             VI.
             
          
           
             21.
             
             All
             Triangles
             are
             either
             with
             three
             equal
             sides
             ,
             as
             Figure
             IIII
             ,
             or
             two
             equal
             sides
             ,
             as
             Figure
             V
             ,
             or
             all
             unequal
             sides
             ,
             as
             Figure
             VI
             ;
             the
             first
             of
             which
             is
             called
             Equilateral
             ,
             the
             second
             Isosceles
             ,
             the
             third
             Scalenum
             .
          
           
             22.
             
             Again
             ,
             they
             may
             be
             sometimes
             named
             from
             their
             Angles
             ;
             thus
             :
             Orthigonium
             ,
             with
             one
             Right
             Angle
             ,
             and
             two
             Acute
             Angles
             .
             Ambligonium
             ,
             with
             one
             Obtuce
             Angle
             ,
             and
             two
             Acute
             Angles
             .
             Oxigonium
             ,
             with
             three
             Acute
             Angles
             only
             .
          
           
           
             23.
             
             The
             three
             Angles
             of
             every
             
               Plain
               Triangle
            
             ,
             are
             equal
             to
             two
             Right
             Angles
             .
          
           
             24.
             
             All
             
               Four-sided
               Figures
            
             are
             either
             Squares
             ,
             with
             four
             sides
             ,
             and
             four
             right
             Angles
             all
             equal
             ;
             or
             long
             Squares
             (
             or
             Oblongs
             )
             with
             the
             two
             opposite
             sides
             equal
             ,
             or
             the
             same
             crushed
             together
             ,
             or
             not
             Right-Angled
             ,
             as
             the
             Rombus
             ,
             and
             Rombords
             ,
             or
             else
             with
             four
             unequal
             Sides
             ,
             called
             Trapeziaes
             .
          
           
             25.
             
             Lastly
             ,
             many
             sided-Figures
             ,
             are
             some
             Regular
             ,
             having
             every
             side
             alike
             ,
             as
             5
             ,
             6
             ,
             7
             ,
             8
             ,
             9
             ,
             10
             ,
             &c.
             
             Or
             else
             unlike
             ,
             as
             Fields
             ,
             and
             Woods
             ,
             and
             Meadows
             ,
             which
             being
             infinite
             ,
             cannot
             be
             comprehended
             under
             any
             Regular
             Order
             or
             Rule
             .
          
           
             26.
             
             Multiplicator
             ,
             is
             a
             term
             used
             in
             Multiplication
             ,
             by
             which
             any
             Number
             is
             to
             be
             multiplied
             ,
             as
             is
             saying
             5
             times
             6
             ,
             5
             is
             the
             Multiplicator
             of
             6.
             
          
           
             27.
             
             Multiplicand
             ,
             is
             the
             Number
             to
             be
             multiplied
             ,
             as
             6
             by
             5
             ,
             as
             above
             named
             .
          
           
             28.
             
             The
             Product
             ,
             is
             the
             Issue
             or
             Result
             of
             two
             Numbers
             multiplied
             one
             by
             the
             other
             ,
             as
             30
             is
             the
             Product
             of
             6
             multiplied
             by
             5
             ;
             for
             6
             times
             5
             is
             30.
             
          
           
             29.
             
             Divisor
             ,
             is
             a
             term
             used
             in
             Division
             ,
             and
             is
             the
             Number
             by
             which
             another
             Number
             is
             to
             be
             divided
             ;
             as
             to
             say
             ,
             How
             many
             times
             5
             in
             30
             ?
             5
             here
             is
             the
             Divisor
             .
          
           
           
             30.
             
             Dividend
             ,
             is
             the
             Number
             to
             be
             divided
             ,
             as
             30
             abovesaid
             .
          
           
             31.
             
             Quotient
             ,
             is
             the
             Answer
             to
             the
             How
             many
             times
             (
             as
             in
             the
             abovesaid
             )
             5
             is
             in
             30
             ?
             6
             times
             :
             6
             then
             is
             the
             Quotient
             .
          
           
             32.
             
             Square
             ,
             is
             the
             Product
             of
             two
             Numbers
             multiplied
             together
             ,
             as
             the
             Square
             of
             6
             multiplied
             by
             6
             ,
             is
             36.
             
          
           
             33.
             
             Square-root
             of
             any
             Number
             ,
             is
             that
             number
             ,
             which
             being
             multiplied
             by
             it self
             ,
             shall
             have
             a
             Product
             or
             Square
             equal
             to
             the
             given
             Number
             ;
             thus
             the
             Square-root
             of
             36
             is
             6
             ;
             for
             6
             multiplied
             by
             6
             ,
             is
             36
             ,
             equal
             to
             the
             first
             given
             Number
             .
          
           
             But
             if
             it
             be
             a
             Number
             that
             cannot
             be
             squared
             ,
             as
             72
             ,
             the
             content
             of
             half
             a
             Foot
             of
             Board
             ;
             whose
             near
             Square-root
             is
             8
             :
             4852811
             of
             10000000
             ,
             then
             is
             the
             Square-root
             to
             be
             exprest
             as
             near
             as
             you
             may
             (
             or
             care
             for
             )
             as
             here
             the
             Square-root
             of
             72
             ,
             which
             is
             called
             a
             Surd
             Number
             ,
             that
             will
             not
             be
             squared
             .
          
           
             34.
             
             Cube
             ,
             is
             a
             second
             Product
             ,
             or
             power
             of
             two
             Numbers
             increasing
             or
             multiplied
             together
             ,
             as
             thus
             ;
             the
             Square
             of
             6
             is
             36
             ,
             the
             first
             power
             :
             and
             the
             Cube
             of
             6
             is
             216
             ,
             that
             is
             to
             say
             6
             times
             36
             ,
             the
             second
             power
             .
          
           
           
             In
             Mr.
             Windgate's
             Book
             of
             Arithmetick
             ,
             is
             the
             way
             of
             doing
             it
             by
             Numbers
             or
             Figures
             ,
             being
             one
             of
             the
             hardest
             Lessons
             in
             Arithmetick
             .
          
        
         
           
           
             CHAP.
             III.
             Certain
             Geometrical
             Propositions
             ,
             fit
             to
             be
             known
             as
             Preparatory
             Rudiments
             for
             the
             following
             Work.
             
          
           
             
               1.
               
               To
               draw
               a
               Right
               Line
               between
               two
               Points
               .
            
             
               EXtend
               a
               Thred
               or
               Hair
               ,
               from
               one
               Point
               to
               the
               other
               ,
               and
               that
               shall
               be
               the
               Line
               required
               .
               But
               if
               you
               use
               a
               Rule
               (
               being
               the
               fittest
               Instrument
               )
               to
               try
               your
               Rule
               ,
               do
               thus
               ;
               apply
               one
               end
               to
               one
               Point
               as
               to
               A
               ,
               and
               the
               other
               end
               to
               the
               other
               Point
               at
               B
               ,
               and
               close
               to
               the
               edge
               draw
               the
               Line
               required
               :
               then
               turn
               the
               Rule
               ,
               and
               lay
               the
               first
               end
               to
               the
               last
               Point
               (
               yet
               keeping
               the
               same
               side
               of
               the
               Rule
               toward
               the
               Paper
               )
               and
               draw
               the
               Line
               again
               ,
               and
               if
               the
               two
               Lines
               appear
               as
               one
               ,
               the
               Rule
               is
               streight
               ,
               or
               else
               not
               .
               Note
               the
               Figure
               I.
               
            
          
           
             
             
               2.
               
               To
               draw
               a
               Line
               Perpendiculer
               to
               another
               ,
               on
               the
               middle
               of
               a
               Line
               .
            
             
               On
               the
               Point
               E
               on
               the
               Line
               AB
               ,
               I
               would
               raise
               the
               Perpendiculer
               Line
               CE
               ,
               set
               one
               point
               of
               the
               Compasses
               in
               E
               ,
               and
               open
               them
               to
               any
               distance
               ,
               as
               EB
               and
               EA
               ,
               and
               note
               the
               Points
               A
               and
               B
               ,
               then
               open
               the
               Compasses
               wider
               ,
               and
               setting
               one
               Point
               in
               A
               ,
               make
               the
               part
               of
               the
               Arch
               by
               C
               upwards
               ;
               and
               if
               you
               have
               room
               do
               the
               like
               downwards
               ,
               near
               D
               :
               Then
               the
               Compasses
               not
               stirring
               ,
               set
               one
               Point
               in
               B
               ,
               and
               with
               the
               other
               ,
               cross
               the
               former
               Arks
               ,
               near
               C
               &
               D
               :
               a
               Rule
               laid
               ,
               and
               a
               Line
               drawn
               ,
               by
               those
               two
               crossings
               ,
               shall
               cut
               the
               Line
               AB
               perpendiculerly
               just
               in
               the
               Point
               E
               ,
               which
               was
               required
               .
            
          
           
             
               3.
               
               To
               let
               fall
               a
               Perpendiculer
               from
               a
               Point
               to
               a
               Line
               .
            
             
               But
               if
               the
               Point
               C
               had
               been
               given
               from
               whence
               to
               let
               fall
               the
               Perpendiculer
               to
               the
               Line
               AB
               ,
               do
               thus
               :
               First
               ,
               set
               one
               Point
               of
               Compasses
               in
               C
               ;
               open
               the
               other
               to
               any
               distance
               ,
               as
               suppose
               to
               A
               and
               B
               ;
               and
               then
               (
               if
               you
               have
               room
               upon
               A
               and
               B
               ,
               strike
               both
               the
               Arks
               by
               D
               ,
               which
               finds
               the
               Point
               D
               ,
               if
               
               not
               )
               the
               middle
               between
               A
               and
               B
               ,
               give
               ;
               the
               Point
               E
               ;
               by
               which
               to
               draw
               CED
               ,
               the
               Perpendiculer
               from
               C
               desired
               .
               Note
               Figure
               2.
               
            
             
               Note
               ,
               That
               if
               you
               can
               come
               to
               find
               the
               Point
               D
               ,
               by
               the
               crossing
               ,
               it
               doth
               readily
               and
               exactly
               divide
               the
               Line
               AB
               in
               two
               equal
               parts
               ,
               by
               the
               Point
               E.
               
            
          
           
             
               4.
               
               To
               raise
               a
               Perpendiculer
               on
               the
               end
               of
               a
               Line
               .
            
             
               On
               the
               end
               of
               the
               Line
               AB
               ,
               at
               B
               ,
               I
               would
               raise
               a
               Perpendiculer
               :
               First
               ,
               set
               one
               Point
               of
               the
               Compasses
               in
               B
               ;
               open
               them
               to
               any
               distance
               ,
               as
               suppose
               to
               C
               ;
               and
               set
               the
               other
               Point
               any
               where
               about
               the
               middle
               ,
               between
               D
               and
               E
               ,
               as
               suppose
               at
               C
               ,
               then
               keep
               that
               Point
               fixed
               there
               ;
               turn
               the
               other
               till
               it
               cut
               the
               Line
               ,
               as
               at
               D
               ,
               and
               keep
               both
               Points
               fixed
               there
               ,
               and
               lay
               a
               streight
               Rule
               close
               to
               both
               Points
               ▪
               and
               there
               keep
               it
               ;
               then
               keep
               the
               middle-Point
               still
               fixed
               at
               C
               ,
               and
               turn
               the
               other
               neatly
               close
               to
               the
               other
               end
               and
               edge
               of
               the
               Rule
               ,
               to
               find
               the
               Point
               E
               ;
               then
               a
               Rule
               laid
               to
               the
               Points
               E
               and
               B
               ,
               shall
               draw
               the
               Perpendiculer
               required
               .
            
             
               Or
               else
               ,
               when
               you
               have
               set
               the
               Compasses
               in
               the
               Point
               C
               ,
               prick
               the
               Point
               D
               in
               
               the
               Line
               ,
               and
               make
               the
               touch
               of
               an
               Ark
               near
               to
               E
               ;
               then
               a
               Rule
               laid
               to
               DC
               cuts
               the
               Ark
               last
               made
               ,
               at
               or
               by
               E
               ,
               in
               the
               Point
               E
               :
               There
               are
               other
               wayes
               ,
               but
               none
               better
               than
               this
               .
               Note
               the
               Figure
               3.
               
            
          
           
             
               5.
               
               From
               a
               Point
               given
               ,
               to
               let
               fall
               a
               Perpendiculer
               to
               the
               end
               of
               a
               Line
               ,
               being
               the
               converse
               of
               the
               former
               .
            
             
               First
               ,
               from
               the
               Point
               E
               ,
               draw
               the
               Line
               ED
               ,
               of
               which
               Line
               find
               the
               middle
               between
               E
               and
               D
               ,
               viz.
               the
               Point
               C
               :
               then
               the
               extent
               CE
               ,
               or
               CD
               ,
               keeping
               one
               Point
               in
               C
               ,
               shall
               cross
               the
               Ground-line
               in
               the
               Point
               B
               ,
               by
               which
               ,
               and
               E
               ,
               you
               may
               draw
               the
               perpendiculer
               Line
               EB
               ,
               which
               is
               but
               the
               converse
               of
               the
               former
               .
            
          
           
             
               6.
               
               To
               draw
               a
               Line
               Parallel
               to
               another
               ,
               at
               any
               distance
               .
            
             
               To
               the
               Line
               AB
               ,
               I
               would
               have
               another
               Parallel
               thereto
               ;
               to
               the
               distance
               of
               AI
               ,
               take
               AI
               between
               your
               Compasses
               ,
               and
               setting
               one
               Point
               in
               one
               end
               of
               the
               Line
               ,
               as
               at
               A
               ,
               sweep
               the
               Ark
               EIF
               ;
               then
               set
               the
               Compasses
               in
               the
               other
               end
               ,
               as
               at
               B
               ,
               and
               sweep
               the
               Ark
               GDH
               ;
               then
               just
               by
               the
               Round-side
               of
               those
               Arks
               ,
               draw
               a
               Line
               ,
               
               which
               shall
               be
               the
               parallel-Line
               required
               .
            
             
               
                 Or
                 thus
              
               ,
            
             
               Take
               BC
               ,
               the
               measure
               from
               the
               Point
               that
               is
               to
               cut
               the
               Parallel-line
               ,
               and
               one
               end
               of
               the
               given-Line
               ,
               viz.
               B
               ;
               with
               this
               distance
               ,
               set
               one
               foot
               in
               A
               ,
               at
               the
               other
               end
               of
               the
               given-Line
               ,
               and
               draw
               the
               Arch
               at
               K
               ;
               then
               take
               all
               AB
               ,
               the
               given-Line
               ,
               and
               setting
               one
               Point
               in
               C
               ,
               cross
               the
               Ark
               at
               K
               ,
               then
               C
               and
               K
               shall
               be
               Points
               to
               draw
               the
               Parallel-line
               by
               .
               Note
               the
               Figure
               4.
               
            
          
           
             
               7.
               
               To
               make
               one
               Angle
               equal
               to
               another
               .
            
             
               The
               Angle
               BAC
               ,
               being
               given
               ,
               and
               I
               would
               have
               another
               Angle
               equal
               unto
               it
               ;
               set
               one
               point
               of
               the
               Compasses
               in
               A
               ,
               and
               draw
               the
               Arch
               CB
               ;
               then
               on
               the
               Line
               DE
               from
               the
               Point
               D
               ,
               draw
               the
               like
               Ark
               EF
               ;
               then
               in
               that
               Ark
               make
               EF
               equal
               to
               CB
               ,
               then
               draw
               the
               Line
               DF
               ,
               it
               shall
               make
               the
               Angle
               EDF
               ,
               equal
               to
               the
               Angle
               BAC
               ▪
               which
               was
               required
               .
            
          
           
             
               8.
               
               To
               divide
               a
               Line
               into
               any
               Number
               of
               parts
               .
            
             
               Let
               AB
               represent
               a
               Line
               to
               be
               divided
               
               
               into
               Eight
               parts
               :
               On
               one
               end
               ,
               viz.
               A
               ,
               draw
               a
               Line
               ,
               as
               AD
               ,
               to
               any
               Angle
               ;
               and
               from
               the
               other
               end
               B
               ,
               draw
               another
               Line
               Parallel
               to
               AD
               ,
               as
               BE
               ,
               then
               open
               the
               Compasses
               to
               any
               convenient
               distance
               ,
               and
               from
               A
               and
               B
               ,
               divide
               the
               Lines
               AD
               ,
               and
               BE
               ,
               into
               eight
               parts
               ;
               then
               Lines
               drawn
               by
               a
               Ruler
               ,
               laid
               to
               every
               division
               ,
               in
               the
               Lines
               AD
               ,
               and
               BE
               ,
               shall
               divide
               the
               Line
               AB
               in
               the
               parts
               required
               .
               Note
               the
               Figure
               marked
               VI.
               
            
             
               This
               Proposition
               is
               much
               easier
               wrought
               by
               the
               Line
               of
               Lines
               on
               the
               Sector
               ,
               thus
               ;
               Take
               AB
               between
               your
               Compasses
               ,
               and
               fit
               it
               over
               parrally
               in
               8
               ,
               and
               8
               of
               the
               Line
               of
               Lines
               ;
               then
               the
               Parallel
               distance
               between
               1
               and
               1
               ,
               shall
               divide
               AB
               into
               8
               parts
               required
               .
            
          
           
             
               9.
               
               Any
               three
               Points
               given
               ,
               to
               bring
               them
               into
               a
               Circle
               .
            
             
               Let
               ABC
               be
               three
               Points
               to
               be
               brought
               into
               a
               Circle
               ;
               first
               set
               one
               Point
               on
               A
               ,
               and
               open
               the
               other
               above
               half-way
               to
               C
               ,
               and
               sweep
               the
               part
               of
               a
               Circle
               above
               and
               below
               the
               Point
               A
               ,
               as
               the
               two
               Arches
               at
               D
               and
               E
               ;
               not
               moving
               the
               Compasses
               ,
               do
               the
               like
               on
               C
               ,
               as
               the
               Arks
               F
               and
               G
               ;
               then
               set
               the
               Compass-point
               in
               B
               ,
               and
               cross
               those
               
               Arks
               in
               DEF
               and
               G
               ;
               then
               a
               Rule
               laid
               from
               D
               to
               E
               ,
               and
               from
               F
               to
               G
               ,
               and
               Lines
               drawn
               do
               inter-sect
               at
               H
               ,
               the
               true
               Center
               ,
               to
               bring
               ABC
               into
               a
               Circle
               .
            
          
           
             
               10.
               
               Any
               two
               Points
               given
               in
               a
               Circle
               ,
               to
               draw
               part
               of
               a
               Circle
               ,
               which
               shall
               cut
               them
               ,
               and
               the
               Circumference
               first
               given
               into
               two
               equal
               parts
               .
            
             
               Let
               A
               and
               B
               be
               two
               Points
               in
               a
               Circle
               ,
               by
               which
               two
               Points
               ,
               I
               would
               draw
               an
               Arch
               ,
               which
               shall
               cut
               the
               whole
               Circumference
               into
               two
               equal
               parts
               .
               First
               ,
               draw
               a
               Line
               from
               A
               ,
               the
               Point
               remotest
               from
               the
               Center
               ,
               through
               the
               Center
               ,
               and
               beyond
               the
               Circumference
               ,
               as
               AD
               ;
               then
               draw
               another
               Line
               from
               A
               ,
               to
               a
               Point
               in
               the
               Circumference
               ,
               perpendiculer
               to
               AD
               ,
               (
               and
               cutting
               the
               Center
               C
               )
               as
               the
               Line
               AE
               :
               Then
               on
               the
               Point
               E
               ,
               draw
               another
               Line
               perpendiculer
               to
               the
               Line
               AE
               ,
               till
               it
               inter-sect
               AD
               at
               D
               ;
               then
               these
               three
               Points
               ABD
               brought
               into
               a
               Circle
               ,
               or
               Arch
               ,
               by
               the
               last
               Rule
               ,
               shall
               divide
               the
               Circumference
               into
               two
               equal
               parts
               .
               Note
               the
               Figure
               8
               ,
               where
               the
               first
               Circle
               is
               cut
               into
               two
               equal
               parts
               at
               F
               and
               G
               ,
               by
               part
               of
               a
               Circle
               passing
               through
               the
               Points
               A
               and
               B.
               
            
          
           
             
             
               11.
               
               Any
               Segment
               of
               a
               Circle
               given
               ,
               to
               find
               the
               Diameter
               and
               Center
               of
               the
               Circle
               belonging
               to
               it
               .
            
             
               Let
               ABC
               be
               the
               Segment
               of
               a
               Circle
               ,
               to
               which
               I
               would
               find
               a
               Center
               ;
               any
               where
               about
               the
               middest
               of
               the
               Segment
               ,
               set
               one
               point
               of
               the
               Compasses
               at
               pleasure
               ,
               as
               at
               B
               ;
               on
               the
               point
               B
               (
               at
               any
               meet
               distance
               )
               describe
               a
               Circle
               ,
               and
               note
               where
               the
               Circle
               doth
               cross
               the
               Segment
               ,
               as
               at
               D
               and
               E
               ,
               then
               (
               not
               stirring
               the
               Compasses
               )
               set
               one
               point
               in
               D
               ,
               and
               cross
               the
               Circle
               twice
               ,
               as
               at
               F
               and
               I
               ;
               and
               again
               set
               one
               point
               in
               E
               ,
               and
               cross
               the
               Circle
               twice
               in
               G
               and
               H
               :
               Lastly
               ,
               by
               the
               Points
               GH
               ,
               and
               FI
               ,
               draw
               two
               Lines
               ,
               which
               will
               meet
               in
               the
               point
               O
               ,
               the
               center
               required
               .
            
          
           
             
               12.
               
               Or
               else
               to
               find
               the
               Diameter
               ,
               thus
               .
            
             
               Multiply
               the
               Chord
               (
               or
               flat-side
               )
               of
               the
               half-Segment
               ,
               viz
               AK
               ,
               12
               by
               it self
               (
               which
               is
               called
               Squaring
               )
               which
               makes
               144
               ;
               then
               divide
               that
               Product
               144
               by
               8
               ,
               the
               Line
               KB
               ,
               called
               a
               Sine
               ,
               the
               Quotient
               which
               comes
               out
               will
               be
               found
               to
               be
               18
               ;
               then
               if
               you
               adde
               8
               the
               Sine
               ,
               and
               18
               the
               Quotient
               together
               ,
               it
               shall
               make
               26
               for
               
               the
               Diameter
               required
               to
               be
               found
               .
            
          
           
             
               13.
               
               Any
               Segment
               of
               a
               Circle
               given
               ,
               to
               find
               the
               Length
               of
               the
               Arch
               of
               the
               Segment
               .
            
             
               Lay
               the
               Chord
               of
               the
               whole
               Segment
               ,
               and
               twice
               the
               Chord
               of
               half
               the
               Segment
               ,
               from
               one
               Point
               severally
               ;
               and
               to
               the
               greatest
               extent
               ,
               adde
               one
               third
               part
               of
               the
               difference
               between
               the
               Extents
               ,
               and
               that
               sum
               of
               Extents
               shall
               be
               equal
               to
               the
               Arch.
               
            
             
               Example
               .
            
             
          
           
             
               14.
               
               To
               draw
               a
               Helical
               Line
               from
               any
               Three
               Points
               ,
               to
               several
               Radiusses
               without
               much
               gibbiosity
               ;
               useful
               for
               Architect
               ,
               Shipwrights
               ,
               and
               others
               .
            
             
               Let
               ABCDE
               be
               five
               Points
               ,
               to
               be
               brought
               into
               a
               Helical-Line
               ,
               smoothly
               ,
               and
               even
               without
               gibbiosity
               or
               bunches
               ,
               as
               the
               under-side
               of
               an
               Arch
               ,
               or
               the
               bending
               of
               a
               Ship
               ,
               or
               the
               like
               .
            
             
               First
               ,
               between
               the
               two
               remote
               Points
               of
               3
               ,
               as
               A
               and
               C
               ,
               draw
               the
               Line
               AC
               ,
               then
               let
               fall
               a
               Perpendiculer
               from
               B
               ,
               to
               cut
               the
               Line
               AC
               at
               Right
               Angles
               ,
               and
               produce
               it
               to
               F
               :
               draw
               the
               like
               perpendiculer-Line
               from
               the
               point
               D
               ,
               to
               cut
               the
               Line
               CE
               at
               Right-Angles
               produced
               to
               F.
               I
               say
               ,
               the
               Center
               both
               for
               the
               Arches
               AB
               the
               lesser
               ,
               and
               BC
               the
               greater
               ,
               will
               be
               found
               to
               be
               in
               the
               Line
               BF
               ;
               the
               like
               on
               the
               other-side
               for
               DE
               and
               CD
               ,
               the
               Helical-Circle
               ,
               or
               Arch
               required
               .
            
             
             
               But
               if
               you
               divide
               the
               Arch
               ABCDE
               into
               24
               or
               more
               parts
               ,
               the
               several
               Centers
               of
               the
               splay-Lines
               are
               thus
               found
               ;
               Take
               the
               measure
               AG
               ,
               and
               lay
               it
               from
               B
               ,
               or
               D
               ,
               or
               C
               ,
               on
               the
               Line
               GF
               ;
               and
               those
               Points
               on
               GF
               ,
               shall
               be
               the
               several
               Points
               to
               draw
               the
               splay-Lines
               of
               the
               Arch
               ,
               and
               Key-stone
               by
               .
            
          
        
         
           
             CHAP.
             IV.
             Of
             the
             Explanation
             of
             certain
             Terms
             used
             in
             this
             following
             Book
             .
          
           
             1.
             
             RAdius
             ,
             or
             Sine
             of
             90
             ,
             
             or
             Tangent
             of
             45
             ,
             or
             Secant
             of
             00
             ,
             are
             all
             one
             and
             the
             same
             thing
             ,
             yet
             taken
             respectively
             in
             their
             proper
             places
             ,
             and
             is
             the
             whole
             Line
             of
             Sines
             ,
             or
             Tangents
             ,
             to
             45
             ;
             or
             more
             particulary
             that
             point
             at
             the
             end
             of
             the
             Natural-Sines
             ,
             on
             the
             Sector-side
             ,
             and
             at
             90
             and
             45
             on
             the
             edge
             of
             the
             Rule
             for
             the
             Artificial
             Sines
             and
             Tangents
             ,
             or
             10
             on
             the
             Line
             of
             Numbers
             ,
             and
             10
             and
             90
             on
             the
             Line
             of
             Lines
             ,
             and
             Sines
             ,
             on
             the
             Quadrantal-side
             of
             the
             Instrument
             .
          
           
           
             
             2.
             
             A
             
               Right
               Sine
            
             of
             any
             Ark
             ,
             or
             Angle
             ,
             is
             the
             measure
             from
             the
             beginning
             of
             the
             Line
             of
             Natural-Sines
             ,
             to
             that
             Point
             on
             that
             Line
             of
             Sines
             ,
             which
             represents
             the
             degrees
             and
             minutes
             contained
             in
             that
             Ark
             or
             Angle
             required
             .
             But
             on
             the
             Artificial-Sines
             we
             respect
             not
             any
             measure
             but
             the
             Point
             only
             .
          
           
             
             3.
             
             The
             same
             account
             is
             used
             both
             for
             the
             Right-Tangent
             ,
             and
             Secant
             also
             ;
             the
             Natural-Tangent
             taken
             from
             the
             beginning
             to
             the
             degree
             and
             minute
             required
             ;
             the
             Artificial
             respecting
             the
             Point
             only
             .
          
           
             
             4.
             
             In
             the
             same
             manner
             count
             for
             the
             Secants
             ,
             and
             Chords
             ,
             Lines
             ,
             or
             versed
             Sines
             .
          
           
             
             5.
             
             A
             Cosine
             ,
             or
             
               Sine
               Complement
            
             of
             any
             Ark
             or
             Angle
             ,
             is
             the
             measure
             from
             the
             Point
             representing
             the
             Ark
             or
             Angle
             ,
             counted
             from
             90
             ,
             to
             the
             beginning
             of
             the
             Line
             of
             Sines
             ,
             being
             in
             effect
             the
             Right-Sine
             of
             the
             Cosine
             of
             the
             Ark
             or
             Angle
             required
             :
             As
             for
             Example
             ;
             I
             would
             take
             out
             the
             Cosine
             of
             the
             Latitude
             of
             London
             ,
             which
             is
             51
             ,
             32
             ;
             Count
             51
             32
             from
             90
             toward
             the
             beginning
             ,
             and
             you
             shall
             find
             your
             account
             to
             end
             at
             the
             Right-Sine
             of
             38
             28
             ,
             which
             is
             the
             Complement
             of
             51
             32
             ;
             for
             both
             put
             together
             ,
             makes
             90
             ,
             the
             whole
             Sine
             or
             Radius
             .
          
           
             But
             on
             the
             Artificial-Lines
             count
             backward
             
             to
             the
             Point
             required
             ,
             without
             minding
             any
             distance
             or
             measure
             ,
             till
             you
             come
             to
             Proportion
             .
          
           
             6.
             
             A
             
               Lateral
               Sine
            
             ,
             Tangent
             ,
             or
             Secant
             ,
             
             or
             Scale
             of
             Equal
             Parts
             ,
             is
             any
             Sine
             Tangent
             or
             Secant
             ,
             taken
             along
             the
             length
             of
             any
             Line
             ,
             from
             the
             beginning
             onwards
             ,
             being
             a
             term
             used
             only
             in
             operation
             with
             a
             Sector
             ,
             or
             one
             Line
             and
             a
             Thred
             ,
             and
             opposed
             to
             a
             parallel-Sine
             ,
             Tangent
             ,
             or
             Secant
             ,
             the
             thing
             next
             to
             be
             explained
             .
          
           
             7.
             
             A
             Parallel
             Sine
             ,
             Tangent
             ,
             or
             Secant
             ,
             
             is
             any
             Sine
             ,
             Tangent
             ,
             or
             Secant
             ,
             taken
             across
             from
             one
             Leg
             to
             the
             other
             of
             a
             Sector
             ;
             or
             from
             any
             degree
             and
             minute
             on
             one
             Line
             to
             a
             Thred
             drawn
             streight
             with
             the
             other
             hand
             ,
             or
             any
             other
             fixed
             Line
             whatsoever
             ,
             at
             the
             nearest
             distance
             .
          
           
             8.
             
             The
             
               Nearest
               Distance
            
             to
             any
             Line
             ,
             
             is
             thus
             taken
             ;
             When
             one
             Point
             of
             the
             Compasses
             stands
             in
             any
             one
             Point
             ,
             and
             the
             Line
             being
             laid
             ,
             I
             open
             or
             close
             my
             Compasses
             till
             the
             other
             moveable-Foot
             ,
             being
             turned
             about
             ,
             will
             but
             just
             touch
             or
             cleave
             the
             Thred
             .
             But
             if
             you
             are
             to
             lay
             the
             Thred
             to
             the
             nearest
             distance
             ,
             then
             one
             Point
             of
             the
             Compasses
             being
             set
             fast
             ,
             the
             other
             is
             to
             be
             turned
             about
             ,
             and
             the
             Thred
             also
             slipped
             to
             and
             fro
             ,
             till
             the
             Compass-point
             shall
             just
             cleave
             the
             Thred
             in
             the
             middest
             .
          
           
           
             
             9.
             
             To
             adde
             one
             Sine
             or
             Tangent
             ,
             to
             a
             Sine
             or
             Tangent
             ,
             is
             to
             take
             the
             Right-Sine
             ,
             or
             Tangent
             of
             any
             Ark
             or
             Angle
             between
             your
             Compasses
             ,
             and
             setting
             one
             Point
             of
             the
             Compasses
             in
             the
             Point
             of
             the
             other
             Number
             ,
             and
             then
             to
             see
             how
             far
             the
             other
             Point
             will
             extend
             Laterally
             .
             Example
             .
             To
             adde
             the
             Sine
             of
             20
             ,
             to
             the
             Sine
             of
             30
             ,
             take
             the
             Sine
             of
             20
             between
             your
             Compasses
             ,
             and
             then
             putting
             one
             Point
             in
             30
             ,
             the
             other
             shall
             reach
             to
             the
             Sine
             of
             51
             21
             ;
             therefore
             the
             distance
             from
             the
             beginning
             to
             51
             21
             ,
             is
             the
             sum
             of
             the
             Sine
             of
             30
             and
             20
             added
             together
             .
             The
             like
             way
             is
             to
             add
             Tangents
             .
          
           
             
             10.
             
             To
             Substract
             a
             Sine
             from
             a
             Sine
             ,
             or
             a
             Tangent
             from
             a
             Tangent
             ,
             is
             but
             to
             take
             the
             Lateral
             least
             Right-Sine
             or
             Tangent
             between
             your
             Compasses
             ,
             and
             setting
             one
             Point
             in
             the
             term
             of
             the
             greatest
             turn
             ,
             the
             other
             toward
             the
             beginning
             ,
             and
             note
             the
             degree
             and
             minute
             that
             the
             other
             Point
             stayes
             in
             ,
             for
             that
             is
             the
             difference
             or
             remainder
             .
          
           
             Example
             .
          
           
             Suppose
             I
             would
             take
             the
             Sine
             of
             10
             degrees
             from
             25
             ;
             Take
             the
             distance
             10
             between
             your
             Compasses
             ,
             and
             setting
             one
             foot
             in
             25
             ,
             and
             the
             other
             turned
             toward
             
             the
             beginning
             ,
             shall
             reach
             to
             14
             23
             ,
             the
             residue
             or
             difference
             required
             .
          
           
             Or
             ,
             you
             may
             sometimes
             take
             the
             distance
             between
             the
             greater
             and
             the
             less
             ,
             and
             lay
             this
             from
             the
             beginning
             ,
             shall
             give
             the
             remainder
             in
             distance
             on
             the
             Sines
             as
             before
             .
          
           
             11.
             
             The
             Rectifying-Point
             ,
             
             is
             a
             Point
             or
             Hole
             on
             the
             Head
             of
             the
             
               Trianguler
               Quadrant
            
             in
             the
             inter-secting
             of
             the
             hour
             and
             Azimuth-line
             ,
             and
             the
             common
             Line
             to
             the
             Lines
             and
             Sines
             on
             the
             Head-leg
             ;
             in
             which
             Point
             you
             are
             ,
             when
             the
             Rule
             is
             open
             ,
             to
             stick
             a
             small
             Pin
             to
             look
             to
             the
             object
             whose
             Altitude
             above
             the
             Horizon
             you
             would
             have
             in
             degrees
             and
             minutes
             .
          
           
             
               Of
               Terms
               used
               in
               DIALLING
               .
            
             
               PLain
               ,
               is
               that
               Board
               ,
               Glass
               ,
               or
               flat
               Superficies
               you
               intend
               to
               draw
               the
               Dial
               upon
               ,
               either
               single
               of
               it self
               ,
               
               or
               joyned
               to
               some
               other
               .
            
             
               
                 Pole
                 of
                 the
                 Plain
              
               ,
               
               is
               an
               imaginary
               Point
               in
               the
               Horizon
               (
               for
               all
               upright
               Dials
               )
               directly
               
               opposite
               to
               the
               Plain
               ,
               or
               in
               all
               Plains
               ,
               a
               Point
               every
               way
               90
               degrees
               from
               the
               Plain
               .
            
             
               
               
                 Declination
                 of
                 a
                 Plain
              
               ,
               is
               only
               the
               number
               of
               degrees
               and
               minutes
               ,
               that
               the
               Pole-point
               of
               the
               Plain
               is
               distant
               from
               the
               North
               and
               South-points
               of
               the
               Horizon
               .
            
             
               
               The
               
                 Perdendiculer-Line
                 on
                 the
                 Plain
              
               ,
               is
               a
               Line
               Square
               to
               a
               Horizontal-line
               ,
               being
               part
               of
               a
               Circle
               passing
               through
               the
               Zenith
               ,
               and
               Nadir
               ,
               and
               Pole-point
               of
               the
               Plain
               .
            
             
               
               The
               Horizontal-line
               ,
               is
               a
               Line
               drawn
               on
               any
               Plain
               ,
               exactly
               parallel
               to
               the
               true
               Horizon
               of
               the
               place
               you
               dwell
               in
               .
            
             
               
               Reclination
               ,
               is
               when
               a
               Plain
               beholdeth
               the
               Zenith-point
               over
               our
               heads
               :
               But
               Inclination
               ,
               is
               ,
               when
               a
               Plain
               beholdeth
               the
               Nadier
               ;
               as
               in
               a
               Roof
               of
               a
               House
               ,
               the
               Tiled-part
               reclines
               ,
               and
               the
               Celid-part
               inclines
               .
            
             
               
               The
               Meridian-line
               ,
               on
               all
               Plains
               is
               the
               Hour-line
               of
               12
               ;
               but
               the
               Meridian
               of
               the
               Plain
               ,
               is
               the
               great
               Circle
               of
               Azimuth
               perpendiculer
               to
               the
               Plain
               ,
               bing
               the
               same
               with
               the
               Perpendiculer-line
               on
               the
               Plain
               ,
               passing
               through
               the
               Points
               of
               Declination
               .
            
             
               
               The
               Substile-line
               on
               all
               Dials
               ,
               is
               that
               Line
               wherein
               the
               Stile
               ,
               Gnomon
               ,
               or
               Cock
               of
               the
               Dial
               doth
               stand
               ,
               usually
               counted
               from
               12
               ,
               the
               Meridian-line
               ,
               or
               from
               the
               Perpendiculer-line
               ,
               
               which
               in
               all
               erect
               Dials
               is
               12.
               
            
             
               The
               Stile
               of
               a
               Dial
               ,
               is
               the
               Angle
               ,
               
               between
               the
               common
               Axis
               of
               the
               World
               and
               the
               Plain
               ,
               upon
               the
               Substile-line
               on
               the
               Plain
               ,
               on
               all
               Dials
               .
            
             
               The
               
                 Angle
                 between
              
               12
               and
               6
               ,
               
               is
               onely
               the
               number
               of
               degrees
               and
               minutes
               contained
               between
               the
               Hour-line
               of
               12
               ,
               and
               the
               Hour-line
               of
               6
               a
               clock
               ,
               on
               any
               kind
               of
               Plain
               ;
               especially
               those
               having
               Centers
               .
            
             
               The
               
                 Inclination
                 of
                 Meridians
              
               ,
               
               is
               the
               number
               of
               degrees
               and
               minutes
               ,
               counted
               on
               the
               Aequinoctial
               ,
               between
               the
               Meridian
               or
               Hour-line
               of
               12
               :
               and
               the
               Substile
               being
               the
               distance
               ,
               between
               the
               Meridian
               of
               the
               place
               ,
               viz.
               12
               a
               clock
               ,
               and
               the
               Meridian
               of
               the
               Plain
               ,
               but
               counted
               on
               the
               Aequinoctial
               ;
               and
               doth
               serve
               to
               make
               the
               Table
               of
               Hour-Arks
               at
               the
               Pole
               ,
               and
               to
               prove
               your
               work
               .
            
             
               The
               
                 Lines
                 Parallel
              
               to
               12
               ,
               
               are
               two
               Lines
               peculiar
               to
               this
               way
               of
               Dialling
               by
               the
               Sector
               ,
               and
               are
               only
               two
               Lines
               drawn
               equidistant
               from
               ,
               and
               parallel
               to
               the
               Hour-line
               of
               12.
               
            
             
               The
               Contingent
               or
               Touch-line
               in
               this
               way
               of
               Dialling
               with
               Centers
               ,
               
               is
               a
               Line
               drawn
               parallel
               to
               the
               Hour-line
               of
               6
               ;
               but
               in
               those
               without
               Centers
               ,
               it
               is
               drawn
               alwayes
               perpendiculer
               to
               the
               Substile
               ,
               and
               so
               may
               it
               be
               also
               ,
               if
               you
               please
               ,
               in
               those
               with
               Centers
               also
               .
            
             
             
               
               The
               
                 Vertical
                 Line
              
               on
               the
               Plain
               ,
               is
               the
               same
               with
               the
               Perpendiculer-line
               on
               the
               Plain
               ,
               being
               perpendiculer
               to
               the
               Horizontal-line
               .
            
             
               
               By
               the
               word
               Nodus
               ,
               is
               meant
               a
               Knot
               or
               Ball
               ,
               on
               the
               Axis
               or
               Stile
               of
               the
               Dial
               ,
               to
               make
               a
               black-shaddow
               on
               the
               Dial
               ,
               to
               trace
               out
               the
               Suns
               motion
               in
               the
               Heavens
               ;
               or
               sometimes
               an
               open
               or
               hollow-place
               in
               the
               Stile
               ,
               to
               leave
               a
               light-place
               to
               do
               the
               same
               office
               .
            
             
               But
               by
               Apex
               is
               meant
               the
               same
               thing
               ,
               when
               the
               Top-end
               ,
               or
               Point
               of
               an
               upright
               Stile
               shall
               shew
               the
               Hour
               and
               Suns
               place
               ,
               as
               the
               Spot
               doth
               in
               Celing-Dials
               ,
               where
               the
               Hours
               and
               Quarters
               are
               all
               of
               one
               length
               ,
               and
               distinguished
               by
               their
               tullours
               or
               greatness
               only
               .
            
             
               
               The
               
                 Perpendiculer
                 height
                 of
                 the
                 Stile
              
               ,
               is
               nothing
               else
               but
               the
               nearest
               distance
               from
               the
               Nodus
               or
               Apex
               to
               the
               Plain
               .
            
             
               The
               
                 Foot
                 of
                 the
                 Stile
              
               is
               properly
               right
               under
               the
               Nodus
               or
               Apex
               at
               the
               nearest
               distance
               .
               
            
             
               
               The
               Vertical-Point
               ,
               is
               a
               Point
               only
               used
               in
               Recliners
               and
               Incliners
               ,
               being
               a
               Point
               right
               over
               ,
               or
               under
               the
               Apex
               ;
               and
               yet
               in
               the
               Meridian
               ,
               being
               let
               fall
               from
               the
               Zenith
               ,
               by
               or
               through
               the
               Apex
               or
               Nodus
               ,
               to
               the
               Plain
               in
               the
               Meridian-line
               .
            
             
             
               The
               
                 Axis
                 of
                 the
                 Horizon
              
               ,
               
               is
               only
               the
               measure
               from
               the
               Apex
               to
               the
               Vertical-point
               last
               spoken
               to
               ,
               being
               the
               Secant
               of
               the
               complement
               of
               the
               Reclination
               to
               the
               Radius
               of
               the
               Perpendiculer
               height
               of
               the
               Stile
               .
            
             
               Erect
               ,
               is
               when
               Plains
               are
               upright
               ,
               
               as
               all
               Walls
               are
               intended
               to
               be
               .
            
             
               Direct
               ,
               
               is
               when
               the
               Dial-plain
               beholdeth
               one
               of
               the
               Four
               Cardinal
               Points
               of
               the
               Horizon
               ,
               as
               South
               or
               North
               ,
               East
               or
               West
               ,
               that
               is
               to
               say
               ,
               when
               the
               Pole
               of
               the
               Plain
               ,
               being
               90
               degrees
               every
               way
               from
               the
               Plain
               ,
               doth
               lie
               precisely
               in
               one
               of
               those
               Four
               Cardinal
               Azimuths
               :
               Which
               in
               an
               Erect
               and
               Direct-Plain
               will
               be
               in
               the
               Horizon
               .
            
             
               
                 Declining
                 ,
                 and
                 Reclining
              
               ,
               
               
                 or
                 Inclining-Plaines
              
               ,
               are
               as
               the
               upper
               or
               under-side
               of
               Roofs
               at
               any
               Oblique
               Scituation
               from
               the
               Cardinal
               Points
               of
               the
               Horizon
               .
            
             
               Oblique
               ,
               is
               only
               a
               wry
               ,
               slanting
               ,
               
               crooked
               ;
               contrary
               to
               direct
               ,
               right
               ,
               plain
               ,
               flat
               ,
               or
               perpendiculer
               ;
               and
               applied
               variously
               ,
               as
               to
               the
               Sphear
               ,
               to
               Triangles
               ,
               to
               Dial-plains
               ,
               to
               Discourse
               and
               Conversation
               .
            
             
               
                 Circles
                 of
                 Position
              
               ,
               
               or
               rather
               Semi-circles
               making
               12
               Houses
               ,
               are
               Circles
               ,
               whose
               Pole
               or
               Meeting-point
               is
               in
               the
               Meridian
               and
               Horizon
               of
               every
               Country
               ,
               dividing
               the
               Aequinoctial
               into
               12
               equal
               parts
               ,
               being
               then
               called
               Houses
               ,
               when
               used
               in
               Astrologie
               ,
               and
               
               some
               times
               drawn
               on
               Sun-Dials
               .
            
             
               But
               when
               they
               are
               used
               in
               Astronomy
               ,
               they
               require
               a
               more
               near
               account
               ,
               as
               to
               degrees
               and
               minutes
               .
            
          
           
             
               Of
               certain
               Terms
               in
               Astronomy
               ,
               and
               Spherical
               Definitions
               of
               Points
               and
               Lines
               in
               the
               Sphear
               .
            
             
               NOt
               to
               be
               curious
               in
               this
               matter
               ,
               a
               Sphear
               may
               be
               understood
               to
               be
               a
               united
               Spherical
               Superficies
               ,
               or
               round
               Body
               ,
               contained
               under
               one
               Surface
               ;
               in
               the
               middle
               whereof
               is
               a
               Point
               or
               Center
               ,
               from
               whence
               all
               Lines
               drawn
               to
               the
               Circumference
               are
               equal
               :
               Or
               you
               may
               conceive
               a
               Sphear
               to
               be
               an
               Instrument
               ,
               consisting
               of
               several
               Rings
               or
               Circles
               ,
               whereby
               ,
               the
               sensible
               motion
               of
               the
               Heavenly
               Bodies
               are
               conveniently
               represented
               .
            
             
               For
               the
               better
               Explanation
               whereof
               ,
               Astronomers
               have
               contrived
               thereon
               ,
               viz.
               on
               the
               Sphear
               ,
               ten
               imaginary
               Points
               ,
               and
               ten
               Circles
               ,
               which
               are
               usually
               drawn
               on
               Globes
               and
               Sphears
               ;
               besides
               others
               not
               usually
               drawn
               ,
               but
               apprehended
               in
               the
               fancy
               ,
               for
               
               Demonstrations-sake
               ,
               in
               Spherical
               Conclusions
               .
            
             
               The
               ten
               Points
               are
               ,
               the
               two
               Poles
               of
               the
               World
               ,
               the
               two
               Poles
               of
               the
               Zodiack
               ,
               the
               two
               Aequinoctial
               Points
               ,
               the
               two
               Solstitial
               Points
               ,
               the
               Zenith
               ,
               and
               Nadir
               .
            
             
               The
               ten
               Circles
               are
               ,
               The
               Horizon
               ,
               the
               Meridian
               ,
               the
               Aequinoctial
               ,
               the
               Zodiack
               ;
               the
               two
               Colures
               ,
               viz.
               that
               of
               the
               Equinoxes
               ,
               and
               that
               of
               the
               Solstices
               ;
               the
               Tropick
               of
               Cancer
               ,
               and
               the
               Tropick
               of
               Capricorn
               ;
               and
               the
               two
               Polar-Circles
               ,
               viz.
               The
               Artick
               or
               North
               ,
               the
               Antartick
               or
               South
               ,
               Polar-Circle
               .
            
             
               The
               
                 first
                 six
              
               ,
               are
               great
               Circles
               ,
               cutting
               the
               Sphear
               into
               two
               equal
               parts
               :
               And
               the
               
                 four
                 last
              
               are
               lesser
               Circles
               ,
               dividing
               the
               Sphear
               unequally
               .
            
             
               All
               which
               Points
               and
               Circles
               shall
               be
               represented
               by
               the
               Figure
               of
               the
               Analemma
               ,
               from
               whence
               the
               
                 Trianguler
                 Quadrant
              
               is
               derived
               ,
               as
               a
               general
               Instrument
               ,
               and
               also
               by
               the
               Horizontal
               projection
               of
               the
               Sphear
               fitted
               for
               London
               ,
               being
               better
               for
               the
               fancy
               to
               apprehend
               the
               Mystery
               of
               Dialling
               ,
               one
               thing
               mainly
               intended
               in
               this
               Discourse
               .
            
          
           
             
             
               Of
               the
               10
               Points
               in
               the
               Sphear
               .
            
             
               
               THe
               two
               Poles
               of
               the
               World
               ,
               are
               the
               two
               Points
               P
               and
               P
               in
               the
               Analemma
               ,
               
               being
               directly
               opposite
               one
               to
               another
               ;
               
               about
               which
               two
               Points
               ,
               the
               whole
               frame
               of
               the
               Heavens
               moveth
               from
               East
               to
               West
               ;
               one
               of
               which
               Poles
               may
               alwayes
               be
               seen
               by
               us
               ,
               called
               the
               Artick
               or
               North-Pole
               ;
               represented
               in
               the
               particular
               Scheam
               by
               the
               Point
               P.
               The
               other
               being
               not
               seen
               ,
               is
               not
               represented
               in
               the
               particular
               Scheam
               ;
               but
               the
               Line
               PEP
               ,
               in
               the
               general
               Scheam
               ,
               drawn
               from
               Pole
               to
               Pole
               ,
               is
               called
               the
               Axis
               ▪
               or
               Axeltree
               of
               the
               World
               ,
               because
               the
               whole
               Sphear
               appears
               to
               move
               round
               about
               it
               .
            
             
               
               The
               Poles
               of
               the
               Zodiack
               are
               two
               Points
               diametrically
               opposite
               also
               ,
               upon
               which
               Points
               the
               Heavens
               move
               slowly
               from
               West
               
               to
               East
               ,
               represented
               by
               the
               two
               Points
               ,
               I
               
               and
               K
               ,
               23
               degrees
               and
               31
               minutes
               distant
               from
               the
               two
               former
               Poles
               ,
               in
               the
               Analemma
               ,
               and
               by
               the
               Point
               PZ
               in
               the
               Horizontal
               projection
               ;
               but
               the
               other
               Pole
               of
               the
               Zodiack
               cannot
               be
               represented
               in
               that
               particular
               Scheam
               .
            
             
               
               The
               Equinoctial
               Points
               ,
               are
               the
               Points
               of
               Aries
               and
               Libra
               ;
               to
               which
               two
               Points
               ,
               
               
               
               when
               the
               Sun
               cometh
               along
               the
               Ecliptick
               ,
               it
               maketh
               the
               Dayes
               and
               Nights
               equal
               in
               all
               places
               ;
               at
               Aries
               March
               10th
               or
               11th
               ;
               to
               Libra
               about
               the
               13th
               of
               September
               ,
               where
               the
               Spring
               ,
               and
               Autumn
               begins
               ;
               being
               represented
               in
               the
               Analemma
               by
               the
               Point
               ●
               ,
               and
               in
               the
               particular
               Scheam
               by
               the
               Points
               E
               ,
               and
               W.
               
            
             
               The
               two
               Solsticial
               Points
               ,
               are
               represented
               one
               by
               the
               Point
               ♋
               ,
               
               and
               the
               other
               by
               the
               Point
               ♑
               ,
               in
               both
               Scheams
               ;
               to
               which
               
               Points
               when
               the
               Sun
               cometh
               along
               the
               Ecliptick
               ,
               
               it
               makes
               the
               Dayes
               in
               Cancer
               ♋
               ,
               longest
               ;
               in
               Capricorn
               ♑
               ,
               shortest
               ;
               ♋
               being
               about
               the
               11th
               of
               Iune
               ;
               and
               ♑
               about
               the
               11th
               of
               December
               .
            
             
               The
               Zenith
               is
               an
               imaginary
               Point
               right
               over
               our
               heads
               ,
               
               being
               every
               way
               90
               degrees
               
               distant
               form
               the
               Horizon
               ;
               in
               which
               Point
               all
               Azimuth
               Lines
               do
               meet
               ,
               represented
               by
               the
               Points
               Z
               ,
               in
               both
               Scheams
               .
            
             
               The
               Nadir
               is
               an
               imaginary
               Point
               under
               our
               feet
               ,
               directly
               opposite
               to
               the
               Zenith
               ,
               
               
               represented
               by
               the
               Point
               N
               in
               the
               Analemma
               ,
               but
               not
               in
               the
               particular
               Scheam
               ,
               because
               it
               is
               not
               seen
               at
               any
               time
               .
            
          
           
             
             
               Of
               the
               Circles
               of
               the
               Sphear
               .
            
             
               
               THe
               Horizon
               is
               twofold
               ,
               
                 viz.
                 Rational
              
               ,
               and
               Sensible
               :
               The
               Rational
               Horizon
               ,
               is
               an
               imaginary
               great
               Circle
               of
               the
               Sphear
               ,
               every
               where
               90
               degrees
               distant
               from
               the
               Zenith
               ,
               and
               Nadir
               ;
               Points
               cutting
               ,
               or
               dividing
               the
               whole
               Sphear
               into
               two
               equal
               parts
               ,
               the
               one
               called
               ,
               The
               upper
               or
               visible
               Hemisphear
               ;
               the
               other
               the
               lower
               or
               invinsible
               Hemisphear
               .
            
             
               This
               Rational
               Horizon
               ,
               is
               distinguished
               also
               into
               Right
               ,
               Oblique
               ,
               and
               Parallel-Horizon
               .
            
             
               1.
               
               The
               Right
               Horizon
               is
               when
               the
               two
               Poles
               of
               the
               World
               lie
               in
               the
               Horizon
               ,
               and
               the
               Equinoctial
               at
               Right
               Angles
               to
               it
               ;
               which
               Horizon
               is
               peculiar
               to
               those
               that
               live
               under
               the
               Equinoctial
               ,
               who
               have
               their
               Dayes
               and
               Nights
               alwayes
               equal
               ,
               and
               all
               the
               Stars
               to
               Rise
               and
               Set
               ,
               and
               the
               Sun
               to
               pass
               twice
               in
               a
               year
               by
               their
               Zenith-point
               ,
               thereby
               making
               two
               Winters
               ,
               and
               two
               Summers
               ;
               Their
               Winters
               being
               in
               Iune
               and
               December
               ,
               and
               their
               Summers
               ,
               in
               March
               and
               September
               .
            
             
               2.
               
               The
               Oblique
               Horizon
               is
               when
               one
               Pole-point
               is
               visible
               ,
               and
               (
               the
               other
               not
               )
               having
               E●evation
               above
               ,
               and
               depression
               below
               
               the
               North
               or
               South
               part
               of
               the
               Horizon
               ,
               according
               to
               the
               Latitude
               of
               the
               place
               :
               in
               which
               Horizon
               when
               the
               Sun
               cometh
               to
               the
               Equinoctial
               ,
               the
               Dayes
               and
               Nights
               are
               only
               then
               equal
               ;
               and
               the
               nearer
               the
               Sun
               comes
               to
               the
               visible
               Pole
               ,
               the
               Dayes
               are
               the
               longer
               ,
               and
               the
               contrary
               ;
               also
               some
               Stars
               never
               set
               ,
               and
               some
               never
               rise
               in
               that
               Horizon
               :
               And
               all
               Horizons
               but
               two
               ,
               are
               in
               a
               strict
               sense
               Oblique-Horizons
               ,
               viz.
               The
               Right
               Horizon
               already
               spoken
               to
               :
               And
            
             
               The
               Parallel
               Horizon
               ,
               is
               that
               Horizon
               which
               hath
               the
               Equinoctial
               for
               its
               Horizon
               ,
               and
               one
               of
               the
               Pole-points
               for
               its
               Zenith
               ;
               peculiar
               only
               to
               those
               Inhabitants
               under
               the
               Pole
               ,
               (
               if
               any
               be
               there
               .
               )
               In
               which
               Horizon
               ,
               ona
               half
               of
               the
               Sphear
               doth
               only
               alwayes
               appear
               ,
               and
               the
               other
               half
               alwayes
               is
               hid
               ;
               and
               the
               Sun
               ,
               for
               one
               half
               year
               ,
               doth
               go
               round
               about
               like
               a
               Skrew
               ,
               making
               it
               continual
               Day
               ,
               and
               the
               other
               half
               year
               is
               continual
               Night
               ,
               and
               cold
               enough
               ;
               which
               Circle
               in
               the
               Analemma
               is
               represented
               by
               the
               Line
               HES
               ,
               but
               in
               the
               particular
               Scheam
               by
               the
               Circle
               NESW
               .
            
             
             
               The
               Meridian
               is
               a
               great
               Circle
               which
               passeth
               through
               the
               two
               Pole-points
               ,
               the
               Zenith
               and
               Nadir
               ,
               and
               the
               North
               and
               South-points
               of
               the
               Horizon
               ,
               and
               is
               called
               Meridian
               ,
               becuse
               when
               the
               Sun
               (
               or
               Stars
               )
               cometh
               to
               that
               Circle
               ,
               it
               maketh
               Mid-day
               ,
               or
               Mid-night
               ,
               which
               is
               twice
               in
               every
               24
               hours
               :
               Also
               all
               places
               ,
               North
               and
               South
               ,
               have
               the
               same
               Meridian
               ;
               but
               places
               that
               lie
               Eastward
               ,
               or
               Westwards
               ,
               have
               several
               Meridians
               .
               Also
               ,
               when
               the
               Sun
               or
               Stars
               come
               to
               the
               South
               ,
               or
               North-part
               of
               the
               Meridian
               ,
               their
               Altitudes
               are
               then
               highest
               ,
               and
               lowest
               .
               And
               the
               difference
               of
               Meridians
               is
               the
               difference
               of
               Longitudes
               of
               Places
               ,
               noted
               by
               the
               Circle
               ZHNS
               in
               the
               Analemma
               ;
               and
               NZ
               ♋
               S
               in
               Horizontal-projection
               .
            
             
               The
               Equinoctial
               is
               a
               great
               Circle
               ,
               every
               where
               90
               degrees
               distant
               from
               the
               two
               Poles
               of
               the
               World
               ,
               dividing
               the
               Sphear
               into
               two
               halfs
               ,
               called
               the
               North
               and
               South
               Hemisphear
               ;
               and
               is
               called
               also
               the
               Aequator
               ,
               because
               when
               the
               Sun
               passeth
               by
               it
               twice
               a
               year
               ,
               it
               makes
               the
               day
               and
               nights
               equal
               in
               all
               places
               ;
               noted
               by
               WAEE
               ,
               and
               AEEAE
               in
               both
               .
            
             
             
               The
               Zodiack
               ,
               or
               Signifer
               ,
               
               is
               another
               great
               Circle
               that
               divides
               the
               Sphear
               and
               Equinoctial
               into
               two
               equal
               parts
               ,
               
               whose
               Poles
               are
               the
               Poles
               of
               the
               Zodiack
               ,
               being
               90
               degrees
               from
               it
               ;
               and
               it
               inter-sects
               the
               Equinoctial
               in
               the
               two
               Points
               of
               Aries
               and
               Libra
               ;
               and
               one
               part
               of
               it
               doth
               decline
               Northward
               ,
               and
               the
               other
               Southward
               ,
               23
               degrees
               31
               minutes
               ,
               as
               the
               Poles
               of
               the
               Zodiack
               decline
               from
               the
               North
               and
               South-Poles
               of
               the
               World
               :
               The
               breadth
               of
               this
               Zodiack
               ,
               or
               Girdle
               ,
               is
               counted
               14
               or
               16
               degrees
               ,
               to
               allow
               for
               the
               wandring
               of
               
                 Luna
                 ,
                 Mars
              
               ,
               and
               Venus
               ;
               the
               middle
               of
               which
               breadth
               is
               the
               Ecliptick-Line
               ,
               because
               all
               Eclipses
               are
               in
               ,
               or
               very
               near
               in
               this
               Line
               .
               And
               this
               Circle
               is
               divided
               into
               12
               Signs
               ,
               and
               each
               Sign
               into
               30
               degrees
               ,
               according
               to
               their
               Names
               and
               Characters
               ,
               ♈
               
                 Aries
                 ,
                 ♉
                 Taurus
                 ,
                 ♊
                 Gemini
                 ,
                 ♋
                 Cancer
                 ;
                 ,
                 ♌
                 Leo
                 ,
                 ♍
                 Virgo
                 ,
                 ♎
                 Libra
                 ,
                 ♏
                 Scorpio
                 ,
                 ♐
                 Sagittarius
                 ,
                 ♑
                 Capricornius
                 ,
                 ♒
                 Aquarius
                 ,
                 ♓
                 Pisces
                 .
              
               6
               being
               Northern
               ,
               and
               the
               6
               latter
               Southern
               .
            
             
               The
               two
               Colures
               are
               only
               two
               Meridians
               ,
               or
               great
               Circles
               ,
               
               crossing
               one
               another
               at
               Right
               Angles
               ;
               
               the
               one
               Colure
               passing
               through
               the
               Poles
               of
               the
               World
               ,
               and
               the
               Points
               of
               Aries
               and
               Libra
               ,
               there
               cutting
               the
               Equinoctial
               and
               Ecliptick
               :
               And
               the
               other
               
               Colure
               passeth
               by
               the
               Poles
               of
               the
               World
               also
               ,
               and
               cuts
               the
               Ecliptick
               in
               ♋
               ,
               and
               ♑
               ,
               making
               the
               Four
               Seasons
               of
               the
               year
               ;
               that
               is
               ,
               the
               equal
               Dayes
               ,
               called
               the
               Equinoctial-Colure
               ;
               and
               the
               unequal
               Dayes
               ,
               in
               Iune
               and
               December
               ,
               called
               the
               Solsticial-Colures
               ,
               represented
               in
               the
               Analemma
               by
               ZP
               ♋
               NS
               ,
               and
               PEP
               ;
               and
               in
               the
               particular
               Scheam
               by
               WPE
               ,
               and
               NPS
               ,
               the
               Solsticial-Colure
               .
            
             
               
               The
               lesser
               Circles
               are
               the
               Tropicks
               of
               ♋
               ,
               and
               ♑
               ;
               
               being
               the
               Lines
               of
               the
               Suns
               motion
               in
               the
               longest
               and
               shortest
               dayes
               ,
               noted
               in
               the
               Scheams
               by
               ♋
               ,
               ☉
               ,
               ♋
               ,
               and
               6
               ♋
               E
               ,
               and
               ♑
               ,
               ♑
               ;
               and
               W
               ♑
               6
               ;
               to
               which
               two
               Circles
               when
               the
               Sun
               cometh
               ,
               it
               is
               on
               the
               11th
               of
               Iune
               ,
               and
               the
               11th
               of
               December
               ,
               making
               the
               Summer
               and
               Winter
               Solstice
               .
            
             
               
               The
               
                 Polar
                 Circles
              
               ,
               are
               two
               Circles
               drawn
               about
               the
               Poles
               of
               the
               World
               ,
               as
               far
               off
               as
               the
               Poles
               of
               the
               Zodiack
               are
               ,
               
               viz.
               23
               degrees
               ,
               31
               minutes
               ;
               That
               about
               the
               North-Pole
               is
               called
               the
               Artick
               ,
               and
               that
               about
               the
               South
               the
               Antartick
               ,
               being
               opposite
               thereunto
               ,
               shewed
               in
               the
               Analemma
               by
               II
               ,
               and
               KK
               ;
               and
               by
               the
               small
               Circle
               about
               P
               in
               the
               particular
               Scheam
               .
            
          
           
             
             
               Of
               the
               other
               Circle
               imagined
               ,
               but
               not
               described
               on
               Sphears
               or
               Globes
               .
            
             
               1.
               
               HOurs
               are
               great
               Circles
               ,
               
               passing
               through
               the
               two
               Poles
               ,
               and
               cutting
               the
               Equinoctial
               in
               24
               equal
               parts
               ,
               as
               the
               Lines
               P1
               ,
               P2
               ,
               P3
               ,
               &c.
               in
               the
               Particular
               ;
               and
               P
               ☉
               H
               in
               the
               Analemma
               ;
               such
               also
               are
               degrees
               of
               Longitude
               ,
               and
               Meridians
               ;
               the
               Meridian
               being
               the
               hour
               of
               12.
               
            
             
               2.
               
               Azimuths
               are
               great
               Circles
               ,
               
               passing
               through
               ,
               or
               meeting
               in
               the
               Zenith
               and
               Nadir-points
               ,
               numbred
               and
               counted
               on
               the
               Horizon
               ,
               from
               the
               Four
               Cardinal
               Points
               of
               North
               and
               South
               ,
               East
               and
               West
               ,
               according
               to
               Four
               90ties
               ,
               or
               180
               degrees
               ,
               or
               according
               to
               the
               32
               Rombs
               or
               Points
               of
               the
               Compass
               ,
               as
               Z
               ☉
               A
               ,
               and
               ZE
               ,
               the
               Azimuth
               of
               East
               and
               West
               ,
               being
               called
               the
               prime
               Virtical
               ,
               viz.
               SE
               ,
               WZ
               .
            
             
               3.
               
               Almicanters
               ,
               or
               Circles
               of
               Altitude
               ,
               
               are
               lesser
               Circles
               ,
               all
               parallel
               to
               the
               Horizon
               ,
               counted
               on
               any
               Azimuth
               from
               the
               Horizon
               to
               the
               Zenith
               ,
               to
               measure
               the
               Altitude
               of
               the
               Sun
               ,
               Moon
               ,
               or
               Stars
               above
               the
               Horizon
               ,
               being
               the
               Portion
               of
               some
               Azimuth
               ,
               between
               the
               Center
               of
               the
               Sun
               ,
               or
               Star
               ;
               and
               the
               
               Horizon
               ,
               commonly
               called
               its
               Altitude
               above
               the
               Horizon
               ,
               showed
               by
               A
               ☉
               in
               the
               Analemma
               ,
               and
               SAE
               in
               the
               particular
               Scheam
               .
            
             
               
               4.
               
               Parallels
               of
               Declination
               ,
               are
               parallels
               to
               the
               Equinoctial
               ,
               as
               the
               Almicanters
               were
               parallel
               to
               the
               Horizon
               ,
               as
               ♋
               ☉
               ♋
               ,
               the
               greatest
               Declination
               or
               Circle
               of
               ♋
               :
               These
               parallels
               have
               the
               2
               Poles
               of
               the
               World
               for
               their
               Centers
               ,
               and
               in
               respect
               of
               the
               Sun
               or
               Stars
               are
               called
               degrees
               of
               Declination
               ;
               but
               in
               respect
               of
               the
               Earth
               ,
               degrees
               of
               Latitude
               ;
               being
               the
               Arch
               on
               the
               Meridian
               of
               any
               place
               ,
               between
               the
               Pole
               and
               Horizon
               ,
               as
               4
               ♋
               4
               in
               the
               Particular
               ,
               and
               HP
               ,
               in
               the
               Analemma
               .
            
             
               
               5.
               
               Parallels
               of
               Latitude
               ,
               in
               respect
               of
               the
               Stars
               ,
               are
               Lines
               drawn
               parallel
               to
               the
               Ecliptick
               ,
               as
               the
               Almicanters
               were
               parallel
               to
               the
               Horizon
               ;
               so
               that
               the
               Latitude
               of
               a
               Star
               is
               counted
               from
               the
               Ecliptick
               toward
               the
               Poles
               of
               the
               Zodiack
               ;
               but
               the
               Sun
               being
               alwayes
               in
               the
               Ecliptick
               ,
               is
               said
               to
               have
               no
               Latitude
               .
            
             
               6.
               
               Degrees
               of
               Longitude
               ,
               in
               respect
               of
               the
               Heavens
               ,
               
               are
               measured
               by
               the
               degrees
               on
               the
               Ecliptick
               ,
               from
               the
               first
               point
               of
               Aries
               forward
               ,
               according
               to
               the
               succession
               of
               the
               12
               Signs
               of
               the
               Zodiack
               .
            
             
             
               But
               Longitude
               on
               the
               Earth
               ,
               is
               counted
               on
               the
               Equinoctial
               Eastwards
               ,
               from
               some
               principal
               Meridian
               on
               the
               Earth
               ,
               as
               the
               Isles
               of
               Azores
               ,
               or
               the
               Peak
               of
               Tenneriff
               ,
               or
               the
               like
               .
            
             
               7.
               
               
                 Right
                 Ascention
              
               is
               an
               Arch
               of
               the
               Equinoctial
               (
               counted
               from
               the
               first
               Point
               of
               Aries
               )
               that
               cometh
               to
               the
               Meridian
               with
               the
               Sun
               ,
               Moon
               ,
               or
               Stars
               ,
               at
               any
               day
               ,
               
               or
               time
               of
               the
               year
               ,
               being
               much
               used
               in
               the
               following
               discourse
               ,
               noted
               in
               the
               Analemma
               by
               EH
               ,
               or
               the
               like
               ;
               but
               counted
               as
               afterward
               is
               shewed
               .
            
             
               8.
               
               
                 Oblique
                 Ascention
              
               is
               an
               Arch
               of
               the
               Equinoctial
               ,
               
               between
               the
               beginning
               of
               Aries
               ,
               and
               that
               part
               of
               the
               Equinoctial
               that
               riseth
               with
               the
               Center
               of
               a
               Star
               ,
               or
               any
               portion
               of
               the
               Ecliptick
               in
               an
               Oblique-Sphear
               .
            
             
               9.
               
               
                 Ascentional
                 Difference
              
               ,
               is
               the
               difference
               between
               the
               Right
               and
               Oblique
               Ascention
               ,
               
               to
               find
               the
               Sun
               or
               Stars
               rising
               before
               or
               after
               6.
               
            
             
               10.
               
               Amplitude
               is
               an
               Arch
               of
               the
               Horizon
               ,
               
               between
               the
               Center
               of
               the
               Sun
               and
               the
               true
               East-point
               ,
               at
               the
               very
               moment
               of
               Rising
               ,
               represented
               by
               ♋
               F
               ,
               in
               the
               particular
               Scheam
               ,
               and
               GE
               ,
               and
               FE
               in
               the
               Analemma
               :
               useful
               at
               Sea.
               
            
             
             
               
               11.
               
               A
               
                 Circle
                 of
                 Position
              
               is
               one
               of
               the
               12
               Houses
               in
               Astronomy
               or
               Astrology
               .
            
             
               12.
               
               An
               
                 Angle
                 of
                 Position
              
               ,
               is
               the
               Angle
               made
               in
               the
               Center
               of
               the
               Sun
               ,
               between
               his
               Meridian
               ,
               or
               Hour
               ,
               and
               some
               Azimuth
               ,
               as
               the
               Prime
               ,
               Vertical
               ,
               or
               the
               Meridian
               ,
               or
               any
               other
               Azimuth
               ,
               being
               useful
               in
               Astronomy
               ,
               and
               sometimes
               in
               Calculation
               ,
               represented
               bp
               P
               ☉
               Z
               in
               the
               Analemma
               .
            
             
               Thus
               much
               for
               Astronomical
               terms
               .
            
          
        
         
           
             CHAP.
             V.
             Some
             Uses
             of
             the
             
               Trianguler
               Quadrant
            
             .
          
           
             
               Use
               I.
               And
               first
               to
               rectifie
               the
               Rule
               ,
               or
               make
               it
               a
               Trianguler
               Quadrant
               .
            
             
               FIrst
               open
               the
               Rule
               ,
               and
               put
               in
               the
               loose
               piece
               into
               the
               two
               Mortice-holes
               ,
               (
               which
               putting
               together
               makes
               it
               a
               
                 Trianguler
                 Quadrant
              
               )
               but
               if
               you
               do
               not
               use
               the
               loose-piece
               ,
               then
               open
               it
               to
               an
               Angle
               ▪
               of
               60
               degrees
               ,
               which
               is
               thus
               exactly
               done
               :
               Measure
               from
               the
               Rectifying-point
               ,
               to
               any
               Number
               
               on
               the
               Sines
               or
               Lines
               ;
               then
               keeping
               the
               Point
               of
               the
               Compasses
               still
               fixed
               in
               the
               Rectifying-point
               ,
               turn
               the
               other
               to
               the
               Common-Line
               of
               the
               Hour
               and
               Azimuth-Line
               ,
               that
               cuts
               the
               Rectifying-point
               ,
               and
               there
               keep
               it
               ;
               then
               removing
               the
               Point
               of
               the
               Compasses
               from
               the
               Rectifying-point
               ,
               open
               or
               close
               the
               Rule
               till
               the
               other
               Point
               shall
               touch
               the
               distance
               first
               measured
               in
               the
               Line
               of
               Sines
               or
               Lines
               ,
               then
               shall
               you
               see
               the
               Lines
               on
               the
               Head
               ,
               and
               Moveable-leg
               ,
               to
               meet
               ;
               and
               also
               see
               quite
               through
               the
               Rectifying-point
               ,
               to
               thrust
               a
               Pin
               quite
               through
               ;
               and
               thus
               is
               it
               set
               to
               an
               Angle
               of
               60
               degrees
               ,
               without
               the
               help
               of
               the
               loose-piece
               ,
               or
               to
               an
               Angle
               of
               45
               ,
               or
               whatsoever
               else
               the
               Rule
               is
               made
               for
               .
            
          
           
             
               Use
               II.
               To
               observe
               the
               Sun
               or
               a
               Stars
               Altitude
               above
               the
               Horizon
               .
            
             
               Put
               a
               Pin
               in
               the
               Center-hole
               on
               the
               Head-Leg
               ,
               and
               another
               in
               the
               Rectifying-point
               ,
               and
               a
               third
               (
               if
               you
               please
               )
               in
               the
               end
               of
               the
               Hour-line
               on
               the
               Moving-leg
               .
               Then
               on
               the
               Pin
               in
               the
               Leg-center
               ,
               hang
               a
               Thred
               and
               Plummet
               ;
               then
               if
               the
               object
               be
               low
               ,
               viz.
               under
               25
               degrees
               high
               ;
               Look
               along
               by
               the
               two
               Pins
               in
               the
               Rectifying-point
               ,
               
               and
               the
               Moving-leg
               ,
               and
               see
               that
               the
               Plummet
               playeth
               evenly
               and
               steady
               ,
               then
               the
               degrees
               cut
               by
               the
               Thred
               ▪
               shall
               be
               the
               Altitude
               required
               ,
               counting
               from
               60
               /
               0
               toward
               the
               Head
               ,
               as
               the
               smaller
               Figures
               shew
               .
            
             
               But
               if
               the
               Object
               be
               above
               25
               degrees
               high
               ,
               then
               look
               by
               the
               Pin
               in
               the
               Rectifying-point
               ,
               and
               that
               on
               which
               the
               Plummet
               hangeth
               ;
               and
               observe
               as
               before
               ,
               and
               the
               Thred
               shall
               shew
               the
               Altitude
               required
               ,
               as
               the
               Figures
               before
               the
               Line
               sheweth
               ;
               If
               you
               have
               Sights
               ,
               use
               them
               instead
               of
               Pins
               ;
               and
               by
               Practice
               learn
               to
               be
               accurate
               in
               this
               Work
               ,
               the
               ground
               and
               foundation
               in
               every
               Observation
               ;
               and
               according
               to
               your
               exactness
               herein
               ,
               is
               the
               following
               Work
               also
               .
            
             
               Note
               also
               ,
               that
               this
               looking
               up
               toward
               the
               Sun
               ,
               is
               only
               then
               when
               the
               Sun
               is
               in
               a
               cloud
               ,
               and
               may
               be
               seen
               in
               the
               Abiss
               ,
               but
               will
               not
               give
               a
               clear
               shadow
               :
               Or
               else
               you
               must
               use
               a
               piece
               of
               Red
               ,
               or
               a
               Blue
               ,
               or
               Green
               Glass
               ,
               to
               darken
               the
               luster
               that
               it
               offend
               not
               the
               eyes
               .
            
             
               But
               if
               the
               Sun
               be
               clear
               and
               bright
               ,
               then
               you
               need
               not
               look
               up
               toward
               it
               ,
               but
               hold
               the
               
                 Trianguler
                 Quadrant
              
               so
               ,
               that
               the
               shadow
               of
               the
               Pin
               in
               the
               Center
               may
               fall
               just
               on
               the
               shadow
               of
               the
               Pin
               in
               the
               Rectifying-point
               ,
               and
               both
               those
               shadows
               on
               your
               
               finger
               beyond
               them
               ,
               and
               the
               Plummet
               being
               somewhat
               heavy
               ,
               and
               the
               Thred
               small
               and
               playing
               evenly
               by
               the
               Rule
               ,
               then
               is
               the
               Observation
               so
               made
               ,
               likely
               to
               be
               near
               the
               very
               truth
               .
            
             
               Note
               also
               further
               ,
               That
               the
               shaking
               of
               the
               hand
               ,
               you
               shall
               find
               will
               hinder
               exactness
               ;
               therefore
               ,
               when
               you
               may
               ,
               find
               some
               place
               to
               lean
               your
               Body
               ,
               or
               Arm
               ,
               or
               the
               Instrument
               against
               ,
               that
               you
               may
               be
               the
               more
               steady
               .
            
             
               But
               the
               surest
               and
               best
               way
               is
               with
               a
               Ball-socket
               ,
               and
               a
               Three-leg-staff
               ,
               such
               as
               Land
               Surveighers
               use
               to
               support
               their
               Instruments
               withal
               ,
               then
               you
               will
               be
               at
               liberty
               to
               move
               and
               remove
               it
               ,
               to
               and
               fro
               ,
               till
               the
               Sights
               or
               Pins
               ,
               and
               Plummet
               and
               Thred
               play
               to
               exactness
               ;
               without
               which
               care
               and
               exactness
               ,
               you
               cannot
               certainly
               and
               knowingly
               attain
               the
               Sun's
               or
               a
               Star's
               Altitude
               to
               a
               minute
               ,
               either
               by
               this
               nor
               any
               other
               Instrument
               whatsoever
               ,
               though
               they
               be
               never
               so
               truly
               made
               :
               Yet
               I
               dare
               affirm
               to
               do
               it
               ,
               or
               it
               may
               be
               done
               as
               well
               by
               this
               ,
               as
               by
               any
               other
               graduated
               Instrument
               whatsoever
               :
               The
               Line
               of
               degrees
               on
               this
               ,
               being
               only
               two
               thirty
               degrees
               of
               a
               Tangent
               laid
               together
               ;
               of
               which
               ,
               that
               on
               the
               in-side
               of
               the
               loose-piece
               is
               the
               largest
               ,
               and
               consequently
               the
               best
               ,
               to
               distinguish
               the
               minuts
               of
               a
               degree
               withal
               .
            
          
           
             
             
               Use
               III.
               To
               try
               if
               any
               thing
               be
               Level
               ,
               or
               Upright
               .
            
             
               Set
               the
               Moving-leg
               of
               the
               
                 Trianguler
                 Quadrant
              
               on
               the
               thing
               you
               would
               have
               to
               be
               Level
               ;
               then
               if
               the
               Thred
               play
               just
               on
               60
               degrees
               ,
               or
               the
               stroke
               by
               600
               ,
               then
               is
               it
               Level
               ,
               or
               else
               not
               .
            
             
               But
               to
               try
               if
               a
               thing
               be
               upright
               or
               not
               ▪
               apply
               the
               Head-leg
               to
               the
               Wall
               or
               Post
               ,
               and
               if
               it
               be
               upright
               ,
               the
               Thred
               will
               play
               just
               on
               the
               common
               Line
               between
               the
               Lines
               and
               Sines
               on
               the
               Head-leg
               ,
               and
               cut
               the
               stroke
               by
               90
               on
               the
               Head
               of
               the
               Instrument
               ,
               or
               else
               not
               .
            
          
           
             
               Use
               IV.
               To
               find
               readily
               what
               Angle
               the
               Sector
               stands
               at
               ,
               at
               any
               opening
               .
            
             
               First
               ,
               on
               the
               Sector
               side
               ,
               about
               the
               Head
               ,
               is
               180
               degrees
               ,
               or
               twice
               90
               graduated
               to
               every
               two
               degrees
               ;
               so
               that
               opening
               the
               Rule
               to
               any
               Angle
               ,
               the
               in-side
               of
               the
               Moving-leg
               ,
               passing
               about
               the
               semi-circle
               of
               the
               Head
               ,
               sheweth
               the
               Angle
               of
               opening
               to
               one
               degree
               .
               But
               to
               do
               it
               more
               exactly
               ,
               do
               thus
               :
            
             
             
               The
               two
               Lines
               of
               Sines
               that
               issue
               from
               the
               Center
               in
               Rules
               of
               a
               Foot
               ,
               shut
               ,
               are
               drawn
               usually
               just
               5
               degrees
               assunder
               ;
               or
               rather
               the
               two
               innermost
               Lines
               ,
               on
               each
               Leg
               ,
               are
               always
               just
               one
               degree
               from
               the
               inside
               ,
               so
               that
               if
               you
               put
               a
               Center-pin
               in
               the
               Line
               of
               Tangents
               ,
               just
               against
               the
               Sine
               of
               30
               ,
               it
               makes
               the
               two
               innermost
               Lines
               that
               come
               from
               the
               Center
               ,
               just
               2
               degrees
               assunder
               ,
               which
               is
               easie
               to
               remember
               either
               in
               adding
               or
               substracting
               as
               followeth
               ,
               two
               wayes
               .
            
             
               1.
               
               Take
               the
               Latteral
               Sine
               of
               30
               ,
               viz.
               the
               measure
               from
               the
               Center
               to
               30
               :
               the
               Compasses
               so
               set
               ,
               set
               one
               Point
               in
               the
               Center-pin
               in
               the
               Tangents
               just
               against
               30
               ;
               and
               turn
               the
               other
               till
               it
               cut
               the
               common
               Line
               ,
               in
               the
               Line
               of
               Sines
               on
               the
               other
               Leg
               ,
               and
               there
               it
               shall
               shew
               what
               Angle
               the
               two
               innermost-Lines
               make
               ,
               counting
               from
               the
               end
               toward
               the
               Head
               ,
               and
               two
               degrees
               less
               is
               the
               Angle
               the
               Sector
               stands
               at
               ,
               both
               on
               the
               in-side
               and
               out-side
               ,
               the
               Legs
               being
               parallel
               ;
               which
               Number
               must
               nearly
               agree
               with
               what
               the
               in-side
               of
               the
               Leg
               cuts
               on
               the
               Head-semicircle
               ,
               or
               there
               is
               a
               mistake
               .
            
             
             
               
                 As
                 thus
                 for
                 Example
              
               .
            
             
               Suppose
               I
               open
               the
               Rule
               at
               all
               adventures
               ▪
               and
               taking
               the
               Latteral
               Sine
               of
               30
               from
               the
               Sines
               on
               the
               Sector-side
               ,
               and
               putting
               one
               Point
               of
               the
               Compass
               in
               the
               Center
               on
               the
               Tangents
               ,
               right
               against
               the
               Sine
               of
               30
               on
               the
               other
               Leg
               (
               or
               the
               beginning
               of
               the
               Secants
               on
               the
               same
               Leg
               )
               and
               turning
               the
               other
               Point
               to
               the
               Line
               of
               Sines
               on
               the
               other
               Leg
               ,
               it
               cuts
               the
               Sine
               of
               60
               on
               the
               innermost
               Line
               that
               comes
               from
               the
               Center
               ;
               then
               I
               say
               ,
               that
               the
               Lines
               of
               Sines
               and
               Tangents
               are
               just
               30
               degrees
               assunder
               ,
               and
               the
               in-side
               or
               out-side
               of
               the
               Legs
               but
               28
               ,
               viz.
               two
               degrees
               less
               ,
               as
               a
               glance
               of
               your
               eye
               to
               the
               Head
               will
               plainly
               shew
               .
            
             
               2.
               
               This
               way
               will
               serve
               very
               well
               for
               all
               Angles
               above
               20
               ,
               and
               under
               80
               :
               But
               for
               all
               under
               20
               ,
               and
               above
               80
               ,
               to
               120
               ,
               this
               is
               a
               better
               way
               ;
            
             
               Open
               the
               Rule
               to
               any
               Angle
               at
               pleasure
               ,
               and
               take
               the
               distance
               parallelly
               (
               that
               is
               ,
               across
               from
               one
               Leg
               to
               the
               other
               )
               between
               the
               Center-pin
               at
               30
               in
               the
               Sines
               ,
               and
               that
               in
               the
               Tangents
               right
               against
               it
               ,
               and
               measure
               it
               latterally
               from
               the
               Center
               ,
               and
               it
               shall
               shew
               the
               Sine
               of
               half
               the
               Angle
               the
               Sines
               and
               Tangents
               stand
               at
               ;
               and
               one
               degree
               less
               
               is
               the
               Sine
               of
               half
               the
               Angle
               the
               Sector
               stands
               at
               .
            
             
               Example
               .
            
             
               Suppose
               that
               opening
               the
               Sector
               at
               adventures
               ,
               or
               to
               the
               Level
               of
               any
               thing
               ,
               I
               would
               know
               the
               Angle
               it
               stands
               at
               :
               I
               take
               the
               parallel
               Distance
               between
               the
               two
               Centers
               ;
               and
               measuring
               it
               latterally
               from
               the
               Center
               ,
               I
               find
               it
               gives
               me
               the
               Sine
               of
               51
               degrees
               ,
               viz.
               the
               half
               Angle
               the
               Lines
               stand
               at
               ;
               or
               50
               ,
               the
               Angle
               the
               Rule
               stands
               at
               ;
               which
               doubled
               ,
               is
               102
               for
               the
               Lines
               ,
               or
               100
               for
               the
               Legs
               of
               the
               Sector
               ,
               as
               a
               glance
               of
               the
               eye
               presently
               resolves
               by
               the
               inner-edge
               of
               the
               Moving-leg
               ,
               and
               the
               divided
               semi-circle
               .
            
             
               3.
               
               On
               the
               contrary
               ,
               Would
               you
               set
               the
               Legs
               or
               Lines
               to
               any
               Angle
               ,
               take
               the
               half
               thereof
               latterally
               ,
               or
               one
               degree
               less
               in
               the
               half
               for
               the
               Legs
               ,
               and
               make
               it
               a
               Parallel
               in
               the
               two
               Centers
               ,
               and
               the
               Sector
               is
               so
               set
               accordingly
               .
            
             
               Example
               .
            
             
               I
               would
               set
               the
               Legs
               to
               90
               degrees
               ,
               or
               a
               just
               Square
               :
               take
               out
               the
               Latteral
               Sine
               of
               44
               ,
               one
               degree
               less
               than
               45
               ,
               the
               half
               of
               90
               ,
               and
               make
               it
               a
               Parallel
               in
               the
               two
               Centers
               abovesaid
               ,
               and
               you
               shall
               find
               the
               Legs
               set
               
               just
               to
               a
               Square
               ,
               or
               Right-Angle
               ,
               as
               by
               looking
               to
               the
               Head
               you
               may
               nearly
               see
               .
            
             
               At
               the
               same
               time
               if
               you
               take
               Latteral
               30
               ,
               and
               lay
               it
               from
               the
               Center
               ,
               according
               to
               the
               first
               Rule
               ,
               you
               shall
               see
               a
               great
               deficiency
               therein
               ,
               as
               above
               is
               hinted
               .
            
          
           
             
               Use
               V.
               The
               Day
               of
               the
               Month
               being
               given
               ,
               to
               find
               the
               Suns
               Declination
               ,
               true
               place
               in
               the
               Zodiack
               ,
               Right
               Ascention
               ,
               Ascentional
               Difference
               ,
               or
               Rising
               and
               Setting
               .
            
             
               1.
               
               Lay
               the
               Thred
               to
               the
               Day
               of
               the
               Month
               (
               in
               the
               upper
               Line
               of
               Months
               ,
               where
               the
               length
               of
               the
               Dayes
               are
               increasing
               ;
               or
               in
               the
               lower-Line
               ,
               when
               the
               Dayes
               are
               decreasing
               ,
               according
               to
               the
               time
               of
               the
               year
               )
               then
               in
               the
               Line
               of
               degrees
               you
               have
               his
               Declination
               ;
               wherein
               note
               ,
               that
               if
               the
               Thred
               lie
               on
               the
               right
               hand
               of
               600
               ,
               then
               the
               Suns
               Declination
               is
               Northwards
               ;
               the
               contrary-way
               is
               Southwards
               :
               Also
               on
               the
               Line
               of
               the
               Sun
               's
               Right
               Ascention
               ,
               you
               have
               his
               Right
               Ascention
               ,
               in
               degrees
               and
               hours
               ,
               (
               counting
               one
               Hour
               for
               15
               degrees
               )
               as
               the
               Months
               proceed
               from
               March
               the
               10th
               ,
               or
               Equinoctial
               ,
               the
               Right
               Ascention
               being
               then
               00
               ,
               and
               so
               forward
               to
               24
               hours
               ,
               or
               360
               degrees
               ,
               as
               the
               Months
               and
               Dayes
               proceed
               .
            
             
             
               Again
               ,
               on
               the
               Line
               o●
               the
               Sun
               's
               true
               place
               ,
               you
               have
               the
               sign
               and
               degree
               of
               his
               place
               in
               the
               Ecliptick
               ,
               Aries
               ,
               or
               the
               Equinoctial-point
               being
               the
               place
               to
               begin
               ,
               and
               then
               proceeding
               forward
               as
               the
               Months
               and
               Dayes
               go
               .
            
             
               Lastly
               ,
               on
               the
               Hour-line
               you
               have
               the
               Ascentional-difference
               ,
               in
               degrees
               and
               minutes
               ,
               counting
               from
               6
               ;
               or
               the
               Suns
               Rising
               ,
               counting
               as
               the
               morning
               hours
               proceed
               ;
               or
               his
               Setting
               ,
               counting
               as
               the
               afternoon
               hours
               proceed
               .
            
             
               
                 Of
                 all
                 which
                 ,
                 take
                 two
                 or
                 three
                 Examples
                 .
              
            
             
               1.
               
               For
               March
               the
               12th
               ,
               lay
               the
               Thred
               to
               the
               Day
               ,
               and
               extend
               it
               streight
               ;
               then
               on
               the
               Line
               of
               degrees
               ,
               it
               sheweth
               near
               1
               degree
               ,
               or
               54
               minutes
               Northward
               .
            
             
               2.
               
               The
               
                 Suns
                 Right
                 Ascention
              
               ,
               is
               in
               time
               8
               minutes
               and
               better
               ,
               or
               in
               degrees
               ,
               2
               deg
               .
               5
               minutes
               .
            
             
               3.
               
               The
               
                 Suns
                 Place
              
               ,
               is
               2
               degrees
               and
               16
               minutes
               in
               Aries
               ,
               ♈
               .
            
             
               4.
               
               The
               
                 Ascentional
                 Difference
              
               ,
               is
               1
               degree
               and
               10
               minutes
               ;
               or
               the
               Sun
               riseth
               4
               minutes
               before
               ,
               and
               sets
               4
               minutes
               after
               6.
               
            
             
             
               Again
               ,
               for
               May
               the
               10th
               ,
               the
               Thred
               laid
               thereon
               ,
               cuts
               in
               the
               degrees
               ,
               20
               deg
               .
               9
               min.
               for
               Northern
               Declination
               ;
               and
               57
               deg
               .
               24
               min.
               or
               3
               hours
               52
               min.
               Right
               Ascention
               ;
               and
               29
               37
               in
               ♉
               Taurus
               for
               his
               true
               place
               ;
               and
               27
               12
               for
               difference
               of
               Ascentions
               ,
               or
               riseth
               11
               minutes
               after
               4
               ,
               and
               sets
               49
               minutes
               after
               7.
               
            
             
               Again
               ,
               on
               the
               
                 last
                 of
                 October
              
               ,
               or
               the
               21
               of
               Ianuary
               ,
               near
               the
               Declination
               ,
               is
               17
               22
               Southwards
               ,
               the
               Right
               Ascention
               for
               October
               31
               ,
               is
               225
               ,
               53
               ,
               for
               Ianuary
               21
               ,
               314
               21
               :
               The
               true
               place
               for
               October
               31
               ,
               is
               ♏
               Scorpio
               ,
               18
               deg
               .
               22
               min
               ;
               but
               for
               Ianuary
               21
               ,
               ♒
               Aquarius
               11
               ,
               52
               ;
               according
               as
               the
               Months
               go
               to
               the
               end
               at
               ♑
               ,
               and
               then
               back
               again
               ;
               but
               the
               Ascentional
               difference
               ,
               and
               Rising
               and
               Setting
               ,
               is
               very
               near
               the
               same
               at
               both
               times
               ,
               viz.
               23
               10
               ,
               and
               Riseth
               32
               minutes
               ,
               and
               more
               ,
               after
               7
               ;
               and
               Sets
               28
               minutes
               less
               after
               4.
               
            
          
           
             
               Use
               VI.
               The
               Declination
               of
               the
               Sun
               ,
               or
               a
               Star
               ,
               given
               ,
               to
               find
               his
               Amplitude
               .
            
             
               Take
               the
               Declination
               ,
               being
               counted
               on
               the
               particular
               Scale
               of
               Altitudes
               ,
               between
               your
               Compasses
               ;
               and
               with
               this
               distance
               ,
               set
               one
               foot
               in
               90
               on
               the
               Azimuth-Line
               ,
               the
               
               other
               Point
               applied
               to
               the
               same
               Line
               ,
               shall
               give
               the
               Amplitude
               ,
               counting
               from
               90.
               
            
             
               Example
               .
            
             
               The
               Declination
               being
               12
               North
               ,
               the
               Amplitude
               is
               19
               deg
               .
               15
               min.
               Northwards
               .
               Or
               the
               Declination
               being
               20
               South
               ,
               the
               Amplitude
               is
               34
               deg
               .
               10
               min.
               Southwards
               .
            
          
           
             
               Use
               VII
               .
               The
               Right
               Ascention
               and
               Ascentional-difference
               being
               given
               ,
               to
               find
               his
               Oblique-Ascention
               .
            
             
               When
               the
               Declination
               is
               North
               ,
               then
               the
               difference
               between
               the
               Right
               Ascention
               ,
               and
               the
               Ascentional-difference
               ,
               is
               the
               Oblique-Ascention
               .
            
             
               But
               in
               
                 Southern
                 declinations
              
               ,
               the
               sum
               of
               the
               Right
               Ascention
               ,
               and
               difference
               of
               Asscentions
               ,
               is
               the
               Oblique
               Ascention
               .
            
             
               Example
               .
            
             
               On
               or
               between
               the
               25
               and
               26
               of
               Iuly
               ,
               the
               Oblique-Ascention
               is
               by
               Substraction
               112
               ,
               15
               :
               On
               the
               30th
               of
               October
               ,
               the
               Oblique-Ascention
               is
               337
               ,
               45
               by
               Addition
               .
            
          
           
             
             
               Use
               VIII
               .
               The
               Day
               of
               the
               Month
               ,
               or
               Sun's
               Declination
               and
               Altitude
               being
               given
               ,
               to
               find
               the
               Hour
               of
               the
               Day
               .
            
             
               Take
               the
               Suns
               Altitude
               ,
               from
               the
               particular
               Scale
               of
               Altitudes
               ,
               setting
               one
               Point
               of
               the
               Compasses
               in
               the
               Center
               ,
               at
               the
               beginning
               of
               that
               Line
               ;
               and
               opening
               the
               other
               to
               the
               degree
               and
               minute
               of
               the
               Sun's
               Altitude
               ,
               counted
               on
               that
               Line
               ;
               then
               lay
               the
               Thred
               on
               the
               Day
               of
               the
               Month
               (
               or
               Declination
               )
               and
               there
               keep
               it
               :
               Then
               carry
               the
               Compasses
               (
               set
               at
               the
               former
               distance
               )
               along
               the
               Line
               of
               Hours
               ,
               perpendiculer
               to
               the
               Thred
               ,
               till
               the
               other
               Point
               ,
               being
               turned
               about
               ,
               will
               but
               just
               touch
               the
               Thred
               ;
               the
               Compasses
               standing
               between
               the
               Thred
               and
               the
               Hour
               12
               ,
               then
               the
               fixed
               Point
               in
               the
               Hour-Line
               shall
               shew
               the
               hour
               and
               minute
               required
               ;
               but
               whether
               it
               be
               the
               Fore
               or
               Afternoon
               ,
               your
               judgment
               ,
               or
               a
               second
               observation
               must
               determine
               .
            
             
               Example
               .
            
             
               On
               the
               
                 first
                 of
                 August
              
               in
               the
               morning
               ,
               at
               20
               degrees
               of
               Altitude
               ,
               you
               shall
               find
               it
               to
               be
               just
               52
               minutes
               past
               6
               ;
               but
               at
               the
               same
               
               Altitude
               in
               the
               afternoon
               ,
               it
               is
               7
               minutes
               past
               5
               at
               night
               ,
               in
               the
               Latitude
               of
               51
               32
               for
               London
               .
            
          
           
             
               Use
               IX
               .
               The
               Suns
               Declination
               and
               Altitude
               given
               ,
               to
               find
               the
               Suns
               Azimuth
               from
               the
               South-part
               of
               the
               Horizon
               .
            
             
               First
               ,
               by
               the
               4th
               Use
               ,
               find
               the
               Suns
               Declination
               ,
               count
               the
               same
               on
               the
               particular
               Scale
               ,
               and
               take
               the
               distance
               between
               your
               Compasses
               ;
               then
               lay
               the
               Thred
               to
               the
               Suns
               Altitude
               ,
               counted
               the
               same
               way
               as
               the
               Southern-Declination
               is
               from
               600
               ,
               toward
               the
               loose-piece
               ;
               and
               when
               need
               requires
               on
               the
               loose-piece
               ,
               then
               carry
               the
               Compasses
               along
               the
               Azimuth-line
               ,
               on
               the
               right-side
               of
               the
               Thred
               ,
               that
               is
               ,
               between
               the
               Thred
               and
               the
               Head
               ,
               when
               the
               Declination
               is
               Northward
               ;
               and
               on
               the
               left-side
               of
               the
               Thred
               ,
               that
               is
               ,
               between
               the
               Thred
               and
               the
               End
               ,
               when
               the
               Declination
               is
               Southward
               .
               So
               as
               the
               Compasses
               set
               to
               the
               Declination
               ,
               as
               before
               ,
               and
               one
               Point
               staying
               on
               the
               Azimuth-line
               ,
               and
               the
               other
               turned
               about
               ,
               shall
               but
               just
               touch
               the
               Thred
               at
               the
               nearest
               distance
               ;
               then
               ,
               I
               say
               ,
               the
               fixed-Point
               shall
               ,
               in
               the
               Azimuth-line
               ,
               shew
               the
               Suns-Azimuth
               required
               .
            
             
             
               Example
               1.
               
            
             
               The
               Sun
               being
               in
               the
               Equinoctial
               ,
               and
               having
               no
               Declination
               ,
               you
               have
               nothing
               to
               take
               with
               your
               Compasses
               ,
               but
               only
               lay
               the
               Thred
               to
               the
               Altitude
               counted
               from
               600
               toward
               the
               loose-piece
               ,
               and
               in
               the
               Azimuth-line
               it
               cuts
               the
               Azimuth
               required
               .
            
             
               Example
               .
               At
               25
               degrees
               high
               ,
               you
               shall
               find
               the
               Suns
               Azimuth
               to
               be
               54
               ,
               10
               ;
               at
               32
               degrees
               high
               ,
               you
               shall
               find
               38
               ,
               20
               ,
               the
               Azimuth
               .
            
             
               Again
               ,
               At
               20
               degrees
               of
               Declination
               ,
               take
               20
               from
               the
               particular
               Scale
               ,
               and
               at
               10
               degrees
               of
               Altitude
               ,
               lay
               the
               Thred
               to
               10
               counted
               as
               before
               ;
               then
               if
               you
               carry
               the
               Compasses
               on
               the
               right-side
               for
               North-Declination
               ,
               you
               shall
               find
               109
               ,
               30
               ,
               from
               South
               ;
               but
               if
               you
               carry
               them
               on
               the
               left-side
               for
               South-Declination
               ,
               you
               shall
               find
               38
               ,
               30
               ,
               from
               South
               .
            
             
               The
               rest
               of
               the
               Vses
               you
               shall
               have
               more
               amply
               afterwards
               .
            
          
        
         
           
           
             CHAP.
             VI.
             The
             Use
             of
             the
             Line
             of
             Numbers
             on
             the
             Edge
             ,
             and
             the
             Line
             of
             Lines
             on
             the
             Quadrantal-side
             ,
             or
             on
             the
             Sector-side
             ,
             being
             all
             as
             one
             .
          
           
             HAving
             shewed
             the
             way
             of
             Numeration
             on
             the
             Lines
             ,
             as
             in
             Chapter
             the
             first
             .
             Also
             to
             add
             or
             substract
             one
             Line
             or
             Number
             to
             or
             from
             one
             another
             ,
             as
             in
             Chapter
             4th
             ,
             Explanation
             the
             9th
             .
             I
             come
             now
             to
             work
             the
             Rules
             of
             Multiplication
             and
             Division
             ,
             and
             the
             Rule
             of
             Three
             ,
             direct
             and
             reverse
             ,
             both
             by
             the
             Artificial
             and
             Natural-Lines
             ;
             and
             first
             by
             the
             Artificial
             ,
             being
             the
             most
             easie
             ;
             and
             then
             by
             the
             Natural-lines
             both
             on
             the
             Sector
             and
             Trianguler
             Quadrant
             ,
             being
             alike
             :
             and
             I
             work
             them
             together
             ;
             First
             ,
             because
             I
             would
             avoid
             tautology
             :
             Secondly
             ,
             because
             thereby
             is
             better
             seen
             the
             harmony
             between
             them
             ,
             and
             which
             is
             best
             and
             speediest
             .
             Thirdly
             ,
             because
             it
             is
             a
             way
             not
             yet
             ,
             as
             I
             know
             of
             ,
             gone
             
             by
             any
             other
             .
             And
             last
             of
             all
             ,
             because
             one
             may
             explain
             the
             other
             ;
             the
             Geometrical
             Figure
             being
             the
             same
             with
             the
             Instrumental-work
             by
             the
             Natural
             way
             .
          
           
             
               Sect.
               I.
               To
               multiply
               one
               Number
               by
               another
               .
            
             
               1.
               
               By
               the
               Line
               of
               Numbers
               on
               the
               Edge
               Artificially
               ,
               thus
               :
            
             
               Extend
               the
               Compasses
               from
               1
               to
               the
               Multiplicator
               ;
               the
               same
               extent
               applied
               the
               same
               way
               from
               the
               Multiplicand
               ,
               will
               cause
               the
               other
               Point
               to
               fall
               on
               the
               Product
               required
               .
            
             
               Example
               .
            
             
               Let
               8
               be
               given
               to
               be
               multiplied
               by
               6
               ;
               If
               you
               set
               one
               Point
               of
               the
               Compasses
               in
               1
               ,
               (
               either
               at
               the
               beginning
               ,
               or
               at
               the
               middle
               ,
               or
               at
               the
               end
               ,
               it
               matters
               not
               which
               ;
               yet
               the
               middle
               1
               on
               the
               Head-leg
               ,
               is
               for
               the
               most
               part
               the
               most
               convenient
               )
               and
               open
               the
               other
               to
               6
               ,
               (
               or
               8
               ,
               it
               matters
               not
               which
               ,
               for
               6
               times
               8
               ,
               and
               8
               times
               6
               ,
               are
               alike
               ;
               (
               but
               yet
               you
               may
               mind
               the
               Precept
               if
               you
               will
               )
               the
               same
               Extent
               ,
               laid
               the
               same
               way
               from
               8
               ,
               shall
               reach
               to
               48
               ,
               the
               Product
               required
               ;
               
               which
               ,
               without
               these
               Parenthesis
               ,
               is
               thus
               :
               
                 
                   The
                   Extent
                   from
                   1
                   to
                   6
                   ,
                   shall
                   reach
                   the
                   same
                   way
                   from
                   8
                   to
                   48.
                   
                   Or
                   ,
                
                 
                   The
                   Extent
                   from
                   1
                   to
                   8
                   ,
                   shall
                   reach
                   the
                   same
                   way
                   from
                   6
                   to
                   48.
                   
                
              
               the
               Product
               required
               .
            
             
               
                 By
                 the
                 Natural-Lines
                 on
                 the
                 Sector-side
                 ,
                 or
                 Trianguler
                 Quadrant
                 with
                 a
                 Thred
                 and
                 Compasses
                 ,
                 the
                 work
                 is
                 thus
              
               ;
            
             
               1.
               
               For
               the
               most
               part
               it
               is
               wrought
               by
               changing
               the
               terms
               from
               the
               Artificial
               way
               ,
               as
               thus
               ;
            
             
               The
               former
               way
               was
               ,
               as
               1
               to
               6
               ,
               so
               is
               8
               to
               48
               ;
               or
               as
               1
               to
               8
               ,
               so
               is
               6
               to
               48
               ;
               but
               by
               the
               Sector
               it
               is
               thus
               :
               As
               the
               Latteral
               6
               taken
               from
               the
               Center
               toward
               the
               end
               ,
               is
               to
               the
               Parallel
               10
               &
               10
               ,
               set
               over
               from
               10
               to
               10
               ,
               at
               the
               end
               counted
               as
               1
               ;
               so
               is
               the
               Parallel-distance
               between
               8
               &
               8
               ,
               on
               the
               Line
               of
               Lines
               taken
               a-cross
               from
               one
               Leg
               to
               the
               other
               ,
               to
               the
               Latteral-distance
               from
               the
               Center
               to
               48
               ,
               the
               Product
               required
               .
            
             
               
                 Or
                 shorter
                 thus
              
               .
            
             
               As
               the
               Latteral
               8
               ,
               to
               the
               Parallel
               10
               ;
            
             
               So
               is
               the
               Parallel
               6
               ,
               to
               the
               Latteral
               48.
               
            
             
               
                 See
                 Figure
              
               I.
               
            
             
             
               2.
               
               Another
               way
               may
               you
               work
               without
               altering
               the
               terms
               from
               the
               Artificial
               way
               ,
               as
               thus
               ,
               by
               a
               double
               Radius
               ;
               Take
               the
               Latteral-Extent
               from
               the
               Center
               to
               1
               ,
               (
               or
               from
               10
               to
               9
               ,
               if
               the
               beginning
               be
               defective
               )
               make
               this
               a
               Parallel
               in
               6
               &
               6
               ,
               then
               the
               Latteral-Extent
               from
               the
               Center
               to
               8
               of
               the
               10
               parts
               between
               Figure
               and
               Figure
               ,
               shall
               reach
               across
               from
               48
               to
               48
               ,
               as
               before
               .
               
                 See
                 Fig.
              
               II.
               
            
             
               The
               same
               work
               as
               was
               done
               by
               the
               Sector
               ,
               is
               done
               by
               the
               Line
               of
               Lines
               ,
               and
               Thred
               on
               the
               Quadrant-side
               ,
               that
               if
               your
               Sector
               be
               put
               together
               as
               a
               Trianguler
               Quadrant
               ,
               you
               may
               work
               any
               thing
               by
               it
               ,
               as
               well
               as
               by
               the
               Sector
               ,
               in
               this
               manner
               ;
               (
               or
               by
               the
               Scale
               and
               Compass
               ,
               as
               in
               the
               Figure
               I.
               )
               and
               first
               ,
               as
               above
               ,
               Sector-wise
               .
            
             
               Take
               the
               Extent
               from
               the
               Center
               to
               6
               latterally
               ,
               between
               your
               Compasses
               ;
               set
               one
               Point
               in
               10
               ,
               and
               with
               the
               other
               lay
               the
               Thred
               in
               the
               nearest
               distance
               ,
               turning
               the
               Compass-Point
               about
               ,
               till
               it
               will
               but
               just
               touch
               the
               Thred
               ,
               then
               there
               keep
               it
               ;
               then
               set
               one
               Point
               of
               the
               Compasses
               in
               8
               ,
               and
               take
               the
               nearest
               distance
               to
               the
               Thred
               ;
               this
               distance
               laid
               latterally
               from
               the
               Center
               ,
               shall
               reach
               to
               48
               the
               Product
               .
            
             
               Or
               as
               Latteral
               6
               ,
               to
               Parallel
               10
               ;
               so
               is
               the
               Parallel
               8
               ,
               to
               the
               Latteral
               48
               ,
               the
               Product
               required
               .
            
             
             
               Or
               the
               last
               way
               by
               a
               double
               Radius
               ,
               or
               a
               greater
               and
               a
               smaller
               Scale
               ,
               as
               the
               Latteral
               Extent
               from
               the
               Center
               to
               1
               ,
               is
               to
               the
               Parallel
               6
               ,
               laying
               the
               Thred
               to
               the
               nearest
               distance
               ;
               so
               is
               the
               latteral
               Extent
               from
               the
               Center
               to
               8
               parts
               ,
               less
               than
               the
               1
               before
               taken
               ,
               carried
               parallelly
               along
               the
               common-line
               ,
               till
               the
               other
               Point
               will
               but
               just
               touch
               the
               Thred
               ,
               it
               shall
               on
               those
               conditions
               stay
               only
               at
               48
               ,
               the
               Product
               required
               .
               Observe
               and
               note
               the
               Figure
               ,
               by
               protraction
               ,
               with
               Scale
               and
               Compass
               .
            
             
               3.
               
               But
               if
               you
               have
               an
               Index
               and
               a
               Square
               ,
               as
               is
               used
               in
               the
               Demonstrative
               Work
               of
               
                 Plain
                 Sayling
              
               ,
               as
               you
               shall
               have
               afterwards
               ,
               then
               the
               representation
               of
               this
               Natural-way
               will
               most
               evidently
               appear
               ,
               as
               thus
               :
            
             
             
               From
               hence
               you
               may
               observe
               ,
               That
               the
               first
               and
               third
               Numbers
               must
               alwayes
               be
               accounted
               alike
               ,
               and
               on
               like
               Scales
               ;
               and
               the
               second
               and
               the
               fourth
               in
               like
               manner
               on
               like
               Scales
               and
               counting
               ;
               and
               the
               Latteral-first
               Number
               ,
               must
               alwayes
               be
               less
               than
               the
               Parallel-second
               ,
               in
               length
               or
               quantity
               ,
               or
               you
               cannot
               work
               it
               ;
               which
               you
               must
               make
               so
               ,
               either
               by
               changing
               the
               terms
               ,
               or
               using
               a
               less
               Scale
               ,
               to
               begin
               and
               end
               upon
               .
            
             
               Here
               you
               must
               except
               a
               Decimal
               gradation
               ,
               as
               thus
               ;
               sometimes
               the
               same
               place
               which
               is
               called
               10
               in
               the
               first
               ,
               may
               be
               counted
               100
               in
               the
               third
               ,
               and
               the
               contrary
               ;
               or
               more
               or
               less
               differing
               in
               a
               Decimal
               Account
               .
            
             
               But
               if
               you
               would
               see
               a
               Figure
               of
               the
               Sector-way
               of
               operation
               ,
               then
               it
               is
               thus
               ;
               Let
               the
               Line
               C
               6
               ,
               represent
               one
               Leg
               of
               the
               Sector
               ;
               and
               the
               other
               Line
               C
               6
               ,
               represent
               the
               other
               Leg
               of
               the
               Sector
               ;
               then
               take
               1
               out
               of
               any
               Scale
               ,
               as
               1
               inch
               ,
               or
               one
               tenth
               part
               of
               a
               foot
               ,
               or
               what
               you
               please
               :
               or
               the
               distance
               from
               the
               Center
               to
               1
               ,
               or
               2
               ,
               on
               the
               Line
               of
               Lines
               between
               your
               Compasses
               :
               put
               this
               distance
               over
               in
               6
               &
               6
               ,
               of
               the
               Line
               of
               Lines
               .
               Then
               is
               the
               Sector
               set
               to
               its
               due
               Isosocles
               Angle
               .
            
             
               Then
               take
               8
               parts
               ,
               or
               rather
               8
               tenth
               
               parts
               of
               the
               former
               1
               ,
               from
               the
               Scale
               from
               whence
               you
               took
               the
               first
               Latteral
               distance
               ,
               and
               carry
               it
               parallel
               between
               the
               Line
               of
               Lines
               till
               it
               stay
               in
               like
               parts
               ,
               which
               you
               shall
               find
               to
               be
               at
               48
               ,
               the
               Product
               required
               .
            
             
               Or
               to
               get
               the
               Answer
               in
               a
               Latteral-line
               ,
               is
               generally
               most
               convenient
               ,
               by
               changing
               the
               terms
               ;
               work
               thus
               :
               Take
               the
               Latteral-distance
               from
               the
               Center
               to
               8
               ,
               on
               the
               Line
               of
               Lines
               ,
               make
               it
               a
               Parallel
               in
               10
               &
               10
               ;
               then
               the
               Sector
               being
               so
               set
               ,
               take
               the
               Parallel-distance
               between
               6
               &
               6
               ,
               and
               lay
               it
               latterally
               from
               the
               Center
               ,
               and
               it
               shall
               reach
               to
               48
               ,
               the
               Product
               required
               .
               
                 See
                 Fig.
              
               IIII.
               
            
             
               Thus
               you
               see
               that
               the
               way
               of
               the
               Sector-side
               ,
               and
               of
               the
               Quadrant-side
               ,
               is
               in
               a
               manner
               all
               one
               ;
               and
               the
               laying
               of
               the
               Thred
               ,
               or
               Index
               to
               the
               nearest
               distance
               ,
               is
               the
               same
               with
               setting
               the
               Legs
               of
               the
               Sector
               to
               their
               Angle
               ;
               and
               the
               taking
               the
               nearest
               distance
               from
               any
               Point
               or
               Number
               to
               the
               Thred
               ,
               is
               the
               same
               with
               taking
               parallelly
               from
               Point
               to
               Point
               ,
               or
               from
               Number
               to
               Number
               :
               So
               that
               having
               thus
               fully
               explained
               the
               Latteral
               and
               Parallel-Extent
               ,
               and
               laying
               of
               the
               Thred
               ,
               and
               setting
               of
               the
               Sector
               ,
               the
               following
               Propositions
               will
               be
               more
               easie
               ,
               and
               ready
               ;
               and
               to
               that
               purpose
               ,
               these
               brief
               Marks
               for
               Latteral
               ,
               Parallel
               ,
               and
               Nearest-distance
               ,
               
               will
               frequently
               be
               used
               ;
               as
               thus
               ,
               for
               Latteral
               ,
               thus
               —
               ;
               for
               Parallel
               ,
               thus
               =
               ;
               for
               Nearest-distance
               ,
               thus
               ‑
               ,
               or
               ND
               ,
               thus
               :
               for
               the
               Sine
               of
               90
               ,
               or
               Radius
               ,
               or
               Tangent
               of
               45
               ,
               thus
               R
               ;
               &c.
               
            
             
               In
               all
               which
               wayes
               you
               may
               see
               ,
               that
               for
               the
               want
               of
               several
               Radiusses
               ,
               which
               do
               properly
               express
               the
               unites
               ,
               tens
               ,
               hundreds
               ,
               and
               thousands
               ,
               and
               ten
               thousands
               of
               Numbers
               ,
               there
               must
               a
               due
               and
               rational
               account
               ,
               or
               consideration
               ,
               go
               along
               with
               this
               Instrumental
               manner
               of
               work
               ,
               else
               you
               may
               give
               an
               erroneous
               answer
               to
               the
               question
               propounded
               ;
               to
               prevent
               which
               ,
               observe
               ,
               that
               in
               Multiplication
               there
               must
               be
               for
               the
               most
               part
               ,
               as
               many
               figures
               in
               the
               Product
               ,
               as
               is
               in
               the
               Multiplicator
               and
               Multiplicand
               ,
               put
               together
               ;
               except
               when
               the
               first
               figures
               of
               the
               Product
               ,
               be
               greater
               than
               any
               of
               the
               first
               figures
               of
               the
               Multiplicator
               ,
               or
               the
               Multiplicand
               ,
               and
               then
               there
               is
               one
               less
               ;
            
             
               
                 As
                 for
                 Example
              
               .
            
             
               2
               times
               2
               makes
               4
               ,
               being
               only
               one
               figure
               ,
               because
               4
               is
               greater
               than
               2
               ;
               but
               2
               times
               5
               is
               10
               ,
               being
               two
               figures
               ;
               wherein
               1
               ,
               the
               first
               figure
               ,
               is
               less
               than
               5.
               
               Again
               ,
               in
               a
               bigger
               sum
               ;
               52
               multiplied
               by
               23
               ,
               makes
               1300
               ,
               consisting
               of
               four
               figures
               ,
               as
               many
               as
               
               is
               in
               the
               Multiplier
               and
               Multiplicand
               put
               together
               ;
               but
               if
               you
               multiply
               42
               by
               22
               ,
               it
               makes
               but
               924
               ,
               which
               is
               but
               three
               figures
               ;
               because
               the
               first
               figure
               9
               is
               greater
               than
               2
               or
               4
               ,
               as
               in
               the
               former
               ,
               the
               first
               figure
               1
               was
               less
               than
               5
               or
               2
               :
               And
               this
               Rule
               is
               general
               ,
               as
               to
               the
               number
               of
               places
               ,
               or
               figures
               ,
               in
               any
               Multiplication
               whatsoever
               ;
               but
               note
               ,
               that
               no
               Instrument
               extant
               ,
               and
               in
               ordinary
               use
               ,
               is
               capable
               to
               express
               above
               5
               or
               6
               places
               :
               Yet
               with
               this
               help
               you
               may
               come
               true
               to
               5
               places
               ,
               with
               a
               good
               Line
               of
               Numbers
               .
            
             
               
                 As
                 thus
              
               ;
            
             
               Suppose
               I
               was
               by
               a
               Line
               of
               Numbers
               to
               multiply
               168
               by
               249
               ;
               the
               extent
               from
               1
               to
               168
               ,
               will
               reach
               the
               same
               way
               from
               249
               ,
               to
               41832
               :
               Now
               by
               the
               Line
               of
               Numbers
               ,
               you
               can
               only
               see
               but
               the
               418
               ,
               and
               estimate
               at
               the
               3
               ;
               but
               the
               last
               figure
               2
               ,
               I
               cannot
               see
               by
               any
               Line
               usually
               put
               on
               two
               foot
               Rules
               ,
               therefore
               the
               168
               ,
               and
               249
               being
               before
               you
               ,
               say
               (
               according
               to
               the
               vulgar
               Rules
               of
               Multiplication
               )
               9
               times
               8
               is
               72
               ;
               therefore
               2
               must
               needs
               be
               the
               last
               figure
               ;
               and
               if
               you
               can
               see
               the
               former
               4
               ,
               you
               have
               the
               Product
               infallibly
               true
               :
               if
               not
               ,
               multiply
               a
               figure
               more
               :
               But
               by
               this
               help
               you
               shall
               be
               sure
               
               to
               come
               right
               alwayes
               to
               4
               figures
               ,
               or
               places
               ,
               in
               any
               Multiplication
               whatsoever
               .
            
             
               4.
               
               Also
               ,
               by
               these
               operations
               ,
               you
               may
               plainly
               see
               ,
               That
               the
               Line
               of
               Numbers
               ,
               or
               Gunters-Line
               ,
               as
               it
               is
               usually
               called
               ,
               is
               the
               easiest
               and
               exactest
               for
               Arithmetical
               operations
               ,
               being
               performed
               with
               an
               Extent
               of
               the
               Compasses
               only
               ,
               without
               any
               opening
               or
               shutting
               of
               the
               Rule
               ,
               or
               laying
               a
               Thred
               or
               Index
               :
               But
               in
               Questions
               of
               Geometry
               ,
               where
               a
               lively
               draught
               or
               representation
               is
               required
               ,
               as
               to
               the
               reason
               of
               the
               work
               ,
               there
               the
               Natural-Lines
               are
               more
               demonstrative
               .
            
             
               In
               which
               Natural
               Work
               ,
               Note
               ,
               That
               the
               Parallel-distance
               must
               alwayes
               be
               the
               greatest
               ,
               or
               you
               cannot
               work
               it
               ,
               unless
               you
               make
               use
               of
               a
               greater
               ,
               and
               a
               lesser
               Scale
               ;
               to
               which
               purpose
               ,
               this
               Instrument
               is
               well
               furnished
               ,
               with
               three
               or
               four
               Radiusses
               ,
               bigger
               and
               less
               ,
               both
               of
               Sines
               ,
               Tangents
               ,
               and
               Secants
               ,
               and
               Equal-Parts
               ,
               as
               in
               their
               due
               places
               shall
               be
               observed
               ;
               and
               taken
               notice
               of
               ,
               in
               the
               Astronomical-Work
               .
            
             
               And
               note
               also
               ,
               That
               if
               the
               Line
               of
               Lines
               were
               repeated
               to
               4
               Radiusses
               ,
               or
               400
               instead
               of
               10
               ,
               you
               might
               work
               right-on
               to
               4
               figures
               ;
               but
               then
               the
               Radiusses
               
               must
               be
               very
               small
               ,
               or
               the
               Instrument
               very
               large
               .
            
             
               Therefore
               this
               of
               10
               ,
               being
               the
               most
               usual
               ,
               I
               shall
               make
               use
               of
               ,
               and
               work
               every
               Question
               the
               most
               convenient
               way
               ;
               that
               by
               a
               frequent
               Practice
               ,
               the
               young
               Beginners
               ,
               for
               whom
               only
               I
               write
               ,
               may
               see
               the
               Reason
               and
               Nature
               of
               the
               Work
               ,
               and
               the
               sooner
               understand
               it
               .
            
             
               For
               a
               Conclusion
               of
               this
               Rule
               of
               Multiplication
               ,
               take
               three
               or
               four
               Examples
               more
               ,
               both
               by
               the
               Line
               of
               Numbers
               ,
               and
               Equal-Parts
               also
               ;
            
             
               1.
               
               First
               15
               foot
               ,
               8
               tenths
               ,
               multiplied
               9
               foot
               and
               7
               tenths
               ,
               by
               the
               Line
               of
               Numbers
               ;
               the
               Extent
               of
               the
               Compasses
               from
               1
               to
               9
               foot
               ,
               7
               tenths
               ,
               shall
               reach
               the
               same
               way
               from
               15
               —
               8
               ,
               to
               153
               ,
               26.
               
            
             
               
                 By
                 the
                 Line
                 of
                 Lines
                 ,
                 or
                 Equal
                 Parts
                 .
              
            
             
               As
               the
               Latteral
               —
               15
               —
               8
               ,
               to
               the
               Parallel
               =
               10
               ,
               at
               the
               End
               counted
               as
               1
               ;
            
             
               So
               is
               the
               Parallel
               =
               9
               —
               7
               ,
               to
               the
               Latteral
               —
               153
               —
               26.
               
            
             
               Where
               you
               may
               observe
               ,
               that
               the
               first
               and
               fourth
               ,
               are
               measured
               on
               like
               Scales
               ;
               and
               the
               second
               and
               third
               ,
               also
               on
               like
               Scales
               .
            
             
             
               But
               note
               ;
               that
               as
               you
               diminished
               the
               account
               in
               the
               third
               =
               work
               ,
               counting
               9
               —
               7
               less
               than
               10
               reckoned
               as
               1
               :
            
             
               So
               likewise
               in
               the
               —
               fourth
               ,
               you
               count
               153
               —
               26
               ,
               which
               is
               less
               in
               Extent
               than
               the
               15
               —
               8
               ,
               first
               taken
               Latterally
               ;
               yet
               is
               to
               be
               read
               as
               before
               ,
               viz.
               153
               —
               26
               ,
               because
               9
               times
               15
               ,
               must
               needs
               be
               above
               100.
               
            
             
               2.
               
               As
               1
               ,
               to
               9
               foot
               10
               inches
               ;
               So
               is
               10
               foot
               9
               inches
               ,
               to
               105
               foot
               6
               inches
               ½
               .
            
             
               To
               work
               this
               properly
               by
               the
               Line
               of
               ●umbers
               ,
               you
               are
               first
               by
               the
               inches
               and
               ●ot-measure
               on
               the
               in-side
               of
               the
               Rule
               ,
               to
               ●educe
               the
               inches
               into
               decimals
               of
               a
               foot
               ;
               as
               ●hus
               :
               Right
               against
               10
               inches
               ,
               in
               the
               Line
               〈◊〉
               Foot-measure
               ,
               you
               shall
               find
               83
               ●
               /
               ●
               .
               Also
               ,
               ●ight
               against
               9
               inches
               ,
               on
               the
               Foot-measure
               ,
               ●ou
               shall
               find
               75
               ;
               this
               being
               done
               ,
               which
               〈◊〉
               with
               a
               glance
               of
               your
               eye
               only
               ,
               on
               those
               two
               Lines
               ,
               then
               the
               work
               is
               thus
               :
            
             
               As
               1
               ,
               to
               10
               75
               ;
               so
               is
               9
               83
               ,
               to
               105
               ½
               ,
               or
               105
               foot
               ,
               6
               inches
               :
               Now
               for
               the
               odd
               6
               square
               inches
               ,
               you
               cannot
               see
               them
               on
               the
               Rule
               ,
               but
               must
               find
               them
               by
               the
               help
               before
               
               mentioned
               ,
               as
               thus
               ,
               Having
               set
               down
               the
               9
               —
               10
               ,
               &
               the
               10
               —
               9
               ,
               as
               in
               the
               Margent
               ,
               say
               by
               vulgar
               Arithmetick
               thus
               ,
               Ten
               times
               
               9
               is
               90
               ;
               for
               which
               you
               must
               set
               down
               7
               foot
               and
               6
               inches
               ,
               which
               is
               the
               6
               inches
               you
               could
               not
               see
               on
               the
               Line
               of
               Numbers
               ,
               and
               there
               must
               needs
               be
               105
               foot
               ,
               and
               not
               10
               foot
               and
               an
               half
               ,
               and
               better
               ,
               which
               is
               as
               to
               the
               right
               Number
               of
               Figures
               .
            
             
               But
               by
               the
               Line
               of
               Lines
               as
               the
               —
               10
               —
               75
               ,
               from
               any
               Scale
               ,
               as
               the
               Line
               of
               Lines
               doubled
               ,
               or
               foot-measure
               ,
               or
               the
               like
               ,
               is
               to
               the
               =
               10
               ,
               so
               is
               the
               =
               9
               —
               83
               ,
               to
               the
               Latteral
               105
               —
               6
               ½
               ,
               as
               before
               ,
               though
               not
               so
               quick
               or
               plain
               ,
               as
               by
               the
               Line
               of
               Numbers
               .
            
             
               3.
               
               As
               1
               ,
               to
               1528
               ,
               so
               is
               3522
               ,
               to
               5371616
               ,
               the
               true
               answer
               ;
               which
               indeed
               is
               to
               more
               places
               than
               possible
               the
               Rule
               can
               come
               to
               without
               the
               help
               last
               mentioned
               :
               But
               if
               the
               Question
               had
               been
               thus
               ,
               with
               the
               same
               Figures
               ,
               15
               foot
               100
               /
               28
               parts
               ,
               by
               35
               foot
               100
               /
               22
               parts
               ,
               as
               so
               many
               feet
               ,
               or
               yards
               ,
               and
               hundred
               parts
               :
               Then
               the
               answer
               would
               be
               as
               before
               ,
               537
               foot
               ;
               and
               cutting
               off
               four
               figures
               for
               the
               four
               figures
               of
               Fractions
               ,
               both
               in
               the
               Multiplicator
               and
               Multiplicand
               ,
               viz.
               the
               1616
               ,
               which
               makes
               near
               2
               inches
               of
               a
               foot
               more
               ,
               or
               16
               /
               100
               parts
               of
               a
               yard
               more
               ,
               which
               in
               ordinary
               measuring
               is
               not
               considerable
               .
            
             
             
               
                 By
                 the
                 Line
                 of
                 Lines
              
               .
            
             
               4.
               
               As
               —
               15
               —
               28
               ,
               to
               =
               1
               ,
               next
               the
               Center
               :
            
             
               So
               is
               =
               35
               —
               22
               ,
               to
               537
               —
               1616
               /
               100000
            
             
               To
               multiply
               3
               pound
               ,
               6
               shillings
               ,
               and
               3
               pence
               ,
               by
               it self
               ;
               the
               Product
               is
               ,
               10.
               l.
               —
               19.
               s.
               —
               5.
               d.
               —
               1.
               f.
               —
               7
               /
               10
               :
               For
               the
               Extent
               from
               1
               ,
               to
               3
               —
               3125
               ,
               the
               Decimal
               number
               for
               3
               ●
               .
               —
               6.
               s.
               —
               3.
               d.
               shall
               reach
               from
               thence
               to
               10
               —
               9726
               ,
               which
               reduced
               again
               ,
               is
               as
               before
               ,
               10
               —
               19
               —
               5
               —
               2
               ;
               as
               followeth
               .
            
             
               Note
               ,
               That
               in
               this
               way
               of
               Multiplication
               by
               the
               Pen
               ,
               works
               thus
               ;
               You
               must
               first
               multiply
               Pounds
               by
               Pounds
               ,
               one
               over
               the
               other
               ,
               as
               3
               by
               3
               :
               Then
               the
               Shillings
               by
               the
               Pounds
               cross-wise
               both-wayes
               ,
               as
               the
               black-line
               sheweth
               .
               Then
               Pounds
               by
               Pence
               ,
               as
               the
               long
               Prick-lines
               sheweth
               both-wayes
               also
               .
               Then
               Shillings
               by
               Shillings
               ,
               as
               the
               6
               by
               6.
               
               Then
               Shillings
               by
               Pence
               ,
               both-wayes
               ,
               as
               the
               short
               Prick-lines
               sheweth
               .
               Then
               lastly
               ,
               the
               Pence
               by
               the
               Pence
               ,
               as
               3
               by
               3
               ;
               whose
               true
               power
               ,
               or
               denomination
               ,
               is
               somewhat
               hard
               to
               conceive
               ▪
               which
               is
               thus
               :
            
             
             
               First
               ,
               3
               times
               3
               (
               next
               the
               left-hand
               )
               is
               9
               Pounds
               .
            
             
               Secondly
               ,
               3
               times
               6
               ,
               is
               18
               Shillings
               .
            
             
               Thirdly
               ,
               3
               times
               6
               ,
               is
               18
               shillings
               again
               .
            
             
               Fourthly
               ,
               3
               times
               3
               ,
               is
               9
               Pence
               ,
               as
               the
               long
               Prick-line
               sheweth
               .
            
             
               Fiftly
               ,
               3
               times
               3
               ,
               is
               9
               Pence
               more
               .
            
             
               Sixtly
               ,
               6
               times
               6
               is
               36
               ,
               every
               20
               whereof
               is
               1
               Shilling
               ;
               and
               every
               5
               thereof
               is
               3
               Pence
               ;
               and
               every
               1
               is
               2
               Farthings
               and
               4
               /
               10ths
               of
               a
               Farthing
               :
               So
               that
               36
               make
               1
               Shilling
               ,
               9
               Pence
               ,
               2
               Farthings
               ,
               4
               /
               10ths
               of
               a
               Farthing
               .
            
             
               Seventhly
               ,
               6
               times
               3
               is
               18
               ;
               every
               5
               whereof
               is
               a
               Farthing
               ,
               and
               every
               1
               is
               two
               tenths
               of
               a
               Farthing
               ,
               as
               the
               short
               Prick-line
               sheweth
               .
            
             
               Eightly
               ,
               6
               times
               3
               is
               18
               ,
               or
               3
               Farthings
               and
               6
               tenths
               ,
               as
               before
               .
            
             
               Ninthly
               and
               lastly
               ;
               to
               the
               right-hand
               ,
               3
               times
               3
               is
               9
               ;
               where
               note
               ,
               that
               there
               goes
               60
               to
               make
               1
               Farthing
               ;
               therefore
               6
               makes
               one
               tenth
               of
               a
               Farthing
               :
               So
               that
               here
               is
               1
               tenth
               and
               ½
               :
               Consider
               the
               Scheam
               and
               the
               Decimal-work
               ,
               to
               prove
               it
               exactly
               to
               the
               hundreds
               of
               millions
               of
               a
               Pound
               ,
               and
               you
               will
               find
               it
               to
               be
               very
               near
               .
            
             
               Example
               .
            
             
             
               
            
             
               Product
               is
               10.
               19.
               5.
               1.
               7.
               
               ½
               .
            
             
               
                 The
                 same
                 Decimally
              
               .
            
             
               
            
             
             
               Which
               Sum
               ,
               being
               brought
               to
               Shillings
               ,
               Pence
               ,
               and
               Farthings
               ,
               and
               tenths
               of
               Farthings
               ,
               is
               just
               as
               aforesaid
               ,
               viz.
               
               
            
             
               Or
               else
               find
               the
               Square
               of
               the
               least
               Denomination
               in
               20
               s.
               and
               divide
               the
               Product
               of
               the
               Sums
               being
               brought
               to
               that
               least
               Denomination
               thereby
               ,
               and
               the
               Quotient
               shall
               be
               the
               Answer
               required
               .
            
             
               Example
               .
            
             
               960
               ,
               the
               Farthings
               in
               20
               s.
               squared
               ,
               is
               921600.
               
               The
               sum
               of
               3
               l.
               —
               6
               s.
               —
               3
               d.
               in
               Farthings
               ,
               is
               3180
               ;
               multiplied
               by
               it self
               is
               10112400
               :
               This
               Product
               divided
               by
               921600
               ,
               the
               square
               of
               the
               Farthings
               in
               20
               s.
               makes
               10
               l.
               896400
               /
               ●216●●
               in
               the
               Quotient
               ,
               which
               reduced
               ,
               is
               10
               l.
               —
               19
               s.
               —
               5
               d.
               —
               1
               7
               /
               10.
               
            
             
               To
               find
               this
               Decimal
               Fraction
               is
               very
               easie
               thus
               ,
               by
               the
               Line
               of
               Numbers
               ;
               for
               if
               20
               shillings
               be
               1000
               ,
               what
               shall
               6
               shillings
               and
               3
               pence
               be
               ?
               Set
               one
               Point
               in
               2
               ,
               representing
               20
               ;
               and
               the
               other
               in
               1
               ,
               representing
               1000
               :
               then
               the
               same
               Extent
               laid
               the
               same
               way
               from
               6
               and
               ¼
               ,
               shall
               reach
               to
               3.125
               ,
               the
               decimal
               fraction
               for
               6
               shillings
               and
               3
               pence
               .
            
             
             
               Or
               by
               inches
               and
               foot-measure
               ;
               for
               if
               you
               account
               every
               8th
               of
               an
               inch
               a
               farthing
               ,
               then
               every
               inch
               is
               2
               d.
               and
               6
               inches
               is
               12
               d.
               right
               against
               which
               ,
               in
               the
               Foot-measure
               ,
               is
               the
               Decimal
               Fraction
               required
               :
               So
               that
               right
               against
               12
               farthings
               ,
               or
               1
               inch
               and
               ½
               on
               the
               Line
               of
               Foot-measure
               ,
               is
               125
               ,
               the
               Decimal
               Fraction
               sought
               for
               .
            
             
               Or
               if
               one
               Pound
               be
               the
               Integer
               ,
               or
               whole
               Number
               ,
               then
               every
               10th
               part
               is
               2
               shillings
               ;
               and
               every
               5th
               is
               1
               shilling
               :
               and
               the
               inter-mediate
               pence
               and
               farthings
               is
               very
               near
               the
               5th
               part
               ;
               for
               if
               you
               conceive
               a
               5th
               part
               ,
               or
               50
               of
               an
               hundred
               to
               contain
               one
               shilling
               ▪
               or
               48
               farthings
               ;
               then
               one
               of
               50
               is
               very
               near
               one
               farthing
               ,
               for
               12
               and
               ⅓
               is
               just
               3
               d.
               and
               25
               is
               just
               6
               d.
               37
               ½
               is
               just
               9
               d.
               and
               50
               just
               12
               d.
               So
               that
               to
               set
               the
               Compass-point
               to
               3
               l.
               6
               s.
               and
               3
               d.
               is
               to
               set
               the
               Point
               on
               3.3125
               ,
               as
               before
               ,
               which
               a
               little
               practice
               will
               make
               easie
               .
            
             
               By
               the
               Line
               of
               Lines
               on
               the
               Trianguler-Quadrant
               ,
               or
               Sector
               .
            
             
               As
               the
               Latteral
               3.3125
               is
               to
               the
               Parallel
               10
               ,
               So
               is
               the
               Parallel
               3.3125
               to
               the
               Latteral
               10
               l.
               —
               19
               s.
               —
               5
               d.
               —
               2
               farthings
               ferè
               ,
               or
               10.973
               .
            
          
           
             
             
               Sect.
               II.
               To
               divide
               one
               Number
               by
               another
               .
            
             
               
                 First
                 ,
                 by
                 the
                 Line
                 of
                 Numbers
                 the
                 Rule
                 is
                 ,
              
            
             
               Extend
               the
               Compasses
               from
               the
               Divisor
               to
               1
               ,
               then
               the
               same
               extent
               of
               the
               Compasses
               ,
               applied
               the
               same
               way
               from
               the
               Dividend
               ,
               shall
               reach
               the
               Quotient
               required
               .
            
             
               Or
               the
               Extent
               from
               the
               Divisor
               to
               the
               Dividend
               ,
               shall
               reach
               the
               same
               way
               from
               1
               ,
               to
               the
               quotient
               required
               .
            
             
               
                 Example
                 the
                 first
              
               .
            
             
               Let
               40
               be
               a
               Number
               given
               ,
               to
               be
               divided
               by
               5
               ;
               here
               40
               is
               the
               Dividend
               ,
               and
               5
               the
               Divisor
               ;
               and
               the
               answer
               to
               how
               many
               ,
               viz.
               8
               is
               the
               Quotient
               .
            
             
               Extend
               the
               Compasses
               from
               5
               to
               1
               ,
               the
               same
               Extent
               shall
               reach
               the
               same
               way
               from
               40
               to
               8
               the
               Quotient
               required
               ;
               or
               the
               Extent
               from
               5
               to
               40
               ,
               shall
               reach
               the
               same
               way
               from
               1
               to
               8
               the
               quotient
               required
               .
            
             
             
               
                 But
                 by
                 the
                 Line
                 of
                 Lines
                 ,
                 the
                 work
                 is
                 thus
              
               ;
            
             
               As
               the
               —
               Latteral
               40
               ,
               to
               the
               =
               Parallel
               5
               ;
            
             
               So
               is
               the
               =
               10
               counted
               as
               1
               to
               the
               —
               8.
               
            
             
               Or
               so
               is
               the
               =
               1
               ,
               to
               the
               —
               8
               of
               the
               smaller
               part
               .
            
             
               
                 Observe
                 the
                 Figure
                 with
                 the
                 Line
              
               AB
               .
            
             
               Or
               ,
               as
               the
               —
               5
               ,
               to
               the
               =
               10
               ;
            
             
               So
               is
               the
               —
               40
               ,
               to
               the
               —
               8
               ;
               As
               the
               Line
               CB
               in
               the
               Figure
               doth
               demonstrate
               ,
               being
               the
               manner
               of
               working
               by
               the
               Trianguler-Quadrant
               ,
               the
               way
               of
               the
               Sector
               being
               the
               same
               .
            
             
               
                 A
                 second
                 Example
              
               .
            
             
               Let
               1668
               ,
               be
               divided
               by
               19.
               
            
             
               As
               19
               to
               1
               ,
               so
               is
               1668
               to
               87
               ●8
               /
               100
               ;
            
             
               Or
               ,
               15
               of
               19
               :
               Or
               ,
            
             
               As
               19
               to
               1668
               ,
               so
               is
               1
               to
               87
               7●
               /
               100
               ;
            
             
               Or
               ,
               15
               of
               19
               ,
               as
               before
               .
            
             
               For
               the
               Extent
               from
               19
               to
               1668
               ,
               shall
               reach
               the
               same
               way
               from
               1
               to
               87
               —
               78
               /
               100
               ;
               the
               work
               by
               the
               Lines
               is
               as
               before
               .
            
             
               In
               this
               work
               of
               Division
               ,
               for
               most
               ordinary
               questions
               ,
               where
               there
               is
               not
               above
               four
               figures
               in
               the
               quotient
               ,
               you
               may
               come
               very
               near
               with
               a
               good
               Line
               of
               Numbers
               ,
               as
               that
               on
               Serpentine-lines
               ,
               and
               the
               like
               ;
               
               but
               the
               difficulty
               is
               ,
               to
               know
               the
               Number
               of
               Figures
               ,
               which
               is
               thus
               most
               certainly
               done
               :
               Write
               down
               the
               Dividend
               ,
               and
               set
               the
               Divisor
               under
               it
               ,
               as
               in
               the
               vulgar
               way
               of
               Division
               ;
               and
               there
               must
               alwayes
               be
               as
               many
               Figures
               as
               the
               Dividend
               hath
               more
               than
               the
               Divisor
               ;
               and
               one
               more
               also
               ,
               when
               the
               first
               figure
               of
               the
               Dividend
               is
               greater
               than
               the
               first
               figure
               of
               the
               Divisor
               ;
               as
               if
               152178●365
               were
               to
               be
               divided
               by
               365
               ,
               then
               there
               would
               be
               3
               figures
               in
               the
               Quotient
               ;
               for
               the
               Divisor
               would
               be
               written
               3
               times
               under
               the
               Dividend
               ,
               in
               the
               usual
               way
               of
               Division
               ;
               and
               those
               figures
               be
               417
               almost
               :
               But
               if
               9172318
               ,
               is
               divided
               by
               8231
               ,
               you
               will
               have
               4
               figures
               ,
               viz.
               1115
               ,
               being
               one
               figure
               more
               than
               3
               ,
               the
               difference
               of
               places
               .
               In
               this
               Rule
               also
               you
               may
               see
               the
               excellency
               of
               the
               Artificial-Lines
               of
               Numbers
               ,
               before
               and
               above
               the
               Natural-Lines
               .
            
          
           
             
             
               Sect.
               III.
               To
               two
               Lines
               or
               Numbers
               given
               ,
               to
               find
               a
               third
               in
               continual
               Proportion
               Geometrical
               .
            
             
               
                 By
                 the
                 Line
                 of
                 Numbers
                 work
                 thus
                 :
              
            
             
               The
               Extent
               from
               one
               Number
               to
               the
               other
               ,
               shall
               reach
               the
               same
               way
               from
               that
               second
               to
               a
               third
               ,
               &c.
               
            
             
               Example
               .
            
             
               As
               5
               to
               7
               ,
               so
               is
               7
               to
               9
               —
               82
               ;
            
             
               So
               is
               9
               —
               82
               to
               13
               —
               76
               ,
               &c.
               
                 ad
                 infinitum
              
               .
            
             
               
                 By
                 the
                 Trianguler-Quadrant
                 ,
                 or
                 Sector
                 .
              
            
             
               As
               the
               Lateral
               first
               Number
               to
               the
               Parallel
               second
               ,
               laying
               the
               thred
               to
               the
               nearest
               distance
               ,
               there
               keep
               it
               :
            
             
               Then
               so
               is
               the
               Latteral
               second
               ,
               to
               the
               Parallel
               third
               .
            
          
           
             
             
               Sect.
               IV.
               Any
               one
               side
               of
               a
               Geometrical
               Figure
               being
               given
               ,
               to
               find
               all
               the
               rest
               ,
               or
               to
               find
               a
               Proportion
               between
               two
               or
               more
               Right
               Lines
               .
            
             
               
                 This
                 Proposition
                 is
                 most
                 proper
                 to
                 the
                 Line
                 of
                 Lines
                 ,
                 and
                 not
                 to
                 the
                 Line
                 of
                 Numbers
                 ;
                 and
                 done
                 thus
                 :
              
            
             
               Take
               the
               Line
               given
               ,
               and
               make
               it
               a
               Parallel
               in
               its
               respective
               Numbers
               ;
               the
               Thred
               so
               laid
               to
               the
               nearest
               distance
               ,
               or
               the
               Sector
               so
               set
               ,
               there
               keep
               it
               :
               then
               take
               out
               all
               the
               rest
               severally
               ,
               and
               carry
               the
               Compasses
               parallelly
               till
               they
               stay
               in
               like
               parts
               ,
               which
               shall
               be
               the
               Numbers
               required
               .
               Note
               the
               Figure
               .
            
             
               Note
               also
               ,
               That
               the
               Line
               of
               Sines
               ,
               and
               the
               Thred
               will
               readily
               lie
               on
               all
               the
               Angles
               ,
               and
               be
               removed
               from
               Radius
               to
               Radius
               more
               nimbly
               than
               any
               Sector
               whatsoever
               ,
               only
               by
               drawing
               the
               Thred
               streight
               ,
               and
               observing
               on
               what
               degree
               and
               part
               it
               cuts
               being
               so
               laid
               .
            
             
               Let
               ABCDEFG
               be
               the
               Plot
               of
               a
               Field
               ,
               whose
               side
               ED
               is
               only
               given
               to
               be
               
               9
               Chains
               ;
               and
               I
               would
               know
               all
               the
               rest
               :
               Take
               ED
               ,
               make
               it
               a
               =
               in
               9
               ;
               lay
               the
               Thred
               to
               ND
               ,
               or
               set
               the
               Sector
               to
               that
               gage
               ,
               and
               there
               keep
               it
               :
               then
               measure
               every
               side
               severally
               ,
               and
               you
               shall
               find
               what
               every
               one
               is
               in
               the
               same
               proportional
               parts
               ,
               by
               carrying
               the
               Compasses
               parallelly
               ,
               till
               it
               stay
               in
               like
               parts
               by
               the
               Sector
               ,
               or
               ND
               ,
               by
               the
               Quadrant
               .
            
          
           
             
               Sect.
               V.
               To
               lay
               down
               any
               Number
               of
               parts
               in
               a
               Line
               ,
               to
               any
               Scale
               less
               than
               the
               the
               Line
               of
               Lines
               .
            
             
               Take
               10
               ,
               or
               any
               other
               Number
               ,
               out
               of
               your
               given
               Scale
               ,
               or
               design
               any
               distance
               to
               be
               so
               much
               as
               you
               please
               ,
               and
               set
               one
               Point
               in
               the
               same
               Number
               ,
               on
               the
               Line
               of
               Lines
               ;
               and
               with
               the
               other
               ,
               lay
               the
               Thred
               to
               the
               nearest
               distance
               ,
               and
               there
               keep
               it
               ,
               by
               noting
               the
               degree
               cut
               by
               ;
               then
               take
               out
               any
               other
               Number
               that
               you
               would
               have
               ,
               setting
               one
               Point
               in
               that
               Number
               on
               the
               Line
               of
               Lines
               ;
               and
               opening
               the
               Compasses
               ,
               till
               the
               other
               Point
               will
               but
               just
               touch
               the
               middle
               ●f
               the
               Thred
               ,
               at
               ND
               ,
               and
               that
               shall
               be
               the
               
               other
               part
               required
               ;
               or
               the
               length
               of
               so
               much
               ,
               according
               to
               the
               first
               Scale
               given
               .
            
             
               
                 Example
                 .
                 Figure
                 I.
              
               
            
             
               Let
               AB
               represent
               a
               Line
               which
               is
               100
               parts
               ,
               and
               I
               would
               lay
               down
               65
               ,
               30
               ,
               42
               ,
               83
               parts
               of
               that
               100.
               
            
             
               First
               ,
               take
               all
               AB
               between
               your
               Compasses
               ,
               and
               set
               one
               Point
               in
               100
               at
               10
               with
               the
               other
               ,
               lay
               the
               Thred
               to
               ND
               ,
               then
               take
               out
               65
               ,
               30
               ,
               42
               ,
               83
               ,
               &c.
               parallelly
               ,
               and
               lay
               them
               down
               for
               the
               parts
               required
               ,
               as
               here
               you
               see
               .
            
             
               The
               like
               work
               is
               by
               the
               Sector
               ,
               making
               AB
               a
               =
               in
               100
               &
               100
               ;
               then
               take
               out
               =
               30
               ,
               42
               ,
               65
               ,
               83
               ,
               or
               any
               thing
               else
               for
               the
               parts
               required
               .
            
             
               But
               note
               ,
               If
               the
               Line
               be
               too
               large
               for
               your
               Scale
               ,
               or
               Line
               of
               Lines
               ,
               then
               take
               half
               ,
               or
               one
               third
               ,
               or
               fourth
               part
               of
               the
               given
               Line
               ;
               then
               if
               you
               take
               half
               ,
               you
               must
               at
               last
               turn
               the
               Compasses
               two
               times
               :
               If
               you
               take
               one
               third
               ,
               then
               turn
               the
               Compasses
               three
               times
               ;
               which
               may
               prove
               a
               very
               convenient
               help
               in
               many
               cases
               ,
               in
               Surveying
               and
               Dialling
               .
            
             
          
           
             
               Sect.
               VI.
               To
               divide
               a
               Line
               into
               any
               Number
               of
               Parts
               .
            
             
               Take
               the
               whole
               length
               of
               the
               Line
               between
               your
               Compasses
               ,
               and
               setting
               one
               Point
               in
               the
               Number
               of
               Parts
               ,
               you
               would
               have
               the
               Line
               divided
               into
               ;
               with
               the
               other
               ,
               lay
               the
               Thred
               to
               ND
               ,
               and
               there
               keep
               it
               ;
               then
               take
               the
               ND
               from
               1
               to
               the
               Thred
               ,
               and
               that
               shall
               divide
               the
               Line
               into
               the
               parts
               required
               .
            
             
               Example
               .
            
             
               Let
               AB
               be
               to
               be
               divided
               into
               7
               parts
               :
               Take
               AB
               ,
               make
               it
               a
               Parallel
               in
               7
               ,
               laying
               the
               Thred
               to
               the
               ND
               ,
               there
               keep
               it
               ;
               then
               the
               =
               1
               shall
               divide
               the
               Line
               into
               7
               parts
               .
               But
               if
               the
               Line
               were
               to
               be
               divided
               into
               many
               parts
               ,
               as
               suppose
               73
               :
               Then
               first
               ,
               fit
               the
               whole
               Line
               in
               =
               73
               ;
               then
               take
               out
               the
               =
               72
               ,
               71
               ,
               &
               70
               ,
               for
               the
               odd
               3
               ;
               then
               the
               =
               10
               s.
               for
               every
               10th
               division
               ,
               then
               the
               =
               1
               for
               the
               smaller
               parts
               ;
               or
               else
               you
               shall
               find
               it
               almost
               an
               impossible
               thing
               ,
               to
               take
               at
               
               once
               any
               distance
               ,
               which
               ,
               being
               turned
               above
               50
               times
               over
               ,
               shall
               not
               at
               last
               happen
               to
               be
               more
               or
               less
               than
               the
               desired
               Number
               required
               .
            
             
               Note
               ,
               That
               if
               the
               given
               Number
               happen
               to
               be
               such
               ,
               that
               the
               Part
               will
               fall
               too
               near
               the
               Center
               ;
               as
               suppose
               11
               ,
               12
               ,
               or
               any
               Number
               under
               30
               ;
               then
               you
               may
               double
               ,
               treble
               ,
               or
               quadruple
               the
               Number
               ,
               and
               then
               count
               2
               ,
               3
               ,
               or
               4
               ,
               for
               one
               of
               the
               Numbers
               required
               .
            
             
               
                 As
                 for
                 Example
              
               .
            
             
               Suppose
               I
               would
               divide
               a
               Line
               into
               15
               parts
               ;
               multiply
               15
               by
               6
               ,
               and
               it
               makes
               90
               :
               Now
               in
               regard
               you
               have
               multiplied
               15
               by
               6
               ,
               you
               must
               take
               the
               =
               6
               ,
               instead
               of
               the
               =
               1
               ,
               to
               divide
               the
               Line
               into
               15
               parts
               ,
               between
               your
               Compasses
               ,
               because
               the
               whole
               Line
               is
               set
               in
               =
               90
               ,
               instead
               of
               =
               15
               ;
               which
               is
               6
               times
               as
               much
               as
               15.
               
            
             
               Note
               also
               ,
               if
               the
               Line
               be
               too
               big
               for
               your
               Scale
               ,
               then
               take
               half
               ,
               or
               a
               third
               ,
               and
               make
               it
               a
               =
               in
               the
               given
               Line
               ;
               then
               take
               out
               the
               =
               1
               ,
               and
               turn
               two
               or
               three
               times
               ,
               to
               divide
               the
               Line
               according
               to
               your
               mind
               ,
               when
               it
               is
               too
               large
               for
               your
               Scale
               .
            
             
               These
               two
               last
               are
               not
               to
               be
               done
               by
               the
               Line
               of
               Numbers
               ,
               but
               proper
               for
               the
               Line
               of
               Sines
               only
               ;
               unless
               you
               turn
               your
               Lines
               
               to
               be
               divided
               into
               Numbers
               ,
               and
               then
               work
               by
               Proportion
               ,
               as
               thus
               ;
            
             
               As
               the
               whole
               Number
               of
               Parts
               ,
               is
               to
               the
               whole
               Line
               ,
               in
               any
               other
               parts
               ;
            
             
               So
               is
               1
               ,
               to
               as
               many
               of
               those
               Parts
               as
               belongs
               to
               1.
               
            
          
           
             
               Sect.
               VII
               .
               To
               find
               a
               mean
               Proportion
               ,
               between
               two
               Lines
               ,
               or
               Numbers
               given
               .
            
             
               A
               mean
               Proportion
               between
               two
               Lines
               or
               Numbers
               ,
               is
               that
               Number
               ,
               which
               being
               multiplied
               by
               it self
               ,
               shall
               produce
               a
               Number
               equal
               to
               the
               Product
               of
               the
               two
               Numbers
               given
               ,
               when
               they
               are
               multiplied
               the
               one
               by
               the
               other
               .
            
             
               Example
               .
            
             
               Let
               4
               and
               9
               be
               two
               Numbers
               ,
               between
               which
               a
               Geometrical
               mean
               is
               required
               .
            
             
               4
               and
               9
               ,
               multiplied
               together
               ,
               make
               36
               :
            
             
               So
               also
               6
               ,
               multiplied
               by
               it self
               ,
               is
               36
               :
               Therefore
               6
               is
               a
               mean
               Proportional
               between
               4
               and
               9.
               
               To
               find
               this
               by
               Arithmetick
               ,
               is
               by
               finding
               the
               Square-root
               of
               36.
               
            
             
             
               
                 But
                 by
                 the
                 Line
                 of
                 Numbers
                 ,
                 thus
              
               ;
            
             
               Divide
               the
               distance
               between
               4
               and
               9
               into
               two
               equal
               parts
               ,
               and
               the
               middle-point
               will
               be
               found
               to
               be
               6
               ,
               the
               Geometrical
               mean
               proportional
               required
               .
            
             
               
                 But
                 to
                 do
                 it
                 by
                 the
                 Line
                 of
                 Lines
                 ,
                 do
                 thus
              
               ;
            
             
               First
               ,
               joyn
               the
               Lines
               ,
               or
               Numbers
               ,
               together
               ,
               to
               get
               the
               sum
               of
               them
               ,
               and
               also
               the
               half
               sum
               ;
               and
               substract
               one
               from
               the
               other
               to
               get
               the
               difference
               ,
               and
               half
               the
               difference
               ;
               then
               count
               the
               half
               difference
               from
               the
               Center
               down-wards
               ;
               and
               note
               where
               it
               ends
               :
               then
               taking
               the
               half
               sum
               between
               your
               Compasses
               ,
               lay
               your
               Thred
               to
               00
               on
               the
               loose-piece
               ;
               then
               ,
               setting
               one
               Point
               in
               the
               half-difference
               ,
               on
               the
               Line
               of
               Lines
               :
               See
               where
               ,
               on
               the
               loose-piece
               ,
               the
               other
               Point
               shall
               touch
               the
               Thred
               ;
               and
               mark
               the
               place
               ,
               with
               a
               Bead
               on
               the
               Thred
               ,
               or
               a
               speck
               of
               Ink
               ,
               or
               otherwise
               :
               for
               the
               measure
               from
               thence
               to
               the
               Center
               is
               the
               mean
               Proportional
               required
               .
            
             
               
                 Or
                 else
                 use
                 this
                 most
                 excellent
                 way
                 by
                 Geometry
              
            
             
               Draw
               the
               Line
               AB
               ,
               and
               from
               any
               Scale
               
               of
               Equal
               Parts
               ,
               take
               off
               4
               and
               9
               ,
               and
               lay
               them
               from
               C
               ,
               to
               A
               and
               B
               ;
               then
               find
               out
               the
               true
               middle
               between
               A
               and
               B
               ,
               as
               at
               E
               ;
               and
               draw
               the
               half
               Circle
               ADB
               ;
               then
               on
               C
               erect
               a
               perpendiculer
               Line
               ,
               as
               CD
               ;
               then
               if
               you
               take
               CD
               between
               the
               Compasses
               ,
               and
               measure
               it
               on
               the
               same
               Scale
               that
               you
               took
               4
               and
               9
               from
               ,
               and
               you
               shall
               find
               it
               to
               be
               6
               ,
               the
               true
               mean
               proportional
               required
               :
               being
               only
               the
               way
               by
               the
               Line
               of
               Lines
               ,
               as
               by
               considering
               the
               Triangle
               CDE
               will
               appear
               .
            
             
               To
               do
               this
               by
               the
               Sector
               ,
               open
               the
               Line
               of
               Lines
               to
               a
               Right-Angle
               (
               by
               3
               ,
               4
               ,
               &
               5
               ,
               or
               6
               ,
               8
               ,
               &
               10.
               thus
               :
               Take
               10
               Latterally
               between
               your
               Compasses
               ,
               make
               it
               a
               Parallel
               in
               6
               and
               8
               ,
               then
               is
               the
               Line
               of
               Lines
               opened
               to
               a
               Right-Angle
               ;
               or
               if
               your
               Rule
               be
               large
               ,
               and
               your
               Compasses
               small
               ,
               then
               take
               Latteral
               5
               ,
               the
               half
               of
               10
               ,
               and
               make
               it
               a
               Parallel
               in
               3
               and
               4
               ,
               the
               half
               of
               6
               and
               8
               ,
               and
               it
               is
               rectangle
               also
               :
               )
               Then
               set
               half
               the
               difference
               on
               one
               Leg
               from
               the
               Center
               ,
               then
               having
               half
               the
               sum
               between
               your
               Compasses
               ,
               set
               one
               Point
               in
               the
               half-difference
               last
               counted
               ,
               and
               turn
               the
               other
               Point
               to
               the
               other
               Leg
               ,
               and
               there
               it
               shall
               shew
               the
               mean
               proportional
               Number
               required
               .
            
             
             
               
                 1.
                 
                 To
                 make
                 a
                 Square
                 ,
                 equal
                 to
                 an
                 Oblong
                 .
              
            
             
               Find
               a
               mean
               proportion
               between
               the
               length
               and
               the
               breadth
               of
               the
               Oblong
               ,
               and
               that
               shall
               be
               the
               side
               of
               a
               Square
               equal
               to
               the
               Oblong
               .
            
             
               Example
               .
            
             
               Let
               the
               breadth
               of
               the
               Oblong
               be
               4
               ,
               and
               the
               length
               9
               ,
               the
               mean
               proportion
               will
               be
               found
               to
               be
               6
               ;
               Therefore
               a
               Square
               ,
               whose
               side
               is
               6
               ,
               is
               equal
               to
               an
               Oblong
               ,
               whose
               breadth
               is
               4
               ,
               and
               length
               9
               ,
               of
               the
               same
               parts
               .
            
             
               
                 2.
                 
                 To
                 make
                 a
                 Square
                 ,
                 equal
                 to
                 a
                 Triangle
                 .
              
            
             
               Find
               a
               mean
               proportion
               between
               the
               half
               Base
               ,
               and
               the
               whole
               Perpendiculer
               ;
               and
               that
               shall
               be
               the
               side
               of
               a
               Square
               equal
               to
               the
               Triangle
               .
            
             
               Example
               .
            
             
               If
               the
               half-Base
               of
               a
               Gable-end
               be
               10
               ,
               and
               the
               whole
               Perpendiculer
               11-18
               ;
               the
               mean
               proportion
               between
               10
               and
               11-18
               ,
               is
               10-575
               ;
               the
               side
               of
               the
               Square
               equal
               to
               that
               Triangle
               ,
               or
               Gable-end
               required
               .
            
             
             
               
                 3.
                 
                 To
                 find
                 a
                 Proportion
                 between
                 the
                 Superfecies
                 ,
                 though
                 unlike
                 to
                 one
                 another
                 .
              
            
             
               First
               ,
               to
               every
               Superfecies
               ,
               find
               the
               side
               of
               his
               equal
               Square
               ,
               whether
               it
               be
               Circle
               ,
               Oblong
               ,
               Romboides
               ,
               or
               Triangle
               ;
               then
               the
               proportion
               between
               the
               sides
               of
               those
               Squares
               ,
               shall
               be
               the
               Proportion
               one
               to
               another
               .
            
             
               Example
               .
            
             
               Suppose
               I
               have
               a
               Triangle
               ,
               and
               a
               Circle
               ,
               and
               the
               side
               of
               the
               Square
               ,
               equal
               to
               the
               Circle
               ,
               is
               10
               inches
               ;
               and
               the
               side
               of
               the
               Square
               ,
               equal
               to
               the
               Triangle
               ,
               is
               15
               inches
               :
               The
               Proportion
               between
               these
               two
               Squares
               ,
               as
               they
               are
               Lines
               ,
               is
               as
               10
               to
               15
               ;
               but
               as
               Superfecies
               ,
               as
               100
               to
               45
               ;
               being
               thus
               found
               out
               ,
               Take
               the
               Extent
               between
               15
               and
               10
               ,
               on
               the
               Line
               of
               Numbers
               ,
               and
               repeat
               it
               two
               times
               the
               same
               way
               from
               100
               ,
               and
               it
               shall
               reach
               to
               45
               ,
               the
               Proportion
               as
               Superfecies
               ,
               between
               that
               Circle
               and
               Triangle
               ,
               whose
               Squares
               equal
               were
               15
               and
               10.
               
            
             
             
               
                 4.
                 
                 To
                 make
                 one
                 Superfecies
                 ,
                 equal
                 to
                 another
                 Superfecies
                 ,
                 of
                 another
                 shape
                 :
                 but
                 like
                 to
                 the
                 first
                 Superfecies
                 given
                 .
              
            
             
               First
               find
               a
               mean
               proportion
               between
               the
               unequal
               sides
               of
               the
               given
               Superfecies
               ,
               that
               you
               are
               to
               make
               one
               like
               ;
               and
               find
               the
               mean
               proportion
               also
               between
               the
               unequal
               sides
               of
               the
               Figure
               that
               you
               are
               to
               make
               one
               equal
               to
               .
            
             
               
                 As
                 thus
                 for
                 Example
              
               .
            
             
               I
               have
               a
               Romboides
               ,
               whose
               base
               is
               5
               ,
               and
               perpendiculer
               is
               3
               ,
               (
               and
               side
               is
               3-55
               )
               the
               mean
               proportion
               between
               is
               3-866
               :
               Also
               ,
               I
               have
               a
               Triangle
               ,
               whose
               half-base
               is
               8
               ,
               and
               the
               perpendiculer
               4
               ,
               the
               mean
               proportional
               is
               5-6552
               ;
               and
               I
               would
               make
               another
               Romboides
               as
               big
               as
               the
               Triangle
               given
               ,
               whos
               's
               Area
               is
               32
               :
               Then
               by
               the
               Line
               of
               Numbers
               ,
               say
               ,
               As
               3-866
               ,
               the
               one
               mean
               proportion
               ,
               is
               to
               5-6552
               ,
               the
               other
               mean
               proportion
               ;
               so
               is
               the
               Sides
               of
               the
               Romboides
               ,
               whose
               like
               I
               am
               to
               make
               ,
               to
               the
               sides
               and
               perpendiculer
               of
               the
               Romboides
               required
               ,
               to
               make
               a
               Romboides
               equal
               to
               a
               Triangle
               given
               ,
               and
               like
               to
               another
               Romboides
               first
               given
               .
            
             
             
               
                 As
                 thus
                 for
                 Example
              
               .
            
             
               As
               3-866
               is
               to
               5-6552
               ;
               so
               is
               5
               ,
               the
               base
               of
               the
               Romboides
               given
               ,
               to
               7-30
               ,
               the
               base
               of
               the
               Romboides
               required
               .
               And
               so
               is
               3
               ,
               the
               given
               perpendiculer
               of
               the
               Romboides
               ,
               to
               4-38
               ,
               the
               perpendiculer
               of
               the
               Romboides
               required
               :
               So
               also
               is
               3-55
               ,
               the
               side
               of
               the
               Romboides
               given
               ,
               to
               5-19
               ,
               the
               side
               of
               the
               Romboides
               required
               :
               for
               ,
               if
               you
               multiply
               7-30
               ,
               the
               base
               thus
               found
               ,
               by
               4-38
               ,
               the
               perpendiculer
               now
               found
               ,
               it
               will
               make
               a
               Romboides
               ,
               whose
               Area
               is
               equal
               to
               32
               ,
               the
               Area
               of
               the
               Triangle
               ,
               that
               I
               was
               to
               make
               the
               Romboides
               equal
               to
               ;
               and
               making
               the
               side
               to
               be
               3-55
               ,
               it
               will
               be
               like
               the
               first
               Romboides
               propounded
               .
            
             
               If
               it
               had
               been
               a
               Trapesia
               ,
               or
               other
               formed
               Figure
               ,
               it
               might
               have
               been
               resolved
               into
               Triangles
               ,
               and
               then
               brought
               into
               Squares
               ,
               as
               before
               :
               Then
               all
               them
               Squares
               added
               into
               one
               sum
               ,
               whose
               Square-root
               is
               the
               mean
               proportional
               or
               side
               of
               a
               Square
               ,
               equal
               to
               that
               many-sided
               Figure
               ,
               whose
               like
               or
               equal
               is
               desired
               to
               be
               made
               and
               produced
               .
            
             
             
               
                 5.
                 
                 One
                 Diameter
                 and
                 Content
                 of
                 a
                 Circle
                 given
                 ,
                 to
                 find
                 the
                 Content
                 of
                 another
                 Circle
                 ,
                 by
                 having
                 the
                 Diameter
                 thereof
                 only
                 given
                 .
              
            
             
               The
               Extent
               from
               one
               Diameter
               to
               the
               other
               ,
               being
               twice
               repeated
               the
               right-way
               from
               the
               given
               Area
               ,
               shall
               reach
               to
               the
               Area
               required
               .
            
             
               If
               the
               Area's
               of
               two
               Circles
               be
               given
               ,
               and
               the
               Diameter
               required
               ;
               then
               the
               half-distance
               on
               the
               Numbers
               ,
               between
               the
               two
               Area's
               ,
               shall
               reach
               from
               the
               one
               Diameter
               to
               the
               other
               .
            
          
           
             
               Sect.
               VIII
               .
               To
               find
               the
               Square-root
               of
               a
               Number
               .
            
             
               To
               do
               this
               by
               the
               Line
               of
               Numbers
               ,
               you
               must
               first
               consider
               ,
               whether
               the
               Figures
               ,
               whereby
               the
               Number
               ,
               whose
               Root
               you
               would
               have
               ,
               is
               expressed
               ,
               be
               even
               or
               odd
               figures
               ,
               that
               is
               ,
               consist
               of
               2
               ,
               4
               ,
               6
               ,
               8
               ,
               or
               10
               ;
               or
               1
               ,
               3
               ,
               5
               ,
               7
               ,
               or
               9
               figures
               .
            
             
               For
               if
               it
               be
               of
               even
               figures
               ,
               then
               you
               must
               count
               the
               10
               at
               the
               end
               for
               the
               unite
               ;
               and
               the
               Root
               and
               Square
               are
               backwards
               toward
               1.
               
            
             
             
               But
               if
               it
               consist
               of
               odd
               Figures
               ,
               then
               the
               1
               ,
               in
               the
               middle
               of
               the
               Line
               ,
               is
               the
               unite
               ;
               and
               the
               Root
               and
               Square
               is
               forwards
               towards
               10
               :
               for
               the
               Square-root
               of
               any
               Number
               ,
               is
               alwayes
               the
               mean
               proportional
               ,
               or
               middle
               space
               between
               1
               ,
               and
               the
               Number
               propounded
               ;
               counting
               the
               unite
               according
               to
               the
               Rule
               abovesaid
               :
               So
               that
               the
               Square-Root
               of
               1728
               ,
               consisting
               of
               four
               figures
               ,
               it
               is
               at
               41
               and
               ●
               /
               ●
               ,
               counting
               10
               for
               the
               unite
               ;
               for
               the
               Number
               42
               &
               ●
               /
               ●●
               ,
               is
               just
               in
               the
               middest
               between
               1728
               and
               10.
               
            
             
               And
               to
               find
               the
               Square-root
               of
               144
               ,
               consisting
               of
               three
               figures
               ;
               divide
               the
               space
               between
               the
               middle
               1
               and
               144
               ,
               counted
               forwards
               ,
               into
               two
               equal
               parts
               ,
               and
               the
               Point
               shall
               rest
               at
               12
               ,
               the
               Square-root
               required
               .
            
             
               
                 To
                 do
                 this
                 by
                 the
                 Line
                 of
                 Lines
                 ,
                 or
                 Sector
                 .
              
            
             
               First
               ,
               find
               out
               a
               Number
               ,
               that
               may
               part
               the
               Number
               given
               evenly
               ,
               or
               as
               even
               as
               may
               be
               ;
               then
               the
               Divisor
               shall
               be
               one
               extream
               ,
               and
               the
               Quotient
               another
               extream
               ;
               the
               mean
               proportional
               between
               which
               two
               ,
               shall
               be
               the
               Square-root
               required
               ,
               working
               by
               the
               last
               Rule
               .
            
             
             
               Example
               .
            
             
               To
               find
               the
               Square-root
               of
               144.
               
               If
               you
               divide
               144
               by
               9
               ,
               you
               shall
               find
               16
               in
               the
               Quotient
               :
               Now
               a
               mean
               proportion
               between
               9
               the
               Divisor
               ,
               and
               16
               the
               Quotient
               ,
               is
               12
               the
               Root
               ,
               found
               by
               the
               last
               Rule
               ,
               viz.
               the
               7th
               .
            
          
           
             
               Sect.
               IX
               .
               To
               find
               the
               Cubick-Root
               of
               a
               Number
               .
            
             
               The
               Cubick-root
               of
               a
               Number
               ,
               is
               alwayes
               the
               first
               of
               two
               mean
               proportionals
               between
               1
               ,
               and
               the
               Number
               given
               ;
               counting
               the
               unite
               with
               the
               following
               cautions
               :
               Set
               the
               Number
               down
               ,
               and
               put
               a
               Point
               under
               the
               1st
               ,
               the
               4th
               ,
               the
               7th
               ,
               and
               the
               10th
               figure
               ;
               and
               look
               how
               many
               Points
               you
               have
               ,
               so
               many
               figures
               shall
               you
               have
               in
               the
               Root
               .
            
             
               Then
               if
               the
               last
               Point
               fall
               on
               the
               last
               Figure
               ,
               then
               the
               middle
               1
               must
               be
               the
               unite
               ,
               and
               the
               Root
               ,
               the
               Square
               ,
               and
               Cube
               will
               fall
               forwards
               toward
               10.
               
            
             
               But
               if
               the
               last
               Point
               fall
               on
               the
               last
               but
               one
               ,
               then
               the
               unite
               may
               be
               placed
               at
               either
               end
               ,
               viz.
               at
               1
               at
               the
               beginning
               ,
               or
               at
               10
               at
               the
               end
               ;
               and
               then
               the
               Cube
               will
               be
               one
               Radius
               beyond
               the
               unite
               ,
               either
               forwards
               or
               backwards
               .
            
             
             
               But
               if
               it
               fall
               on
               the
               last
               but
               two
               ,
               then
               10
               at
               the
               end
               of
               the
               Line
               must
               be
               the
               unite
               ;
               and
               the
               Root
               ,
               the
               Square
               ,
               and
               Cube
               will
               alwayes
               be
               in
               the
               same
               Radius
               ,
               that
               is
               between
               10
               at
               the
               end
               ,
               and
               the
               middle
               1.
               
            
             
               So
               that
               by
               these
               Rules
               ,
               the
               Cubick-root
               of
               8
               is
               2
               ;
               for
               putti●g
               a
               Point
               under
               8
               ,
               being
               but
               one
               figure
               it
               hath
               but
               one
               Point
               ,
               therefore
               but
               one
               figure
               in
               the
               Root
               :
               Secondly
               ,
               the
               Point
               being
               under
               the
               last
               figure
               ,
               the
               middle
               1
               is
               the
               unite
               ;
               then
               dividing
               the
               space
               between
               1
               and
               8
               ,
               into
               three
               equal
               parts
               ;
               the
               first
               part
               ends
               at
               2
               ,
               the
               Root
               required
               .
            
             
               So
               likewise
               in
               1331
               ,
               there
               is
               two
               Points
               ,
               therefore
               two
               figures
               in
               the
               Root
               ;
               and
               the
               last
               Point
               being
               under
               the
               last
               Figure
               ,
               the
               middle
               1
               is
               the
               unite
               ;
               and
               the
               space
               between
               1
               and
               1331
               ,
               being
               divided
               into
               three
               equal
               parts
               ,
               the
               first
               part
               doth
               end
               at
               11
               ,
               the
               Cubick-root
               of
               1331.
               
            
             
               Again
               ,
               for
               64
               there
               is
               one
               Point
               ,
               and
               it
               falls
               on
               the
               last
               figure
               but
               one
               ;
               therefore
               the
               Root
               contains
               but
               one
               figure
               ,
               and
               1
               at
               the
               beginning
               ,
               or
               10
               at
               the
               end
               ,
               which
               you
               please
               ,
               may
               be
               the
               unite
               .
            
             
               But
               yet
               with
               this
               Caution
               ,
               That
               the
               Cube
               must
               be
               in
               the
               next
               Radius
               beyond
               that
               which
               belongs
               to
               the
               unite
               ;
               so
               that
               dividing
               the
               space
               between
               10
               and
               64
               ,
               beyond
               the
               
               middle
               1
               ,
               towards
               the
               beginning
               ,
               into
               three
               equal
               parts
               ;
               the
               first
               part
               falls
               on
               4
               ,
               the
               Cubick-root
               required
               :
               Or
               ,
               if
               you
               divide
               the
               space
               between
               1
               and
               64
               ,
               near
               the
               10
               ,
               into
               three
               equal
               parts
               ,
               the
               first
               part
               falls
               on
               4
               also
               .
            
             
               Again
               ,
               for
               729
               ,
               there
               is
               but
               one
               Point
               ;
               therefore
               but
               one
               figure
               :
               Again
               ,
               it
               falls
               on
               the
               last
               but
               2
               ;
               therefore
               10
               at
               the
               end
               is
               the
               unite
               ;
               and
               between
               10
               and
               the
               middle
               1
               backwards
               ,
               you
               shall
               have
               both
               Root
               ,
               Square
               ,
               and
               Cube
               ,
               for
               the
               Number
               required
               ,
               which
               will
               be
               at
               9
               ;
               For
               if
               you
               divide
               the
               space
               between
               10
               ,
               and
               729
               ,
               into
               three
               equal
               parts
               ,
               the
               first
               part
               will
               stay
               at
               9
               ,
               the
               Cubick-root
               required
               .
            
             
               Note
               ,
               if
               it
               be
               a
               surd
               Number
               that
               cannot
               be
               cubed
               exactly
               ;
               yet
               the
               Number
               of
               figures
               to
               be
               accounted
               as
               Integers
               is
               as
               before
               ;
               and
               the
               residue
               discoverable
               by
               the
               Line
               ,
               is
               a
               Decimal
               Fraction
               .
            
             
               Example
               .
            
             
               For
               1750
               the
               Root
               resulting
               ,
               is
               12
               04●
               /
               1000
               ,
               or
               12
               ,
               and
               near
               5
               of
               a
               100.
               
            
             
               Thus
               you
               have
               a
               very
               good
               and
               ready
               way
               for
               this
               hard
               question
               in
               Arithmetick
               ▪
               and
               will
               come
               near
               enough
               for
               most
               uses
               .
            
             
               But
               to
               perform
               this
               by
               the
               Natural-lines
               ,
               at
               the
               best
               it
               is
               very
               troublesome
               ,
               and
               cannot
               come
               to
               no
               such
               exactness
               ,
               as
               by
               the
               Line
               of
               
               Numbers
               ;
               and
               therefore
               I
               shall
               omit
               it
               a●
               inconvenient
               .
            
             
               For
               Application
               or
               Use
               of
               this
               last
               Rule
               of
               finding
               the
               Cube-Root
               ,
               observe
               with
               me
               as
               followeth
               :
            
             
               
                 1.
                 
                 Between
                 two
                 Numbers
                 ,
                 or
                 Lines
                 given
                 ,
                 to
                 find
                 two
                 mean
                 proportional
                 Numbers
                 ,
                 or
                 Lines
                 required
                 .
              
            
             
               Divide
               the
               space
               on
               the
               Line
               of
               Numbers
               ,
               between
               the
               two
               Numbers
               given
               ,
               into
               three
               equal
               parts
               ;
               and
               the
               Numbers
               where
               the
               Points
               of
               the
               Compasses
               stay
               at
               each
               repetition
               ,
               (
               or
               turning
               )
               shall
               be
               the
               two
               mean
               proportional
               Numbers
               required
               .
            
             
               Example
               .
            
             
               Let
               4
               and
               32
               ,
               be
               two
               extream
               Numbers
               (
               or
               the
               measure
               of
               two
               extream
               Lines
               )
               between
               which
               I
               would
               have
               ,
               two
               mean
               proportional
               Numbers
               (
               or
               Lines
               )
               required
               .
            
             
               In
               dividing
               the
               space
               on
               the
               Line
               of
               Numbers
               ,
               between
               4
               and
               32
               ,
               into
               three
               equal
               parts
               ,
               you
               shall
               find
               the
               Compasses
               to
               stay
               first
               at
               8
               ,
               secondly
               at
               16
               ;
               the
               two
               mean
               proportionals
               ,
               between
               8
               and
               32
               ,
               the
               two
               extream
               Numbers
               first
               given
               .
            
             
               For
               the
               Square
               or
               Product
               of
               4
               and
               32
               ,
               the
               two
               extreams
               ,
               128
               ,
               is
               equal
               to
               the
               Square
               
               or
               Product
               of
               8
               and
               16
               ,
               the
               two
               means
               multiplied
               together
               ,
               being
               128
               also
               .
            
             
               2.
               
               To
               apply
               it
               then
               thus
               for
               Example
               :
               If
               I
               have
               the
               solid
               content
               of
               a
               Cube
               to
               be
               1728
               Cubick
               inches
               ,
               and
               the
               side
               thereof
               be
               12
               inches
               ;
               I
               would
               know
               what
               shall
               the
               side
               of
               the
               Cube
               be
               ,
               whose
               solid
               content
               is
               3456
               ,
               the
               double
               of
               1728
               ?
               Divide
               the
               space
               between
               1728
               ,
               and
               3456
               ,
               into
               three
               equal
               parts
               ;
               then
               ,
               lay
               the
               same
               distance
               the
               Compasses
               stand
               at
               from
               12
               ,
               the
               side
               of
               the
               Cube
               given
               ,
               and
               it
               shall
               reach
               to
               15-12
               ,
               the
               side
               of
               the
               Cube
               required
               ,
               whose
               solid
               content
               is
               3456
               inches
               .
            
             
               Also
               ,
               If
               I
               have
               a
               Shot
               of
               Iron
               ,
               whose
               weight
               is
               3
               pound
               ,
               and
               the
               diameter
               thereof
               2
               inches
               ,
               and
               780
               parts
               of
               an
               inch
               in
               a
               1000
               ;
               what
               shall
               the
               diameter
               of
               a
               Shot
               be
               ,
               whose
               weight
               is
               71
               pound
               ?
               One
               third
               part
               of
               the
               distance
               ,
               on
               the
               Line
               of
               Numbers
               ,
               between
               3
               pound
               ,
               and
               71
               pound
               ,
               shall
               reach
               from
               2
               inches
               ,
               780
               parts
               ,
               the
               given
               diameter
               ,
               to
               8
               inches
               ,
               the
               true
               diameter
               of
               a
               cast
               Iron
               Bullet
               ,
               whose
               weight
               is
               71
               pound
               .
            
             
             
               
                 
                   2
                   Secondly
                
                 ,
                 on
                 the
                 contrary
                 ,
                 if
                 the
                 Diameter
                 and
                 Content
                 of
                 one
                 Globe
                 ,
                 or
                 Cube
                 be
                 given
                 ,
                 and
                 the
                 Diameter
                 of
                 another
                 Globe
                 or
                 Cube
                 ,
                 to
                 find
                 the
                 content
                 thereof
                 .
              
            
             
               As
               the
               diameter
               of
               the
               Globe
               ,
               whose
               content
               is
               also
               given
               ,
               is
               to
               the
               diameter
               of
               the
               Globe
               whose
               content
               is
               required
               ;
               so
               is
               the
               content
               given
               ,
               to
               the
               content
               required
               ;
               by
               repeating
               the
               Extent
               the
               same
               way
               three
               times
               .
               Example
               .
               Suppose
               the
               capassity
               or
               content
               of
               a
               Globe
               ,
               whose
               diameter
               is
               10
               inches
               ,
               be
               523
               inches
               solid
               ,
               and
               80
               parts
               ;
               what
               shall
               the
               content
               of
               that
               Globe
               be
               whose
               diameter
               is
               20
               inches
               ?
               the
               Extent
               from
               10
               to
               20
               ,
               being
               turned
               three
               times
               from
               523-8
               ,
               the
               content
               of
               a
               Globe
               of
               10
               inches
               diameter
               ,
               shall
               reach
               to
               4190
               10
               /
               1
               ,
               the
               Cubick
               inches
               contained
               in
               a
               Globe
               of
               20
               inches
               diameter
               ,
               being
               8
               times
               as
               much
               as
               the
               former
               .
            
             
               
                 3.
                 
                 The
                 Proportion
                 between
                 the
                 weights
                 and
                 magnitudes
                 of
                 several
                 Metals
                 ,
                 are
                 as
                 followeth
                 ,
                 according
                 to
                 
                   Marinus
                   Ghetaldi
                
                 .
              
            
             
               If
               7
               pieces
               of
               the
               7
               Metals
               ,
               are
               all
               of
               one
               shape
               ,
               and
               bigness
               ,
               either
               Sphears
               ,
               or
               Cubes
               ,
               or
               Cillenders
               ,
               or
               Parallelepipedons
               ;
               then
               their
               weights
               are
               in
               proportion
               as
               followeth
               ,
               according
               to
               
                 Marinus
                 Ghetaldi
              
               .
            
             
             
               
                 The
                 Shape
                 and
                 Magnitudes
                 equal
                 ,
                 The
                 Weights
                 are
                 in
                 proportion
                 ,
                 as
                 ,
              
            
             
               
                 
                   
                     ♃
                     Tinn
                  
                   
                     1554
                  
                
                 
                   
                     ♂
                     Iron
                  
                   
                     1680
                  
                
                 
                   
                     ♀
                     Copper
                  
                   
                     1890
                  
                
                 
                   
                     ☽
                     Silver
                  
                   
                     2030
                  
                
                 
                   
                     ♄
                     Lead
                  
                   
                     2415
                  
                
                 
                   
                     ☿
                     Quicksilver
                  
                   
                     2850
                  
                
                 
                   
                     ☉
                     Gold
                  
                   
                     3990
                  
                
              
            
             
               So
               that
               if
               a
               Cillender
               of
               Tinn
               ,
               whose
               side
               is
               one
               inch
               ,
               weigh
               1824
               grains
               ;
               What
               shall
               a
               Cillender
               of
               Gold
               weigh
               ,
               the
               height
               and
               diameter
               being
               just
               one
               inch
               ,
               viz.
               4682.
               
            
             
               For
               as
               1554
               ,
               is
               to
               3990
               ;
               So
               is
               1824
               ,
               to
               4682
               ,
               the
               grains
               in
               one
               inch
               of
               Gold.
               
            
             
             
               
                 The
                 Shapes
                 and
                 Weights
                 of
                 the
                 pieces
                 of
                 the
                 seven
                 several
                 Metals
                 being
                 equal
                 ,
                 then
                 the
                 Magnitudes
                 of
                 the
                 sides
                 are
                 as
                 followeth
                 ,
                 according
                 to
                 Mr.
                 Gunter
                 .
              
            
             
               
                 
                   
                     ☉
                     Gold
                  
                   
                     3895
                  
                
                 
                   
                     ☿
                     Quicksilver
                  
                   
                     5433
                  
                
                 
                   
                     ♄
                     Lead
                  
                   
                     6435
                  
                
                 
                   
                     ☽
                     Silver
                  
                   
                     7161
                  
                
                 
                   
                     ♀
                     Copper
                  
                   
                     8222
                  
                
                 
                   
                     ♂
                     Iron
                  
                   
                     9250
                  
                
                 
                   
                     ♃
                     Tinn
                  
                   
                     10000
                  
                
              
            
             
               So
               that
               if
               I
               have
               a
               Sphear
               of
               Iron
               ,
               weigheth
               9
               pound
               ,
               whose
               Diameter
               is
               4
               inches
               ;
               What
               must
               the
               Diameter
               of
               a
               Leaden
               Sphear
               ,
               or
               Bullet
               ,
               be
               of
               the
               same
               weight
               ?
            
             
               
                 Say
                 thus
              
               ;
            
             
               One
               third
               part
               of
               the
               space
               between
               9250
               ,
               and
               6435
               ,
               shall
               reach
               from
               4
               the
               Diameter
               of
               the
               Iron
               Bullet
               ,
               to
               3
               inches
               54
               parts
               ,
               the
               diameter
               of
               the
               Leaden
               Bullet
               ,
               that
               weighs
               9
               pound
               .
            
             
             
               
                 4.
                 
                 So
                 that
                 if
                 I
                 have
                 the
                 weight
                 and
                 magnitude
                 ,
                 of
                 a
                 body
                 of
                 one
                 kind
                 of
                 Metal
                 ,
                 and
                 would
                 know
                 the
                 magnitude
                 of
                 a
                 body
                 of
                 another
                 Metal
                 ,
                 having
                 the
                 same
                 weight
                 :
                 work
                 thus
              
               ;
            
             
               The
               first
               of
               two
               mean
               proportionals
               ,
               between
               the
               two
               Points
               on
               the
               Line
               of
               Numbers
               ,
               representing
               the
               Numbers
               in
               the
               last
               Table
               ,
               for
               the
               two
               Metals
               ,
               shall
               reach
               the
               right
               way
               ,
               from
               the
               Magnitude
               given
               ,
               to
               the
               Magnitude
               required
               .
            
             
               
                 As
                 in
                 the
                 Example
                 before
                 ,
                 and
                 illustrated
                 by
                 another
                 thus
              
               ;
            
             
               Suppose
               a
               Cube
               of
               Gold
               ,
               whose
               side
               is
               2
               inches
               ,
               weigh
               29000
               grains
               ;
               What
               shall
               the
               side
               of
               a
               Cube
               of
               Tinn
               be
               ,
               having
               the
               same
               weight
               ?
               Divide
               the
               space
               on
               the
               Line
               of
               Numbers
               ,
               between
               3895
               ,
               the
               Point
               on
               the
               Numbers
               for
               Gold
               ;
               and
               10000
               ,
               the
               Point
               for
               Tinn
               ;
               and
               this
               extent
               parted
               into
               3
               equal
               parts
               ,
               and
               that
               distance
               laid
               from
               2
               ,
               the
               side
               of
               the
               Cube
               of
               Gold
               ,
               shall
               reach
               to
               2-74
               ;
               the
               side
               of
               the
               Cube
               of
               Tinn
               required
               .
            
             
               
                 
                 5.
                 
                 The
                 Magnitudes
                 of
                 two
                 bodies
                 of
                 several
                 Metals
                 being
                 given
                 ,
                 and
                 the
                 weight
                 of
                 the
                 one
                 ,
                 to
                 find
                 the
                 weight
                 of
                 the
                 other
                 .
              
            
             
               Take
               the
               Extent
               between
               their
               Points
               ,
               on
               the
               Line
               of
               Numbers
               ,
               according
               to
               the
               last
               Table
               ,
               for
               each
               several
               Metal
               ;
               and
               this
               Extent
               laid
               from
               the
               given
               weight
               ,
               shall
               reach
               to
               the
               enquired
               weight
               ,
               of
               the
               other
               Metal
               propounded
               .
            
             
               Example
               .
            
             
               If
               a
               Bullet
               of
               Iron
               ,
               of
               4
               inches
               Diameter
               ,
               will
               weigh
               9
               pound
               ;
               a
               Bullet
               of
               Lead
               ,
               of
               the
               same
               Diameter
               ,
               will
               weigh
               13
               pound
               .
            
             
               
                 6.
                 
                 A
                 body
                 of
                 one
                 Metal
                 being
                 given
                 ,
                 to
                 make
                 another
                 body
                 like
                 unto
                 it
                 of
                 another
                 Metal
                 ,
                 and
                 any
                 other
                 weight
                 ,
                 to
                 find
                 the
                 Diameters
                 and
                 Magnitudes
                 thereof
                 .
              
            
             
               First
               ,
               by
               the
               4th
               last
               past
               ,
               find
               the
               Magnitude
               of
               the
               side
               ,
               or
               Diameter
               of
               the
               Sphear
               ,
               having
               equal
               weight
               ;
               and
               note
               that
               down
               ,
               or
               keep
               it
               .
            
             
               Then
               find
               out
               two
               mean
               proportions
               ,
               between
               the
               weights
               given
               ;
               and
               setting
               this
               distance
               ,
               on
               the
               Line
               of
               Numbers
               ,
               the
               right
               way
               ,
               either
               increasing
               ,
               or
               diminishing
               from
               
               the
               side
               given
               ,
               shall
               shew
               the
               side
               or
               diameter
               required
               .
            
             
               Example
               .
            
             
               Suppose
               I
               have
               a
               Bullet
               of
               Iron
               weighing
               60
               pound
               ,
               and
               the
               diameter
               thereof
               is
               7
               inches
               ,
               57
               parts
               ;
               I
               would
               have
               a
               Bullet
               of
               Lead
               like
               unto
               it
               ,
               to
               weigh
               one
               third
               part
               more
               ;
               what
               must
               his
               diameter
               be
               ?
            
             
               First
               by
               finding
               out
               of
               two
               mean
               proportionals
               ,
               between
               the
               Points
               ,
               on
               the
               Line
               of
               Numbers
               ,
               for
               each
               Metal
               ;
               and
               laying
               it
               the
               right
               way
               from
               the
               side
               of
               the
               body
               given
               ,
               it
               gives
               the
               diameter
               of
               the
               other
               body
               of
               equal
               weight
               ,
               with
               the
               first
               given
               body
               ;
               as
               the
               Iron
               Bullet
               being
               7
               inches
               ,
               57
               parts
               diameter
               ,
               the
               Leaden
               Bullet
               of
               the
               same
               weight
               is
               but
               6
               inches
               ,
               and
               70
               parts
               .
            
             
               Then
               find
               two
               mean
               proportionals
               between
               60
               ,
               the
               weight
               given
               ;
               and
               80
               ,
               one
               third
               part
               more
               :
               and
               repeat
               this
               the
               right
               way
               (
               viz.
               increasing
               )
               from
               6
               inches
               ,
               70
               parts
               ,
               and
               it
               shall
               shew
               on
               the
               Line
               of
               Numbers
               7-37
               ,
               the
               diameter
               of
               a
               Leaden
               Bullet
               ,
               that
               shall
               weigh
               one
               third
               part
               more
               than
               an
               Iron
               Bullet
               of
               60
               pound
               weight
               ,
               viz.
               80
               pound
               .
            
             
               The
               performance
               of
               these
               Propositions
               by
               the
               Sector
               ,
               without
               the
               help
               of
               a
               Line
               of
               Superfecies
               ,
               and
               solids
               ,
               is
               very
               troublesome
               to
               do
               :
               And
               if
               you
               have
               a
               right
               Gunters
               Sector
               ,
               you
               may
               have
               also
               his
               Book
               of
               the
               
               Use
               thereof
               ,
               but
               this
               way
               by
               the
               Numbers
               is
               as
               quick
               ,
               and
               more
               certain
               ,
               having
               your
               proportional
               Points
               or
               Numbers
               true
               ;
               but
               the
               way
               by
               the
               Pen
               is
               best
               .
            
          
           
             
               Sect.
               X.
               To
               divide
               a
               Line
               ,
               or
               Number
               ,
               by
               extream
               and
               mean
               proportion
               .
            
             
               To
               part
               ,
               or
               divide
               a
               Line
               or
               Number
               ,
               by
               extream
               and
               mean
               proportion
               ,
               is
               to
               part
               it
               so
               ,
               that
               the
               greatest
               part
               shall
               bear
               such
               proportion
               to
               the
               lesser
               ,
               as
               the
               whole
               doth
               to
               the
               greatest
               part
               ;
               so
               that
               the
               square
               of
               the
               greater
               part
               ,
               shall
               be
               equal
               to
               the
               Product
               of
               the
               whole
               ,
               and
               the
               lesser
               part
               multiplied
               together
               :
               Or
               ,
               if
               you
               add
               the
               Square
               of
               the
               whole
               ,
               and
               the
               square
               of
               the
               lesser-part
               together
               ,
               it
               is
               3
               times
               as
               much
               as
               the
               square
               of
               the
               greater-part
               .
               
                 See
                 Dig's
                 Theorems
              
               .
            
             
               As
               if
               you
               would
               part
               12
               ,
               by
               extream
               and
               mean
               proportion
               ,
               the
               greater-part
               will
               be
               7-42
               ,
               &
               the
               lesser-part
               4-58
               near
               ;
               and
               the
               whole
               is
               12
               :
               as
               by
               squaring
               and
               dividing
               you
               shall
               find
               .
            
             
               For
               the
               extent
               of
               the
               Compasses
               on
               the
               Line
               of
               Numbers
               ,
               from
               12
               to
               7-42
               ,
               being
               turned
               once
               ,
               the
               same
               way
               ,
               from
               7-42
               ,
               shall
               reach
               to
               4-58
               ,
               the
               lesser
               part
               .
            
             
             
               
                 To
                 find
                 this
                 by
                 Arithmetick
                 ,
                 do
                 thus
              
               ;
            
             
               First
               ,
               square
               the
               given
               Number
               (
               that
               is
               ,
               multiply
               it
               by
               it self
               )
               then
               multiply
               the
               Product
               by
               5
               ,
               and
               divide
               this
               Product
               by
               4
               ;
               then
               find
               the
               Square-root
               of
               the
               Quotient
               ,
               and
               from
               it
               take
               half
               the
               given
               Number
               ,
               the
               residue
               is
               the
               greater
               portion
               ,
               then
               the
               greater
               part
               taken
               from
               the
               whole
               ,
               leaves
               the
               lesser-part
               .
            
             
               
                 By
                 the
                 Sector
                 ,
                 work
                 thus
              
               ;
            
             
               Open
               the
               Sector
               to
               a
               Right
               Angle
               ,
               in
               the
               Line
               of
               Lines
               (
               making
               Latteral
               10
               ,
               a
               Parallel
               in
               8
               and
               6
               )
               or
               else
               make
               the
               Latteral
               90
               ,
               a
               Parallel
               Sine
               of
               45
               (
               or
               the
               Latteral
               Sine
               of
               45
               ,
               a
               Parallel
               Sine
               of
               30
               )
               then
               upon
               both
               Legs
               count
               the
               given
               Number
               ;
               then
               take
               the
               Parallel
               Ex●ent
               from
               the
               whole
               Number
               on
               one
               Leg
               ,
               to
               the
               half
               on
               the
               other
               Leg
               ,
               and
               lay
               this
               from
               the
               Center
               Latterally
               ;
               and
               whatsoever
               the
               Point
               reacheth
               beyond
               the
               whole
               Number
               ,
               must
               be
               added
               to
               the
               half
               Number
               ,
               to
               make
               up
               the
               greater
               Number
               ;
               or
               taken
               from
               the
               half
               to
               make
               the
               lesser
               .
            
             
               Example
               .
            
             
               Let
               the
               Number
               given
               be
               12
               ,
               which
               may
               be
               represented
               at
               6
               on
               the
               Line
               of
               Lines
               ;
               
               then
               the
               Sector
               standing
               at
               Right
               Angles
               ,
               take
               the
               Parallel-distance
               from
               3
               ,
               the
               half
               of
               6
               (
               counted
               as
               12
               )
               on
               one
               Leg
               ,
               to
               6
               on
               the
               other
               Leg
               ,
               and
               you
               shall
               find
               it
               reach
               to
               6-71
               ,
               which
               doubled
               is
               13-42
               ;
               from
               which
               if
               you
               take
               6
               ,
               the
               half
               sum
               ,
               rest
               7-42
               for
               the
               greater
               part
               :
               and
               if
               you
               take
               7-42
               from
               12
               ,
               there
               remains
               4-58
               ,
               the
               lesser-part
               .
            
             
               
                 But
                 by
                 the
                 Line
                 of
                 Numbers
                 work
                 thus
                 :
                 following
                 the
                 Arithmetical
                 way
                 .
              
            
             
               Extend
               the
               Compasses
               from
               1
               to
               12
               ,
               and
               that
               extent
               shall
               reach
               from
               12
               to
               144
               ;
               then
               next
               the
               extent
               from
               1
               to
               144
               ,
               shall
               reach
               from
               5
               to
               720
               ;
               then
               the
               extent
               from
               4
               to
               1
               ,
               shall
               reach
               from
               720
               to
               180
               :
               then
               to
               find
               the
               Square-root
               of
               180
               ;
               the
               half-distance
               between
               180
               and
               1
               ,
               you
               will
               find
               to
               be
               at
               13-42
               ,
               as
               before
               ;
               which
               used
               as
               abovesaid
               ,
               gives
               the
               extream
               ,
               and
               the
               mean
               proportional
               parts
               of
               12
               required
               .
            
             
               
                 Another
                 Example
                 of
                 26.
              
               
            
             
               Extend
               the
               Compasses
               from
               1
               to
               26
               ,
               and
               repeat
               the
               same
               again
               forward
               from
               26
               ,
               shall
               reach
               to
               676.
               
            
             
               Again
               ,
               the
               Extent
               from
               1
               to
               676
               ,
               shall
               reach
               from
               5
               to
               3380.
               
            
             
             
               Lastly
               ,
               the
               Extent
               from
               4
               to
               1
               ,
               shall
               reach
               from
               3380
               ,
               to
               845
               ;
               and
               the
               Square-root
               of
               845
               ,
               is
               29-07
               :
               from
               which
               Number
               or
               Root
               ,
               if
               you
               take
               half
               the
               given
               Number
               ,
               viz.
               13
               ;
               then
               there
               will
               remain
               16-07
               ,
               one
               extream
               :
               then
               16-07
               taken
               from
               26-0
               ,
               rest
               9-93
               ,
               the
               other
               extream
               required
               .
            
             
               For
               the
               Extent
               from
               26
               to
               16-07
               ,
               will
               reach
               from
               16-07
               ,
               to
               9-93
               .
            
             
               
                 Another
                 way
                 by
                 the
                 Line
                 of
                 Sines
                 ,
                 Geometrically
                 .
              
            
             
               The
               best
               and
               quickest
               way
               is
               by
               the
               Line
               of
               Sines
               ,
               thus
               ;
               Make
               the
               given
               Line
               a
               Parallel-Sine
               of
               90
               ;
               then
               take
               out
               the
               parallel-Sine
               of
               38
               degrees
               ,
               10
               minutes
               ,
               and
               that
               shall
               be
               the
               greater
               part
               .
            
             
               Also
               ,
               take
               out
               the
               Parallel-Sine
               of
               22
               degrees
               ,
               27
               minutes
               ,
               and
               that
               shall
               be
               the
               lesser
               extream
               required
               :
               Or
               ,
               according
               to
               Mr.
               Gunter
               ,
               use
               54
               for
               the
               whole
               Line
               ,
               30
               for
               the
               greater
               part
               ,
               and
               18
               for
               the
               less
               .
            
             
               Also
               by
               consequence
               ,
               having
               the
               mean
               ,
               or
               greater
               part
               ,
               make
               it
               a
               parallel-Sine
               of
               38
               degrees
               ,
               10
               minutes
               ;
               then
               Parallel
               90
               shall
               be
               the
               whole
               Line
               ,
               and
               Parallel
               22
               degrees
               ,
               27
               minutes
               ,
               shall
               be
               the
               lesser
               part
               .
            
             
               And
               lastly
               ,
               having
               the
               least
               part
               ,
               make
               
               it
               a
               Parallel
               in
               22
               degrees
               ,
               27
               minutes
               ;
               then
               Parallel
               90
               deg
               .
               10
               min.
               shall
               be
               the
               whole
               Line
               ;
               and
               Parallel
               38-10
               ,
               the
               greater
               part
               .
            
             
               The
               Use
               whereof
               ,
               you
               shall
               have
               afterwards
               in
               the
               11
               
                 th
                 Chapter
              
               ,
               about
               the
               cutting
               off
               the
               
                 Platonical
                 Bodies
              
               .
            
          
           
             
               Sect.
               XI
               .
               Three
               Lines
               or
               Numbers
               given
               ,
               to
               find
               a
               Fourth
               ,
               in
               Geometrical
               Proportion
               ;
               or
               ,
               the
               Rule
               of
               Three
               direct
               .
            
             
               1.
               
               In
               all
               Questions
               of
               the
               Rule
               of
               Three
               ,
               there
               be
               three
               terms
               propounded
               ,
               viz.
               two
               of
               Supposition
               ,
               and
               one
               of
               Demand
               .
            
             
               2.
               
               Also
               note
               ,
               that
               two
               of
               the
               terms
               propounded
               ,
               are
               of
               one
               denomination
               ,
               (
               or
               at
               least
               to
               be
               reduced
               to
               one
               denomination
               )
               and
               one
               of
               another
               denomination
               .
            
             
               3.
               
               Of
               the
               three
               termes
               propounded
               ,
               (
               in
               direct
               proportion
               )
               that
               of
               Demand
               is
               alwayes
               the
               third
               term
               ,
               and
               one
               of
               the
               terms
               of
               Supposition
               ,
               viz.
               that
               of
               the
               same
               Denomination
               ,
               with
               the
               term
               of
               Demand
               ,
               is
               alwayes
               the
               first
               ;
               then
               the
               other
               of
               Supposition
               left
               ,
               must
               needs
               be
               the
               second
               term
               in
               the
               Question
               .
            
             
             
               4.
               
               In
               direct
               proportion
               Alwayes
               ;
               As
               the
               first
               term
               is
               to
               the
               second
               ,
               so
               is
               the
               third
               to
               the
               fourth
               term
               required
               .
            
             
               5.
               
               Having
               discovered
               which
               be
               the
               first
               ,
               second
               ,
               and
               third
               terms
               ;
               If
               the
               first
               and
               third
               term
               be
               of
               divers
               Denominations
               ,
               they
               must
               be
               reduced
               to
               one
               Denomination
               ,
               if
               it
               cannot
               be
               done
               on
               the
               Line
               in
               the
               operation
               ,
               as
               many
               times
               it
               may
               ;
               
                 As
                 thus
                 for
                 instance
              
               :
            
             
               If
               one
               pound
               cost
               two
               shillings
               ,
               what
               shall
               30
               ounces
               cost
               ?
            
             
               Here
               you
               see
               that
               the
               term
               of
               Demand
               ,
               30
               ounces
               ▪
               viz.
               the
               third
               term
               ,
               is
               not
               directly
               of
               the
               same
               Denomination
               with
               one
               pound
               ,
               the
               first
               term
               ;
               but
               is
               thus
               to
               be
               reduced
               to
               ounces
               :
               Saying
               ;
            
             
               If
               16
               ounces
               cost
               2
               shillings
               ,
               what
               shall
               30
               ounces
               cost
               ?
               3
               s
               —
               9
               d.
               
            
             
               Thus
               the
               first
               and
               third
               terms
               ,
               are
               brought
               to
               one
               Denomination
               :
               Also
               you
               see
               that
               the
               Demand
               or
               Question
               ,
               viz.
               What
               shall
               30
               ounces
               cost
               ?
               is
               joyned
               to
               the
               third
               term
               ;
               and
               also
               that
               16
               ounces
               the
               first
               term
               ,
               is
               of
               the
               same
               Denomination
               ;
               therefore
               the
               3
               s.
               must
               needs
               be
               the
               second
               term
               ,
               and
               the
               Answer
               to
               the
               Question
               is
               the
               fourth
               .
            
             
               6.
               
               Having
               thus
               discovered
               ,
               which
               are
               the
               first
               ,
               second
               ,
               and
               third
               terms
               ,
               and
               reduced
               
               the
               first
               and
               third
               to
               the
               like
               Denominations
               ;
               then
               the
               work
               by
               the
               Line
               of
               Numbers
               is
               alwayes
               thus
               ;
            
             
               As
               the
               first
               ,
               to
               the
               second
               ;
               so
               is
               the
               third
               ,
               to
               the
               fourth
               .
            
             
               Or
               the
               Extent
               of
               the
               Compasses
               upon
               the
               Line
               of
               Numbers
               ,
               from
               the
               first
               ,
               to
               the
               second
               ;
               shall
               reach
               the
               same
               way
               ,
               from
               the
               third
               ,
               to
               the
               fourth
               required
               .
            
             
               As
               16
               ounces
               is
               to
               2
               s
               ▪
               so
               is
               30
               ounces
               to
               2
               s
               ¾
               ,
               or
               9
               pence
               .
               Or
               ,
            
             
               As
               16
               ounces
               is
               to
               24.
               d
               ;
               so
               is
               30
               ounces
               to
               45
               d
               ;
               which
               is
               3
               s
               ▪
               —
               9
               d.
               as
               before
               .
            
             
               7.
               
               But
               by
               the
               Line
               of
               Lines
               or
               Sector
               ,
               if
               you
               will
               work
               on
               one
               Scale
               only
               ,
               you
               must
               consider
               which
               term
               of
               the
               first
               or
               second
               is
               biggest
               ;
               for
               you
               must
               alwayes
               order
               it
               so
               ,
               that
               the
               Parallel
               wor●
               must
               be
               the
               largest
               ,
               (
               or
               at
               least
               so
               as
               it
               may
               be
               wrought
               )
               and
               as
               much
               as
               may
               be
               ,
               that
               the
               fourth
               term
               may
               be
               a
               Latteral
               Extent
               ,
               as
               the
               first
               alwayes
               is
               ;
               for
               then
               it
               is
               wrought
               the
               soonest
               ,
               and
               also
               the
               exactest
               .
            
             
               Yet
               by
               this
               Instrument
               ,
               you
               need
               not
               much
               care
               for
               these
               Cautions
               ,
               having
               several
               Scales
               of
               Equal-Parts
               ,
               to
               begin
               and
               end
               the
               work
               on
               ,
               you
               are
               freed
               from
               that
               trouble
               .
            
             
             
               
                 As
                 thus
                 for
                 Example
              
               .
            
             
               
                 When
                 the
                 second
                 term
                 is
                 greater
                 then
                 the
                 first
                 ,
                 then
                 the
                 Work
                 is
                 well
                 performed
                 thus
                 ,
                 two
                 wayes
                 .
              
            
             
               
                 As
                 the
                 Latteral
                 first
                 16
                 or
                 else
                 As
                 Latteral
                 third
                 30
                 from
                 a
                 lesser
                 Scale
                 .
              
               
                 To
                 the
                 Parallel
                 second
                 02
                 or
                 else
                 To
                 Parallel
                 first
                 16
              
               
                 So
                 is
                 the
                 Latteral
                 third
                 30
                 or
                 else
                 So
                 Parallel
                 second
                 2
              
               
                 To
                 the
                 Parallel
                 fourth
                 3
                 75
                 /
                 ●00
                 or
                 else
                 To
                 Latteral
                 fourth
                 3
                 75
                 /
                 100
                 by
                 the
                 same
                 Scale
                 .
              
            
             
               Or
               ,
               As
               —
               second
               ,
               to
               =
               first
               ;
               So
               =
               third
               ,
               to
               —
               fourth
               ;
               by
               a
               less
               Scale
               also
               ,
               if
               need
               be
               .
            
             
               
                 But
                 when
                 the
                 second
                 term
                 is
                 less
                 than
                 the
                 first
                 ,
                 then
                 the
                 work
                 is
                 performed
                 thus
                 :
              
            
             
               If
               50
               Foot
               of
               Timber
               cost
               40
               s.
               what
               shall
               20
               Foot
               cost
               ?
            
             
               
                 As
                 the
                 Latteral
                 second
                 40
                 Or
                 as
                 before
                 ;
                 As
                 —
                 3d
                 20
                 foot
              
               
                 To
                 the
                 Parallel
                 first
                 50
                 Or
                 as
                 before
                 ;
                 To
                 =
                 1st
                 50
                 foot
                 .
              
               
                 So
                 is
                 the
                 Parallel
                 third
                 20
                 Or
                 as
                 before
                 ;
                 So
                 =
                 2d
                 40
                 shill.
                 
              
               
                 To
                 the
                 Latteral
                 fourth
                 16
                 Or
                 as
                 before
                 ;
                 To
                 —
                 4th
                 16
                 shill.
                 
              
            
             
             
               8.
               
               Thus
               you
               see
               several
               wayes
               of
               working
               :
               but
               for
               Beginners
               ,
               I
               would
               advise
               thus
               ,
               briefly
               .
            
             
               First
               ,
               either
               to
               observe
               this
               Rule
               of
               changing
               the
               terms
               ,
               from
               the
               first
               to
               the
               second
               ,
               viz.
               To
               take
               the
               second
               Latterally
               ,
               and
               make
               it
               a
               Parallel
               in
               the
               first
               ;
               then
               the
               Parallel
               third
               gives
               you
               a
               Latteral
               Answer
               .
               Or
               else
               to
               work
               directly
               ,
               as
               the
               first
               to
               the
               second
               ,
               and
               so
               be
               content
               with
               a
               Parallel
               Answer
               ,
               which
               you
               may
               alwayes
               do
               with
               the
               help
               of
               a
               smaller
               Scale
               ,
               when
               need
               requires
               it
               .
            
             
               Note
               the
               Figures
               of
               Operation
               ,
               by
               the
               
                 Trianguler
                 Quadrant
              
               .
            
          
           
             
               Sect.
               XII
               .
               The
               Rule
               of
               Three
               inversed
               .
            
             
               1.
               
               The
               Rule
               of
               Three
               inversed
               ,
               or
               the
               back-Rule
               of
               Three
               ,
               is
               ,
               when
               the
               term
               required
               ,
               or
               fourth
               term
               ,
               ought
               to
               proceed
               from
               the
               second
               term
               ,
               according
               to
               the
               same
               proportion
               ,
               that
               the
               first
               term
               proceeds
               from
               the
               third
               .
            
             
             
               
                 As
                 thus
                 for
                 Example
              
               .
            
             
               
                 
                   
                     Car.
                     
                  
                   
                     Hour
                     .
                  
                   
                     Car.
                     
                  
                   
                     Hour
                     .
                  
                
                 
                   
                     20.
                     
                  
                   
                     16.
                     
                  
                   
                     10.
                     
                  
                   
                     32.
                     
                  
                
                 
                   
                     1.
                     
                  
                   
                     2.
                     
                  
                   
                     3.
                     
                  
                   
                     4.
                     
                  
                
              
            
             
               If
               20
               Carts
               carry
               60
               Square
               yards
               of
               Earth
               in
               60
               hours
               ,
               how
               many
               Square
               yards
               of
               Earth
               shall
               10
               Carts
               carry
               in
               16
               hours
               ?
            
             
               Here
               it
               is
               apparent
               that
               fewer
               Carts
               must
               have
               a
               longer
               time
               to
               carry
               the
               like
               quantity
               ;
               therefore
               to
               the
               same
               time
               must
               less
               work
               be
               allotted
               ,
               as
               in
               the
               work
               doth
               follow
               .
            
             
               
                 
                   
                     Car.
                     
                  
                   
                     Hour
                     .
                  
                   
                     Car.
                     
                  
                   
                     Hour
                     .
                  
                
                 
                   
                     20.
                     
                  
                   
                     16.
                     
                  
                   
                     10.
                     
                  
                   
                     32.
                     
                  
                
                 
                   
                     1.
                     
                  
                   
                     2.
                     
                  
                   
                     3.
                     
                  
                   
                     4.
                     
                  
                
              
            
             
               For
               if
               you
               extend
               the
               Compasses
               from
               10
               to
               20
               ,
               terms
               of
               like
               Denomination
               ,
               viz.
               that
               of
               Carts
               ;
               the
               same
               extent
               applyed
               the
               contrary
               way
               ,
               from
               16
               ,
               the
               time
               required
               ,
               by
               20
               Carts
               ,
               shall
               reach
               to
               32
               ,
               the
               time
               required
               by
               10
               Carts
               ,
               to
               carry
               60
               Load
               .
            
             
               For
               Note
               ,
               as
               in
               the
               former
               Rule
               of
               Three
               direct
               :
            
             
               Look
               how
               much
               the
               third
               term
               is
               greater
               
               than
               the
               first
               ;
               so
               much
               the
               fourth
               is
               greater
               than
               the
               second
               .
            
             
               
                 And
                 contrarily
              
               ,
            
             
               Look
               how
               much
               the
               third
               term
               is
               less
               than
               the
               first
               ,
               by
               so
               much
               is
               the
               fourth
               term
               less
               than
               the
               second
               .
            
             
               
                 As
                 thus
                 in
                 Numbers
              
               .
            
             
               As
               2
               is
               to
               4
               ,
               so
               is
               6
               to
               12
               ,
               for
               as
               6
               the
               third
               term
               ,
               is
               thrice
               as
               much
               as
               2
               ,
               the
               first
               term
               ;
               so
               is
               12
               ,
               the
               fourth
               term
               ,
               thrice
               as
               much
               as
               4
               ,
               the
               second
               .
            
             
               
                 And
                 contrarily
                 decreasing
              
               .
            
             
               As
               12
               is
               to
               6
               ,
               so
               is
               4
               to
               2
               ;
               For
               as
               4
               is
               one
               third
               part
               of
               12
               ,
               so
               is
               2
               one
               third
               part
               of
               6.
               
            
             
               2.
               
               But
               now
               in
               this
               Rule
               of
               Three
               inversed
               ,
               or
               the
               back-Rule
               of
               three
               ;
               it
               is
               contrarily
               ordered
               ,
               
                 as
                 thus
              
               ;
            
             
               Look
               how
               much
               the
               third
               term
               is
               greater
               (
               or
               lesser
               )
               than
               the
               first
               ,
               by
               so
               much
               is
               the
               fourth
               term
               lesser
               (
               or
               greater
               )
               than
               the
               second
               .
            
             
               
                 As
                 thus
                 in
                 Numbers
              
               .
            
             
               As
               5
               is
               to
               60
               ,
               so
               is
               30
               to
               2
               1
               /
               ●
               ;
               that
               is
               ,
               If
               5
               s.
               is
               60
               d.
               How
               many
               shillings
               is
               30
               
               pence
               ?
               The
               Answer
               is
               ,
               2
               s.
               ½
               .
               For
               as
               30
               is
               greater
               than
               5●
               ;
               so
               is
               2
               ½
               less
               than
               60.
               
            
             
               Again
               ,
               as
               2
               ½
               is
               to
               30
               ;
               so
               is
               60
               to
               5
               ,
               in
               the
               like
               manner
               .
            
             
               
                 
                   
                     Pion.
                     
                  
                   
                     Dayes
                     .
                  
                   
                     Pion.
                     
                  
                   
                     Dayes
                     .
                  
                
                 
                   
                     18.
                     
                  
                   
                     40.
                     
                  
                   
                     15.
                     
                  
                   
                     48.
                     
                  
                
                 
                   
                     1.
                     
                  
                   
                     2.
                     
                  
                   
                     3.
                     
                  
                   
                     4.
                     
                  
                
              
            
             
               If
               18
               Pioneers
               make
               a
               Trench
               in
               40
               days
               ,
               how
               many
               Pioneers
               is
               needful
               to
               perform
               the
               same
               in
               15
               dayes
               ?
            
             
               As
               40
               to
               15
               ,
               so
               is
               18
               to
               48
               :
               Here
               ,
               as
               the
               third
               is
               lesser
               than
               the
               first
               ;
               so
               is
               th●
               fourth
               greater
               than
               the
               second
               .
            
             
               
                 
                   
                     Hors.
                     
                  
                   
                     Dayes
                     .
                  
                   
                     Hors.
                     
                  
                   
                     Dayes
                     .
                  
                
                 
                   
                     12.
                     
                  
                   
                     30.
                     
                  
                   
                     24.
                     
                  
                   
                     15.
                     
                  
                
                 
                   
                     1.
                     
                  
                   
                     2.
                     
                  
                   
                     3.
                     
                  
                   
                     4.
                     
                  
                
              
            
             
               Again
               ,
               if
               12
               Horses
               eat
               20
               bushels
               of
               Provender
               ,
               in
               30
               dayes
               ;
               how
               soon
               will
               24
               Horses
               eat
               up
               the
               like
               quantity
               of
               Provender
               ?
               The
               Answer
               is
               in
               15
               dayes
               .
            
             
               3.
               
               The
               manner
               of
               working
               this
               Rule
               on
               the
               Line
               of
               Numbers
               ,
               
                 is
                 thus
              
               .
            
             
               Extend
               the
               Compasses
               from
               one
               term
               to
               the
               other
               of
               like
               Denomination
               ;
               the
               same
               
               extent
               laid
               the
               contrary
               way
               from
               the
               other
               term
               ,
               shall
               reach
               to
               the
               Answer
               required
               .
            
             
               As
               in
               the
               
                 last
                 Example
              
               ;
               the
               extent
               from
               12
               to
               24
               ,
               the
               terms
               under
               the
               denomination
               of
               Horses
               ,
               shall
               reach
               the
               contrary
               way
               from
               30
               to
               15
               ,
               the
               number
               of
               dayes
               required
               .
            
             
               4.
               
               Note
               ,
               That
               by
               due
               consideration
               ,
               this
               back-Rule
               may
               be
               wro●ght
               by
               the
               Precepts
               ,
               for
               the
               direct-Rule
               ,
            
             
               Thus
               :
            
             
               In
               all
               Questions
               of
               this
               nature
               ,
               there
               be
               three
               terms
               given
               to
               find
               a
               fourth
               ;
               of
               which
               three
               terms
               ,
               two
               are
               of
               one
               Denomination
               ,
               and
               one
               of
               a
               different
               Denomination
               ;
               of
               which
               ,
               the
               fourth
               must
               alwayes
               be
               ;
               which
               in
               the
               first
               Rule
               of
               the
               tenth
               Section
               before
               going
               ,
               are
               called
               two
               termes
               of
               Supposition
               ,
               and
               one
               of
               Demand
               .
               Now
               here
               you
               are
               to
               consider
               ,
               That
            
             
               5.
               
               When
               the
               fourth
               term
               required
               ,
               ought
               to
               be
               greater
               than
               that
               of
               Demand
               ;
               which
               by
               reason
               you
               may
               certainly
               know
               ;
            
             
               
                 Then
                 say
              
               ,
            
             
               As
               the
               lesser
               term
               of
               Supposition
               is
               to
               the
               greater
               ;
            
             
               So
               is
               the
               term
               of
               Demand
               ,
               to
               his
               Answer
               ,
               the
               fourth
               .
            
             
             
               Example
               .
            
             
               
                 
                   
                     Men.
                     
                  
                   
                     Dayes
                     .
                  
                   
                     Men.
                     
                  
                   
                     Dayes
                     .
                  
                
                 
                   
                     80.
                     
                  
                   
                     12.
                     
                  
                   
                     40.
                     
                  
                   
                     24.
                     
                  
                
                 
                   
                     1.
                     
                  
                   
                     2.
                     
                  
                   
                     3.
                     
                  
                   
                     4.
                     
                  
                
              
            
             
               If
               80
               Men
               do
               a
               Work
               in
               12
               dayes
               ,
               how
               soon
               may
               40
               Men
               do
               the
               like
               Work
               ?
               Here
               Reason
               tells
               me
               ,
               that
               fewer
               Men
               must
               have
               longer
               time
               ;
               therefore
               the
               fourth
               term
               required
               must
               be
               greater
               .
               Therefore
               ,
            
             
               As
               40
               to
               80
               ,
               viz.
               As
               the
               lesser
               term
               of
               Supposition
               40
               ,
               to
               the
               greater
               80
               ;
            
             
               So
               is
               12
               ,
               the
               term
               of
               Demand
               ,
               to
               24
               ,
               the
               Answer
               required
               .
            
             
               6.
               
               But
               if
               the
               required
               term
               ought
               to
               be
               lesser
               ,
               which
               Reason
               will
               discover
               in
               like
               manner
               ;
               
                 Then
                 thus
              
               :
            
             
               As
               the
               greater
               term
               of
               Supposition
               ,
               is
               to
               the
               lesser
               ;
               so
               is
               the
               term
               of
               Demand
               to
               the
               fourth
               term
               required
               .
            
             
               As
               80
               to
               40
               ,
               so
               is
               24
               to
               12
               ;
               extending
               the
               Compasses
               the
               same
               way
               from
               the
               third
               to
               the
               fourth
               ,
               as
               from
               the
               first
               to
               the
               second
               .
            
             
               But
               Note
               here
               ,
               That
               you
               are
               not
               tyed
               to
               observe
               which
               is
               the
               first
               ,
               second
               ,
               or
               third
               term
               ;
               but
               to
               consider
               only
               the
               nature
               of
               
               the
               Question
               ,
               that
               you
               may
               Answer
               accordingly
               ;
               and
               indeed
               this
               way
               will
               ,
               generally
               ,
               take
               in
               the
               direct
               Rule
               also
               .
               For
               alwayes
               in
               Direct
               Proportion
               ,
               you
               may
               as
               well
               say
               ,
               As
               the
               third
               term
               is
               to
               the
               first
               ,
               so
               is
               the
               second
               to
               the
               fourth
               ;
               as
               to
               say
               ,
               As
               the
               first
               to
               the
               second
               ,
               so
               is
               the
               third
               to
               the
               fourth
               .
            
             
               Also
               backwards
               ,
               or
               inversly
               ;
               As
               the
               third
               to
               the
               first
               ,
               so
               is
               the
               second
               to
               the
               fourth
               ;
               extending
               the
               Compasses
               the
               contrary
               way
               .
            
             
               As
               80
               to
               40
               ;
               So
               is
               12
               to
               24.
               
            
             
               8.
               
               To
               perform
               this
               by
               the
               Sector
               ,
               or
               general
               Scale
               and
               Thred
               ,
               on
               the
               Quadrantal-side
               ,
               you
               may
               generally
               observe
               this
               Rule
               ;
               Enter
               the
               second
               term
               taken
               Latterally
               ,
               Parallelly
               in
               the
               first
               ▪
               keeping
               the
               Sector
               ,
               or
               Thred
               ,
               at
               that
               Angle
               ;
               then
               the
               Parallel-third
               ,
               shall
               give
               the
               Latteral-fourth
               ,
               LATTERALLY
               .
            
             
               9.
               
               Or
               else
               ,
               As
               the
               Latteral-first
               ,
               to
               the
               Parallel-second
               ;
               so
               is
               the
               Latteral-third
               ,
               to
               the
               Parallel-fourth
               ,
               PARALLELLY
               .
            
             
               And
               if
               the
               second
               be
               less
               than
               the
               first
               ,
               make
               use
               of
               a
               smaller
               Scale
               ;
               or
               change
               the
               terms
               ,
               as
               is
               shewed
               before
               ▪
            
          
           
             
             
               Sect.
               XIII
               .
               The
               Double
               or
               Compound
               Rule
               of
               Three
               ,
               Direct
               and
               Reverse
               .
            
             
               Having
               premised
               the
               way
               to
               bring
               the
               back
               (
               or
               inversed
               )
               Rule
               of
               Three
               ,
               to
               be
               performed
               by
               the
               Rules
               for
               the
               Direct
               ;
               and
               considering
               that
               the
               Double
               and
               Compound
               Rules
               of
               Three
               are
               alike
               by
               the
               Line
               of
               Numbers
               ;
               I
               have
               therefore
               joyned
               them
               together
               in
               one
               Section
               .
            
             
               1.
               
               The
               Compound
               ,
               or
               Double
               (
               Golden
               )
               Rule
               of
               Three
               ;
               is
               ,
               when
               more
               than
               three
               terms
               are
               propounded
               ,
               or
               given
               .
            
             
               2.
               
               The
               Double
               Rule
               of
               Three
               ,
               is
               when
               five
               terms
               are
               propounded
               ,
               and
               a
               sixt
               term
               proportional
               unto
               them
               is
               demanded
               .
            
             
               
                 As
                 thus
              
               ;
            
             
               If
               6
               Men
               spend
               18
               l.
               in
               three
               months
               ;
               How
               much
               will
               serve
               12
               Men
               for
               6
               months
               ?
            
             
               
                 Or
                 ,
                 again
              
               .
            
             
               If
               two
               Barrels
               of
               Beer
               serve
               12
               Men
               for
               14
               dayes
               ;
               How
               many
               dayes
               will
               4
               Barrels
               serve
               24
               Men
               ?
            
             
               3.
               
               The
               five
               terms
               given
               consi●t
               of
               two
               
               parts
               ,
               viz.
               a
               Supposition
               ,
               and
               Demand
               ;
               as
               in
               the
               Rule
               of
               Three
               direct
               .
            
             
               The
               Supposition
               lies
               in
               these
               three
               Numbers
               first
               propounded
               ,
               viz.
               If
               6
               Men
               spend
               18
               l.
               in
               3
               months
               ;
               and
               the
               Demand
               lies
               in
               the
               two
               remaining
               ▪
               viz.
               How
               much
               will
               serve
               12
               Men
               6
               mon●●s
               ?
            
             
               Or
               in
               the
               other
               Example
               ,
               viz.
               If
               2
               Barrels
               of
               Beer
               serve
               12
               Men
               14
               dayes
               ,
               are
               the
               terms
               of
               Supposition
               ;
               and
               ,
               how
               many
               dayes
               will
               4
               Barrels
               serve
               24
               Men
               ,
               are
               the
               terms
               of
               Demand
               ?
            
             
               4.
               
               The
               next
               work
               is
               to
               rank
               the
               three
               terms
               of
               Supposition
               ,
               and
               the
               two
               of
               Demand
               ,
               in
               their
               due
               and
               proper
               order
               ,
               for
               convenience
               of
               Operation
               ;
               which
               may
               be
               thus
               :
            
             
               Of
               the
               three
               terms
               of
               Supposition
               ,
               that
               which
               hath
               the
               same
               Denomination
               with
               the
               term
               required
               ,
               pla●e
               in
               the
               second
               place
               ;
               and
               the
               other
               two
               ,
               one
               above
               another
               in
               the
               first
               place
               :
               
                 
                   Thus
                   ;
                
                 
                   
                     6
                  
                   
                     18
                  
                   
                     12
                  
                
                 
                   
                     3
                  
                   
                      
                  
                   
                     6
                  
                
              
            
             
               And
               then
               place
               the
               two
               terms
               of
               Demand
               one
               above
               another
               in
               the
               third
               place
               ,
               only
               observing
               to
               keep
               the
               Numbers
               of
               like
               Denomination
               
               in
               the
               same
               ranks
               ;
               as
               6
               Men
               ,
               and
               12
               Men
               in
               the
               upper
               rank
               ;
               and
               3
               Months
               ,
               and
               6
               Months
               in
               the
               lower
               rank
               ;
               as
               in
               the
               Work
               is
               exprest
               .
            
             
               5.
               
               When
               Questions
               of
               this
               nature
               are
               resolved
               by
               two
               single
               Rules
               ,
               then
               the
               Analogy
               ,
               or
               Proportion
               ,
               is
               thus
               ;
            
             
               Operation
               I.
               
            
             
               As
               the
               first
               term
               ,
               in
               the
               upper
               Rank
               ,
               is
               to
               the
               second
               ;
            
             
               So
               is
               the
               third
               ,
               in
               the
               same
               Rank
               ,
               to
               a
               fourth
               .
               Again
               ,
            
             
               Operation
               II.
               
            
             
               As
               the
               first
               term
               in
               the
               lower
               Rank
               ,
               is
               to
               the
               fourth
               last
               found
               ;
            
             
               So
               is
               the
               other
               term
               in
               the
               lower
               Rank
               ,
               to
               the
               term
               required
               .
            
             
               As
               in
               the
               first
               Example
               ;
               As
               6
               to
               18
               ;
               so
               is
               12
               to
               36
               a
               fourth
               .
            
             
               Again
               ,
               as
               3
               to
               36
               ;
               so
               is
               6
               to
               72
               ,
               the
               term
               required
               .
            
             
               
                 Which
                 by
                 the
                 Line
                 of
                 Numbers
                 ,
                 is
                 thus
                 wrought
              
               ;
            
             
               Extend
               the
               Compasses
               from
               6
               to
               18
               ;
               the
               same
               extent
               applyed
               the
               same
               way
               from
               12
               ,
               shall
               reach
               to
               36.
               
               Then
               again
               ,
               extend
               the
               Compasses
               from
               3
               to
               36
               ,
               the
               same
               extent
               applied
               the
               same
               way
               from
               6
               ,
               shall
               reach
               to
               72
               ,
               the
               term
               required
               .
            
             
             
               
                 By
                 the
                 Trianguler
                 Quadrant
                 ,
                 or
                 Sector
                 ,
                 thus
              
               ;
            
             
               6.
               
               As
               —
               18
               to
               =
               6
               ;
               so
               is
               =
               12
               to
               —
               36.
               
               Again
               ,
            
             
               As
               —
               36
               to
               =
               3
               ;
               so
               is
               =
               6
               to
               —
               72
               ,
               the
               term
               required
               .
            
             
               Or
               else
               work
               it
               Parallelly
               ,
               observing
               the
               same
               order
               ,
               as
               by
               the
               Line
               of
               Numbers
               ,
               thus
               ;
            
             
               As
               —
               6
               to
               =
               18
               ;
               so
               is
               —
               12
               to
               =
               36
               ,
               the
               fourth
               term
               .
               Again
               ,
            
             
               As
               —
               3
               to
               =
               36
               ;
               so
               is
               —
               6
               ,
               to
               =
               72
               ,
               the
               sixt
               term
               required
               .
            
             
               
                 The
                 Double
                 Rule
                 of
                 Three
                 inversed
                 .
              
            
             
               7.
               
               In
               the
               other
               Example
               ,
               is
               comprehended
               the
               double
               Rule
               of
               Three
               inverse
               ;
               which
               runs
               thus
               ;
            
             
               If
               two
               Barrels
               of
               Beer
               ,
               serve
               12
               Men
               14
               dayes
               ;
               How
               many
               dayes
               will
               4
               Barrels
               serve
               24
               Men
               ?
            
             
               If
               you
               Rank
               the
               terms
               ,
               according
               to
               the
               former
               Precept
               ,
               they
               will
               stand
               thus
               :
            
             
               2
               —
               14
               —
               4
               12
               24
               or
               thus
               ,
               12
               —
               14
               —
               24
               2
               4
            
             
             
               8.
               
               Which
               if
               you
               work
               according
               to
               the
               back-Rule
               ,
               the
               way
               is
               thus
               ;
            
             
               Operation
               I.
               
            
             
               Extend
               the
               Compasses
               from
               2
               to
               4
               ,
               term●
               of
               like
               Denomination
               ,
               viz.
               of
               Barrels
               ;
               th●●
               same
               Extent
               applied
               the
               contrary
               w●y
               from
               14
               ,
               shall
               reach
               to
               7
               ,
               for
               a
               Fourth
               Proportional
               .
            
             
               Operation
               II.
               
            
             
               Again
               ,
               Extend
               the
               Compasses
               from
               12
               to
               7
               ,
               the
               fourth
               last
               found
               ;
               the
               same
               Extent
               shall
               reach
               the
               contrary
               way
               ▪
               from
               24
               to
               14
               ,
               the
               number
               of
               dayes
               required
               .
            
             
               9.
               
               But
               if
               you
               would
               reduce
               this
               ,
               to
               be
               wrought
               by
               two
               single
               direct
               Rules
               ;
               you
               must
               consider
               the
               Precept
               Rule
               ,
               the
               5th
               and
               6th
               ,
               of
               the
               Eleventh
               Section
               ;
               and
               the
               terms
               of
               Supposition
               and
               Demand
               ;
               and
               the
               increasing
               ,
               or
               decreasing
               of
               the
               fourth
               term
               ,
               which
               is
               required
               .
            
             
               
                 As
                 thus
              
               ;
            
             
               First
               ,
               I
               part
               this
               into
               two
               single
               Rules
               ,
               thus
               :
            
             
               Operation
               I.
               
            
             
               If
               12
               Men
               drink
               2
               Barrels
               in
               14
               dayes
               ,
               then
               24
               Men
               may
               drink
               2
               Barrels
               in
               7
               dayes
               .
            
             
             
               Operation
               II.
               
            
             
               Again
               ,
               If
               2
               Barrels
               last
               24
               Men
               7
               dayes
               ,
               ●
               Barrels
               will
               last
               them
               14
               dayes
               ;
               the
               Answer
               to
               the
               Question
               required
               .
            
             
               Here
               by
               the
               6th
               Rule
               ,
               where
               the
               Number
               sought
               is
               to
               be
               less
               ;
               As
               24
               ,
               the
               greater
               term
               of
               Men
               ,
               is
               to
               12
               the
               less
               of
               the
               same
               Denonomination
               ;
               So
               is
               14
               to
               7
               ,
               the
               fourth
               .
            
             
               Again
               .
            
             
               As
               2
               the
               lesser
               term
               ,
               is
               to
               4
               the
               greater
               of
               the
               same
               Denomination
               ;
               so
               is
               7
               to
               14
               ,
               the
               Answer
               required
               ,
               by
               the
               5th
               Rule
               of
               the
               11th
               Section
               .
            
             
               
                 Or
                 else
                 thus
              
               ;
            
             
               As
               2
               to
               7
               ,
               so
               is
               4
               to
               14
               ;
               that
               is
               ,
               the
               Extent
               from
               2
               to
               7
               ,
               shall
               reach
               the
               same
               way
               from
               4
               to
               14
               ,
               the
               term
               required
               .
            
             
               To
               work
               this
               by
               the
               Trianguler
               Quadrant
               ,
               or
               Sector
               ,
               the
               general
               Rule
               in
               this
               Section
               ,
               Rule
               6
               and
               7
               ,
               giveth
               sufficient
               direction
               .
            
             
               10.
               
               The
               Rule
               of
               Three
               ,
               compounded
               of
               five
               Numbers
               ,
               is
               no
               other
               than
               the
               double
               Rule
               of
               Three
               ;
               and
               is
               ,
               or
               for
               the
               most
               part
               ,
               may
               be
               wrought
               by
               one
               Operation
               ,
               having
               prepared
               the
               Numbers
               by
               Multiplication
               ,
               for
               that
               purpose
               :
               Which
               two
               Multiplications
               by
               the
               Line
               of
               Numbers
               ,
               though
               they
               
               are
               presently
               wrought
               ,
               yet
               the
               two
               Rules
               of
               Three
               are
               done
               as
               soon
               ;
               so
               that
               the
               Compound
               Rule
               ,
               is
               here
               of
               no
               advantage
               at
               all
               ,
               therefore
               I
               might
               wave
               it
               ;
               yet
               because
               the
               only
               difficulty
               lies
               in
               the
               ordering
               the
               Question
               ,
               I
               shall
               propound
               it
               ,
               for
               the
               addition
               sake
               of
               another
               Example
               ,
               which
               is
               this
               ;
            
             
               If
               the
               Carriage
               of
               2
               hundred
               weight
               ,
               30
               miles
               ,
               cost
               4
               s.
               What
               will
               the
               Carriage
               of
               5
               hundred
               weight
               cost
               for
               100
               miles
               ?
               The
               Numbers
               Ranked
               ,
               according
               to
               the
               first
               Precept
               ,
               will
               stand
               thus
               ,
               as
               followeth
               .
            
             
               
            
             
               11.
               
               Then
               for
               the
               Operation
               ,
               multiply
               the
               two
               first
               Numbers
               one
               by
               the
               other
               ;
               as
               2
               times
               30
               is
               60
               ,
               which
               is
               the
               first
               term
               ;
               and
               let
               the
               middle
               Number
               be
               the
               second
               term
               ;
               and
               the
               Product
               of
               the
               two
               last
               (
               multiplyed
               together
               )
               for
               the
               third
               term
               :
               Then
               the
               Numbers
               being
               so
               prepared
               ,
               say
               ,
            
             
               As
               60
               ,
               the
               Product
               of
               the
               two
               first
               Numbers
               ,
               is
               to
               4
               ,
               the
               middle
               Number
               ;
            
             
               So
               is
               500
               ,
               the
               Product
               of
               the
               two
               last
               ,
               to
               33
               ●
               /
               ●
               ,
               the
               Answer
               required
               .
            
             
             
               By
               the
               Line
               of
               Numbers
               ,
               the
               Extent
               from
               60
               to
               4
               ,
               will
               reach
               the
               same
               way
               from
               500
               to
               33
               ⅓
               ,
               or
               ,
               thirty
               three
               shillings
               and
               four
               pence
               ,
               the
               price
               of
               5
               hundred
               weight
               ,
               carried
               100
               miles
               .
            
             
               Note
               ,
               This
               Rule
               serves
               when
               it
               is
               performed
               by
               the
               Compound
               Rule
               of
               Three
               direct
               .
            
             
               12.
               
               But
               if
               the
               inverse
               ,
               or
               backer
               Rule
               of
               Three
               ,
               be
               used
               in
               the
               work
               ;
               then
               Operate
               thus
               :
            
             
               
                 As
                 in
                 this
                 following
                 Example
                 ,
                 is
                 manifest
                 .
              
            
             
               A
               Merchant
               hath
               received
               10
               l.
               10
               s.
               for
               the
               Interest
               of
               a
               certain
               sum
               of
               Money
               for
               six
               Months
               ;
               and
               he
               received
               after
               the
               rate
               of
               6
               l.
               for
               the
               use
               of
               an
               hundred
               pound
               in
               a
               year
               ;
               the
               Question
               is
               ,
               how
               much
               Mone●
               was
               Principal
               to
               10
               l.
               —
               10
               s.
               for
               6
               Months
               ?
            
             
               First
               ,
               I
               range
               the
               Numbers
               ,
               according
               to
               the
               order
               first
               propounded
               ,
               in
               the
               4th
               Rule
               of
               the
               12th
               Section
               ,
               as
               followeth
               .
            
             
               
            
             
             
               Then
               I
               observe
               diligently
               ,
               whether
               the
               inverse
               Proportion
               be
               in
               the
               first
               or
               second
               Operation●
               or
               Line
               ,
               as
               thus
               in
               this
               Question
               it
               is
               in
               the
               lower
               Line
               ;
               therefore
               after
               the
               Cross
               Multiplication
               ,
               it
               is
               to
               be
               wrought
               by
               the
               single
               inversed
               Rule
               of
               Three
               ;
               but
               when
               the
               inverse
               Proportion
               is
               in
               the
               upper
               Line
               ,
               it
               is
               wrought
               by
               the
               single
               Rule
               direct
               .
            
             
               
            
             
               Then
               I
               multiply
               the
               double
               terms
               across
               ;
               that
               is
               ,
               the
               lowest
               on
               the
               right-hand
               by
               the
               uppermost
               on
               the
               left
               ;
               and
               the
               uppermost
               on
               the
               right
               ,
               by
               the
               lowest
               on
               the
               left
               ;
               
                 As
                 thus
              
               :
            
             
               6
               by
               6
               ,
               which
               makes
               36
               ,
               to
               be
               set
               under
               6
               ;
               and
               12
               by
               10-5
               ,
               or
               10
               l.
               (
               which
               is
               126
               ,
               and
               10
               s.
               )
               and
               set
               it
               under
               10
               :
               then
               say
               by
               the
               inversed
               Rule
               ,
               thus
               ▪
               *
            
             
               As
               126
               to
               36
               ,
               so
               is
               100
               to
               350
               ,
               the
               Answer
               demanded
               ;
               So
               that
               350
               l.
               as
               Principal
               will
               yield
               10
               l.
               —
               10
               s.
               in
               6
               Months
               ;
               Or
               ,
               the
               Extent
               from
               126
               to
               36
               ,
               shall
               reach
               the
               contrary
               way
               from
               100
               to
               350
               ,
               the
               Principal
               Money
               required
               .
            
             
             
               Which
               you
               may
               more
               readily
               prove
               by
               reasoning
               thus
               :
            
             
               13.
               
               If
               3
               l.
               be
               the
               Interest
               of
               100
               l.
               in
               6
               Months
               ,
               to
               how
               much
               Money
               shall
               10
               l.
               10
               s.
               be
               interest
               in
               6
               Months
               ?
               
                 work
                 thus
              
               ;
            
             
               The
               Extent
               of
               the
               Compasses
               from
               3
               to
               100
               ,
               shall
               reach
               the
               same
               way
               from
               10
               l.
               10
               s.
               to
               350
               ,
               the
               Principal
               Money
               answering
               to
               10
               l.
               —
               10
               s.
               the
               Answer
               required
               .
            
             
               
                 By
                 the
                 Line
                 of
                 Lines
                 ,
                 work
                 thus
              
               ;
            
             
               As
               —
               3
               to
               =
               10
               ,
               counted
               100
               ;
            
             
               So
               is
               —
               10
               ½
               at
               the
               first
               1
               next
               the
               Center
               ,
               to
               =
               350.
               
            
             
               Or
               ,
            
             
               As
               —
               100
               ,
               to
               =
               3
               ;
               so
               is
               =
               10
               1
               /
               ●
               to
               —
               350.
               
            
          
           
             
               Sect.
               XIV
               .
               The
               Rule
               of
               Fellowship
               .
            
             
               1.
               
               Rules
               of
               Plural
               Proportion
               are
               those
               ,
               by
               which
               those
               Questions
               are
               resolved
               ,
               which
               requi●e
               more
               Golden
               Rules
               than
               one
               ;
               and
               yet
               cannot
               be
               Resolved
               by
               the
               double
               (
               Rule
               of
               Three
               ,
               or
               )
               Golden
               〈…〉
               was
               last
               mentioned
               .
            
             
             
               2.
               
               Of
               these
               Rules
               there
               be
               ●ivers
               kinds
               and
               varieties
               ,
               according
               to
               the
               nature
               of
               the
               Question
               propounded
               ;
               for
               here
               the
               terms
               given
               ,
               are
               sometimes
               four
               ,
               five
               ,
               or
               six
               ,
               or
               more
               ;
               and
               the
               terms
               required
               also
               more
               than
               one
               ,
               two
               ,
               or
               three
               .
            
             
               3.
               
               The
               Rule
               of
               Fellowship
               ,
               is
               to
               discover
               the
               Gain
               or
               Loss
               of
               every
               Partner
               in
               the
               Stock
               ,
               by
               their
               several
               Stocks
               ,
               and
               the
               whole
               gain
               or
               loss
               of
               the
               whole
               Stock
               .
            
             
               Also
               observe
               ,
               That
               the
               Rule
               of
               Fellowship
               may
               be
               either
               single
               or
               double
               ;
               of
               both
               which
               in
               order
               .
            
             
               4.
               
               The
               single
               Rule
               of
               Fellowship
               is
               ,
               when
               the
               Stocks
               propounded
               are
               single
               Numbers
               .
            
             
               
                 As
                 thus
                 for
                 Example
              
               .
            
             
               ABC
               and
               D
               ,
               representing
               the
               Names
               of
               4
               Men
               ,
               put
               into
               one
               common
               Stock
               100
               l.
               to
               trade
               withal
               :
               A
               puts
               in
               10
               l.
               B
               puts
               in
               20
               l.
               C
               30
               l.
               and
               D
               40
               l
               ;
               and
               with
               this
               Stock
               ,
               in
               a
               certain
               time
               ,
               they
               gained
               10
               l.
               or
               200
               s
               ;
               Now
               the
               Question
               is
               ,
               what
               ought
               each
               man
               to
               have
               of
               the
               200
               s.
               that
               may
               be
               proportionable
               to
               his
               particular
               Stock
               ?
            
             
               5.
               
               The
               Rule
               of
               Operation
               is
               ,
               first
               ,
               by
               Addition
               find
               the
               total
               of
               all
               the
               particular
               Stocks
               ,
               for
               the
               first
               term
               ;
               the
               whole
               gain
               (
               or
               loss
               )
               ,
               for
               the
               second
               term
               ;
               and
               each
               
               particular
               Stock
               for
               a
               third
               term
               ;
               and
               repeating
               the
               Rule
               of
               Three
               as
               often
               as
               there
               be
               particular
               Stocks
               in
               the
               Question
               ,
               you
               shall
               bring
               forth
               ,
               or
               find
               out
               ,
               as
               many
               fourths
               for
               the
               particular
               gains
               (
               or
               losses
               )
               of
               each
               particular
               Man
               required
               .
            
             
               
                 As
                 thus
                 for
                 Instance
              
               .
            
             
               The
               sum
               of
               the
               four
               Stocks
               are
               100
               l.
               
            
             
               The
               whole
               gain
               is
               10
               l.
               or
               200
               s.
               
            
             
               Then
               ,
            
             
               
            
             
               For
               the
               Extent
               from
               100
               to
               200
               ,
               shall
               reach
               from
               10
               to
               20
               ,
               and
               from
               20
               to
               40
               ,
               and
               from
               30
               to
               60
               ,
               and
               from
               40
               to
               80
               ;
               the
               particular
               gains
               due
               to
               ABCD
               ,
               which
               was
               required
               .
            
             
               6.
               
               For
               proof
               whereof
               ,
               if
               you
               add
               20
               ,
               40
               ,
               60
               ,
               and
               80
               together
               ,
               they
               make
               up
               200
               s
               ,
               or
               10
               l
               ;
               the
               whole
               gain
               of
               the
               whole
               Stock
               .
            
             
             
               7.
               
               The
               double
               Rule
               of
               Fellowship
               is
               ,
               when
               the
               Stocks
               propounded
               are
               double
               Numbers
               .
            
             
               
                 As
                 thus
                 for
                 Example
              
               .
            
             
               AB
               and
               C
               ,
               holds
               a
               Field
               in
               common
               ,
               for
               which
               they
               pay
               50
               l.
               a
               year
               ;
               and
               in
               this
               Field
               ,
               A
               had
               25
               Oxen
               went
               30
               dayes
               ;
               B
               had
               15
               Oxen
               there
               40
               dayes
               ;
               and
               C
               had
               29
               Oxen
               went
               there
               40
               dayes
               :
               What
               ought
               each
               man
               to
               pay
               for
               his
               part
               of
               the
               Rent
               ,
               viz.
               50
               l
               ?
               Here
               you
               see
               the
               Stocks
               propounded
               are
               double
               Numbers
               ,
               as
               of
               Oxen
               ,
               and
               their
               dayes
               ,
               or
               time
               of
               feeding
               ;
               as
               25
               &
               30
               ,
               15
               &
               40
               ,
               20
               &
               40
               ,
               being
               double
               Numbers
               .
            
             
               8.
               
               The
               Rule
               of
               Operation
               is
               thus
               ,
               in
               the
               double
               Rule
               of
               Fellowship
               :
            
             
               Multiply
               the
               double
               Numbers
               ,
               severally
               one
               by
               the
               other
               ,
               one
               after
               another
               ,
               and
               take
               the
               sum
               of
               their
               several
               Products
               ,
               for
               the
               first
               term
               ;
               and
               the
               whole
               gain
               or
               loss
               ,
               for
               the
               second
               term
               ;
               and
               the
               particular
               Products
               of
               every
               double
               Number
               ,
               for
               the
               third
               term
               ,
               one
               after
               another
               :
               This
               done
               ,
               repeating
               the
               Rule
               of
               Three
               ,
               as
               often
               as
               there
               be
               double
               Numbers
               ,
               the
               4th
               term
               produced
               from
               those
               Operations
               ,
               shall
               be
               Answers
               to
               the
               Questions
               required
               ,
               viz.
               the
               quantity
               of
               each
               mans
               gain
               or
               loss
               .
            
             
             
               Example
               .
            
             
               
                 
                   
                     25
                     &
                     30
                     ,
                     A's
                     Oxen
                     and
                     time
                     of
                     feeding
                     ,
                     multiplied
                     ,
                     is
                  
                   
                     750
                  
                
                 
                   
                     15
                     &
                     40
                     ,
                     B's
                     Oxen
                     and
                     time
                     of
                     feeding
                     ,
                     multiplied
                     ,
                     is
                  
                   
                     600
                  
                
                 
                   
                     20
                     &
                     40
                     ,
                     C's
                     Oxen
                     and
                     dayes
                     of
                     feeding
                     ,
                     multiplied
                     ,
                     is
                  
                   
                     800
                  
                
                 
                   
                     The
                     Sum
                  
                   
                     2150
                  
                
              
               
                 
                   
                     AS
                     215
                     ,
                     to
                     50
                     ;
                     so
                     is
                  
                   
                     750
                     A's
                  
                   
                     Stock
                     to
                  
                   
                     17-9
                     A's
                  
                   
                     Rent
                  
                
                 
                   
                      
                  
                   
                     600
                     B's
                  
                   
                     13-19
                     B's
                  
                
                 
                   
                      
                  
                   
                     800
                     C's
                  
                   
                     18-12
                     C's
                  
                
              
            
             
               9.
               
               To
               work
               by
               the
               Line
               of
               Numbers
               ,
               the
               Extent
               of
               the
               Compasses
               from
               1
               to
               25
               ,
               shall
               reach
               the
               same
               way
               from
               30
               to
               750
               ,
               the
               first
               Product
               of
               a
               A's
               double
               Number
               ,
               or
               Stock
               .
            
             
               And
               as
               1
               to
               15
               ,
               so
               is
               40
               to
               600
               ,
               the
               Product
               of
               B's
               double
               Number
               ,
               and
               Stock
               .
            
             
               And
               as
               1
               to
               20
               ,
               so
               is
               40
               to
               800
               ,
               the
               Product
               of
               C's
               double
               Number
               ,
               and
               Stock
               .
            
             
               Which
               three
               Products
               added
               ,
               make
               2150
               ,
               the
               first
               term
               ;
               and
               50
               is
               the
               second
               term
               ;
               and
               750
               ,
               600
               ,
               and
               800
               ,
               the
               three
               Products
               severally
               ,
               the
               third
               term
               .
               Then
               ,
            
             
               The
               Extent
               from
               2150
               to
               50
               ,
               shall
               reach
               the
               same
               way
               from
               750
               to
               17-45
               ,
               or
               17
               l.
               9
               s.
               And
               from
               600
               to
               13-95
               ,
               or
               13
               l.
               —
               19
               s.
               
               And
               from
               800
               to
               18-60
               ,
               or
               18
               l.
               =
               12
               s.
               the
               several
               Answers
               required
               ;
               which
               being
               added
               together
               ,
               make
               up
               50
               l.
               the
               whole
               Rent
               to
               be
               paid
               among
               them
               .
            
             
               There
               be
               other
               Rules
               of
               Arithmetick
               ,
               as
               the
               Rule
               called
               
                 Allegation
                 ,
                 Medial
              
               ,
               and
               Alternate
               ,
               and
               the
               Rule
               of
               Position
               or
               Falsehood
               ;
               in
               the
               working
               of
               which
               ,
               are
               so
               many
               Cautions
               in
               ordering
               the
               Numbers
               ,
               before
               you
               come
               to
               the
               proportional
               work
               ,
               that
               it
               would
               make
               the
               Book
               more
               bulky
               than
               useful
               ;
               therefore
               I
               shall
               wave
               it
               ,
               and
               refer
               you
               to
               the
               particular
               Books
               of
               Arithmetick
               ,
               as
               that
               of
               Mr.
               
                 Record
                 ,
                 Dee
              
               ,
               and
               Mellis
               ;
               or
               that
               of
               Mr.
               Wingate
               Natural
               and
               Artificial
               ,
               having
               in
               it
               plenty
               of
               Examples
               ;
               and
               others
               also
               ,
               as
               
                 Iohnf●us
                 ,
                 Iaggers
              
               ,
               or
               Moores
               Arithmetick
               ,
               any
               of
               which
               exceed
               the
               bounds
               I
               intend
               for
               this
               whole
               discourse
               ;
               I
               shall
               therefore
               pass
               on
               to
               the
               Rules
               of
               Practice
               ,
               in
               several
               kinds
               ,
               as
               measuring
               Superfecies
               ,
               and
               Solids
               ,
               and
               Rules
               of
               double
               and
               treble
               Proportion
               and
               Questions
               of
               Interest
               ;
               which
               are
               tedious
               by
               the
               Pen
               ,
               without
               the
               help
               of
               particular
               Tables
               ,
               and
               very
               easie
               by
               the
               Line
               of
               Numbers
               ,
               as
               will
               fully
               appear
               in
               the
               next
               Chapters
               .
            
          
        
         
           
           
             CHAP.
             VII
             .
             The
             use
             of
             the
             Line
             of
             Numbers
             in
             measuring
             of
             any
             kind
             of
             Superficial
             Measure
             .
          
           
             THe
             Measure
             that
             is
             commonly
             used
             in
             this
             Work
             ,
             is
             a
             Foot-Rule
             ,
             divided
             into
             100
             parts
             ;
             or
             else
             into
             12
             inches
             ,
             and
             those
             inches
             into
             halves
             ,
             and
             quarters
             ,
             or
             8
             parts
             ;
             or
             inches
             and
             10
             parts
             ▪
             but
             in
             regard
             that
             the
             Numbers
             do
             most
             fitly
             agree
             to
             the
             100
             parts
             of
             a
             Foot
             ,
             it
             will
             be
             convenient
             here
             to
             shew
             how
             to
             reduce
             them
             ,
             or
             any
             other
             Fraction
             ,
             from
             12
             s.
             to
             10
             s.
             or
             any
             other
             whatsoever
             ,
             from
             one
             Fraction
             to
             the
             other
             ,
             which
             by
             the
             Line
             of
             Numbers
             is
             quickly
             done
             ;
             as
             thus
             ,
             from
             12●
             to
             10●
             .
          
           
             Reduction
             .
          
           
             Extend
             the
             Compasses
             from
             one
             Denominator
             to
             the
             other
             ,
             the
             same
             Extent
             shall
             reach
             the
             same
             way
             from
             one
             Numerator
             to
             the
             other
             .
          
           
           
             Example
             .
          
           
             As
             12
             to
             10
             ,
             so
             is
             6
             half
             of
             12
             ,
             to
             5
             half
             of
             10.
             
          
           
             Again
             .
          
           
             As
             120
             to
             100
             ,
             so
             is
             30
             a
             4th
             of
             120
             ,
             to
             25
             a
             4th
             of
             100.
             
          
           
             Which
             two
             Lines
             of
             Inches
             ,
             and
             Foot-Measure
             ,
             are
             usually
             set
             together
             on
             Rules
             ,
             for
             the
             ready
             way
             of
             Reduction
             by
             Occular
             inspection
             ,
             only
             in
             this
             manner
             ,
             
               as
               in
               the
               Figure
            
             ;
             And
             the
             like
             may
             be
             for
             any
             thing
             whatsoever
             ,
             as
             Mr.
             
               Edmond
               Windgate
            
             hath
             largely
             shewed
             in
             his
             Arithmetick
             .
             Which
             Line
             being
             next
             to
             the
             Line
             of
             Numbers
             on
             your
             Rule
             ,
             will
             be
             very
             plain
             and
             ready
             in
             the
             use
             of
             the
             Line
             of
             Numbers
             for
             feet
             and
             inches
             ,
             or
             shillings
             and
             pence
             ;
             and
             the
             same
             Rule
             of
             Reduction
             ,
             serves
             for
             all
             manner
             of
             Fractions
             :
             For
             as
             the
             Denominator
             of
             one
             Fraction
             is
             to
             the
             Denominator
             of
             the
             other
             ,
             (
             which
             in
             the
             Decimal
             work
             is
             alwayes
             a
             unite
             ,
             with
             one
             ,
             two
             ,
             or
             more
             Cyphers
             )
             so
             is
             the
             Numerator
             of
             one
             ,
             to
             the
             Numerator
             of
             the
             other
             .
          
           
             
               And
               Note
            
             ,
             That
             the
             operation
             of
             Decimal
             Numbers
             ,
             and
             their
             Fractions
             ,
             is
             no
             other
             than
             whole
             Numbers
             ,
             except
             only
             the
             cutting
             off
             so
             many
             Figures
             as
             there
             is
             Fractions
             
             in
             the
             Multiplicator
             and
             Multiplicand
             ,
             after
             any
             Multiplication
             ;
             as
             in
             the
             following
             Examples
             will
             appear
             .
          
           
             This
             being
             premised
             ,
             I
             come
             next
             to
             the
             Work.
             
          
           
             
               Problem
               I.
               The
               breadth
               of
               an
               Oblong
               Superficies
               given
               in
               Foot
               ▪
               Measure
               ,
               to
               find
               how
               much
               in
               length
               makes
               one
               Foot.
               
            
             
               The
               Extent
               of
               the
               Compasses
               from
               the
               breadth
               to
               1
               ,
               shall
               reach
               the
               same
               way
               from
               1
               ,
               to
               the
               length
               required
               .
            
             
               Example
               at
               
                 7
                 10th
              
               broad
               .
            
             
               As
               7
               to
               1
               ,
               so
               is
               1
               to
               1
               Foot
               and
               43
               parts
            
             
               The
               breadth
               given
               in
               inches
               ,
               to
               find
               how
               much
               make
               a
               Foot.
               
            
             
               As
               the
               breadth
               in
               inches
               to
               12
               ,
               so
               is
               12
               to
               the
               length
               of
               a
               Foot
               in
               inches
               ,
               and
               10
               parts
               .
            
             
               Example
               .
            
             
               At
               8
               inches
               broad
               ,
               you
               must
               have
               18
               inches
               to
               make
               a
               Foot
               ;
               for
               the
               Extent
               from
               8
               to
               12
               ,
               shall
               reach
               the
               same
               way
               from
               12
               to
               18.
               
            
             
             
               To
               work
               these
               two
               by
               the
               Line
               of
               Lines
               .
            
             
               By
               Inches
               .
               As
               —
               1
               to
               =
               7
               ,
               so
               is
               =
               1
               to
               —
               1
               —
               43
               ,
               the
               length
               in
               Foot-measure
               ;
            
             
               By
               Inches
               .
               As
               —
               12
               to
               =
               8
               ,
               so
               is
               =
               12
               to
               —
               18
               ;
               
                 Or
                 else
              
               ,
            
             
               By
               Inches
               .
               As
               —
               8
               to
               =
               12
               ,
               so
               is
               —
               12
               to
               =
               18
               ,
               the
               length
               in
               inches
               .
            
          
           
             
               Problem
               II.
               Having
               the
               breadth
               of
               an
               Oblong
               Superficies
               given
               in
               Foot-measure
               ,
               to
               find
               how
               much
               is
               in
               a
               Foot
               long
               .
            
             
               This
               is
               soon
               wrought
               ;
               for
               in
               every
               Foot
               long
               there
               is
               just
               as
               much
               as
               the
               breadth
               is
               ,
               either
               in
               Foot-measure
               or
               inches
               ;
               for
               a
               piece
               of
               Board
               half
               a
               Foot
               broad
               ,
               and
               a
               Foot
               long
               ,
               is
               just
               half
               a
               Foot.
               
            
          
           
             
               Problem
               III.
               Having
               the
               length
               and
               breadth
               in
               Foot-Measure
               ,
               to
               find
               the
               Content
               in
               Feet
               .
            
             
               The
               Extent
               from
               1
               to
               the
               length
               ,
               shall
               reach
               the
               same
               way
               from
               the
               length
               to
               the
               Content
               in
               Feet
               .
            
             
             
               Example
               .
            
             
               As
               1
               to
               1
               foot
               50
               ,
               the
               breadth
               ;
               so
               is
               11
               foot
               ,
               10
               parts
               ,
               the
               length
               ,
               to
               16
               foot
               and
               65
               parts
               ,
               the
               Coment
               required
               .
            
             
               The
               breadth
               given
               in
               inches
               ,
               and
               the
               length
               in
               feet
               ,
               to
               find
               the
               Content
               in
               feet
               .
            
             
               As
               12
               to
               the
               breadth
               in
               inches
               ,
               so
               is
               the
               length
               in
               feet
               ,
               to
               the
               Content
               in
               feet
               required
               .
            
             
               Example
               at
               9
               inches
               broad
               ,
               and
               11
               foot
               long
               .
            
             
               The
               Extent
               from
               12
               to
               9
               ,
               shall
               reach
               the
               same
               way
               from
               11
               to
               8
               foot
               ,
               3
               inches
               ,
               or
               ¼
               .
            
             
               By
               the
               Line
               of
               Lines
               .
            
             
               As
               —
               11
               ,
               to
               =
               12
               ;
            
             
               So
               is
               —
               9
               ,
               to
               —
               8
               ¼
               .
            
             
               
                 But
                 Note
              
               ,
               That
               in
               working
               this
               ,
               and
               many
               such-like
               ,
               it
               will
               be
               convenient
               to
               double
               your
               Scale
               in
               account
               ,
               calling
               10
               at
               the
               end
               20
               ,
               and
               every
               single
               figure
               as
               much
               more
               ,
               as
               to
               call
               12
               ,
               and
               24
               ,
               &c.
               
            
             
             
               So
               that
               in
               this
               Operation
               ,
               the
               work
               runs
               thus
               ;
            
             
               As
               —
               11
               taken
               from
               the
               Line
               of
               Lines
               ,
               counting
               1
               for
               10
               as
               usually
               ,
            
             
               To
               =
               6
               ,
               the
               half
               of
               12
               reckoned
               double
               for
               12
               :
            
             
               So
               is
               =
               4
               ½
               counted
               for
               9
               ,
               to
               —
               8
               ¼
               between
               the
               Center
               and
               1.
               
            
             
               
                 Or
                 else
                 thus
              
               ;
            
             
               As
               —
               5
               ½
               counted
               for
               11
               ,
               is
               to
               =
               6
               counted
               for
               12
               ;
            
             
               So
               is
               =
               9
               ,
               to
               —
               8
               ¼
               near
               the
               end
               ,
               and
               as
               large
               as
               may
               be
               .
            
             
               Thus
               you
               may
               many
               times
               vary
               the
               manner
               of
               work
               to
               get
               the
               Answer
               latterally
               ,
               and
               as
               large
               as
               may
               be
               on
               the
               Scale
               of
               Lines
               ,
               by
               doubling
               or
               halfing
               the
               Numbers
               ,
               or
               taking
               the
               whole
               Number
               of
               quarters
               ,
               or
               using
               a
               less
               or
               a
               bigger
               Scale
               ,
               as
               hath
               been
               hinted
               ,
               and
               shall
               be
               more
               in
               places
               convenient
               ,
               in
               the
               following
               Discourse
               ,
               to
               attain
               exactness
               and
               ease
               ,
               ●s
               much
               as
               may
               be
               ,
               as
               time
               and
               practice
               will
               demonstrate
               to
               the
               willing
               Practitioner
               in
               these
               Operations
               .
            
          
           
             
             
               Problem
               IV.
               Having
               the
               length
               and
               breadth
               given
               in
               Inches
               to
               find
               the
               Content
               in
               Superficial-Square
               Inches
               .
            
             
               As
               1
               inch
               ,
               to
               the
               breadth
               in
               inches
               ;
               so
               is
               the
               length
               in
               inches
               ,
               to
               the
               Content
               in
               Superficial
               inches
               .
            
             
               
                 Example
                 ,
                 20
                 inches
                 broad
                 ,
                 and
                 36
                 inches
                 long
                 .
              
            
             
               The
               Extent
               of
               the
               Compasses
               from
               1
               on
               the
               Line
               of
               Numbers
               to
               20
               ,
               shall
               reach
               the
               same
               way
               from
               36
               to
               72
               ,
               the
               true
               Number
               of
               Superficial
               Square
               inches
               in
               that
               Oblong
               .
            
             
               
                 By
                 the
                 Line
                 of
                 Lines
              
               .
            
             
               As
               —
               36
               to
               =
               5
               ,
               counted
               as
               1
               ;
            
             
               So
               is
               =
               10
               counted
               as
               20
               ,
               to
               —
               72
               ,
               at
               the
               largest
               Extent
               .
            
             
               
                 For
                 Note
              
               ,
               The
               reason
               that
               the
               Latteral
               72
               and
               36
               ,
               are
               from
               the
               same
               Scale
               in
               account
               ;
               and
               the
               Parallel
               1
               and
               20
               counted
               Decimally
               ,
               are
               from
               the
               same
               Scale
               also
               ,
               or
               else
               according
               to
               the
               Proportion
               by
               the
               Line
               of
               Numbers
               ;
            
             
               As
               —
               1
               to
               =
               20
               ,
               So
               is
               —
               36
               to
               =
               72.
               
               Here
               also
               is
               the
               same
               advance
               Decimally
               from
               1
               to
               20
               ,
               as
               before
               .
            
          
           
             
             
               Problem
               V.
               Having
               the
               length
               and
               breadth
               given
               in
               Inches
               ,
               to
               find
               the
               Content
               in
               Feet
               Superficial
               :
            
             
               As
               144
               to
               the
               breadth
               in
               inches
               ,
               so
               is
               the
               length
               in
               inches
               ,
               to
               the
               Content
               in
               Feet
               Superficial
               .
            
             
               
                 Example
                 at
                 40
                 broad
                 ,
                 and
                 60
                 long
                 .
              
            
             
               For
               the
               Extent
               on
               the
               Line
               of
               Numbers
               from
               144
               ,
               the
               number
               of
               inches
               in
               one
               foot
               ,
               to
               40
               the
               breadth
               ,
               shall
               reach
               from
               60
               inches
               the
               length
               ,
               to
               16
               foot
               ½
               and
               26
               inches
               .
               To
               count
               so
               many
               inches
               on
               the
               Line
               ,
               observe
               with
               me
               this
               way
               of
               Reduction
               ,
               the
               16
               foot
               and
               ½
               is
               very
               plainly
               seen
               .
               
                 And
                 Note
              
               ,
               That
               there
               is
               10
               cuts
               in
               this
               place
               between
               16
               and
               17
               ;
               and
               10
               times
               14
               is
               140
               ,
               which
               is
               near
               144
               ,
               the
               inches
               in
               a
               foot
               ;
               so
               the
               Point
               of
               the
               Compasses
               staying
               at
               near
               2
               10ths
               beyond
               the
               half-foot
               ,
               I
               count
               almost
               twice
               14
               ,
               which
               is
               26
               inches
               for
               the
               Fraction
               above
               16
               foot
               and
               ½
               .
            
             
               
                 By
                 the
                 Line
                 of
                 Lines
              
               .
            
             
               As
               —
               144
               (
               found
               between
               1
               and
               2
               near
               the
               Center
               )
               is
               to
               =
               40
               (
               at
               the
               figure
               4
               )
               
               So
               is
               half
               =
               60
               ,
               or
               the
               measure
               from
               the
               Center
               to
               3
               ,
               to
               =
               8-35
               ,
               which
               is
               the
               half
               of
               16
               foot
               ½
               ,
               and
               26
               inches
               :
               For
               if
               you
               had
               taken
               all
               60
               ,
               it
               would
               have
               exceeded
               the
               whole
               Parallel
               Radius
               ,
               where
               the
               Answer
               would
               have
               been
               right
               16
               ½
               and
               better
               ,
               but
               taking
               the
               half
               ,
               it
               gives
               the
               half
               also
               .
            
             
               
                 Or
                 else
                 work
                 thus
                 with
                 a
                 Latteral
                 Answer
                 .
              
            
             
               As
               ½
               —
               60
               to
               =
               ½
               144
               ,
               So
               is
               all
               =
               40
               to
               —
               16
               7
               /
               10.
               
            
             
               Or
               ,
            
             
               As
               all
               —
               40
               ,
               to
               ½
               =
               144
               ;
               So
               is
               all
               =
               60
               to
               —
               33
               and
               4
               /
               10
               the
               double
               of
               16
               and
               7
               /
               10.
               
            
             
               Note
               ,
               That
               in
               both
               these
               two
               last
               workings
               ,
               the
               144
               is
               at
               72
               ,
               which
               is
               the
               half
               of
               144
               ;
               to
               make
               the
               work
               the
               larger
               .
               By
               these
               the
               excellency
               of
               the
               Line
               of
               Numbers
               ,
               over
               the
               Line
               of
               Lines
               ,
               is
               evident
               in
               these
               kind
               of
               Proportions
               .
               And
               for
               discovering
               the
               Reason
               of
               these
               Proportions
               ,
               read
               the
               beginning
               of
               the
               6th
               Chapter
               ,
               Section
               3d.
               
            
          
           
             
             
               Problem
               VI.
               The
               length
               and
               breadth
               of
               an
               Oblong
               Superficies
               being
               given
               ,
               to
               find
               the
               side
               of
               a
               Square
               equal
               to
               it
               ,
               by
               the
               Line
               of
               Numbers
               .
            
             
               Divide
               the
               space
               between
               the
               length
               and
               breadth
               into
               two
               equal
               parts
               ,
               and
               the
               middle
               Point
               shall
               be
               the
               side
               of
               the
               Square
               equal
               to
               the
               Oblong
               given
               ,
               in
               quantity
               .
            
             
               Example
               .
            
             
               If
               a
               long
               Square
               ,
               or
               Oblong
               ,
               be
               18
               foot
               one
               way
               ,
               and
               12
               foot
               the
               other
               way
               ;
               the
               middle
               Point
               between
               18
               and
               12
               is
               14
               and
               7
               /
               10
               ferè
               ;
               for
               18
               multiplied
               by
               12
               ,
               makes
               216
               ,
               and
               14-7
               multiplied
               by
               14-7
               ,
               is
               near
               216
               also
               .
            
             
               To
               do
               this
               by
               the
               Line
               of
               Lines
               is
               shewed
               at
               large
               in
               the
               7th
               Section
               of
               the
               6th
               Chapter
               .
            
          
           
             
               Problem
               VII
               .
               Having
               the
               Diameter
               ,
               or
               Circumference
               of
               a
               Circle
               ,
               to
               find
               the
               Circumference
               ,
               or
               Diameter
               ,
               or
               Squares
               equal
               ,
               or
               Inscribed
               ,
               and
               Content
               .
            
             
               For
               this
               purpose
               there
               are
               certain
               Proportional
               Numbers
               found
               out
               ,
               
                 As
                 thus
              
               ;
            
             
               If
               the
               Diameter
               of
               a
               Circle
               be
               10
               ,
               then
               
               the
               Perifera
               ,
               or
               Circumference
               ,
               is
               31
               42
               ,
               the
               side
               of
               the
               Square
               equal
               to
               the
               Circle
               ,
               is
               8-862
               ,
               the
               side
               of
               the
               Square
               inscribed
               is
               7-071
               ,
               and
               the
               Superficial
               Content
               is
               78-54
               ;
               so
               that
               any
               one
               of
               these
               being
               given
               ,
               you
               may
               find
               out
               any
               of
               the
               rest
               by
               the
               Line
               of
               Numbers
               .
            
             
               
                 Thus
                 having
                 the
                 Diameter
                 ,
                 to
                 find
                 the
                 Circumference
                 .
              
            
             
               As
               10
               to
               31
               42
               ,
               so
               is
               the
               given
               Diameter
               to
               the
               Circumference
               required
               .
            
             
               Or
               ,
            
             
               As
               10
               to
               8-862
               ,
               so
               is
               the
               given
               Diameter
               to
               the
               Square
               equal
               .
            
             
               As
               10
               to
               7-071
               ,
               so
               is
               the
               given
               Diameter
               to
               the
               Square
               inscribed
               .
            
             
               As
               10
               to
               30
               ,
               so
               is
               78-54
               to
               the
               Square
               of
               the
               Area
               of
               that
               Circle
               ,
               whose
               Diameter
               is
               30.
               
               Or
               ,
               To
               the
               Diameter
               turning
               the
               Compasses
               twice
               .
            
             
             
               
                 As
                 for
                 Example
              
               .
            
             
               Let
               the
               given
               Diameter
               of
               a
               Circle
               be
               30
               ,
               
                 
                   The
                   Circumference
                   is
                   94
                   26
                   Diameter
                   10
                   00
                
                 
                   The
                   qu
                   are
                   equal
                   is
                   26
                   58
                   ½
                   Circumference
                   ●1
                   42
                
                 
                   The
                   Square
                   within
                   is
                   21
                   21
                   Square
                   equal
                   8
                   862
                
                 
                   The
                   Content
                   or
                   Area
                   is
                   707
                   00
                   Square
                   within
                   7
                   071
                
                 
                   The
                   Diameter
                   is
                   030
                   00
                   Area
                   or
                   Content
                   78
                   54
                
              
            
             
               
                 Also
                 Note
              
               ,
               If
               the
               Circumference
               be
               first
               given
               ,
               then
               say
               ,
               As
               31
               42
               is
               to
               the
               Number
               10
               for
               the
               Diameter
               ,
               or
               8-862
               for
               the
               Square
               equal
               ;
               or
               to
               7-071
               for
               the
               inscribed
               Square
               ;
               so
               is
               the
               given
               Circumference
               to
               the
               rest
               .
            
             
               But
               to
               find
               the
               Area
               say
               ,
               As
               the
               fixed
               Diameter
               10
               ,
               is
               to
               the
               given
               Diameter
               30
               ;
               so
               is
               the
               fixed
               Area
               for
               10
               ,
               viz.
               78-54
               ,
               to
               707
               —
               by
               turning
               the
               Compasses
               two
               times
               .
            
             
               
                 Or
                 if
                 the
                 Circumference
                 be
                 given
                 ,
                 and
                 I
                 would
                 find
                 the
                 Area
                 .
              
            
             
               As
               31-42
               ,
               the
               fixed
               Circumference
               ,
               is
               to
               94-26
               the
               given
               Circumference
               ;
               so
               is
               78-54
               ,
               the
               fixed
               Area
               ,
               answering
               the
               fixed
               Circumference
               707
               ,
               the
               Area
               
               required
               ,
               turning
               the
               Compasses
               two
               times
               the
               same
               way
               .
            
             
               Thus
               by
               having
               five
               Centers
               at
               the
               five
               fixed
               Numbers
               ;
               or
               four
               Centers
               answering
               to
               the
               four
               fixed
               Numbers
               ;
               for
               a
               Circle
               whose
               Diameter
               is
               10
               ,
               having
               any
               one
               of
               those
               5
               given
               ,
               you
               may
               find
               any
               of
               the
               other
               required
               .
            
             
               Thus
               you
               have
               eight
               Problems
               couched
               in
               one
               ;
               therefore
               be
               the
               more
               diligent
               to
               understand
               it
               .
               To
               work
               these
               by
               the
               Line
               of
               Lines
               ,
               observe
               the
               former
               directions
               ,
               which
               for
               brevity
               sake
               I
               now
               omit
               .
            
          
           
             
               Problem
               VIII
               .
               The
               Content
               of
               a
               Circle
               being
               given
               ,
               to
               find
               the
               Diameter
               .
            
             
               Divide
               the
               distance
               on
               the
               Line
               of
               Numbers
               ,
               between
               the
               fixed
               Content
               ,
               or
               the
               Point
               78-54
               ,
               and
               the
               given
               Content
               into
               two
               equal
               parts
               ;
               that
               distance
               laid
               the
               same
               way
               ,
               from
               the
               fixed
               Diameter
               ,
               shall
               reach
               to
               the
               required
               Diameter
               .
            
             
               
                 Example
                 .
                 The
                 Content
                 being
                 707-00
              
               .
            
             
               The
               half
               distance
               between
               78-54
               ,
               and
               707
               ,
               shall
               reach
               from
               10
               to
               30
               ,
               the
               Diameter
               required
               .
            
          
           
             
             
               Problem
               IX
               .
               The
               Content
               of
               a
               Circle
               being
               given
               ,
               to
               find
               the
               Circumference
               .
            
             
               Divide
               the
               distance
               between
               the
               fixed
               and
               the
               given
               Contents
               or
               Area's
               into
               two
               equal
               parts
               ,
               the
               distance
               laid
               from
               the
               fixed
               Circumference
               ,
               shall
               reach
               to
               the
               required
               Circumference
               .
            
             
               Example
               .
            
             
               A
               Circle
               ,
               whose
               Area
               is
               707
               ,
               shall
               be
               94-26
               about
               .
               For
               the
               half
               distance
               between
               78-54
               ,
               and
               707-00
               ,
               shall
               reach
               from
               31-42
               the
               fixed
               Circumference
               ,
               to
               94-26
               ,
               the
               enquired
               Circumference
               .
            
             
               And
               from
               8-862
               ,
               the
               fixed
               Square
               ,
               equal
               to
               26-58
               ½
               ,
               the
               inquired
               Square
               Equal
               .
            
             
               And
               from
               7-071
               ,
               the
               fixed
               Square
               inscribed
               ,
               to
               21-21
               ,
               the
               inquired
               Square
               Inscribed
               .
            
          
           
             
               Problem
               X.
               Certain
               Rules
               to
               measure
               several
               Geometrical
               Figures
               Superficially
               .
            
             
               For
               the
               Square
               ,
               the
               long
               Square
               and
               Circle
               ,
               hath
               been
               spoken
               to
               just
               before
               ;
               All
               other
               Figures
               are
               to
               be
               reduced
               to
               a
               Square
               ,
               or
               to
               a
               long
               Square
               ;
               and
               then
               measured
               by
               Multiplication
               ,
               as
               before
               .
            
             
             
               
                 Or
                 thus
              
               .
            
             
               Multiply
               the
               Diameter
               by
               it self
               ;
               and
               then
               that
               Product
               by
               11
               :
               then
               lastly
               ,
               divide
               this
               last
               Product
               by
               14
               ,
               and
               the
               Quotient
               shall
               be
               the
               Area
               ,
               or
               Content
               ,
               of
               the
               Circle
               required
               .
            
             
               
               For
               a
               Circle
               (
               otherwise
               )
               thus
               ;
               Multiply
               half
               the
               Diameter
               ,
               by
               half
               the
               Circumference
               ,
               and
               the
               Product
               shall
               be
               the
               Content
               required
               .
            
             
               
               For
               a
               
                 Half
                 Circle
              
               ;
               Multiply
               half
               the
               Diameter
               of
               the
               whole
               Circle
               ,
               and
               a
               quarter
               of
               the
               whole
               Circumference
               together
               ,
               and
               the
               Product
               shall
               be
               the
               Content
               .
            
             
               
               For
               a
               
                 Quadrant
                 ,
                 or
                 a
                 quarter
              
               of
               a
               Circle
               ;
               Multiply
               half
               the
               Arch
               ,
               by
               the
               half
               Diameter
               ,
               or
               Radius
               of
               the
               Circle
               and
               the
               Product
               shall
               be
               the
               Superficial
               Content
               .
            
             
               
               The
               like
               Rule
               holds
               for
               any
               
                 lesser
                 portion
              
               of
               a
               Circle
               ,
               whose
               Point
               goeth
               to
               the
               Center
               ,
               viz.
               to
               take
               half
               the
               Arch
               ,
               and
               the
               whole
               Radius
               ,
               and
               multiply
               them
               together
               ,
               and
               the
               Product
               shall
               be
               the
               Content
               .
            
             
               
                 Any
                 Segment
                 of
                 a
                 Circle
                 given
                 ,
                 to
                 find
                 the
                 true
                 Diameter
                 .
              
            
             
               Square
               half
               the
               Chord
               ,
               and
               divide
               the
               Product
               by
               the
               Sine
               ,
               then
               add
               the
               Quotient
               and
               Sine
               together
               ;
               the
               sum
               is
               the
               Diameter
               .
               
               Chord
               is
               24
               ½
               12.
               
               Squared
               is
               144
               ;
               divided
               by
               8
               ,
               gives
               18
               in
               the
               Quotient
               ,
               which
               added
               to
               8
               ,
               makes
               26
               for
               the
               Diameter
               .
            
             
               For
               any
               other
               Segment
               of
               a
               Circle
               ,
               
               find
               the
               true
               semi-Diameter
               ,
               and
               measure
               it
               as
               before
               ;
               then
               take
               out
               the
               Triangle
               ,
               and
               the
               remainder
               is
               the
               true
               Content
               of
               the
               Segment
               .
               
                 See
                 Chapter
              
               3.11
               ,
               12.
               
            
             
               
                 Or
                 else
                 thus
                 ,
                 by
                 the
                 Line
                 of
                 Segments
                 joyned
                 to
                 a
                 Line
                 of
                 Numbers
                 ,
                 in
                 this
                 manner
                 .
              
            
             
               To
               the
               Segment
               given
               ,
               find
               the
               true
               Diameter
               ,
               by
               Chap.
               III.
               11
               ,
               12.
               
               Then
               having
               the
               Diameter
               ,
               find
               out
               the
               Area
               ,
               or
               Content
               of
               the
               Circle
               ,
               by
               any
               of
               the
               former
               Rules
               ,
               then
               the
               Proportion
               or
               Analogy
               is
               thus
               ;
            
             
               As
               the
               whole
               Diameer
               is
               to
               100
               on
               the
               Segments
               ;
               So
               is
               the
               Altitude
               of
               the
               Segment
               ,
               whose
               Area
               is
               required
               to
               a
               4th
               Number
               on
               the
               Line
               of
               Segments
               ,
               which
               you
               must
               keep
               .
            
             
               
                 Then
                 Secondly
              
               ,
            
             
               As
               1
               ,
               to
               the
               whole
               Content
               of
               the
               whole
               Circle
               given
               ;
               So
               is
               the
               4th
               Number
               ,
               kept
               ,
               counted
               on
               the
               Numbers
               ,
               to
               the
               Area
               of
               the
               Segment
               required
               .
            
             
               If
               the
               Line
               of
               Segments
               is
               not
               on
               your
               Rule
               ,
               then
               this
               Table
               annexed
               ,
               will
               supply
               the
               defect
               ,
               reasonably
               well
               thus
               :
               
                 
                 
                   A
                   Table
                   to
                   divide
                   a
                   Line
                   of
                   Segments
                   ,
                   making
                   the
                   whole
                   Circle
                   10000
                   parts
                   .
                
                 
                   
                     A.
                     Table
                     of
                     Segments
                     .
                  
                
                 
                   
                     Seg.
                     
                  
                   
                     par
                  
                   
                     Seg.
                     
                  
                   
                     par
                  
                   
                     Seg.
                     
                  
                   
                     par
                  
                   
                     Seg.
                     
                  
                   
                     par
                     .
                  
                
                 
                   
                     70
                  
                   
                     1
                  
                   
                     1127
                  
                   
                     2
                  
                   
                     3484
                  
                   
                      
                  
                   
                     6682
                  
                   
                      
                  
                
                 
                   
                     112
                  
                   
                     2
                  
                   
                     1151
                  
                   
                     4
                  
                   
                     3566
                  
                   
                      
                  
                   
                     6786
                  
                   
                      
                  
                
                 
                   
                     147
                  
                   
                     3
                  
                   
                     1177
                  
                   
                     6
                  
                   
                     3645
                  
                   
                      
                  
                   
                     6850
                  
                   
                      
                  
                
                 
                   
                     178
                  
                   
                     4
                  
                   
                     1201
                  
                   
                     8
                  
                   
                     3729
                  
                   
                      
                  
                   
                     6935
                  
                   
                      
                  
                
                 
                   
                     206
                  
                   
                     5
                  
                   
                     1224
                  
                   
                     7
                  
                   
                     3810
                  
                   
                     35
                  
                   
                     7020
                  
                   
                     75
                  
                
                 
                   
                     233
                  
                   
                     6
                  
                   
                     1248
                  
                   
                     2
                  
                   
                     3892
                  
                   
                      
                  
                   
                     7106
                  
                   
                      
                  
                
                 
                   
                     258
                  
                   
                     7
                  
                   
                     1272
                  
                   
                     4
                  
                   
                     3971
                  
                   
                      
                  
                   
                     7193
                  
                   
                      
                  
                
                 
                   
                     282
                  
                   
                     8
                  
                   
                     1296
                  
                   
                     6
                  
                   
                     4050
                  
                   
                      
                  
                   
                     7281
                  
                   
                      
                  
                
                 
                   
                     307
                  
                   
                     9
                  
                   
                     1318
                  
                   
                     8
                  
                   
                     4131
                  
                   
                      
                  
                   
                     7370
                  
                   
                      
                  
                
                 
                   
                     329
                  
                   
                     1
                  
                   
                     1341
                  
                   
                     8
                  
                   
                     4211
                  
                   
                     40
                  
                   
                     7460
                  
                   
                     80
                  
                
                 
                   
                     350
                  
                   
                     1
                  
                   
                     1365
                  
                   
                     2
                  
                   
                     4290
                  
                   
                      
                  
                   
                     7550
                  
                   
                      
                  
                
                 
                   
                     371
                  
                   
                     2
                  
                   
                     1388
                  
                   
                     4
                  
                   
                     4369
                  
                   
                      
                  
                   
                     7642
                  
                   
                      
                  
                
                 
                   
                     392
                  
                   
                     3
                  
                   
                     1411
                  
                   
                     6
                  
                   
                     4448
                  
                   
                      
                  
                   
                     7735
                  
                   
                      
                  
                
                 
                   
                     412
                  
                   
                     4
                  
                   
                     1433
                  
                   
                     8
                  
                   
                     4527
                  
                   
                      
                  
                   
                     7829
                  
                   
                      
                  
                
                 
                   
                     431
                  
                   
                     5
                  
                   
                     1455
                  
                   
                     9
                  
                   
                     4606
                  
                   
                     45
                  
                   
                     7924
                  
                   
                     85
                  
                
                 
                   
                     451
                  
                   
                     6
                  
                   
                     1478
                  
                   
                     2
                  
                   
                     4686
                  
                   
                      
                  
                   
                     8022
                  
                   
                      
                  
                
                 
                   
                     469
                  
                   
                     7
                  
                   
                     1500
                  
                   
                     4
                  
                   
                     4766
                  
                   
                      
                  
                   
                     8119
                  
                   
                      
                  
                
                 
                   
                     487
                  
                   
                     8
                  
                   
                     1522
                  
                   
                     6
                  
                   
                     4844
                  
                   
                      
                  
                   
                     82●2
                  
                   
                      
                  
                
                 
                   
                     507
                  
                   
                     9
                  
                   
                     1544
                  
                   
                     8
                  
                   
                     4922
                  
                   
                      
                  
                   
                     8327
                  
                   
                      
                  
                
                 
                   
                     524
                  
                   
                     2
                  
                   
                     1565
                  
                   
                     10
                  
                   
                     5000
                  
                   
                     50
                  
                   
                     8436
                  
                   
                     90
                  
                
                 
                   
                     558
                  
                   
                     2
                  
                   
                     1673
                  
                   
                     11
                  
                   
                     5078
                  
                   
                      
                  
                   
                     8552
                  
                   
                      
                  
                
                 
                   
                     592
                  
                   
                     4
                  
                   
                     1778
                  
                   
                     12
                  
                   
                     5156
                  
                   
                      
                  
                   
                     8669
                  
                   
                      
                  
                
                 
                   
                     626
                  
                   
                     6
                  
                   
                     1881
                  
                   
                     13
                  
                   
                     5234
                  
                   
                      
                  
                   
                     8788
                  
                   
                      
                  
                
                 
                   
                     657
                  
                   
                     8
                  
                   
                     1978
                  
                   
                     14
                  
                   
                     5314
                  
                   
                      
                  
                   
                     8908
                  
                   
                      
                  
                
                 
                   
                     688
                  
                   
                     3
                  
                   
                     2076
                  
                   
                     15
                  
                   
                     5394
                  
                   
                     55
                  
                   
                     9029
                  
                   
                     95
                  
                
                 
                   
                     718
                  
                   
                     2
                  
                   
                     2171
                  
                   
                     16
                  
                   
                     5473
                  
                   
                      
                  
                   
                     9172
                  
                   
                      
                  
                
                 
                   
                     749
                  
                   
                     4
                  
                   
                     2265
                  
                   
                     17
                  
                   
                     5552
                  
                   
                      
                  
                   
                     9330
                  
                   
                      
                  
                
                 
                   
                     779
                  
                   
                     6
                  
                   
                     2358
                  
                   
                     18
                  
                   
                     5●31
                  
                   
                      
                  
                   
                     9505
                  
                   
                      
                  
                
                 
                   
                     808
                  
                   
                     8
                  
                   
                     2450
                  
                   
                     19
                  
                   
                     5710
                  
                   
                      
                  
                   
                     97●0
                  
                   
                      
                  
                
                 
                   
                     083●
                  
                   
                     4
                  
                   
                     2540
                  
                   
                     20
                  
                   
                     5789
                  
                   
                     60
                  
                   
                     10000
                  
                   
                     100
                  
                
                 
                   
                     0864
                  
                   
                     2
                  
                   
                     2630
                  
                   
                     21
                  
                   
                     5869
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                
                 
                   
                     0892
                  
                   
                     4
                  
                   
                     2719
                  
                   
                     22
                  
                   
                     5950
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                
                 
                   
                     0918
                  
                   
                     6
                  
                   
                     2807
                  
                   
                     23
                  
                   
                     6029
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                
                 
                   
                     0948
                  
                   
                     8
                  
                   
                     2894
                  
                   
                     24
                  
                   
                     6108
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                
                 
                   
                     0970
                  
                   
                     ●
                  
                   
                     2980
                  
                   
                     25
                  
                   
                     6190
                  
                   
                     65
                  
                   
                      
                  
                   
                      
                  
                
                 
                   
                     1000
                  
                   
                     2
                  
                   
                     3065
                  
                   
                      
                  
                   
                     6271
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                
                 
                   
                     1027
                  
                   
                     4
                  
                   
                     3150
                  
                   
                      
                  
                   
                     6355
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                
                 
                   
                     1051
                  
                   
                     6
                  
                   
                     3214
                  
                   
                      
                  
                   
                     6434
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                
                 
                   
                     1077
                  
                   
                     8
                  
                   
                     3318
                  
                   
                      
                  
                   
                     6516
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                
                 
                   
                     1102
                  
                   
                     6
                  
                   
                     3402
                  
                   
                     30
                  
                   
                     6598
                  
                   
                     70
                  
                   
                      
                  
                   
                      
                  
                
              
            
             
             
               
                 The
                 Diameter
                 of
                 the
                 Circle
                 ,
                 answering
                 to
                 the
                 Segment
                 given
                 ,
                 being
                 found
                 out
                 ,
                 Say
                 ,
              
            
             
               As
               the
               whole
               Diameter
               to
               100
               ;
               so
               is
               the
               Altitude
               of
               the
               Segment
               to
               a
               4th
               Number
               ,
               which
               sought
               in
               the
               Table
               of
               Segments
               ,
               or
               the
               nearest
               to
               it
               ,
               gives
               in
               the
               parts
               the
               Number
               to
               be
               kept
               .
               Then
               again
               ,
            
             
               As
               the
               whole
               Content
               of
               the
               Circle
               fixed
               ,
               viz.
               100
               ,
               is
               to
               the
               whole
               Content
               of
               the
               new
               Circle
               ;
               so
               is
               the
               Number
               kept
               ,
               being
               the
               Content
               ,
               of
               Area
               ,
               of
               the
               fixed
               Segment
               ,
               to
               the
               Area
               of
               the
               Segment
               required
               .
            
             
               Example
               .
            
             
               Let
               the
               Segment
               of
               a
               Circle
               ,
               whose
               whole
               Area
               is
               314-2
               ,
               and
               whose
               Diameter
               is
               20
               ,
               and
               let
               the
               Altitude
               of
               the
               Segment
               be
               5
               ,
               one
               4th
               part
               of
               the
               whole
               Diameter
               .
            
             
               
                 Then
                 say
              
            
             
               As
               2-000
               ,
               the
               whole
               Diameter
               given
               ,
               is
               to
               10000
               :
            
             
               So
               is
               the
               Altitude
               of
               the
               Segment
               5
               ,
               to
               2500
               ,
               the
               4th
               ;
               which
               sought
               in
               the
               Table
               of
               Segments
               ,
               in
               the
               Parts
               ,
               gives
               19-50
               for
               a
               5th
               Number
               to
               be
               kept
               .
            
             
               
                 Then
                 again
              
               ,
            
             
               As
               1
               ,
               to
               314-2
               ,
               the
               whole
               Area
               ;
               So
               is
               19-50
               ,
               to
               61-30
               ,
               the
               Area
               ,
               or
               Content
               of
               the
               Segment
               required
               .
            
             
             
               
               For
               all
               manner
               of
               Triangles
               ,
               multiply
               the
               longest
               side
               (
               being
               properly
               called
               the
               base
               )
               by
               half
               the
               perpendiculer
               ,
               and
               the
               Product
               shall
               be
               the
               Content
               of
               the
               Triangle
               ;
               or
               as
               2
               to
               the
               base
               ,
               so
               is
               the
               perpendiculer
               to
               the
               Content
               .
            
             
               
               For
               a
               Rhombus
               ,
               being
               a
               Figure
               like
               a
               Quarry
               of
               Glass
               ,
               containing
               4
               equal
               sides
               ,
               and
               two
               pair
               of
               equal
               Angles
               :
               And
               any
               Figure
               having
               his
               opposite-sides
               Parallel
               one
               to
               another
               ;
               then
               the
               length
               of
               one
               side
               and
               the
               nearest
               distance
               between
               the
               other
               two
               opposite-sides
               multplied
               together
               ,
               shall
               be
               the
               true
               Area
               required
               .
            
             
               
               For
               all
               other
               four-sided-figures
               ,
               call'd
               Trapeziaes
               ,
               being
               irregular
               Figures
               ;
               draw
               a
               Line
               from
               one
               corner
               to
               the
               other
               ,
               which
               makes
               it
               two
               Triangles
               ;
               then
               multiply
               that
               Line
               ,
               being
               the
               whole
               base
               of
               both
               the
               Triangles
               ,
               by
               the
               half
               sum
               of
               both
               the
               Perpendiculers
               ,
               and
               the
               Product
               shall
               be
               the
               Content
               required
               .
            
             
               Or
               ,
            
             
               
               For
               all
               
                 Regular
                 Polligons
              
               ,
               or
               Figures
               ,
               with
               equal
               sides
               ,
               the
               measure
               from
               the
               Center
               to
               the
               middle
               of
               one
               side
               ,
               and
               the
               half
               sum
               ,
               of
               the
               measure
               of
               all
               the
               sides
               multiplied
               together
               ,
               shall
               be
               the
               true
               Area
               ,
               or
               Content
               thereof
               .
            
             
             
               All
               other
               Figures
               whatsoever
               ,
               of
               how
               many
               sides
               soever
               they
               be
               ,
               may
               be
               reduced
               to
               Triangles
               ,
               or
               to
               Trapeziaes
               ,
               and
               measured
               as
               before
               ;
               which
               kind
               of
               Figure
               ,
               Surveyors
               and
               Builders
               oftentimes
               meet
               withal
               ,
               in
               their
               Operations
               .
            
          
           
             
               Problem
               XI
               .
               For
               the
               measuring
               of
               on
               Oval
               ,
               the
               best
               way
               Ovals
               ▪
               is
               to
               reduce
               it
               to
               a
               Circle
               thus
               ;
            
             
               Divide
               the
               distance
               on
               the
               Line
               of
               Numbers
               ,
               between
               the
               length
               and
               the
               breadth
               of
               the
               Oval
               into
               two
               equal
               parts
               ;
               and
               the
               middle
               Point
               where
               the
               Compass
               stayeth
               on
               ,
               shall
               be
               the
               Diameter
               of
               a
               Circle
               equal
               in
               Area
               to
               the
               Oval
               given
               .
            
             
               Example
               .
            
             
               Suppose
               an
               Oval
               be
               10
               foot
               long
               (
               transverse
               )
               and
               8
               foot
               broad
               (
               conjugate
               )
               ;
               the
               mean
               proportion
               ,
               between
               10
               and
               8
               ,
               is
               8-95
               :
               I
               say
               ,
               that
               a
               Circle
               whose
               Diameter
               is
               8-95
               ,
               is
               equal
               to
               an
               Oval
               of
               8
               broad
               ,
               and
               10
               long
               ;
               And
               how
               to
               measure
               the
               Circle
               ,
               is
               shewed
               before
               .
            
             
               
                 Of
                 these
                 Figures
              
               .
            
             
               If
               the
               Content
               be
               100
               ,
               then
               the
               sides
               of
               these
               Regular
               Figures
               are
               〈…〉
               ,
               and
               
               also
               so
               in
               proportion
               ,
               is
               any
               other
               quantity
               ,
               of
               content
               required
               .
            
             
               
                 
                   
                     Perpendiculer-Triangle
                     ,
                  
                   
                     13.
                     
                  
                   
                     123.
                     
                  
                
                 
                   
                     Trianguler-side
                     ,
                  
                   
                     15.
                     
                  
                   
                     2.
                     
                  
                
                 
                   
                     
                       Square
                       ,
                       its
                       Side
                    
                     ,
                  
                   
                     10.
                     
                  
                   
                     0.
                     
                  
                
                 
                   
                     
                       Pantagon
                       of
                       five
                       Sides
                    
                     ,
                  
                   
                     7.
                     
                  
                   
                     62.
                     
                  
                
                 
                   
                     
                       Hexagon
                       of
                       six
                    
                     ,
                  
                   
                     6.
                     
                  
                   
                     02.
                     
                  
                
                 
                   
                     
                       Heptagon
                       of
                       seven
                    
                     ,
                  
                   
                     5.
                     
                  
                   
                     26.
                     
                  
                
                 
                   
                     
                       Octagon
                       of
                       eight
                    
                     ,
                  
                   
                     4.
                     
                  
                   
                     55.
                     
                  
                
                 
                   
                     
                       Nonagon
                       of
                       nine
                    
                     ,
                  
                   
                     4.
                     
                  
                   
                     03.
                     
                  
                
                 
                   
                     
                       Decagon
                       of
                       ten
                    
                     ,
                  
                   
                     3.
                     
                  
                   
                     06.
                     
                  
                
                 
                   
                     
                       Half
                       Diameter
                       ,
                       or
                       Radius
                    
                     ,
                  
                   
                     5.
                     
                  
                   
                     64.
                     
                  
                
              
            
             
               
                 Example
                 as
                 thus
                 :
                 I
                 would
                 have
                 a
                 Triangle
                 to
                 contain
                 200
                 ,
                 What
                 must
                 the
                 Sides
                 be
                 ?
              
            
             
               The
               half
               distance
               on
               the
               Numbers
               between
               100
               and
               200
               ,
               shall
               reach
               from
               15-2
               to
               21-5
               ,
               the
               side
               required
               .
            
             
               And
               from
               13-123
               the
               fixed
               perpendiculer
               for
               a
               Triangle
               ,
               whose
               Area
               is
               100
               ,
               to
               18-6
               ,
               the
               perpendiculer
               of
               an
               equilatteral
               Triangle
               ,
               whose
               Area
               is
               200.
               
            
             
             
               
                 But
                 if
                 the
                 Sides
                 be
                 given
                 ,
                 and
                 you
                 would
                 find
                 the
                 Area
                 ,
                 work
                 thus
              
               ;
            
             
               The
               Extent
               from
               the
               fixed-side
               ,
               to
               the
               given-side
               ,
               shall
               reach
               at
               two
               turnings
               ,
               from
               the
               fixed
               Area
               ,
               to
               the
               Area
               required
               .
            
             
               The
               Extent
               from
               15-2
               ,
               to
               21-5
               ,
               shall
               reach
               ,
               at
               twice
               repeating
               ,
               from
               100
               to
               200.
               
            
          
           
             
               Problem
               XII
               .
               To
               make
               an
               Oval
               equal
               to
               a
               Circle
               ,
               having
               the
               Diameter
               of
               the
               Circle
               ,
               and
               the
               length
               or
               breadth
               of
               the
               Oval
               given
               .
            
             
               S●t
               one
               Point
               of
               the
               Compasses
               in
               the
               Diameter
               of
               the
               Circle
               found
               out
               on
               the
               Line
               of
               Numbers
               ,
               and
               the
               other
               Point
               to
               the
               Ovals
               length
               ;
               then
               turn
               that
               distance
               the
               contrary
               way
               from
               the
               same
               Diameter-point
               ,
               and
               it
               shall
               reach
               to
               the
               breadth
               of
               the
               Oval
               required
               .
            
             
               Example
               .
            
             
               Let
               the
               Diameter
               of
               a
               Circle
               be
               10
               foot
               ,
               I
               would
               have
               an
               Oval
               to
               contain
               as
               much
               as
               the
               Circle
               ,
               and
               be
               12
               foot
               long
               ;
               the
               Query
               is
               ,
               how
               broad
               must
               it
               be
               ?
            
             
               Set
               one
               Point
               in
               10
               ,
               and
               the
               other
               in
               12
               ,
               that
               Extent
               turned
               the
               other
               way
               from
               10
               ,
               shall
               reach
               to
               8-34
               ,
               the
               breadth
               of
               the
               Oval
               required
               .
            
             
             
               If
               you
               please
               to
               alter
               the
               breadth
               or
               length
               ,
               you
               shall
               soon
               find
               the
               length
               or
               breadth
               accordingly
               .
            
             
               
                 To
                 work
                 this
                 by
                 the
                 Line
                 of
                 Lines
                 ,
                 you
                 must
                 work
                 by
                 the
                 Directions
                 in
                 the
                 7th
                 Section
                 of
                 the
                 6th
                 Chapter
                 ,
                 as
                 thus
              
               ;
            
             
               First
               ,
               To
               find
               the
               Content
               of
               the
               Oval
               ,
               joyn
               the
               length
               and
               breadth
               in
               one
               sum
               ,
               to
               get
               the
               sum
               ,
               the
               half
               sum
               ,
               and
               difference
               ,
               and
               half
               difference
               ;
               then
               open
               the
               Sector
               ,
               (
               or
               lay
               the
               Thred
               on
               600
               )
               to
               a
               Right
               Angle
               ;
               Then
               count
               half
               the
               difference
               from
               the
               Center
               downwards
               ,
               and
               note
               the
               place
               ;
               then
               take
               half
               the
               sum
               between
               your
               Compasses
               ,
               and
               setting
               one
               Point
               in
               the
               half-difference
               ,
               and
               extending
               the
               other
               to
               the
               other
               Leg
               ,
               (
               or
               perpendiculer
               Line
               )
               and
               it
               shall
               shew
               a
               Point
               ,
               whose
               distance
               from
               the
               Center
               is
               the
               mean
               proportional
               required
               ;
               which
               is
               the
               Diameter
               of
               a
               Circle
               ,
               equal
               in
               Area
               ,
               to
               the
               Oval
               ,
               or
               Elipsis
               given
               to
               be
               measured
               ;
               as
               before
               is
               shewed
               .
            
             
               
                 To
                 make
                 an
                 Oval
                 equal
                 to
                 a
                 Circle
                 .
              
            
             
               Take
               the
               guessed
               half-sum
               of
               the
               length
               and
               breadth
               of
               the
               Oval
               ,
               and
               setting
               one
               Point
               in
               the
               Diameter
               of
               the
               Circle
               ;
               and
               on
               the
               other
               Leg
               ,
               set
               at
               a
               Right
               Angle
               ,
               the
               other
               Point
               shall
               shew
               half
               the
               difference
               ,
               
               between
               the
               length
               and
               breadth
               of
               the
               Oval
               ;
               then
               if
               the
               mean
               proportional
               between
               them
               be
               equal
               to
               the
               Diameter
               ,
               you
               have
               wrought
               right
               ;
               if
               not
               ,
               then
               resolving
               upon
               the
               length
               or
               breadth
               of
               the
               Oval
               ,
               take
               more
               or
               less
               ,
               for
               the
               breadth
               or
               length
               accordingly
               :
               Herein
               also
               is
               seen
               the
               excellency
               of
               the
               Line
               of
               Numbers
               ,
               in
               many
               operations
               .
            
          
           
             
               Problem
               XIII
               .
               The
               length
               and
               breadth
               of
               any
               Oblong
               Superficies
               given
               in
               Feet
               ,
               to
               find
               the
               Content
               in
               Yards
               .
            
             
               As
               9
               foot
               (
               the
               number
               of
               feet
               in
               one
               yard
               )
               to
               the
               length
               in
               feet
               and
               parts
               ;
            
             
               So
               is
               the
               breadth
               ,
               in
               feet
               and
               parts
               ,
               to
               the
               Content
               in
               yards
               .
            
             
               
                 Example
                 at
                 13
                 Foot
                 6
                 Inches
                 long
                 ,
                 and
                 7
                 Foot
                 6
                 Inches
                 broad
                 .
              
            
             
               The
               Extent
               of
               the
               Compasses
               from
               9
               to
               13
               ½
               the
               length
               ,
               shall
               reach
               the
               same
               way
               from
               7
               ½
               the
               breadth
               ,
               to
               11
               yards
               and
               a
               quarter
               ,
               the
               Content
               .
            
             
               Note
               ,
               That
               if
               yo●
               measure
               by
               feet
               and
               hundred
               parts
               ,
               you
               shall
               find
               this
               way
               exceeding
               ready
               ;
               the
               Answer
               being
               given
               in
               yards
               ,
               and
               hundred
               parts
               of
               a
               yard
               .
               But
               if
               you
               have
               a
               yard
               divided
               into
               a
               100
               parts
               ,
               
               to
               measure
               withal
               ;
               
                 Then
                 the
                 Rule
                 is
                 thus
                 ▪
              
            
             
               As
               1
               to
               the
               length
               o●
               breadth
               ,
               so
               is
               the
               breadth
               or
               length
               to
               the
               Content
               in
               yards
               .
            
             
               
                 Example
                 at
                 3
                 yards
                 ,
                 72
                 parts
                 broad
                 ,
                 and
                 5
                 yards
                 ,
                 82
                 parts
                 long
                 .
              
            
             
               The
               Extent
               of
               the
               Compasses
               on
               the
               Line
               of
               Numbers
               ,
               from
               1
               to
               3-72
               ,
               shall
               reach
               the
               same
               way
               from
               5-82
               ,
               to
               21
               yards
               65
               parts
               ,
               the
               Content
               in
               square
               yards
               ,
               and
               100
               parts
               .
            
             
               
                 By
                 the
                 Line
                 of
                 Lines
              
               .
            
             
               As
               —
               5-82
               ,
               to
               =
               1
               at
               10
               the
               end
               ;
            
             
               So
               is
               =
               3-72
               ,
               to
               —
               21-65
               yards
               .
            
             
               
                 Or
                 in
                 the
                 Example
                 before
              
               .
            
             
               As
               —
               13-6
               ,
               counting
               6
               ½
               for
               13
               ,
               is
               to
               =
               9
               ;
            
             
               So
               is
               =
               7
               ½
               to
               —
               11
               ¼
               ;
               as
               you
               counted
               at
               first
               .
            
          
           
             
             
               Problem
               XIV
               .
               The
               length
               and
               breadth
               of
               any
               Wall
               ,
               being
               given
               in
               feet
               and
               100
               parts
               ,
               to
               find
               how
               many
               Rods
               of
               Walling
               there
               shall
               be
               at
               a
               Brick
               and
               an
               half
               thick
               .
            
             
               First
               you
               must
               Note
               ,
               That
               272
               foot
               and
               a
               quarter
               ,
               makes
               one
               Rod
               ,
               (
               or
               so
               many
               feet
               is
               in
               a
               Rod
               )
               .
            
             
               Secondly
               ,
               That
               let
               the
               Walls
               be
               half
               a
               Brick
               ,
               one
               Brick
               ,
               two
               Bricks
               ,
               two
               and
               a
               half
               ,
               or
               three
               Bricks
               thick
               ;
               it
               is
               to
               be
               reduced
               to
               Brick
               and
               a
               half
               thick
               ,
               as
               a
               standard
               thickness
               .
            
             
               Thirdly
               Note
               ,
               That
               this
               reducing
               to
               a
               Brick
               and
               half
               thick
               ,
               may
               be
               at
               the
               measuring
               ,
               or
               after
               the
               casting-up
               ,
               as
               you
               please
               ,
               as
               in
               the
               Examples
               following
               will
               plainly
               appear
               .
            
             
               
                 As
                 thus
                 for
                 Instance
              
               ;
            
             
               A
               Front
               ,
               or
               side-Wall
               of
               a
               House
               is
               to
               be
               measured
               ,
               wherein
               the
               Celler-story
               Wall
               is
               2
               Bricks
               and
               a
               half
               thick
               ;
               The
               Shop
               and
               first
               Chamber-story
               is
               two
               bricks
               thicks
               ;
               the
               other
               Stories
               1
               Brick
               and
               a
               half
               thick
               ;
               and
               the
               Gable-ends
               1
               Brick
               thick
               .
            
             
             
               
                 The
                 nearest
                 way
                 to
                 measure
                 this
                 Wall
                 ,
                 I
                 conceive
                 is
                 thus
              
               ;
            
             
               1.
               
               The
               Cellar-story
               is
               10
               foot
               high
               ,
               but
               being
               2
               bricks
               and
               a
               half
               thick
               ,
               I
               make
               it
               16
               foot
               8
               inches
               high
               ,
               by
               adding
               two
               thirds
               of
               10
               foot
               ,
               to
               the
               10
               foot
               high
               ,
               which
               is
               6
               foot
               8
               inches
               ,
               in
               all
               16
               foot
               8
               inches
               .
            
             
               2.
               
               The
               other
               two
               Stories
               ,
               are
               supposed
               22
               foot
               ;
               but
               in
               regard
               they
               are
               two
               bricks
               thick
               ,
               I
               add
               one
               third
               part
               of
               22
               foot
               ,
               which
               is
               7
               foot
               4
               inches
               ,
               to
               22
               ;
               and
               it
               makes
               29
               foot
               and
               4
               inches
               ,
               the
               height
               of
               the
               Shop
               and
               next
               Story
               above
               .
            
             
               3.
               
               The
               other
               two
               Stories
               being
               a
               brick
               and
               half
               thick
               ,
               need
               no
               alteration
               ,
               which
               suppose
               may
               be
               19
               foot
               .
            
             
               4.
               
               The
               Gable-end
               ,
               or
               Garret-story
               ,
               if
               ●ny
               be
               ,
               being
               but
               one
               brick
               thick
               ;
               you
               must
               take
               away
               one
               third
               part
               to
               bring
               it
               to
               a
               brick
               and
               a
               half
               .
               Also
               if
               it
               be
               a
               Gable-end
               ,
               Note
               ,
               it
               is
               a
               Triangle
               ,
               and
               you
               must
               measure
               but
               half
               the
               height
               ,
               and
               the
               whole
               breadth
               ,
               to
               find
               the
               Content
               ;
               which
               here
               may
               be
               5
               foot
               .
            
             
             
               
                 
                   
                     The
                     Cellar
                     Story
                     ,
                  
                   
                     16
                     —
                     8
                  
                
                 
                   
                     Two
                     next
                     Stories
                     ,
                  
                   
                     29
                     —
                     4
                  
                
                 
                   
                     Two
                     next
                     Stories
                     ,
                  
                   
                     19
                     —
                     0
                  
                
                 
                   
                     The
                     Garret
                     ,
                  
                   
                     5
                     —
                     0
                  
                
                 
                   
                      
                  
                   
                     70
                     —
                     0
                  
                
              
            
             
               5.
               
               Add
               all
               these
               sums
               of
               feet
               high
               together
               ,
               and
               they
               make
               70
               ;
               then
               measure
               the
               breadth
               ,
               which
               is
               common
               to
               every
               Room
               ,
               the
               out-side
               going
               upright
               ,
               which
               in
               a
               double
               House
               may
               be
               36
               or
               40
               foot
               .
            
             
               6.
               
               Then
               having
               gotten
               the
               Dimentions
               right
               by
               the
               Line
               of
               Numbers
               ,
               Say
               ,
            
             
               As
               272
               ¼
               (
               the
               feet
               in
               one
               Rod
               )
               is
               to
               40
               foot
               ,
               the
               breadth
               of
               the
               House
               ;
               so
               is
               70
               foot
               ,
               the
               whole
               height
               of
               every
               several
               Story
               ,
               (
               reduced
               )
               to
               10
               Rod
               and
               29
               parts
               ;
               which
               29
               parts
               you
               may
               call
               a
               quarter
               of
               a
               Rod
               ,
               and
               10
               foot
               and
               a
               half
               .
            
             
               
                 The
                 reason
                 whereof
                 is
                 apparent
                 thus
                 :
              
            
             
               As
               100
               is
               to
               272
               ¼
               ;
               so
               is
               29
               to
               near
               79
               ;
               of
               which
               79-68
               is
               a
               quarter
               of
               a
               Rod
               ,
               or
               25
               of
               100
               is
               a
               quarter
               likewise
               ,
               which
               by
               the
               Line
               of
               Numbers
               is
               apparently
               seen
               ;
               then
               every
               10th
               part
               is
               2
               foot
               ,
               and
               72
               of
               a
               hundred
               ,
               which
               is
               near
               two
               and
               three
               quarters
               ;
               so
               that
               here
               25
               being
               a
               quarter
               of
               a
               
               Rod
               ,
               there
               is
               4
               hundred
               parts
               more
               in
               29
               :
               
                 Then
                 thus
              
               ;
               the
               double
               of
               4
               is
               8
               ,
               or
               ,
               twice
               4
               is
               8
               ,
               and
               four
               times
               three
               quarters
               is
               three
               foot
               more
               ;
               of
               which
               you
               must
               abate
               somewhat
               (
               because
               72
               ¼
               is
               not
               75
               ,
               which
               is
               just
               three
               quarters
               )
               and
               all
               put
               together
               ,
               make
               ten
               rod
               ,
               one
               quarter
               ,
               ten
               foot
               and
               a
               half
               :
               for
               if
               you
               shall
               divide
               the
               Product
               of
               40
               ,
               multiplied
               by
               70
               ,
               which
               is
               2800
               by
               272
               ¼
               ,
               you
               shall
               find
               the
               Quotient
               to
               be
               10
               rod
               ,
               78
               ½
               ,
               which
               is
               ,
               as
               before
               ,
               10
               rod
               ,
               1
               quarter
               and
               10
               foot
               and
               a
               half
               .
            
             
               But
               note
               also
               by
               the
               way
               ,
               That
               when
               you
               come
               to
               take
               out
               the
               deductions
               for
               the
               doors
               and
               windows
               ,
               if
               any
               happen
               in
               a
               Wall
               of
               two
               Bricks
               and
               a
               half
               ,
               or
               in
               two
               Bricks
               ;
               you
               must
               add
               two
               thirds
               ,
               or
               one
               third
               more
               to
               the
               length
               or
               bredth
               one
               way
               ;
               and
               then
               casting
               them
               up
               severally
               ,
               when
               they
               be
               of
               several
               lengths
               or
               breadths
               ,
               you
               shall
               do
               no
               wrong
               to
               the
               Work-master
               nor
               Work-man
               :
               For
               true
               Arithmetick
               and
               Geometry
               will
               lie
               for
               no
               man
               ,
               or
               use
               any
               kind
               of
               partiality
               .
            
             
               This
               I
               conceive
               is
               as
               near
               a
               way
               ,
               as
               any
               such
               business
               can
               be
               performed
               .
               But
               if
               you
               will
               measure
               every
               Story
               severally
               ,
               taking
               account
               of
               each
               Story
               severally
               in
               their
               thicknesses
               ;
               then
               ,
               after
               it
               is
               cast
               up
               ,
               the
               best
               way
               ,
               by
               the
               Rule
               ,
               to
               reduce
               it
               ,
               is
               thus
               ;
            
             
             
               As
               3
               half
               bricks
               ,
               for
               a
               brick
               and
               a
               half
               ,
               is
               to
               any
               other
               number
               of
               half
               bricks
               thick
               ,
               over
               or
               under
               3
               ;
               So
               is
               the
               Content
               at
               that
               rate
               accordingly
               ,
               to
               his
               Content
               ,
               at
               a
               brick
               and
               a
               half
               required
               .
            
             
               Example
               .
            
             
               1269
               foot
               at
               5
               half
               bricks
               thick
               is
               2115
               ,
               for
               two
               thirds
               of
               1269
               ,
               which
               is
               846
               ,
               added
               to
               1269
               ,
               makes
               2115
               ;
               For
               the
               Extent
               on
               the
               Line
               of
               Numbers
               ,
               from
               3
               to
               5
               ,
               shall
               reach
               the
               same
               way
               from
               1269
               to
               2115
               ,
               the
               Number
               required
               to
               be
               found
               out
               .
            
             
               
                 Otherwise
                 thus
              
               ;
            
             
               To
               bring
               any
               kind
               of
               thickness
               ,
               to
               one
               brick
               and
               a
               half
               thick
               ,
               at
               one
               operation
               ,
               by
               the
               Line
               of
               Numbers
               .
            
             
               For
               this
               purpose
               ,
               you
               must
               use
               several
               Points
               ,
               as
               so
               many
               gage
               Points
               ,
               as
               in
               the
               short
               Table
               following
               doth
               appear
               .
            
             
               
                 
                   
                     For
                     half
                     a
                     brick
                     ,
                     use
                  
                   
                     3-00000
                  
                
                 
                   
                     For
                     1
                     brick
                     ,
                  
                   
                     1-50000
                  
                
                 
                   
                     For
                     1
                     brick
                     &
                     a
                     half
                     ,
                     use
                  
                   
                     1-0000
                  
                
                 
                   
                     For
                     2
                     bricks
                     ,
                  
                   
                     0-7500
                  
                
                 
                   
                     For
                     2
                     bricks
                     &
                     a
                     half
                     ,
                     use
                  
                   
                     0-6000
                  
                
                 
                   
                     For
                     3
                     bricks
                     ,
                  
                   
                     0-5000
                  
                
                 
                   
                     For
                     3
                     bricks
                     &
                     a
                     half
                     ,
                  
                   
                     0-4285
                  
                
                 
                   
                     For
                     4
                     bricks
                     ,
                  
                   
                     0-3750
                  
                
                 
                   
                     For
                     4
                     bricks
                     and
                     a
                     half
                     ,
                  
                   
                     0-3333
                  
                
                 
                   
                     For
                     5
                     bricks
                     ,
                  
                   
                     0-3000
                  
                
                 
                   
                     
                     For
                     5
                     bricks
                     and
                     a
                     half
                     ,
                  
                   
                     0-2727
                  
                
                 
                   
                     For
                     6
                     bricks
                     ,
                     &c.
                     
                       ad
                       infinitum
                    
                     .
                  
                   
                     0-2500
                  
                
              
            
             
               
                 Example
                 at
                 the
                 6
                 ordinary
                 thicknesses
                 .
              
            
             
               Let
               a
               Wall
               be
               30
               foot
               long
               ,
               and
               10
               foot
               high
               ;
               and
               let
               it
               be
               supposed
               of
               any
               of
               these
               thicknesses
               following
               ,
               from
               half
               a
               bricks
               length
               ,
               to
               three
               bricks
               length
               in
               thickness
               ;
               then
               thus
               in
               order
               ,
               increasing
               ,
               &c.
               
            
             
               
                 First
                 ,
                 at
                 half
                 a
                 Foot.
                 
              
            
             
               For
               ½
               brick
               .
               As
               3
               to
               30
               ;
               so
               is
               10
               to
               100
               foot
               ,
               at
               1
               ½
               .
            
             
               For
               1
               brick
               .
               As
               15
               to
               30
               ;
               so
               is
               10
               to
               200
               foot
               ,
               at
               1
               brick
               .
            
             
               For
               1
               ½
               thick
               .
               As
               10
               to
               30
               ;
               so
               is
               10
               to
               300
               foot
               ,
               at
               1
               ½
               .
            
             
               For
               2
               bricks
               .
               As
               0-75
               to
               30
               ;
               so
               is
               10
               to
               400
               foot
               ,
               at
               1
               ½
               .
            
             
               For
               2
               ½
               thick
               .
               As
               0-60
               to
               30
               ;
               so
               is
               10
               to
               500
               foot
               ,
               at
               ½
               .
            
             
               For
               3
               bricks
               .
               As
               0-50
               to
               30
               ;
               so
               is
               10
               to
               600
               foot
               ,
               at
               1
               ½
               .
            
             
               For
               3
               ½
               thick
               .
               As
               0-4285
               to
               30
               ;
               so
               is
               10
               to
               700
               foot
               ,
               at
               1
               ½
               .
            
             
               For
               4
               bricks
               .
               As
               0-3750
               to
               30
               ;
               so
               is
               10
               to
               800
               foot
               ,
               at
               1
               ½
               .
            
             
             
               And
               so
               for
               any
               other
               thickness
               ,
               as
               far
               as
               you
               please
               ;
               which
               Points
               are
               found
               thus
               ;
            
             
               The
               Exten●
               ▪
               from
               the
               number
               of
               bricks
               ,
               any
               Wall
               is
               thick
               to
               15
               (
               or
               1
               and
               ½
               )
               shall
               reach
               the
               same
               way
               from
               10
               ,
               or
               1
               ,
               to
               the
               Gage-Point
               required
               for
               that
               Wall
               ,
               or
               Walls
               of
               that
               thickness
               .
            
             
               Example
               .
            
             
               As
               2
               to
               1
               ½
               ;
               so
               is
               10
               ,
               to
               0-750
               ,
               for
               2
               bricks
               thick
               ,
               &c.
               
            
             
               Lastly
               ,
               having
               the
               Number
               of
               Feet
               in
               the
               whole
               work
               ,
               to
               find
               how
               many
               Rods
               there
               is
               .
               Say
               ,
            
             
               If
               272
               ¼
               ,
               be
               one
               Rod
               ;
               what
               shall
               any
               other
               Number
               of
               Feet
               make
               in
               Rods
               ?
            
             
               The
               Extent
               of
               the
               Compasses
               from
               172
               ¼
               ,
               to
               1
               ,
               shall
               reach
               the
               same
               way
               ,
               from
               the
               Number
               of
               Feet
               ,
               to
               the
               Number
               of
               Rods
               ,
               and
               hundred
               Parts
               ,
               or
               Rods
               ,
               and
               Quarters
               ,
               and
               Feet
               ;
               as
               by
               the
               6th
               ,
               last
               mentioned
               .
            
             
               
                 Example
                 .
                 In
                 5269
                 Feet
                 ,
                 how
                 many
                 Rods
                 ?
              
            
             
               The
               Extent
               from
               272
               ¼
               ,
               to
               1
               ,
               shall
               reach
               the
               same
               way
               ,
               from
               5269
               ,
               to
               19
               Rod
               ,
               and
               36
               parts
               of
               a
               100
               ;
               or
               ,
               19
               Rod
               1
               quarter
               ,
               and
               29
               foot
               ,
               and
               a
               quarter
               of
               a
               foot
               .
               The
               19
               Rod
               ,
               and
               a
               quarter
               ,
               is
               seen
               
               plainly
               on
               the
               Rule
               ;
               and
               25
               being
               a
               quarter
               ,
               36
               is
               11
               parts
               more
               ;
               for
               which
               11
               parts
               more
               ,
               I
               say
               ,
               2
               times
               11
               is
               22
               foot
               ,
               and
               11
               ,
               3
               quarters
               of
               a
               foot
               is
               near
               8
               foot
               ,
               which
               put
               together
               ,
               makes
               29
               foot
               ,
               as
               before
               :
               Or
               ,
               as
               the
               Compasses
               stand
               ,
               turn
               them
               the
               contrary
               way
               ,
               from
               the
               Decimal
               parts
               ,
               above
               the
               even
               quarter
               ,
               and
               it
               shall
               reach
               to
               the
               odd
               feet
               above
               the
               quarter
               required
               .
            
             
               Example
               .
            
             
               The
               Extent
               from
               272
               ¼
               ,
               to
               100
               ;
               or
               1
               ,
               shall
               reach
               the
               contrary
               way
               from
               10
               ½
               ,
               to
               29
               foot
               ,
               the
               feet
               above
               ¼
               of
               a
               Rod.
               
            
             
               8.
               
               Observe
               ,
               That
               the
               Tyling
               ,
               the
               Roof
               ,
               the
               Floors
               ,
               and
               Partitions
               ,
               are
               measured
               by
               the
               Square
               ;
               which
               is
               10
               foot
               Square
               every
               way
               ,
               or
               100
               foot
               in
               Area
               .
               The
               Chimneys
               are
               usually
               done
               by
               a
               certain
               rate
               for
               a
               Chimney
               ;
               or
               if
               to
               be
               measured
               ,
               thus
               are
               the
               height
               and
               breadths
               taken
               ,
               &c.
               
            
             
               If
               a
               Chimney
               stand
               singly
               and
               alone
               ,
               not
               leaning
               against
               ,
               or
               in
               a
               Wall
               ,
               the
               usual
               way
               is
               to
               girt
               it
               about
               ;
               and
               if
               the
               Jaumes
               are
               but
               a
               brick
               thick
               ,
               and
               wrought
               upright
               over
               the
               Mantle-tree
               to
               the
               Floor
               ;
               then
               I
               say
               ,
               girt
               it
               about
               for
               a
               length
               ,
               and
               the
               height
               of
               the
               Story
               is
               the
               breadth
               ,
               at
               a
               brick
               thick
               ,
               because
               of
               the
               gathering
               together
               ,
               to
               make
               room
               for
               the
               next
               Hearth
               above
               in
               the
               next
               Story
               .
            
             
             
               But
               if
               the
               Chimney-back
               be
               a
               Party-Wall
               ,
               the
               Wall
               being
               first
               measured
               ,
               then
               the
               brest
               and
               the
               depth
               of
               the
               two
               Jaumes
               is
               one
               side
               ,
               and
               the
               height
               of
               the
               Story
               another
               side
               ,
               to
               be
               multiplied
               together
               ,
               at
               a
               brick
               and
               a
               half
               thick
               ,
               or
               a
               brick
               thick
               ,
               according
               as
               the
               Jaumes
               be
               ,
               and
               nothing
               to
               be
               abated
               for
               the
               want
               between
               the
               Hearth
               and
               the
               Mantle-tree
               ,
               because
               of
               the
               Wit
               hs
               and
               thickning
               for
               the
               next
               Hearth
               .
            
             
               
                 For
                 measuring
                 the
                 Shafts
                 of
                 the
                 Chimneys
                 .
              
            
             
               Girt
               with
               a
               Line
               ,
               round
               about
               the
               least
               place
               of
               them
               ,
               for
               one
               side
               ;
               and
               the
               height
               for
               the
               other
               side
               ,
               at
               a
               brick
               thick
               ,
               in
               consideration
               of
               the
               Wit
               hs
               ,
               Pargitting
               ,
               and
               Scaffolds
               .
            
             
               In
               measuring
               of
               Ceiling
               a
               foot
               broad
               ,
               and
               the
               length
               of
               the
               Vallies
               is
               alwayes
               to
               be
               allowed
               ,
               more
               than
               the
               whole
               Roof
               ;
               Also
               the
               length
               of
               the
               Rafter
               feet
               ,
               above
               or
               beyond
               the
               Roof
               .
            
             
               When
               Rafters
               have
               their
               usual
               pitch
               ,
               which
               is
               ,
               when
               the
               breadth
               of
               the
               House
               is
               12
               foot
               ,
               the
               Rafter
               is
               9
               foot
               long
               ,
               which
               is
               3
               quarters
               of
               the
               Floors
               breadth
               ,
               be
               it
               more
               or
               less
               ;
               then
               ,
               I
               say
               ,
               that
               the
               Content
               of
               one
               Floor
               ,
               and
               half
               so
               much
               ,
               is
               the
               Area
               of
               the
               whole
               Roof
               in
               Squares
               ;
               to
               which
               is
               to
               
               be
               added
               ,
               the
               Vallies
               and
               Rafter-Feet
               ,
               or
               Eves
               ,
               in
               Tileing
               .
            
             
               And
               also
               a
               Deduction
               for
               Chimney-room
               ,
               and
               Gutters
               ,
               if
               any
               be
               .
            
             
               
                 Which
                 work
                 by
                 the
                 Line
                 of
                 Numbers
                 ,
                 is
                 done
                 at
                 one
                 Operation
                 ,
                 thus
              
               ;
            
             
               As
               6666
               ,
               is
               to
               the
               length
               of
               the
               House
               ;
            
             
               So
               is
               the
               breadth
               to
               the
               Content
               in
               the
               Roof
               .
            
             
               Example
               .
            
             
               A
               House
               30
               foot
               long
               ,
               and
               20
               foot
               broad
               ,
               is
               900
               foot
               ,
               or
               9
               square
               .
            
             
               For
               the
               Extent
               ,
               from
               6666
               ,
               to
               20
               ,
               shall
               reach
               the
               same
               way
               from
               30
               to
               900.
               
            
             
               Also
               in
               measuring
               of
               the
               Roof
               ,
               as
               to
               Carpenters
               work
               ,
               by
               the
               Square
               ,
               there
               is
               to
               be
               allowance
               for
               those
               Rafters
               in
               the
               Dormers
               ,
               and
               Gable-ends
               ,
               on
               which
               no
               Tiles
               are
               laid
               ,
               as
               over-work
               for
               a
               particular
               use
               and
               convenience
               ,
               more
               than
               need
               be
               in
               a
               bare
               Covering
               ,
               or
               Roof
               .
            
             
               Also
               in
               measuring
               of
               Plasterers
               work
               in
               Partitions
               and
               Walls
               ;
               the
               Timbers
               and
               Quarters
               ,
               are
               not
               to
               be
               deducted
               out
               of
               the
               rendring
               for
               Work
               only
               ,
               except
               when
               the
               Workman
               finds
               the
               Work
               and
               Stuff
               also
               ,
               then
               substract
               a
               6th
               part
               for
               the
               Quarters
               in
               the
               rendring
               Work
               :
               But
               in
               Ceilings
               ,
               the
               
               Summers
               which
               are
               seen
               ,
               are
               alwayes
               abated
               ▪
               and
               Doors
               and
               Windows
               also
               ,
               unless
               by
               a
               due
               considerate
               (
               or
               an
               unconsiderate
               )
               bargain
               of
               running
               measure
               .
            
             
               Thus
               you
               have
               a
               brief
               account
               of
               the
               usual
               order
               ,
               used
               among
               Workmen
               ,
               in
               taking
               the
               Dimentions
               of
               a
               House
               ,
               viz.
               Brick-work
               by
               the
               Rod
               ;
               Tileing
               and
               Carpenters-work
               by
               the
               Square
               ;
               Chimneys
               usually
               by
               the
               Fire
               ;
               And
               Plasterers
               and
               Painters-work
               by
               the
               Yard
               ;
               Glasiers
               ,
               by
               the
               Foot.
               
            
             
               There
               are
               many
               other
               things
               to
               be
               taken
               notice
               of
               in
               the
               Carpenters
               Bill
               ,
               as
               Lintels
               ,
               Mantle-trees
               ,
               and
               Tassels
               ;
               Luthern
               Lights
               ,
               and
               other
               Lights
               ,
               both
               Architrave
               and
               Plain
               Lights
               ,
               Sky-lights
               ,
               or
               Cubiloes
               ,
               Modillean
               Cornish
               ,
               and
               guttering
               Penthouse
               Cornish
               ,
               Timber-Front-Story
               ,
               Cellar-doors
               ,
               and
               Door-cases
               ;
               the
               Plank
               and
               Curb
               at
               the
               Cellar-stairs
               ,
               Dogleg-stairs
               ,
               and
               Open-Newel-stairs
               ,
               Canted-stairs
               ,
               counted
               either
               by
               the
               step
               or
               pair
               ;
               together
               with
               the
               half
               Spaces
               on
               the
               Corners
               of
               the
               open
               Newel-stairs
               ,
               the
               Rayles
               and
               Ballasters
               ,
               small
               and
               great
               Cornish
               ,
               Outside-work
               and
               Partitions
               ,
               Ceiling
               Joysts
               ,
               and
               the
               Ashlering
               ,
               Boarded
               Partitions
               ,
               and
               Chequer-work
               ;
               back-Doors
               ,
               and
               Door-cases
               ;
               Window-boards
               ,
               and
               Wall-timber
               ;
               Planks
               in
               the
               Foundation
               ,
               Palcing
               ,
               Penthouse-floors
               ,
               and
               
               Penthouse-roof
               ,
               furring
               the
               Platform
               ,
               Centerings
               for
               the
               Chimney
               ,
               Trimmers
               ,
               Girders-ends
               ,
               Ends
               of
               Brest-summers
               ,
               and
               Plate
               ;
               and
               more
               the
               like
               ,
               which
               will
               come
               in
               Accompt
               to
               be
               remembred
               and
               set
               down
               according
               as
               the
               Building
               is
               .
            
             
               Also
               ,
               with
               due
               allowance
               into
               the
               Wall
               that
               way
               the
               ends
               of
               the
               Joysts
               are
               entred
               or
               laid
               in
               the
               Wall
               ,
               
                 as
                 thus
              
               ;
            
             
               If
               it
               be
               Framing
               Work
               is
               only
               measured
               ,
               then
               9
               Inches
               ought
               to
               be
               allowed
               into
               each
               Wall
               ,
               that
               way
               the
               Joysts
               ends
               are
               laid
               ;
               because
               every
               Joyst
               ,
               if
               well
               laid
               ,
               should
               have
               9
               inches
               ,
               at
               least
               ,
               hold
               on
               the
               Wall.
               
            
             
               But
               if
               it
               be
               Timber
               ,
               and
               Boarding
               ,
               both
               to
               be
               measured
               ,
               then
               6
               inches
               only
               is
               a
               competent
               allowance
               ;
               because
               the
               Timber
               is
               usually
               vallued
               at
               one
               third
               part
               more
               than
               the
               Boarding
               is
               .
            
             
               Also
               ,
               As
               the
               Workman
               doth
               think
               on
               this
               ,
               the
               Work-master
               may
               not
               forget
               to
               deduct
               for
               Stairs
               ,
               and
               Chimneys
               also
               ,
               where
               Work
               and
               Stuff
               are
               both
               measured
               ;
               though
               for
               Work
               only
               it
               may
               be
               very
               well
               allowed
               ,
               unless
               the
               better
               Price
               make
               an
               allowance
               for
               it
               .
            
             
               
                 Note
                 also
              
               ,
               That
               by
               the
               Line
               of
               Numbers
               ,
               you
               may
               readily
               find
               the
               length
               of
               the
               Hips
               and
               Rafters
               ,
               in
               a
               Roof
               of
               any
               largeness
               ,
               
               at
               true
               pitch
               ,
               by
               this
               following
               Proportion
               and
               Table
               .
            
             
               
                 The
                 Breadth
                 of
                 the
                 House
                 being
                 40
                 Feet
                 ,
                 and
                 the
                 Ends
                 Square
                 ,
                 the
                 Length
                 and
                 Angles
                 are
                 ,
                 as
                 in
                 the
                 Table
                 ,
                 at
                 the
                 usual
                 tru●
                 pitch
                 .
              
            
             
               
                 
                   
                      
                  
                   
                     f●et
                     .
                  
                   
                     100
                     
                       par
                       ▪
                    
                  
                
                 
                   
                     Whole
                     breadth
                  
                   
                     40
                  
                   
                     00
                  
                
                 
                   
                     Half
                     breadth
                  
                   
                     20
                  
                   
                     00
                  
                
                 
                   
                     Rafter
                  
                   
                     30
                  
                   
                     00
                  
                
                 
                   
                     Hip-Rafter
                  
                   
                     36
                  
                   
                     00
                  
                
                 
                   
                     Diagonal
                     Line
                  
                   
                     56
                  
                   
                     57
                  
                
                 
                   
                     Half-Diagonal
                  
                   
                     28
                  
                   
                     28
                  
                
                 
                   
                     Perpendiculer
                  
                   
                     22
                  
                   
                     36
                  
                
              
               
                 
                   
                      
                  
                   
                      
                  
                   
                     deg
                     .
                  
                   
                     min.
                     
                  
                
                 
                   
                     Hip
                     Angles
                  
                   
                     at
                     Foot
                  
                   
                     38
                  
                   
                     22
                  
                
                 
                   
                     at
                     Top
                  
                   
                     51
                  
                   
                     38
                  
                
                 
                   
                     on
                     the
                     Outside
                  
                   
                     116
                  
                   
                     12
                  
                
                 
                   
                     Rafter
                     Angles
                  
                   
                     at
                     Foot
                  
                   
                     48
                  
                   
                     10
                  
                
                 
                   
                     at
                     Top
                  
                   
                     41
                  
                   
                     50
                  
                
              
            
             
               
                 For
                 any
                 other
                 House
                 ,
                 by
                 the
                 Numbers
                 thus
                 :
                 as
                 suppose
                 18
                 Foot
                 broad
                 .
              
            
             
               The
               Extent
               of
               the
               Compasses
               from
               40
               ,
               the
               breadth
               in
               the
               Table
               ,
               to
               18
               the
               breadth
               given
               ,
               shall
               reach
               the
               same
               way
               from
               30
               ,
               the
               Rafter
               in
               the
               Table
               ,
               to
               13-50
               ,
               the
               Rafter
               
               required
               .
               And
               from
               36
               ,
               the
               Hip
               in
               the
               Table
               ,
               to
               16-22
               ,
               the
               Hip
               required
               .
               And
               from
               22-36
               ,
               the
               Perpendiculer
               in
               the
               Table
               ,
               to
               10-06
               ,
               the
               Perpendiculer
               required
               .
               And
               from
               ●6-57
               ,
               the
               Diagonal
               in
               the
               Table
               ,
               to
               ●5-48
               ,
               the
               Diagonal
               required
               .
               The
               Angles
               are
               alwayes
               the
               same
               in
               all
               Rooss
               ,
               small
               or
               great
               ,
               as
               in
               the
               Table
               ,
               being
               Square
               and
               true
               pitch
               .
            
             
               If
               you
               would
               have
               Directions
               for
               Bevel
               or
               Taper
               Frames
               ,
               to
               find
               the
               Lengths
               and
               Angles
               of
               Ra●ters
               and
               Hips
               ,
               you
               may
               have
               it
               at
               large
               ,
               in
               an
               Appendix
               to
               the
               
                 Mirrour
                 of
                 Architecture
              
               ;
               or
               ,
               
                 Vincent
                 Stamo●●i
              
               ,
               Printed
               for
               
                 William
                 Fisher
              
               ,
               at
               the
               Postern-Gate
               ,
               1669.
               
            
             
               By
               which
               Directions
               ,
               and
               the
               Sector
               ,
               you
               may
               find
               any
               thing
               that
               is
               there
               set
               down
               .
               As
               also
               ,
               by
               the
               
                 Trianguler
                 Quadrant
              
               ,
               Thred
               and
               Compasses
               .
            
             
               
                 Note
                 also
              
               ,
               That
               having
               Inches
               and
               Foot-measure
               together
               ,
               you
               may
               presently
               ,
               by
               inspection
               ,
               find
               the
               price
               of
               one
               Foot
               ,
               having
               the
               price
               of
               the
               Square
               ,
               and
               the
               contrary
               .
               Also
               ,
               having
               the
               12
               Inches
               on
               the
               other
               Foot
               ,
               divided
               into
               85
               parts
               (
               near
               )
               ,
               and
               figured
               at
               every
               8
               with
               1
               ,
               2
               ,
               3
               ,
               4
               ,
               5
               ,
               6
               ,
               7
               ,
               8
               ,
               9
               ,
               10.
               shall
               represent
               pence
               and
               half
               farthings
               ;
               then
               at
               any
               price
               the
               Rod
               ,
               you
               have
               the
               price
               of
               one
               Foot
               ,
               &
               the
               contrary
               .
            
             
             
             
             
             
               
                 As
                 thus
              
               ;
            
             
               Let
               every
               Inch
               ,
               represent
               one
               pound
               ;
               〈◊〉
               ●very
               8th
               part
               ,
               2
               shillings
               and
               6
               pence
               ;
               or
               every
               10th
               part
               ,
               2
               shillings
               ;
               because
               8
               half-crowns
               ,
               or
               10
               two
               shillings
               ,
               is
               20
               shillings
               .
            
             
               Example
               .
            
             
               Right
               against
               6
               Inches
               and
               a
               half
               ,
               for
               6
               l.
               ●0
               s.
               on
               this
               other
               Line
               ,
               I
               find
               ●
               pence
               3
               farthings
               ,
               the
               price
               of
               one
               Foot
               ,
               at
               6
               l.
               10
               
                 s.
                 per
              
               Rod
               :
               And
               at
               7
               farthings
               per
               Foot
               ,
               I
               find
               near
               40
               shillings
               ,
               or
               2
               pound
               per
               Rod.
               
            
             
               Also
               ,
               at
               40
               shillings
               per
               Square
               ,
               found
               at
               40
               ,
               on
               Foot-measure
               ,
               is
               4
               pence
               3
               farthings
               1
               /
               ●
               per
               Foot
               ,
               found
               just
               against
               it
               on
               the
               Inches
               .
            
             
          
        
         
           
           
             CHAP.
             VIII
             .
             The
             use
             of
             the
             Line
             of
             Numbers
             ,
             in
             measuring
             of
             Land
             by
             Perches
             and
             Acres
             .
          
           
             
               Problem
               I.
               At
               any
               length
               of
               the
               Land
               ,
               to
               find
               the
               breadth
               of
               the
               Acre
               .
            
             
               IN
               the
               Answering
               of
               this
               Question
               ,
               it
               is
               not
               amiss
               ,
               but
               very
               needful
               to
               premise
               ,
               how
               many
               Square
               Inches
               ,
               Feet
               ▪
               Yards
               ,
               Perches
               ;
               or
               Chains
               I
               mean
               a
               Chain
               of
               ●6
               ●oot
               long
               )
               is
               contained
               in
               a
               Square
               Acre
               of
               Land
               ;
               for
               which
               purpose
               ,
               have
               recourse
               to
               the
               Table
               annexed
               ,
               which
               is
               drawn
               with
               great
               care
               and
               exactness
               for
               that
               purpose
               .
            
             
               By
               which
               Table
               you
               may
               perceive
               ,
               That
               6272640
               Square
               inches
               are
               contained
               in
               one
               Square
               Acre
               .
            
             
             
               And
               (
               100000
               ,
               or
               )
               one
               hundred
               thousand
               Square
               Links
               of
               a
               4
               Pole
               Chain
               ,
               make
               a
               Square
               Acre
               .
            
             
               And
               43560
               Square
               Feet
               ,
               make
               a
               Square
               Acre
               .
            
             
               And
               4840
               Square
               Yards
               ,
               make
               a
               Square
               Acre
               .
            
             
               And
               1742
               ,
               4
               Square
               Paces
               ,
               make
               a
               Square
               Acre
               .
            
             
               And
               160
               Square
               Perch
               ,
               make
               a
               Square
               Acre
               .
            
             
               And
               10
               Square
               4
               Pole
               Chain
               ,
               make
               one
               Acre
               .
               As
               in
               the
               Table
               you
               may
               see
               .
            
             
               And
               3097
               1
               /
               ●
               Square
               Ells
               ,
               make
               one
               Acre
               of
               Land
               ,
               Statute
               measure
               .
            
             
             
               
                 
                   The
                   Table
                   .
                
                 
                   
                      
                  
                   
                     Inch
                  
                   
                     Links
                  
                   
                     Feet
                  
                   
                     Yards
                  
                   
                     Pace
                  
                   
                     Perch
                  
                   
                     Chain
                  
                   
                     Acre
                  
                   
                     Mile
                  
                
                 
                   
                     Inch
                  
                   
                     1
                  
                   
                     7.92
                  
                   
                     12
                  
                   
                     36
                  
                   
                     60
                  
                   
                     198
                  
                   
                     792
                  
                   
                     7920
                  
                   
                     63360
                  
                
                 
                   
                     Link
                  
                   
                     62.720
                  
                   
                     1
                  
                   
                     1.515
                  
                   
                     4.56
                  
                   
                     7.575
                  
                   
                     25
                  
                   
                     100
                  
                   
                     1000
                  
                   
                     8000
                  
                
                 
                   
                     Feet
                  
                   
                     144
                  
                   
                     2.295
                  
                   
                     1
                  
                   
                     3
                  
                   
                     5
                  
                   
                     16.5
                  
                   
                     66
                  
                   
                     660
                  
                   
                     5280
                  
                
                 
                   
                     Yard
                  
                   
                     1296
                  
                   
                     20.755
                  
                   
                     9
                  
                   
                     1
                  
                   
                     1.66
                  
                   
                     550
                  
                   
                     22
                  
                   
                     220
                  
                   
                     1760
                  
                
                 
                   
                     Pace
                  
                   
                     3600
                  
                   
                     57.381
                  
                   
                     25
                  
                   
                     2.778
                  
                   
                     1
                  
                   
                     3.3
                  
                   
                     13.2
                  
                   
                     132
                  
                   
                     1056
                  
                
                 
                   
                     Perch
                  
                   
                     39204
                  
                   
                     625
                  
                   
                     272.25
                  
                   
                     30
                     25
                  
                   
                     10.89
                  
                   
                     1
                  
                   
                     4
                  
                   
                     40
                  
                   
                     320
                  
                
                 
                   
                     Chain
                  
                   
                     627264
                  
                   
                     10000
                  
                   
                     4356
                  
                   
                     484
                  
                   
                     174.24
                  
                   
                     16
                  
                   
                     1
                  
                   
                     10
                  
                   
                     80
                  
                
                 
                   
                     Acre
                  
                   
                     6272640
                  
                   
                     100000
                  
                   
                     43560
                  
                   
                     4840
                  
                   
                     1742.4
                  
                   
                     160
                  
                   
                     10
                  
                   
                     1
                  
                   
                     8
                  
                
                 
                   
                     Mile
                  
                   
                     4014489600
                  
                   
                     64000000
                  
                   
                     27878400
                  
                   
                     3097600
                  
                   
                     1115136
                  
                   
                     102400
                  
                   
                     6400
                  
                   
                     640
                  
                   
                     1
                  
                
                 
                   
                     Square
                  
                   
                     Inches
                  
                   
                     Links
                  
                   
                     Feet
                  
                   
                     Yards
                  
                   
                     Pace
                  
                   
                     Perch
                  
                   
                     Chain
                  
                   
                     Acre
                  
                   
                     Mile
                  
                
              
            
             
             
               Then
               as
               the
               length
               of
               the
               Land
               given
               in
               Feet
               ,
               Yards
               ,
               Paces
               ,
               Perches
               ,
               or
               Chains
               ,
               is
               to
               the
               number
               of
               Square
               Feet
               ,
               Yards
               ,
               Paces
               ,
               Perches
               ,
               or
               Chains
               in
               a
               Square
               Acre
               ;
               So
               is
               1
               to
               the
               breadth
               of
               the
               Land
               (
               in
               that
               measure
               the
               length
               was
               given
               )
               to
               make
               a
               Square
               Acre
               :
               See
               the
               Examples
               of
               all
               these
               measures
               in
               their
               order
               ,
               viz.
               of
               Feet
               ,
               Yards
               ,
               Paces
               ,
               Perches
               ,
               and
               Chains
               .
            
             
               Suppose
               a
               piece
               of
               Land
               be
               660
               Feet
               long
               ,
               or
               220
               Yards
               ,
               or
               132
               Paces
               ,
               or
               40
               Perches
               ,
               or
               10
               Chains
               in
               length
               ▪
               which
               several
               measures
               are
               all
               of
               the
               same
               quantity
               ;
               I
               would
               know
               how
               much
               in
               breadth
               I
               must
               have
               to
               make
               a
               Square
               Acre
               ?
            
             
               Extend
               the
               Compasses
               from
               the
               length
               given
               ,
               viz.
               660
               Feet
               ,
               or
               220
               Yards
               ,
               or
               132
               Paces
               ,
               or
               40
               Perches
               ,
               or
               10
               Chains
               ;
               to
               43560
               ,
               for
               Feet
               ;
               or
               to
               4840
               ,
               for
               Yards
               ;
               or
               to
               1742
               ,
               for
               Paces
               ;
               or
               to
               160
               ,
               for
               Perches
               ;
               or
               to
               10
               ,
               for
               Chains
               ;
               To
               the
               Number
               in
               the
               Table
               for
               that
               measure
               in
               a
               Square
               Acre
               ;
               the
               same
               Extent
               applyed
               the
               same
               way
               from
               1
               ,
               shall
               reach
               to
               the
               Feet
               ,
               Yards
               ,
               Paces
               ,
               Perches
               ,
               or
               Chains
               required
               .
            
             
               
                 Note
                 the
                 Work.
              
               
            
             
               1.
               
               As
               660
               ,
               to
               43563
               ,
               the
               Feet
               in
               a
               Square
               Acre
               ;
            
             
             
               So
               is
               1
               ,
               to
               66
               ,
               the
               breadth
               in
               Feet
               required
               .
            
             
               2.
               
               As
               220
               ,
               the
               length
               in
               Yards
               ,
               to
               4840
               ,
               the
               Square
               Yards
               in
               a
               Square
               Acre
               ;
            
             
               So
               is
               1
               ,
               to
               22
               ,
               the
               breadth
               in
               Yards
               required
               .
            
             
               3.
               
               As
               132
               ,
               the
               length
               in
               Paces
               ,
               to
               1742
               ;
            
             
               So
               is
               1
               ,
               to
               13-2
               ,
               the
               breadth
               in
               Paces
               sought
               .
            
             
               4.
               
               As
               40
               ,
               the
               length
               in
               Perches
               ,
               to
               1●0
               ;
               So
               is
               1
               ,
               to
               4
               ,
               the
               breadth
               in
               Perches
               .
            
             
               5.
               
               As
               10
               ,
               the
               length
               in
               Chains
               ,
               to
               10
               ;
               So
               is
               1
               ,
               to
               1
               ,
               the
               breadth
               in
               Chains
               required
               .
            
             
               6.
               
               As
               176
               ,
               the
               length
               in
               Elles
               ,
               to
               3097
               ⅓
               ;
            
             
               So
               is
               ●
               ,
               to
               17-6
               ,
               the
               breadth
               in
               Elles
               required
               .
            
             
               
                 To
                 work
                 this
                 by
                 the
                 Line
                 of
                 Lines
                 ,
                 say
              
               ;
            
             
               1.
               
               As
               the
               —
               43560
               ,
               to
               =
               660
               ;
               So
               is
               =
               10
               ,
               to
               66
               ,
               Latterally
               .
            
             
             
               2
               :
               As
               the
               Latteral
               220
               ,
               to
               Parallel
               4840
               ;
            
             
               So
               is
               Latteral
               1
               ,
               to
               Parallel
               22
               ,
               (
               single
               ,
               double
               ,
               or
               four-fold
               )
               .
            
             
               3.
               
               As
               —
               132
               doubled
               ,
               is
               to
               =
               1742
               likewise
               doubled
               ,
               because
               it
               falls
               near
               the
               Center
               ;
            
             
               So
               is
               —
               1
               quadrupled
               ,
               viz.
               4
               ,
               to
               =
               13-2
               quadrupled
               ,
               viz.
               52-4
               .
            
             
               4.
               
               As
               —
               160
               ,
               to
               =
               40
               ;
            
             
               So
               is
               =
               10
               for
               1
               ,
               to
               —
               4
               Perch
               .
            
             
               5.
               
               As
               —
               10
               ,
               to
               =
               10
               ;
            
             
               So
               is
               =
               1
               to
               —
               1
               ,
               the
               breadth
               required
               .
            
             
               If
               you
               would
               know
               how
               much
               breadth
               ,
               at
               any
               length
               ,
               shall
               make
               2
               ,
               3
               ,
               or
               4
               Acres
               ;
            
             
               
                 Then
                 say
              
               ,
            
             
               As
               the
               length
               given
               to
               the
               quantity
               of
               one
               Acre
               in
               that
               measure
               ,
               according
               to
               the
               Table
               ;
            
             
               So
               is
               2
               ,
               3
               ,
               4
               ,
               or
               5
               ,
               to
               the
               breadth
               required
               .
            
             
               
                 Example
                 at
                 30
                 Perch
                 in
                 length
                 .
              
            
             
               The
               Extent
               from
               30
               to
               160
               ,
               shall
               reach
               the
               same
               way
               from
               4
               to
               21
               Perch
               ,
               and
               34
               of
               100
               (
               or
               5
               Foot
               ,
               06
               Inches
               )
               the
               breadth
               of
               4
               Acres
               ,
               at
               30
               Perches
               in
               length
               .
            
          
           
             
             
               Problem
               II.
               The
               length
               and
               breadth
               given
               in
               Perches
               ,
               to
               find
               the
               Content
               in
               Perches
               ,
               of
               any
               piece
               of
               Land.
               
            
             
               The
               Extent
               from
               1
               ,
               to
               the
               breadth
               in
               Perches
               ,
               shall
               reach
               the
               same
               way
               ,
               from
               the
               length
               in
               Perches
               ,
               to
               the
               true
               Content
               in
               Square
               Perches
               .
            
             
               Example
               .
            
             
               As
               1
               to
               50
               ,
               so
               is
               179
               to
               8950
               ,
               the
               Content
               in
               Square
               Perches
               .
            
          
           
             
               Problem
               III.
               The
               length
               and
               breadth
               being
               given
               in
               Perches
               ,
               to
               find
               the
               Content
               in
               Square
               Acres
               .
            
             
               The
               Extent
               from
               160
               to
               the
               breadth
               in
               Perches
               ,
               shall
               ●each
               the
               same
               way
               ,
               from
               the
               length
               in
               Perches
               ,
               to
               the
               Content
               in
               Square
               Acres
               .
            
             
               Example
               .
            
             
               As
               160
               to
               50
               ,
               so
               is
               179
               to
               5-58
               Acres
               ,
               or
               5
               A●res
               ,
               2
               Rood
               ,
               and
               13
               Perches
               .
            
          
           
             
             
               Problem
               IV.
               The
               length
               and
               breadth
               of
               a
               piece
               of
               Land
               being
               given
               in
               Chains
               ,
               to
               find
               the
               Content
               in
               Acres
               .
            
             
               The
               Extent
               from
               1
               ,
               to
               the
               breadth
               in
               Chains
               ,
               and
               100
               parts
               ,
               which
               are
               Links
               ,
               shall
               reach
               the
               same
               way
               from
               the
               length
               in
               Chains
               and
               Links
               ,
               to
               the
               Content
               in
               Square
               Acres
               .
            
             
               Example
               .
            
             
               As
               1
               to
               5
               Chains
               ,
               52
               Links
               ,
               the
               breadth
               ;
            
             
               So
               is
               8
               Chains
               ,
               72
               Links
               ,
               to
               48
               Acres
               ,
               and
               3960
               Square
               Links
               .
            
          
           
             
               Problem
               V.
               Having
               the
               Base
               and
               Perpendiculer
               of
               a
               Triangle
               given
               in
               Chains
               or
               Perches
               ,
               to
               find
               the
               Content
               in
               Acres
               .
            
             
               The
               Extent
               from
               2
               ,
               if
               you
               use
               Chains
               ;
               or
               from
               320
               ,
               if
               you
               measure
               by
               Perches
               ,
               to
               the
               whole
               Base
               ,
               shall
               reach
               the
               same
               way
               from
               the
               whole
               Perpendiculer
               ,
               to
               the
               whole
               Content
               of
               the
               Triangle
               ;
               or
               if
               it
               be
               a
               Trapezia
               ,
               joyn
               both
               the
               Perpendiculers
               in
               one
               sum
               .
            
             
             
               Example
               .
            
             
               As
               2
               (
               for
               Chains
               )
               to
               3-63
               ,
               the
               whole
               Perpendiculer
               ;
            
             
               So
               is
               11-80
               ,
               the
               whole
               Base
               ,
               to
               21
               Acres
               ,
               42
               Links
               ,
               the
               Content
               of
               the
               whole
               Triangle
               .
            
             
               
                 Or
                 in
                 Perches
              
               .
            
             
               As
               320
               to
               14-55
               ,
               the
               Perpendiculer
               in
               Perches
               ;
            
             
               So
               is
               47-20
               ,
               the
               length
               ,
               or
               base
               Line
               ,
               in
               Perches
               ,
               to
               21
               Acres
               ,
               24
               Links
               ,
               the
               Content
               in
               Acres
               .
            
          
           
             
               Problem
               VI.
               The
               Area
               ,
               or
               Content
               of
               a
               piece
               of
               Land
               given
               ,
               that
               was
               measured
               by
               Statute-Perches
               ;
               to
               find
               the
               Content
               of
               the
               same
               piece
               of
               Land
               in
               Wood-land
               measure
               ,
               or
               Customary
               Acres
               ,
               or
               Irish
               Acres
               .
            
             
               For
               the
               better
               understanding
               of
               this
               Problem
               ,
               it
               is
               necessary
               to
               describe
               the
               several
               kinds
               and
               quantities
               of
               Perch●s
               ,
               which
               are
               spoken
               of
               by
               Authors
               ,
               and
               used
               in
               several
               places
               ;
               together
               with
               their
               proportion
               to
               the
               Statute
               Perch
               of
               16
               Foot
               and
               a
               half
               square
               ,
               London
               measure
               .
            
             
               The
               kinds
               of
               Perches
               ,
               are
               first
               Statute-measure
               of
               16
               foot
               ½
               to
               the
               Perch
               ,
               according
               
               to
               the
               Standard
               at
               Guild-Hall
               ,
               or
               the
               King's
               Majesties
               Exchequer
               .
               Secondly
               ,
               Woodland-measure
               ,
               a
               Perch
               whereof
               contains
               18
               Foot
               Square
               of
               the
               same
               London
               measure
               .
               Thirdly
               ,
               Irish
               Acres
               ,
               of
               21
               Foot
               to
               the
               Perch
               or
               Pole.
               And
               lastly
               ,
               Three
               sorts
               of
               Customary
               ,
               used
               in
               several
               places
               of
               England
               ,
               of
               20
               ,
               24
               ,
               Cheshire
               measure
               ,
               and
               28
               Foot
               square
               to
               the
               Perch
               .
            
             
               As
               for
               the
               Proportions
               one
               to
               another
               ,
               that
               is
               ,
               as
               16
               ½
               ,
               to
               18
               ,
               20
               ,
               21
               ,
               24
               ,
               28
               ,
               or
               any
               the
               like
               wha●soever
               .
            
             
               But
               to
               find
               their
               difference
               in
               Squares
               or
               Scales
               ,
               the
               Work
               is
               thus
               ;
               
                 By
                 the
                 Line
                 of
                 Numbers
              
               ,
               First
               appoint
               what
               Number
               in
               an
               Inch
               shall
               be
               the
               Scale
               for
               Statute
               measure
               ,
               which
               I
               shall
               appoint
               a
               Scale
               of
               30
               in
               an
               Inch.
               
            
             
               Then
               the
               Extent
               from
               16
               ½
               ,
               to
               18
               ,
               for
               Woodland
               measure
               ,
               shall
               reach
               the
               contrary
               way
               from
               30
               ,
               being
               twice
               repeated
               ,
               to
               25-2
               ;
               so
               ,
               I
               say
               ,
               that
               a
               Scale
               made
               to
               25-2
               in
               an
               Inch
               ,
               shall
               be
               the
               Scale
               for
               a
               Woodland
               Perch
               of
               18
               Foot
               Square
               ;
               and
               in
               proportion
               to
               that
               of
               16
               Foot
               ½
               ,
               at
               30
               parts
               in
               an
               Inch.
               
            
             
               Again
               ,
               For
               Irish
               Acres
               ,
               which
               are
               measured
               by
               a
               Pole
               of
               21
               Foot
               to
               a
               Perch
               ,
               the
               Extent
               on
               the
               Line
               of
               Numbers
               from
               21
               to
               16
               ½
               ,
               shall
               reach
               (
               being
               turned
               twice
               the
               
               same
               way
               )
               from
               30
               ,
               to
               16
               ½
               ,
               the
               quantity
               of
               the
               Scale
               for
               Irish
               Acres
               ,
               to
               be
               in
               proportion
               to
               a
               Scale
               of
               30
               in
               an
               Inch
               for
               Statute-measure
               ;
               and
               so
               for
               the
               rest
               ,
               or
               any
               other
               whatsoever
               ,
               as
               in
               the
               following
               Table
               .
            
             
               
                 
                   
                     16
                     ½
                  
                   
                     
                       The
                       Scale
                       that
                       is
                       to
                       it
                       proportionable
                       to
                    
                     30
                     
                       for
                       Statute
                       measure
                       is
                    
                     ,
                  
                   
                     16
                     ½
                  
                   
                     is
                  
                   
                     30
                     —
                     00
                  
                   
                     Statute-Measure
                     .
                  
                
                 
                   
                     18
                  
                   
                     18
                  
                   
                     25
                     —
                     22
                  
                   
                     Woodland-Measure
                     .
                  
                
                 
                   
                     20
                  
                   
                     20
                  
                   
                     20
                     —
                     42
                  
                   
                     Customary
                     .
                  
                
                 
                   
                     21
                  
                   
                     21
                  
                   
                     18
                     =
                     50
                  
                   
                     
                       In
                       an
                       Inch
                       for
                       Irish.
                    
                     
                  
                
                 
                   
                     22
                  
                   
                     22
                  
                   
                     16
                     —
                     89
                  
                   
                     Customary
                     .
                  
                
                 
                   
                     24
                  
                   
                     24
                  
                   
                     14
                     —
                     20
                  
                   
                     
                       Customary
                       Cheshire-measure
                    
                     .
                  
                
                 
                   
                     28
                  
                   
                     28
                  
                   
                     10
                     —
                     42
                  
                   
                     Customary
                     .
                  
                
                 
                   
                     30
                  
                   
                     30
                  
                   
                     09
                     —
                     08
                  
                   
                     
                       Or
                       any
                       other
                    
                     .
                  
                
              
            
             
               So
               that
               if
               you
               have
               several
               Scales
               made
               upon
               a
               Rule
               (
               to
               draw
               the
               Plot
               of
               your
               Field
               withal
               )
               to
               these
               Proportions
               (
               which
               may
               be
               convenient
               enough
               for
               Difference
               between
               one
               another
               )
               ,
               then
               for
               the
               reducing
               of
               
               the
               quantity
               of
               Acres
               found
               by
               Statute-measure
               ,
               to
               Woodland
               ,
               Irish
               ,
               or
               Customary
               ,
               is
               no
               more
               but
               thus
               :
            
             
               Take
               the
               Acres
               ,
               measured
               by
               Statute-measure
               ,
               out
               of
               the
               Scale
               of
               30
               in
               an
               Inch
               appointed
               for
               Statute
               measure
               ,
               and
               measure
               it
               in
               the
               Scale
               of
               25-22
               in
               an
               Inch
               for
               Woodland
               ;
               or
               by
               the
               Scale
               of
               18-55
               for
               Irish
               Acres
               ;
               or
               by
               the
               Scale
               of
               16-89
               for
               Customary
               ;
               and
               you
               shall
               have
               the
               quantity
               of
               Woodland
               ,
               Irish
               ,
               or
               Customary
               Acres
               required
               .
            
             
               Example
               .
            
             
               Suppose
               I
               have
               30
               Acres
               of
               Statute
               measure
               ,
               how
               many
               Acres
               of
               Woodland
               ,
               Irish
               ,
               or
               Customary
               measure
               will
               they
               make
               ?
               Take
               30
               from
               the
               Scale
               of
               30
               in
               an
               Inch
               ,
               and
               on
               the
               Scale
               of
               25-22
               ,
               it
               shall
               give
               25-22
               ,
               for
               so
               many
               Woodland
               Acres
               ;
               and
               on
               the
               Scale
               of
               18-55
               ,
               for
               Irish
               Acres
               ,
               it
               shall
               give
               18-55
               for
               so
               many
               Irish
               Acres
               ;
               and
               on
               the
               Scale
               of
               16-89
               in
               an
               Inch
               for
               Customary
               Acres
               ,
               it
               shall
               give
               16-89
               for
               so
               many
               Customary
               Acres
               ,
               at
               22
               Foot
               to
               the
               Perch
               or
               Pole
               ,
               &c.
               
            
             
               This
               being
               thus
               fully
               premised
               ,
               to
               work
               these
               Questions
               by
               the
               Line
               of
               Numbers
               only
               :
               the
               Extent
               of
               the
               Compasses
               from
               
               1●-5
               ,
               the
               Feet
               in
               a
               Statute
               Perch
               ,
               to
               18
               the
               Feet
               in
               a
               Woodland
               Perch
               (
               or
               to
               21
               the
               Feet
               in
               an
               Irish
               Perch
               ,
               or
               to
               22
               ,
               24
               ,
               28
               the
               Feet
               in
               a
               Customary
               Perch
               )
               shall
               reach
               from
               30
               ,
               the
               Acres
               in
               Statute
               measure
               ,
               beng
               twice
               repeated
               ,
               to
               25-22
               ,
               the
               Acres
               in
               Woodland
               measure
               required
               ,
               &c.
               it
               being
               a
               larger-Acre
               ▪
               must
               nee●s
               be
               less
               in
               quantity
               .
            
             
               Which
               work
               is
               performed
               by
               the
               back-Rule
               of
               Three
               in
               a
               duplicated
               proportion
               .
            
          
           
             
               Problem
               VII
               .
               Having
               the
               Plot
               or
               Draught
               of
               a
               Field
               ,
               and
               its
               Content
               in
               Acres
               ,
               to
               find
               by
               what
               Scale
               it
               was
               Plotted
               ;
               that
               is
               ,
               by
               what
               parts
               in
               an
               Inch.
               
            
             
               Suppose
               a
               Triangle
               ,
               or
               a
               Parallellagram
               ,
               or
               long
               Square
               ,
               do
               contain
               4
               Acres
               and
               a
               half
               ,
               which
               is
               set
               down
               in
               figures
               thus
               ,
               4-50
               ;
               which
               if
               I
               should
               measure
               by
               a
               Scale
               of
               12
               in
               an
               Inch
               ,
               might
               happen
               to
               be
               2-25
               Chains
               one
               way
               ,
               and
               1
               Chain
               ,
               25
               Links
               ●he
               other
               way
               ;
               which
               two
               sums
               being
               multiplied
               together
               ,
               make
               2-5200
               ,
               whereas
               it
               should
               be
               4-5000
               ;
            
             
               
                 Therefore
                 by
                 th●
                 〈◊〉
                 of
                 Numbers
                 ,
                 to
                 gain
                 〈…〉
                 do
                 thus
              
               ;
            
             
               Divide
               the
               dista●ce
               between
               2-5200
               ,
               and
               
               4-5000
               ,
               into
               two
               equal
               parts
               ;
               that
               distance
               laid
               the
               right
               way
               from
               12
               ,
               the
               Scale
               I
               measured
               by
               ,
               shall
               reach
               to
               16
               ,
               the
               Scale
               the
               Plot
               was
               made
               by
               .
            
             
               
                 For
                 Note
              
               ,
               That
               if
               the
               Scale
               I
               guessed
               a●
               ,
               gives
               more
               than
               I
               should
               have
               ,
               then
               I
               have
               too
               many
               in
               an
               Inch
               ;
               but
               if
               less
               ,
               I
               must
               have
               more
               in
               an
               Inch
               ,
               as
               here
               ,
               which
               infallibly
               sheweth
               which
               way
               ,
               which
               is
               alwayes
               the
               same
               way
               as
               you
               divided
               the
               space
               ,
               from
               the
               guessed
               Sum
               or
               Product
               ,
               to
               the
               true
               Product
               .
            
             
               To
               this
               Rule
               may
               be
               referred
               the
               way
               to
               discover
               the
               true
               size
               of
               Glasiers
               Quarries
               ;
               the
               method
               whereof
               is
               thus
               :
               They
               are
               usually
               cut
               to
               ,
               and
               called
               by
               8s
               ,
               10s
               ,
               12s
               ,
               15s
               ,
               18s
               ,
               and
               20s
               in
               a
               Foot
               ,
               or
               any
               other
               what
               you
               please
               ;
               that
               is
               to
               say
               ,
               8
               quarries
               of
               Glass
               of
               8s
               ,
               make
               a
               Superficial
               Foot
               ;
               and
               10
               quarries
               of
               10s
               ,
               make
               a
               Foot
               Superficial
               ;
               and
               12
               of
               the
               12s
               ,
               &c.
               
            
             
               Also
               they
               are
               cut
               in
               a
               Diamond
               form
               to
               one
               sort
               of
               Angle
               for
               the
               Squ●re
               quarries
               ;
               and
               another
               for
               the
               Long
               quarries
               :
               The
               acute
               Angle
               of
               the
               Squar●
               quarries
               being
               77
               degrees
               ,
               and
               19
               〈…〉
               and
               the
               acute
               Angle
               of
               the
               Long
               quar●●es
               ●7
               degrees
               and
               22
               minutes
               :
               The
               long
               ●as
               being
               just
               6
               inches
               long
               ,
               and
               4
               inches
               broad
               ;
               and
               the
               Square
               10s
               ,
               6
               inches
               long
               ,
               and
               4
               
               inches
               ,
               and
               80
               parts
               of
               a
               100
               broad
               ▪
            
             
               This
               being
               the
               standing
               Rule
               or
               Method
               ,
               and
               those
               two
               sizes
               being
               known
               ,
               I
               would
               find
               out
               any
               other
               ,
               as
               13s
               ,
               or
               14s
               ,
               or
               17s
               and
               the
               like
               .
            
             
               
                 Do
                 thus
              
               ;
            
             
               Divide
               the
               distance
               on
               the
               Line
               of
               Numbers
               ,
               between
               the
               Content
               of
               some
               known
               size
               ,
               and
               the
               Content
               of
               the
               inquired
               size
               into
               two
               equal
               parts
               ;
               and
               that
               distance
               laid
               the
               right
               way
               from
               the
               sides
               of
               the
               known
               size
               (
               increasing
               for
               a
               bigger
               ,
               and
               decreasing
               for
               a
               less
               )
               shall
               give
               the
               reciprocal
               sides
               of
               the
               size
               required
               .
            
             
               
                 Example
                 .
                 The
                 Sides
                 ,
                 Ranges
                 ,
                 Lengths
                 ,
                 and
                 Breadt●
                 of
                 Square
                 10s
                 ,
                 are
                 as
                 in
                 the
                 Table
                 following
                 ;
                 and
                 I
                 would
                 have
                 the
                 Ranges
                 ,
                 Sides
                 ,
                 Length
                 ,
                 and
                 Breadth
                 of
                 14s
                 ,
                 an
                 unusual
                 Size
                 .
              
            
             
               The
               Content
               of
               a
               Square
               quarry
               of
               Glass
               called
               10s
               ,
               is
               a
               just
               10th
               part
               of
               a
               Foot
               ,
               which
               is
               1
               inch
               and
               20
               parts
               ;
               or
               one
               10th
               part
               of
               a
               Superficial
               Foot
               ,
               containing
               12
               long
               inches
               .
            
             
               And
               the
               Content
               of
               the
               size
               called
               14
               s
               ,
               must
               be
               one
               14th
               part
               of
               the
               same
               measure
               ,
               or
               Foot
               Superficial
               ,
               which
               is
               0-85714
               ,
               that
               is
               0-857
               parts
               of
               one
               long
               inch
               in
               a
               1000
               parts
               .
            
             
             
               Then
               ,
               by
               the
               Line
               of
               Numbers
               ,
               divide
               the
               space
               between
               1-2000
               ,
               the
               Content
               of
               the
               10s
               ,
               and
               0-857
               the
               Content
               of
               the
               14s
               into
               two
               equal
               parts
               ;
               that
               Extent
               ,
               I
               say
               ,
               laid
               the
               same
               way
               from
               3-76
               ,
               the
               Ranges
               of
               Square
               10s
               ,
               shall
               reach
               to
               3-18
               ,
               the
               Ranges
               for
               14s
               :
               And
               from
               3-84
               ,
               the
               sides
               of
               Square
               10s
               ,
               to
               3-25
               the
               sides
               of
               Square
               14s
               :
               And
               from
               4-80
               the
               breadth
               of
               Square
               10s
               ,
               to
               4-05
               the
               breadth
               of
               Square
               14s
               :
               And
               from
               6-00
               ,
               the
               length
               of
               Square
               10s
               ,
               to
               5-07
               the
               length
               of
               Square
               14s
               ,
               the
               requisites
               of
               the
               unknown
               Size
               required
               .
               And
               the
               like
               for
               any
               other
               whatsoever
               .
            
             
             
               
                 
                   The
                   true
                   size
                   of
                   Glasiers
                   Quarries
                   ,
                   both
                   Long
                   and
                   Square
                   ,
                   By
                   
                     J.
                     B.
                     1660.
                  
                   
                
                 
                   
                     Square
                     Quarries
                     77
                     deg
                     .
                     19
                     min.
                     
                  
                   
                     Long
                     Quarries
                     67
                     deg
                     .
                     22
                     min.
                     
                  
                
                 
                   
                     Sizes
                  
                   
                     Rang.
                     
                  
                   
                     Sides
                     .
                  
                   
                     bredth
                  
                   
                     length
                  
                   
                     Content
                     .
                  
                   
                     Content
                     .
                  
                   
                     Size
                     .
                  
                   
                     Rang.
                     
                  
                   
                     Sides
                     .
                  
                   
                     bredth
                  
                   
                     length
                  
                   
                     Content
                     .
                  
                   
                     Content
                     .
                  
                
                 
                   
                     In.
                     100
                  
                   
                     In.
                     100
                  
                   
                     In.
                     100
                  
                   
                     In
                     100
                  
                   
                     Inc.
                     100
                  
                   
                     Tens
                     .
                  
                   
                     In.
                     100
                  
                   
                     In.
                     100
                  
                   
                     In.
                     100
                  
                   
                     In.
                     100
                  
                   
                     Inc.
                     100
                  
                   
                     Tens
                     .
                  
                
                 
                   
                     8
                  
                   
                     4
                     20
                  
                   
                     4
                     30
                  
                   
                     5
                     36
                  
                   
                     6
                     70
                  
                   
                     1
                     500
                  
                   
                     1
                     250
                  
                   
                     8
                  
                   
                     4
                     09
                  
                   
                     4
                     41
                  
                   
                     4
                     90
                  
                   
                     7
                     34
                  
                   
                     1
                     500
                  
                   
                     1
                     250
                  
                
                 
                   
                     10
                  
                   
                     3
                     76
                  
                   
                     3
                     84
                  
                   
                     4
                     80
                  
                   
                     6
                     00
                  
                   
                     1
                     200
                  
                   
                     1
                     000
                  
                   
                     10
                  
                   
                     3
                     65
                  
                   
                     3
                     95
                  
                   
                     4
                     38
                  
                   
                     6
                     57
                  
                   
                     1
                     200
                  
                   
                     1
                     000
                  
                
                 
                   
                     12
                  
                   
                     3
                     43
                  
                   
                     3
                     51
                  
                   
                     4
                     38
                  
                   
                     5
                     47
                  
                   
                     1
                     000
                  
                   
                     0
                     833
                  
                   
                     12
                  
                   
                     3
                     34
                  
                   
                     3
                     61
                  
                   
                     4
                     00
                  
                   
                     6
                     00
                  
                   
                     1
                     000
                  
                   
                     0
                     833
                  
                
                 
                   
                     15
                  
                   
                     3
                     07
                  
                   
                     3
                     13
                  
                   
                     3
                     92
                  
                   
                     4
                     90
                  
                   
                     0
                     800
                  
                   
                     0
                     667
                  
                   
                     15
                  
                   
                     2
                     98
                  
                   
                     3
                     23
                  
                   
                     3
                     58
                  
                   
                     5
                     37
                  
                   
                     0
                     800
                  
                   
                     0
                     667
                  
                
                 
                   
                     18
                  
                   
                     2
                     80
                  
                   
                     2
                     86
                  
                   
                     3
                     57
                  
                   
                     4
                     77
                  
                   
                     0
                     666
                  
                   
                     0
                     555
                  
                   
                     18
                  
                   
                     2
                     72
                  
                   
                     2
                     95
                  
                   
                     3
                     26
                  
                   
                     4
                     90
                  
                   
                     0
                     666
                  
                   
                     0
                     555
                  
                
                 
                   
                     20
                  
                   
                     2
                     66
                  
                   
                     2
                     72
                  
                   
                     3
                     39
                  
                   
                     4
                     24
                  
                   
                     0
                     600
                  
                   
                     0
                     500
                  
                   
                     20
                  
                   
                     2
                     58
                  
                   
                     2
                     79
                  
                   
                     3
                     10
                  
                   
                     4
                     65
                  
                   
                     0
                     600
                  
                   
                     0
                     500
                  
                
              
            
          
        
         
           
           
             CHAP.
             IX
             .
             The
             use
             of
             the
             Line
             of
             Numbers
             in
             measuring
             of
             Solid
             measure
             ,
             as
             Timber
             ,
             Stone
             ,
             or
             the
             like
             Solid
             bodies
             .
          
           
             
               Problem
               I.
               A
               piece
               of
               Timber
               being
               broader
               one
               way
               than
               the
               other
               ,
               to
               find
               the
               side
               of
               a
               Square
               that
               shall
               be
               equal
               thereunto
               ,
               being
               called
               ,
               Squaring
               the
               Piece
               .
            
             
               THe
               side
               of
               the
               Square
               ,
               that
               shall
               be
               equal
               to
               the
               square
               of
               the
               Oblong
               ,
               is
               nothing
               else
               but
               a
               mean
               proportion
               between
               the
               length
               and
               breadth
               of
               the
               Oblong
               :
            
             
               
                 As
                 thu●
              
               ;
            
             
               Suppose
               a
               piece
               of
               Timber
               is
               12
               inches
               in
               depth
               ,
               and
               16
               inches
               in
               breadth
               (
               and
               10
               foot
               in
               length
               .
               )
            
             
               16
               the
               breadth
               ,
               and
               12
               the
               depth
               ,
               multiplied
               
               together
               ,
               make
               192
               ;
               the
               Square
               or
               Product
               of
               16
               and
               12
               multiplied
               .
            
             
               Now
               the
               square
               Root
               of
               192
               ,
               which
               is
               near
               13
               ●59
               /
               1000
               ,
               is
               the
               side
               of
               a
               Square
               ,
               equal
               to
               12
               and
               16
               ,
               the
               depth
               and
               thickness
               of
               the
               piece
               of
               Timber
               propounded
               .
               For
               if
               you
               shall
               multiply
               13-859
               by
               13-859
               ,
               you
               shall
               find
               192-071881
               ,
               the
               nearest
               Root
               ,
               you
               can
               express
               in
               5
               figures
               ,
               and
               an
               indifferent
               true
               mean
               proportion
               ,
               between
               12
               and
               16
               ,
               the
               depth
               and
               breadth
               ;
               so
               that
               in
               fine
               ,
               13-86
               ,
               is
               the
               side
               of
               a
               Square
               ,
               nearly
               equal
               to
               12
               and
               16
               ,
               whereas
               the
               doubling
               and
               halfing
               ,
               the
               old
               false
               way
               ,
               gives
               full
               14.
               
            
             
               
                 To
                 work
                 this
                 by
                 the
                 Line
                 of
                 Numbers
                 ,
                 is
                 thus
              
               ;
            
             
               Divide
               the
               distance
               on
               the
               Line
               of
               Numbers
               ,
               between
               12
               and
               16
               ,
               into
               two
               equal
               parts
               ,
               and
               you
               shall
               find
               the
               Point
               to
               stay
               at
               13
               ,
               and
               near
               86
               parts
               ,
               the
               Answer
               required
               .
            
             
               The
               way
               of
               doing
               it
               ,
               
                 by
                 the
                 Line
                 of
                 Lines
              
               ,
               is
               shewed
               in
               the
               VI
               Chapter
               ,
               and
               7th
               Proposition
               ,
               either
               by
               the
               Sector
               ,
               or
               Trianguler
               Quadrant
               ,
               and
               therefore
               needs
               no
               repetition
               in
               this
               pla●e
               .
            
          
           
             
             
               Problem
               II.
               At
               any
               Breadth
               and
               Depth
               ,
               or
               Squareness
               ,
               to
               find
               how
               much
               makes
               a
               Foot
               of
               Timber
               .
            
             
               1.
               
               If
               the
               Timber
               be
               square
               (
               or
               squared
               )
               then
               the
               way
               by
               the
               Line
               of
               Numbers
               ,
               is
               thus
               ;
            
             
               
                 First
                 by
                 Foot
                 measure
              
               .
            
             
               Extend
               the
               Compasses
               from
               the
               side
               of
               the
               square
               to
               the
               middle
               1
               ,
               the
               same
               Extent
               applyed
               ,
               or
               turned
               twice
               the
               same
               way
               from
               1
               ,
               shall
               reach
               to
               the
               length
               that
               makes
               a
               Foot
               of
               Timber
               ,
               at
               that
               squareness
               .
            
             
               Example
               .
            
             
               Suppose
               a
               piece
               of
               Timber
               be
               50
               of
               a
               100
               (
               or
               6
               inches
               )
               or
               half
               a
               Foot
               Square
               (
               which
               is
               all
               as
               one
               )
               Extend
               the
               Compasses
               from
               5
               to
               1
               (
               forwards
               )
               the
               same
               Extent
               being
               turned
               two
               times
               ,
               the
               same
               way
               from
               1
               ,
               shall
               reach
               to
               4
               ,
               being
               4
               Foot
               ,
               or
               40
               such
               parts
               ,
               whereof
               the
               side
               of
               the
               Square
               was
               5.
               
            
             
               
                 
                   2.
                   
                   Secondly
                
                 ,
                 The
                 same
                 again
                 by
                 Inches
                 .
              
            
             
               The
               Extent
               from
               6
               to
               12
               ,
               shall
               reach
               ,
               being
               turned
               two
               times
               the
               same
               way
               from
               12
               ,
               to
               48
               ,
               the
               number
               of
               inches
               in
               length
               that
               makes
               a
               Foot
               ,
               at
               that
               Squareness
               ;
               being
               48
               
               such
               parts
               whereof
               the
               side
               of
               the
               Square
               was
               6.
               
            
             
               
                 So
                 that
              
               ,
            
             
               As
               the
               side
               of
               the
               Square
               ,
               in
               inches
               ,
               is
               to
               12
               :
               so
               is
               12
               to
               a
               4th
               ,
               and
               so
               is
               that
               4th
               to
               the
               length
               of
               a
               Foot
               required
               ,
               turning
               the
               Compasses
               twice
               ,
               the
               same
               way
               as
               you
               turned
               from
               the
               side
               of
               the
               Square
               in
               inches
               to
               12.
               
            
             
               3.
               
               If
               the
               piece
               of
               Timber
               ,
               or
               Stone
               ,
               be
               not
               Square
               or
               Squared
               ,
            
             
               Then
            
             
               The
               Extent
               from
               1
               to
               the
               depth
               ,
               shall
               reach
               the
               same
               way
               from
               the
               breadth
               to
               a
               4th
               Number
               .
            
             
               Again
               ,
            
             
               The
               Extent
               from
               that
               4th
               Number
               to
               1
               ,
               shall
               reach
               ,
               being
               turned
               once
               ,
               the
               same
               way
               from
               1
               ,
               to
               the
               length
               of
               a
               Foot
               in
               Foot-measure
               required
               .
            
             
               Example
               .
            
             
               Suppose
               a
               piece
               of
               Timber
               be
               0-333
               deep
               ,
               and
               0-750
               broad
               in
               Foot-measure
               ;
               or
               4
               inches
               deep
               ,
               and
               9
               inches
               broad
               ,
               as
               with
               a
               glance
               of
               your
               eye
               on
               inches
               and
               foot-measure
               ,
               you
               may
               see
               how
               these
               Numbers
               agree
               .
               The
               Extent
               ,
               I
               say
               ,
               from
               1
               to
               0-333
               ,
               shall
               reach
               the
               same
               way
               from
               0-750
               to
               2-50
               .
            
             
               Again
               ,
            
             
               I
               say
               ,
               The
               Extent
               from
               250
               the
               4th
               ,
               to
               
               1
               ,
               shall
               reach
               the
               same
               way
               ,
               from
               1
               to
               40
               ,
               or
               4
               Foot
               ,
               the
               length
               required
               ,
               to
               make
               a
               Foot
               at
               that
               breadth
               and
               depth
               .
            
             
               
                 4.
                 
                 By
                 Inch-measure
                 ,
                 〈◊〉
                 find
                 the
                 length
                 of
                 a
                 Foot
                 in
                 Inches
                 .
              
            
             
               As
               12
               to
               the
               breadth
               in
               inches
               ,
               so
               is
               the
               depth
               in
               inches
               to
               a
               4th
               ;
               then
               as
               that
               4th
               to
               12
               ,
               so
               is
               12
               to
               the
               length
               in
               inches
               required
               .
            
             
               Example
               .
            
             
               The
               Extent
               from
               12
               to
               9
               the
               breadth
               ,
               shall
               reach
               the
               same
               way
               from
               4
               the
               depth
               ,
               to
               3
               for
               a
               4th
               .
            
             
               Then
               the
               Extent
               from
               3
               the
               4th
               to
               12
               ,
               shall
               reach
               the
               same
               way
               from
               12
               to
               48
               ,
               the
               inches
               in
               length
               required
               ,
               to
               make
               a
               Foot.
               
            
             
               
                 5.
                 
                 The
                 breadth
                 and
                 depth
                 given
                 in
                 Inches
                 ,
                 to
                 find
                 the
                 length
                 of
                 a
                 Foot
                 of
                 Timber
                 ,
                 in
                 Feet
                 and
                 Parts
                 .
              
            
             
               Then
               say
               ,
               As
               1
               ,
               to
               the
               depth
               ;
               so
               is
               the
               breadth
               to
               a
               4th
               .
            
             
               Again
               ,
            
             
               As
               that
               4th
               to
               12
               ,
               so
               is
               12
               to
               the
               length
               in
               feet
               and
               parts
               .
            
             
             
               Example
               .
            
             
               The
               Extent
               from
               1
               to
               4
               ,
               shall
               reach
               the
               same
               way
               from
               9
               to
               36
               ,
               a
               4th
               ;
            
             
               Then
               ,
            
             
               The
               Extent
               from
               36
               to
               12
               ,
               shall
               reach
               the
               same
               way
               from
               12
               to
               4
               foot
               ,
               the
               length
               in
               feet
               required
               .
            
             
               The
               reason
               of
               this
               Order
               and
               Method
               ,
               if
               you
               consider
               ,
               you
               will
               find
               thus
               ;
            
             
               
                 In
                 the
                 4th
                 way
                 of
                 working
                 ,
                 you
                 went
                 thus
              
               ;
            
             
               As
               12
               ,
               the
               inches
               in
               a
               foot
               ,
               is
               to
               the
               breadth
               in
               inches
               ;
            
             
               So
               is
               the
               depth
               to
               3
               Foot.
               
            
             
               
                 But
                 in
                 the
                 5th
                 and
                 last
                 way
                 you
                 say
                 ,
              
            
             
               As
               1
               foot
               to
               the
               depth
               in
               inches
               ;
            
             
               So
               is
               the
               breadth
               to
               36
               inches
               ,
               which
               is
               3
               foot
               also
               .
            
             
               But
               altering
               the
               Order
               in
               the
               beginning
               ,
               alters
               it
               in
               the
               issue
               ,
               though
               the
               same
               truth
               ,
               yet
               in
               or
               under
               divers
               denominations
               ;
               for
               48
               inches
               ,
               and
               4
               foot
               ,
               are
               the
               same
               ;
               yet
               sometimes
               one
               way
               is
               more
               convenient
               than
               another
               .
            
          
           
             
             
               Problem
               III.
               At
               any
               Squareness
               ,
               or
               Breadth
               and
               Depth
               given
               in
               Foot-measure
               ,
               or
               Inches
               ,
               to
               find
               how
               much
               Timber
               is
               in
               a
               Foot
               long
               ,
               in
               Foot-measure
               ,
               or
               feet
               ,
               and
               100
               parts
               or
               inches
               .
            
             
               
                 1.
                 
                 If
                 the
                 piece
                 of
                 Timber
                 be
                 Square
                 (
                 or
                 Squared
                 )
                 then
                 work
                 thus
                 for
                 Foot-measure
                 .
              
            
             
               As
               1
               ,
               to
               the
               side
               of
               the
               Square
               ,
               so
               is
               the
               side
               of
               the
               Square
               to
               the
               quantity
               of
               Timber
               in
               one
               Foot
               long
               ;
               which
               multiplied
               by
               the
               length
               ,
               gives
               the
               whole
               Content
               required
               .
            
             
               
                 Example
                 .
                 At
                 50
                 ,
                 or
                 half
                 a
                 Foot
                 Square
                 ,
                 how
                 much
                 is
                 in
                 a
                 Foot
                 long
                 ?
              
            
             
               Extend
               the
               Compasses
               from
               1
               to
               5
               ,
               the
               same
               Extent
               turned
               the
               same
               way
               from
               5
               ,
               reaches
               to
               25
               ,
               or
               a
               quarter
               of
               a
               Foot
               ;
               then
               if
               the
               Tree
               be
               12
               foot
               long
               ,
               12
               quarters
               will
               make
               3
               foot
               ,
               the
               Content
               .
            
             
               
                 2.
                 
                 The
                 Side
                 of
                 the
                 Square
                 given
                 in
                 Inches
                 ,
                 to
                 find
                 the
                 Quantity
                 ,
                 or
                 Content
                 ,
                 in
                 a
                 Foot.
                 
              
            
             
               As
               12
               ,
               to
               the
               side
               of
               the
               Square
               ,
               so
               is
               the
               
               side
               of
               the
               Square
               to
               3
               twelve
               parts
               of
               a
               Foot
               Solid
               ,
               or
               ¼
               of
               a
               Foot.
               
            
             
               Or
               ,
            
             
               As
               1
               ,
               to
               the
               side
               of
               the
               Square
               ,
               so
               is
               the
               side
               of
               the
               Square
               to
               36
               ,
               144th
               parts
               of
               a
               Foot
               Solid
               .
            
             
               Example
               .
            
             
               The
               Extent
               from
               12
               to
               6
               ,
               the
               inches
               Square
               ,
               shall
               reach
               the
               same
               way
               from
               6
               to
               3
               inches
               in
               a
               Foot
               long
               ,
               which
               is
               3
               12th
               parts
               of
               a
               Foot
               Solid
               .
            
             
               Again
               ,
            
             
               The
               Extent
               from
               1
               to
               6
               ,
               the
               inches
               square
               ,
               shall
               reach
               the
               same
               way
               from
               6
               to
               36
               ,
               the
               number
               of
               long
               inches
               in
               a
               foot
               long
               ;
               or
               pieces
               of
               1
               inch
               square
               ,
               and
               a
               foot
               long
               ,
               144
               of
               which
               makes
               one
               foot
               of
               Timber
               .
            
             
               
                 3.
                 
                 But
                 if
                 the
                 Piece
                 be
                 not
                 square
                 (
                 or
                 squared
                 )
                 then
                 to
                 find
                 how
                 much
                 is
                 in
                 a
                 Foot-long
                 ,
                 work
                 thus
              
               ;
            
             
               As
               1
               to
               the
               depth
               ,
               so
               is
               the
               breadth
               ,
               to
               the
               quantity
               in
               a
               Foot.
               
            
             
               
                 Example
                 3
                 wayes
                 :
                 At
                 9
                 and
                 4
                 breadth
                 and
                 depth
                 .
              
            
             
               1.
               
               The
               Extent
               from
               1
               to
               0-333
               ,
               shall
               reach
               the
               same
               way
               from
               0-75
               ,
               to
               0-25
               ,
               or
               a
               quarter
               of
               a
               Foot
               ;
               for
               Foot-measure
               .
            
             
             
               2.
               
               The
               Extent
               from
               1
               to
               9
               ,
               shall
               reach
               the
               same
               way
               from
               4
               to
               36
               ,
               the
               long
               inches
               in
               a
               foot
               long
               ;
               for
               Inch-measure
               .
            
             
               3.
               
               The
               Extent
               from
               12
               to
               4
               ,
               shall
               reach
               the
               same
               way
               from
               9
               to
               3
               inches
               ,
               or
               3
               12ths
               ,
               viz.
               a
               quarter
               of
               a
               Foot
               ;
               for
               Inch-measure
               .
            
          
           
             
               Problem
               IV.
               The
               side
               of
               the
               Square
               ,
               or
               the
               breadth
               and
               depth
               given
               in
               Inches
               ,
               or
               Foot-measure
               ,
               and
               the
               length
               in
               F●●t
               ,
               to
               find
               the
               Quantity
               ,
               or
               Conten●
               of
               the
               whole
               Piece
               ,
               in
               feet
               and
               parts
               .
            
             
               
                 1.
                 First
                 ,
                 for
                 Foot-measure
              
               ;
               
            
             
               As
               1
               ,
               to
               the
               side
               of
               the
               Square
               ,
               in
               Foot-measure
               ,
               so
               is
               the
               length
               in
               Feet
               to
               a
               4th
               ,
               and
               then
               that
               4
               to
               the
               Content
               in
               feet
               and
               parts
               .
            
             
               Example
               .
            
             
               The
               Extent
               from
               1
               to
               0-833
               ,
               the
               side
               of
               the
               Square
               ,
               shall
               reach
               from
               10
               foot
               25
               parts
               ,
               the
               length
               to
               8-54
               ,
               and
               from
               thence
               to
               7-11
               ,
               the
               Content
               in
               feet
               and
               parts
               required
               .
            
             
               
                 2.
                 For
                 Inch-measure
                 ,
                 Say
              
               ,
               
            
             
               As
               12
               to
               the
               side
               of
               the
               Square
               ,
               in
               inches
               ,
               so
               is
               the
               length
               in
               feet
               ,
               to
               a
               4th
               ;
               and
               then
               
               that
               4th
               to
               the
               Content
               in
               feet
               and
               parts
               .
            
             
               
                 Example
                 at
                 10
                 Inches
                 Square
                 ,
                 and
                 10
                 Foot
                 ,
                 3
                 Inches
                 in
                 length
                 .
              
            
             
               The
               Extent
               of
               the
               Compasses
               ,
               on
               the
               Line
               of
               Numbers
               ,
               from
               12
               ,
               to
               10
               inches
               Square
               ,
               shall
               reach
               the
               same
               way
               from
               10
               foot
               ¼
               ,
               or
               3
               inches
               ,
               to
               85-4
               for
               a
               4th
               ;
               and
               from
               thence
               to
               7-11
               ,
               or
               7
               foot
               1
               inch
               ,
               and
               a
               third
               part
               ,
               the
               Content
               required
               .
               As
               by
               looking
               for
               11
               on
               the
               Line
               of
               Foot-measure
               ,
               right
               against
               which
               ,
               on
               the
               inches
               ,
               is
               1
               inch
               and
               a
               quarter
               ,
               and
               somewhat
               more
               .
            
             
               3.
               
               But
               if
               the
               piece
               of
               Timber
               be
               not
               square
               ,
               and
               you
               would
               measure
               it
               without
               squaring
               ,
               by
               the
               
                 first
                 Problem
              
               ;
            
             
               
                 Then
                 say
                 first
                 by
                 Foot-measure
                 ,
                 thus
              
               ;
            
             
               As
               1
               is
               to
               the
               breadth
               ,
               so
               is
               the
               depth
               to
               a
               4th
               .
               Then
               ,
            
             
               As
               1
               to
               the
               4th
               ,
               so
               is
               the
               length
               in
               feet
               to
               the
               true
               Content
               ,
               in
               feet
               and
               parts
               .
            
             
               Example
               .
            
             
               Let
               a
               Timber-tree
               of
               one
               foot
               25
               ,
               or
               a
               quarter
               one
               way
               ,
               and
               one
               foot
               50
               the
               other
               way
               ,
               and
               12
               foot
               long
               be
               measured
               .
            
             
               The
               Extent
               of
               the
               Compasses
               from
               1
               to
               1-25
               ,
               shall
               reach
               the
               same
               way
               from
               150
               ,
               to
               18-74
               ,
               for
               a
               4th
               .
            
             
             
               Then
               the
               Extent
               from
               1
               to
               18-74
               ,
               shall
               reach
               the
               same
               way
               from
               12
               foot
               ,
               the
               length
               ,
               to
               22-50
               ,
               for
               the
               Content
               ;
               viz.
               22
               foot
               and
               a
               half
               ,
               the
               whole
               Content
               required
               .
            
             
               
                 4.
                 
                 When
                 the
                 breadth
                 and
                 depth
                 is
                 given
                 in
                 Inches
                 ,
                 and
                 the
                 length
                 in
                 Feet
                 ,
                 to
                 find
                 the
                 Content
                 without
                 squaring
                 .
              
            
             
               As
               12
               ,
               to
               the
               breadth
               in
               inches
               ;
            
             
               So
               is
               the
               depth
               in
               inches
               to
               a
               4th
               :
            
             
               Then
               ,
            
             
               As
               12
               to
               that
               4th
               ,
               so
               is
               the
               length
               in
               feet
               and
               parts
               ,
               to
               the
               Content
               in
               feet
               and
               parts
               required
               .
            
             
               
                 Example
                 at
                 15
                 inches
                 deep
                 ,
                 and
                 18
                 inches
                 broad
                 ,
                 and
                 13
                 foot
                 long
                 .
              
            
             
               Extend
               the
               Compasses
               on
               the
               Line
               of
               Numbers
               from
               12
               to
               15
               the
               depth
               ;
               the
               same
               Extent
               applied
               the
               same
               way
               from
               18
               the
               breadth
               ,
               shall
               reach
               to
               22-50
               ,
               for
               a
               4th
               .
            
             
               Then
               the
               Extent
               ,
               from
               12
               to
               22-50
               ,
               the
               4th
               ,
               shall
               reach
               the
               same
               way
               from
               13
               foot
               ,
               the
               length
               ,
               to
               24
               foot
               38
               parts
               ,
               or
               4
               inches
               and
               a
               half
               ,
               as
               a
               glance
               of
               your
               eye
               to
               the
               Inches
               and
               Foot-measure
               will
               plainly
               shew
               .
            
             
             
               Thus
               you
               have
               the
               Solution
               of
               any
               Question
               that
               may
               concern
               proper
               Measuring
               by
               Foot-measure
               ,
               and
               Inches
               ;
               using
               only
               the
               Center
               at
               10
               for
               Foot-measure
               ,
               and
               at
               12
               for
               Inch-measure
               ,
               without
               troubling
               you
               with
               144
               ,
               or
               1728
               ,
               or
               41-57
               ,
               or
               the
               like
               ,
               as
               in
               the
               little
               Book
               of
               the
               
                 Carpenters
                 Rule
              
               ,
               may
               be
               seen
               .
            
             
               To
               work
               these
               Questions
               by
               the
               Line
               of
               Lines
               ,
               though
               it
               may
               be
               done
               several
               ways
               ,
               yet
               no
               way
               so
               soon
               ,
               nor
               so
               exact
               ,
               as
               by
               the
               Line
               of
               Numbers
               :
               Yet
               I
               shall
               shew
               now
               in
               this
               place
               ,
               together
               by
               themselves
               ,
               the
               Three
               principal
               Questions
               ,
               
                 viz.
                 How
                 much
                 makes
                 a
                 Foot
                 in
                 quantity
              
               ;
               And
               ,
               
                 How
                 much
                 is
                 in
                 a
                 Foot
                 long
              
               ;
               And
               ,
               
                 By
                 the
                 length
                 ,
                 breadth
                 and
                 depth
                 ,
                 the
                 Content
                 in
                 Feet
                 :
              
               In
               the
               doing
               whereof
               ,
               you
               must
               conceive
               the
               10
               principal
               parts
               to
               be
               doubled
               ,
               and
               then
               10
               is
               called
               20
               ;
               and
               consequently
               6
               is
               called
               12
               ,
               the
               Point
               so
               often
               used
               ;
               and
               5
               is
               called
               10
               ,
               the
               Point
               used
               for
               Foot-measure
               .
            
             
               
                 1.
                 
                 To
                 find
                 how
                 many
                 Inches
                 makes
                 a
                 Foot
                 at
                 any
                 Squareness
                 .
              
            
             
               As
               the
               —
               side
               of
               the
               Square
               ,
               to
               =
               12
               ;
            
             
               So
               is
               the
               =
               side
               of
               the
               Square
               again
               ,
               to
               a
               —
               4th
               Number
               .
            
             
             
               Again
               ,
            
             
               As
               =
               12
               ,
               to
               that
               =
               4th
               Number
               ;
            
             
               So
               is
               =
               12
               ,
               to
               the
               —
               Number
               of
               Inches
               that
               goes
               to
               make
               a
               Foot
               of
               Timber
               .
            
             
               
                 Example
                 ,
                 at
                 8
                 Inches
                 Square
                 .
              
            
             
               Take
               the
               distance
               from
               the
               Center
               to
               4
               ,
               accounted
               as
               8
               ;
               and
               make
               it
               a
               Parallel
               in
               6
               ,
               counted
               as
               12
               ;
               or
               lay
               the
               Thred
               to
               the
               nearest
               distance
               ,
               and
               there
               keep
               it
               .
               Then
               ,
               take
               the
               nearest
               distance
               from
               4
               to
               the
               Thred
               ,
               and
               that
               shall
               be
               a
               Latteral
               4th
               .
            
             
               Then
               take
               the
               Latteral
               distance
               from
               the
               Center
               to
               12
               ,
               according
               to
               the
               usual
               account
               ,
               and
               make
               it
               a
               Parallel
               in
               the
               4th
               last
               found
               ,
               laying
               the
               Thred
               to
               the
               nearest
               distance
               ,
               and
               there
               keep
               it
               ;
               then
               take
               the
               nearest
               distance
               from
               6
               ,
               counted
               as
               12
               ,
               to
               the
               Thred
               ,
               and
               that
               shall
               reach
               Latterally
               from
               the
               Center
               to
               27
               Inches
               ,
               the
               length
               required
               ,
               to
               make
               a
               Foot
               of
               Timber
               ,
               at
               8
               Inches
               Square
               .
            
             
               
                 Which
                 work
                 I
                 more
                 briefly
                 word
                 thus
                 ,
                 as
                 formerly
                 is
                 done
                 .
              
            
             
               As
               —
               4
               ,
               counted
               as
               8
               ,
               to
               =
               6
               ,
               counted
               for
               12
               ;
            
             
               So
               is
               =
               8
               ,
               to
               a
               —
               4th
               .
            
             
             
               Then
               ,
            
             
               As
               =
               12
               ,
               to
               =
               4th
               ;
               So
               is
               =
               12
               ,
               to
               —
               27
               ,
               the
               length
               in
               inches
               required
               .
            
             
               
                 2.
                 
                 If
                 you
                 would
                 use
                 Foot-measure
                 ,
                 count
                 the
                 5
                 in
                 the
                 midst
                 for
                 10
                 ,
                 or
                 1
                 Foot
                 ;
                 and
                 work
                 all
                 the
                 rest
                 as
                 before
                 :
                 As
                 thus
                 for
                 Example
                 .
              
            
             
               
                 In
                 the
                 same
                 quantity
                 ,
                 Square
                 ,
                 exprest
                 in
                 Decimals
                 :
              
            
             
               As
               —
               0-666
               ,
               counted
               double
               ,
               to
               =
               5
               ,
               counted
               double
               for
               10
               ;
            
             
               So
               is
               =
               0-666
               ,
               to
               —
               22
               1
               /
               ●
               ,
               for
               a
               4th
               .
            
             
               Then
               ,
            
             
               As
               —
               1
               ,
               to
               =
               22
               1
               /
               ●
               ;
               So
               is
               =
               5
               counted
               for
               1
               ,
               to
               225
               ,
               which
               is
               2
               Foot
               ¼
               ,
               as
               by
               the
               Foot-measure
               and
               Inches
               you
               may
               see
               .
            
             
               
                 3.
                 
                 If
                 the
                 Piece
                 be
                 not
                 square
                 ,
                 then
                 say
                 thus
              
               ;
            
             
               As
               —
               breadth
               ,
               to
               =
               12
               ;
            
             
               So
               is
               the
               =
               depth
               ,
               to
               the
               —
               4th
               .
            
             
               Then
               ,
            
             
               As
               —
               12
               ,
               to
               the
               =
               4th
               ;
            
             
               So
               is
               =
               12
               ,
               to
               —
               length
               that
               goes
               to
               make
               1
               Foot.
               
            
             
             
               
                 Example
                 ,
                 at
                 9
                 Inches
                 ,
                 and
                 4
                 Inches
                 ,
                 for
                 breadth
                 and
                 depth
                 .
              
            
             
               As
               —
               9
               ,
               to
               =
               12
               ;
            
             
               So
               is
               =
               4
               ,
               to
               —
               150
               ,
               for
               a
               4th
               .
            
             
               As
               —
               12
               ,
               to
               =
               4th
               ,
               best
               taken
               at
               75
               for
               largeness
               sake
               ;
            
             
               So
               is
               =
               12
               ,
               to
               —
               48
               Inches
               .
            
             
               
                 Or
                 else
                 thus
              
               ;
            
             
               As
               —
               9
               ,
               to
               =
               1
               ;
            
             
               So
               is
               =
               4
               ,
               to
               —
               1-80
               ,
               a
               4th
               .
            
             
               Then
               ,
            
             
               As
               —
               12
               ,
               to
               =
               1-80
               ;
            
             
               So
               is
               =
               12
               ,
               to
               —
               4
               Foot
               ,
               the
               length
               in
               Feet
               ,
               that
               goes
               to
               make
               1
               Foot
               of
               Timber
               .
            
             
               
                 4.
                 
                 To
                 find
                 how
                 much
                 is
                 in
                 a
                 Foot-long
                 ,
                 at
                 any
                 Squareness
                 .
              
            
             
               As
               the
               —
               side
               of
               the
               Square
               is
               to
               =
               1
               ,
               counted
               double
               as
               before
               ;
            
             
               So
               is
               the
               =
               side
               of
               the
               Square
               to
               the
               —
               quantity
               in
               a
               Foot.
               
            
             
             
               
                 Example
                 at
                 6
                 Inches
                 ,
                 or
                 (
                 5
                 )
                 half
                 ●
                 Foot
                 Square
                 .
              
            
             
               As
               —
               5
               ,
               to
               =
               1
               ;
               so
               is
               =
               5
               ,
               to
               —
               125
               for
               Foot-measure
               :
               Or
               ,
            
             
               As
               —
               6
               ,
               to
               =
               12
               ;
               so
               is
               =
               6
               ,
               to
               —
               3
               ,
               for
               3
               inches
               ,
               or
               ¼
               of
               a
               foot
               .
            
             
               
                 5.
                 
                 The
                 side
                 of
                 the
                 Square
                 given
                 in
                 Inches
                 ,
                 and
                 the
                 length
                 in
                 Feet
                 ,
                 to
                 find
                 the
                 Content
                 in
                 Feet
                 .
              
            
             
               As
               —
               side
               of
               the
               Square
               ,
               to
               =
               12
               ;
            
             
               So
               is
               the
               =
               length
               to
               a
               4th
               .
            
             
               Then
               ,
            
             
               As
               —
               4th
               ;
               to
               =
               12
               ;
               so
               is
               =
               side
               of
               the
               Square
               to
               —
               Content
               required
               ,
               in
               feet
               and
               parts
               .
            
             
               
                 Example
                 ,
                 at
                 9
                 Inches
                 Square
                 ,
                 and
                 16
                 Foot
                 long
                 .
              
            
             
               As
               —
               9
               to
               =
               12
               ,
               so
               is
               =
               16
               to
               —
               12.
               
            
             
               Again
               ,
            
             
               As
               —
               4th
               ,
               viz.
               12
               ,
               to
               =
               12
               ;
            
             
               So
               is
               =
               9
               ,
               the
               square
               to
               9
               ,
               the
               true
               Content
               of
               such
               a
               Piece
               in
               feet
               and
               parts
               required
               .
            
             
             
               The
               like
               Work
               serves
               for
               Foot-measure
               ,
               using
               of
               1.
               
            
             
               
                 6.
                 
                 The
                 Length
                 given
                 in
                 Feet
                 ,
                 〈◊〉
                 the
                 Breadth
                 and
                 Depth
                 in
                 Inches
                 ,
                 to
                 find
                 the
                 Content
                 in
                 feet
                 and
                 parts
                 .
              
            
             
               As
               —
               breadth
               ,
               to
               =
               12
               ;
            
             
               So
               is
               =
               depth
               ,
               to
               a
               4th
               .
            
             
               Then
               ,
            
             
               As
               —
               4th
               ,
               to
               =
               12
               ;
            
             
               So
               is
               =
               length
               ,
               to
               —
               Content
               in
               feet
               .
            
             
               
                 Example
                 at
                 5
                 Inches
                 and
                 a
                 half
                 Deep
                 ,
                 and
                 15
                 Inches
                 Broad
                 ,
                 and
                 16
                 Foot
                 Long.
                 
              
            
             
               As
               —
               5
               ½
               ,
               to
               =
               12
               ;
            
             
               So
               is
               =
               15
               ,
               to
               —
               69
               for
               a
               4th
               (
               at
               34
               ½
               )
            
             
               Then
               ,
            
             
               As
               —
               69
               (
               or
               34
               ½
               )
               that
               —
               4th
               ,
               to
               =
               12
               ;
            
             
               So
               is
               =
               16
               ,
               taken
               at
               8
               ,
               to
               —
               9
               foot
               2
               inches
               ;
               the
               Content
               required
               .
            
             
               Thus
               you
               see
               the
               way
               and
               manner
               of
               working
               by
               the
               Line
               of
               Lines
               ,
               either
               on
               the
               Quadrant
               ,
               or
               Sector-side
               ,
               for
               the
               usual
               Questions
               ;
               for
               I
               have
               neglected
               to
               give
               the
               Content
               of
               Pieces
               in
               
                 Cube
                 Inches
              
               ,
               for
               two
               Reasons
               :
               First
               ,
               Because
               it
               is
               very
               seldom
               
               required
               .
               Secondly
               ,
               Because
               the
               Line
               of
               Numbers
               at
               most
               will
               shew
               but
               4
               figures
               ,
               which
               is
               not
               sufficient
               for
               any
               Piece
               above
               6
               Foot
               ,
               therefore
               not
               fit
               for
               Instrumental
               Work.
               
            
             
               And
               withal
               you
               may
               observe
               ,
               That
               alwayes
               the
               Latter●l
               Extent
               first
               taken
               ,
               must
               be
               less
               than
               the
               distance
               from
               the
               Center
               to
               the
               parallel
               Point
               of
               Entrance
               ,
               which
               in
               these
               Examples
               is
               remedied
               by
               calling
               6
               12
               :
               And
               also
               ,
               there
               are
               so
               many
               Cautions
               in
               doubling
               and
               halfing
               of
               Numbers
               ,
               to
               make
               it
               applicable
               ,
               that
               without
               due
               consideration
               ,
               you
               may
               soon
               err
               ;
               Also
               ,
               the
               opening
               and
               shutting
               the
               Rule
               ,
               and
               using
               of
               several
               Scales
               ,
               makes
               it
               far
               inferior
               to
               the
               Line
               of
               Numbers
               ,
               which
               may
               be
               easily
               enlarged
               .
            
          
        
         
           
           
             CHAP.
             X.
             To
             measure
             Round
             Timber
             ,
             or
             Cillenders
             ,
             by
             the
             Line
             of
             Numbers
             .
          
           
             
               Problem
               I.
               Having
               the
               Diameter
               of
               a
               Cillender
               ,
               given
               in
               Inches
               ,
               or
               Foot-measure
               ,
               to
               find
               the
               length
               of
               one
               Foot.
               
            
             
               1.
               
               AS
               the
               Diameter
               in
               inches
               ,
               to
               46-90
               ,
               (
               at
               which
               Diameter
               one
               Inch
               makes
               a
               Foot
               )
               ;
               So
               is
               1
               to
               a
               4th
               ,
               and
               that
               4th
               to
               the
               length
               in
               inches
               .
            
             
               
                 Example
                 at
                 10
                 Inches
                 Diameter
              
               .
            
             
               The
               Extent
               from
               10
               to
               46-90
               ,
               being
               turned
               two
               times
               the
               same
               way
               from
               1
               ,
               shall
               reach
               to
               21
               inches
               ,
               8
               10ths
               ,
               for
               the
               length
               of
               a
               Foot
               ,
               at
               that
               Diameter
               ,
               in
               Inches
               .
            
             
             
               
                 Or
                 rather
                 work
                 thus
              
               ;
            
             
               As
               the
               Inches
               Diameter
               ,
               to
               13-54
               ;
            
             
               So
               is
               12
               twice
               ,
               to
               the
               Inches
               that
               make
               a
               Foot
               of
               Timber
               .
            
             
               Or
               ,
            
             
               The
               Extent
               from
               10
               ,
               to
               13-54
               ,
               turned
               twice
               the
               same
               way
               from
               12
               ,
               shall
               reach
               to
               22
               Inches
               .
            
             
               Or
               ,
            
             
               The
               same
               Extent
               being
               turned
               two
               times
               the
               same
               way
               from
               1
               ,
               shall
               reach
               to
               1-831
               ,
               which
               is
               the
               Decimal
               for
               22
               Inches
               ,
               as
               by
               looking
               on
               Inch
               and
               Foot
               measure
               ,
               you
               may
               plainly
               see
               .
            
             
               Again
               ,
            
             
               2.
               
               For
               the
               same
               Diameter
               in
               Foot-measure
               .
            
             
               The
               Extent
               from
               0-833
               (
               the
               Decimal
               of
               10
               Inches
               )
               to
               1-128
               ,
               being
               turned
               twice
               the
               same
               way
               from
               1
               ,
               shall
               reach
               to
               1-83
               ,
               which
               is
               almost
               22
               Inches
               ,
               as
               by
               comparing
               Inches
               and
               Foot-measure
               together
               ,
               is
               plainly
               seen
               .
            
          
           
             
             
               Problem
               II.
               Having
               the
               Diameter
               given
               in
               Inches
               ,
               or
               Foot-measure
               ,
               to
               find
               how
               much
               is
               in
               a
               Foot
               long
               .
            
             
               1.
               
               As
               13-54
               (
               the
               Inches
               Diameter
               that
               make
               a
               Foot
               of
               Timber
               ,
               at
               one
               Foot
               long
               )
               ,
               to
               the
               Diameter
               in
               Inches
               ;
            
             
               So
               is
               12
               to
               a
               4th
               ,
               and
               so
               is
               that
               4th
               ,
               to
               the
               quantity
               in
               a
               Foot
               long
               .
            
             
               
                 Example
                 at
                 10
                 Inches
                 Diameter
              
               .
            
             
               The
               Extent
               from
               13-54
               to
               10
               ,
               being
               repeated
               two
               times
               the
               same
               way
               from
               12
               ,
               shall
               reach
               to
               6
               Inches
               ½
               ,
               or
               ,
               54
               of
               100
               ,
               being
               somewhat
               more
               than
               a
               half
               Foot
               ,
               for
               the
               true
               Content
               of
               one
               Foot
               long
               .
            
             
               2.
               
               But
               if
               the
               Timber
               is
               great
               ,
               then
               it
               is
               more
               convenient
               to
               have
               the
               quantity
               of
               a
               Foot
               ,
               in
               feet
               and
               parts
               .
            
             
               
                 Then
                 say
              
               ,
            
             
               As
               13-54
               ,
               is
               to
               the
               Diameter
               in
               Inches
               ;
            
             
               So
               is
               1
               ,
               to
               a
               4th
               ,
               and
               that
               4th
               to
               the
               quantity
               in
               a
               Foot
               ,
               in
               feet
               and
               parts
               .
            
             
               
                 Example
                 ,
                 as
                 before
                 ,
                 at
                 10
                 Inches
                 .
              
            
             
               The
               Extent
               from
               13-54
               to
               10
               ,
               the
               Diameter
               
               in
               Inches
               ,
               shall
               reach
               ,
               being
               turned
               twice
               the
               same
               way
               from
               1
               ,
               to
               0-545
               ,
               the
               Content
               of
               a
               Foot
               long
               .
            
             
               
                 Again
                 at
                 30
                 inches
                 Diameter
              
               .
            
             
               The
               Extent
               from
               13-54
               ,
               to
               30
               ,
               being
               turned
               two
               times
               the
               same
               way
               from
               1
               ,
               shall
               reach
               to
               4
               foot
               ,
               93
               parts
               ;
               which
               4-93
               multiplied
               by
               the
               length
               in
               feet
               ,
               shall
               give
               the
               whole
               Content
               of
               the
               Tree
               .
            
             
               3.
               
               To
               perform
               the
               same
               ,
               having
               the
               Diameter
               given
               in
               Foot-measure
               ,
            
             
               
                 Do
                 thus
              
               ;
            
             
               The
               Extent
               of
               the
               Compasses
               from
               1-128
               ,
               (
               the
               feet
               and
               10ths
               Diameter
               that
               make
               a
               Foot
               ,
               at
               one
               foot
               in
               length
               )
               to
               the
               Diameter
               in
               Foot-measure
               ,
               shall
               reach
               ,
               being
               turned
               twice
               the
               same
               way
               from
               1
               ,
               to
               the
               quantity
               in
               a
               Foot
               long
               .
            
             
               
                 Example
                 at
                 1
                 Foot
                 ,
                 
                   50
                   /
                   100
                
                 Diameter
                 .
              
            
             
               The
               Extent
               from
               1-128
               ,
               to
               1-50
               ,
               shall
               reach
               ,
               being
               turned
               twice
               the
               same
               way
               from
               1
               ,
               to
               1-77
               ,
               the
               true
               quantity
               in
               one
               Foot
               long
               .
            
          
           
             
             
               Problem
               III.
               1.
               
               The
               Diameter
               of
               any
               Cillender
               given
               in
               Inches
               ,
               and
               the
               length
               in
               Feet
               ,
               to
               find
               the
               Content
               in
               Feet
               .
            
             
               As
               13-●4
               ,
               to
               the
               Diameter
               in
               Inches
               ;
            
             
               So
               is
               the
               length
               in
               Feet
               to
               a
               4th
               .
            
             
               Then
               ,
            
             
               As
               the
               length
               ,
               to
               the
               4th
               ;
            
             
               So
               is
               the
               4th
               ,
               to
               the
               Content
               in
               Feet
               required
               .
            
             
               
                 Example
                 at
                 8
                 Inches
                 Diameter
                 ,
                 and
                 20
                 Foot
                 long
                 .
              
            
             
               The
               Extent
               from
               13-54
               ,
               to
               8
               ,
               being
               turned
               twice
               the
               same
               way
               from
               20
               ,
               the
               length
               ,
               shall
               stay
               at
               6-94
               ,
               or
               near
               7
               foot
               .
            
             
               
                 2.
                 
                 The
                 Diameter
                 and
                 length
                 of
                 a
                 Cillender
                 given
                 in
                 Inches
                 ,
                 to
                 find
                 the
                 Content
                 in
                 Cube-inches
                 .
              
            
             
               The
               Extent
               from
               1-128
               ,
               to
               the
               Diameter
               in
               Inches
               ,
               being
               turned
               twice
               the
               same
               way
               from
               the
               length
               in
               Inches
               ,
               shall
               reach
               to
               the
               Content
               in
               Inches
               .
            
             
               Thus
               the
               Extent
               from
               1-128
               to
               10
               inches
               Diameter
               ,
               shall
               reach
               from
               24
               inches
               ,
               the
               
               length
               ,
               to
               1888
               ,
               the
               Content
               in
               inches
               .
            
             
               
                 3.
                 
                 The
                 Diameter
                 and
                 Length
                 given
                 in
                 Foot-measure
                 ,
                 to
                 find
                 the
                 Content
                 in
                 Feet
                 .
              
            
             
               The
               Extent
               from
               1-128
               ,
               to
               the
               Diameter
               ,
               shall
               reach
               from
               the
               length
               ,
               being
               twice
               repeated
               the
               same
               way
               ,
               to
               the
               Content
               in
               feet
               required
               .
            
             
               Thus
               the
               Extent
               from
               1-128
               ,
               to
               1-50
               ,
               shall
               reach
               ,
               being
               turned
               twice
               the
               same
               way
               ,
               from
               5-30
               ,
               to
               9-37
               ,
               the
               Content
               in
               feet
               required
               .
            
          
           
             
               Problem
               IV.
               Having
               the
               Circumference
               of
               a
               Cillender
               given
               in
               Inches
               ,
               or
               Foot-measure
               ,
               to
               find
               the
               length
               that
               makes
               one
               Foot
               of
               Solid-measure
               .
            
             
               
                 1.
                 
                 First
                 to
                 find
                 the
                 Inches
                 in
                 length
                 ,
                 that
                 makes
                 a
                 Foot.
                 
              
            
             
               As
               the
               Circumference
               in
               Inches
               ,
               is
               to
               134-50
               ,
               (
               because
               at
               so
               many
               inches
               about
               ,
               one
               of
               a
               Foot
               in
               length
               ,
               is
               a
               Foot
               )
               so
               is
               12
               to
               a
               4th
               ,
               and
               so
               is
               that
               4th
               to
               the
               length
               of
               a
               Foot
               in
               inches
               .
            
             
             
               
                 Example
                 at
                 30
                 Inches
                 about
              
               .
            
             
               The
               Extent
               from
               30
               to
               134-50
               ,
               being
               turned
               twice
               the
               same
               way
               from
               12
               ,
               shall
               reach
               to
               24
               inches
               ,
               13
               parts
               ;
               the
               inches
               and
               parts
               that
               make
               one
               Foot
               Solid
               .
            
             
               
                 2.
                 
                 To
                 find
                 the
                 length
                 of
                 a
                 Foot
                 in
                 feet
                 and
                 parts
                 .
              
            
             
               As
               the
               Circumference
               in
               Inches
               ,
               to
               134-50
               ;
            
             
               So
               is
               1
               to
               a
               4th
               ,
               and
               that
               4th
               to
               the
               length
               in
               feet
               and
               parts
               ,
               that
               makes
               1
               Foot.
               
            
             
               For
               the
               Extent
               of
               the
               Compasses
               from
               30
               to
               134-50
               ,
               being
               turned
               twice
               from
               1
               ,
               the
               same
               way
               ,
               shall
               reach
               to
               two
               foot
               ,
               and
               one
               tenth
               ,
               the
               length
               that
               makes
               one
               Foot
               Solid
               .
            
             
               
                 3.
                 
                 When
                 the
                 Circumference
                 is
                 given
                 in
                 Foot-measure
                 .
              
            
             
               As
               the
               Circumference
               in
               Feet
               ,
               or
               Feet
               and
               parts
               ,
               is
               to
               3-54
               ;
            
             
               So
               is
               that
               Extent
               twice
               repeated
               the
               same
               way
               from
               1
               ,
               to
               the
               length
               that
               makes
               a
               Foot
               Solid
               .
            
             
             
               Example
               .
            
             
               The
               Extent
               from
               2-50
               ,
               to
               3-54
               ,
               being
               turned
               two
               times
               the
               same
               way
               from
               1
               ,
               doth
               reach
               to
               2
               foot
               ,
               001
               ,
               the
               length
               in
               Foot-measure
               .
            
          
           
             
               Problem
               V.
               The
               Circumference
               given
               in
               Inches
               ,
               or
               Foot-measure
               ,
               to
               find
               how
               much
               is
               in
               a
               Foot
               long
               .
            
             
               
                 1.
                 
                 The
                 Circumference
                 of
                 a
                 Tree
                 ,
                 when
                 one
                 Foot
                 long
                 makes
                 a
                 Foot
                 of
                 Timber
                 .
              
            
             
               As
               3
               foot
               ,
               545
               parts
               ,
               to
               the
               feet
               about
               ;
            
             
               So
               is
               1
               foot
               to
               a
               4th
               ,
               and
               that
               4th
               to
               the
               solid
               Content
               in
               one
               foot
               long
               .
            
             
               Example
               .
            
             
               The
               Extent
               of
               the
               Compasses
               from
               3-545
               ,
               to
               2-50
               ,
               the
               feet
               about
               ,
               shall
               reach
               ,
               being
               turned
               twice
               the
               same
               way
               from
               1
               ,
               to
               0-497
               ,
               the
               quantity
               in
               a
               foot
               long
               ,
               viz.
               near
               half
               a
               foot
               .
            
             
               
                 2.
                 
                 The
                 Circumference
                 given
                 in
                 Inches
                 ,
                 to
                 find
                 the
                 Content
                 of
                 one
                 Foot
                 in
                 length
                 ,
                 Solid-measure
                 ,
                 in
                 Inches
                 .
              
            
             
               
                 The
                 Inches
                 a
                 Tree
                 is
                 about
                 ,
                 when
                 one
                 10th
                 of
                 a
                 Foot
                 in
                 length
                 ,
                 makes
                 a
                 Foot
                 of
                 Timber
                 in
                 quantity
                 .
              
            
             
               As
               134-5
               ,
               to
               the
               Inches
               about
               ;
            
             
               So
               is
               12
               to
               a
               4th
               ,
               and
               that
               4th
               to
               the
               Content
               of
               one
               foot
               long
               .
            
             
             
               
                 Example
                 at
                 30
                 inches
                 about
              
               .
            
             
               Th●
               Extent
               from
               134-5
               ,
               to
               30
               ,
               being
               turned
               two
               times
               from
               12
               ,
               shall
               reach
               to
               near
               6
               inches
               for
               the
               Content
               of
               one
               foot
               long
               ,
               at
               30
               inches
               about
               .
            
             
               
                 3.
                 
                 The
                 Circumference
                 of
                 a
                 Cillend●r
                 given
                 in
                 Inches
                 ,
                 to
                 find
                 the
                 quantity
                 of
                 one
                 Foot
                 long
                 in
                 feet
                 and
                 inches
                 .
              
            
             
               As
               134-5
               ,
               to
               the
               Circumference
               ;
            
             
               So
               is
               1
               to
               a
               4th
               ,
               and
               that
               4th
               to
               the
               quantity
               of
               one
               foot
               long
               in
               Feet
               and
               Inches
               .
            
             
               The
               Extent
               from
               134-5
               ,
               to
               30
               ,
               being
               twice
               repeated
               the
               same
               way
               from
               1
               ,
               shall
               reach
               to
               0-497
               ,
               or
               near
               half
               a
               foot
               ,
               the
               Content
               of
               one
               foot
               long
               ,
               at
               that
               Circumference
               ,
               which
               being
               multiplied
               by
               the
               length
               in
               feet
               ,
               gives
               the
               true
               Content
               of
               any
               Cillender
               whatsoever
               .
            
          
           
             
             
               Problem
               VI.
               The
               Circumference
               ,
               and
               length
               of
               any
               Cillender
               given
               in
               Inches
               ,
               or
               Feet
               and
               Inches
               ,
               to
               find
               the
               Content
               .
            
             
               
                 1.
                 
                 The
                 Circumference
                 given
                 in
                 Inches
                 ,
                 and
                 the
                 length
                 in
                 Feet
                 ,
                 to
                 find
                 the
                 Content
                 in
                 feet
                 and
                 parts
                 .
              
            
             
               As
               42-54
               (
               the
               Circumference
               in
               Inches
               ,
               that
               makes
               1
               foot
               long
               ,
               a
               Foot
               )
               is
               to
               the
               Inches
               in
               Circumference
               ;
            
             
               So
               is
               the
               length
               in
               Feet
               to
               a
               4th
               ,
               and
               that
               4th
               to
               the
               Content
               in
               Feet
               .
            
             
               Example
               .
            
             
               The
               Extent
               from
               42-54
               ,
               to
               48
               the
               inches
               about
               ,
               being
               twice
               repeated
               from
               12
               foot
               the
               length
               ,
               shall
               reach
               to
               15-28
               ,
               the
               Content
               in
               feet
               required
               .
            
             
               
                 2.
                 
                 The
                 Circumference
                 and
                 length
                 given
                 in
                 Feet
                 ,
                 to
                 find
                 the
                 Content
                 in
                 feet
                 and
                 parts
                 .
              
            
             
               As
               3-545
               ,
               (
               because
               at
               3
               foot
               and
               a
               half
               about
               ,
               and
               a
               foot
               in
               length
               ,
               is
               a
               Foot
               )
               is
               to
               the
               Circumference
               ;
            
             
               So
               is
               the
               length
               in
               Feet
               to
               a
               4th
               ,
               and
               that
               4th
               to
               the
               Content
               in
               Foot-measure
               .
            
             
             
               Example
               .
            
             
               The
               Extent
               from
               3-545
               ,
               to
               4-0
               ,
               the
               Circumference
               ,
               being
               turned
               two
               times
               from
               12
               foot
               the
               length
               ,
               shall
               reach
               to
               15-28
               ,
               the
               Content
               in
               feet
               required
               .
            
             
               
                 3.
                 
                 The
                 Circumference
                 and
                 length
                 given
                 in
                 Inches
                 ,
                 to
                 find
                 the
                 Content
                 in
                 Inches
                 .
              
            
             
               As
               3-545
               ,
               to
               the
               Circumference
               in
               Inches
               ;
            
             
               So
               is
               the
               length
               in
               Inches
               to
               a
               4th
               :
            
             
               Then
               ,
            
             
               As
               the
               length
               to
               that
               4th
               ;
            
             
               So
               is
               the
               4th
               ,
               to
               the
               Content
               in
               Cube-Inches
               .
            
             
               Example
               .
            
             
               The
               precise
               Extent
               on
               a
               true
               Line
               of
               Numbers
               ,
               from
               3-545
               ,
               to
               48
               ,
               being
               turned
               two
               times
               from
               144
               ,
               the
               length
               in
               Inches
               ,
               shall
               reach
               to
               26383
               ,
               the
               number
               of
               Inches
               ,
               in
               a
               Tree
               48
               inches
               about
               ,
               and
               144
               inches
               in
               length
               .
            
             
               This
               is
               sufficient
               for
               the
               Mensuration
               of
               any
               solid
               body
               ,
               in
               a
               square
               ,
               or
               Cillender-like
               form
               ,
               as
               Timber
               or
               Stone
               usually
               is
               ,
               after
               the
               true
               quantity
               of
               a
               foot
               ,
               or
               1728
               Cubical
               inches
               ;
               but
               there
               is
               a
               custome
               used
               in
               buying
               of
               Oaken-Timber
               ,
               and
               Elm-Timber
               ,
               when
               it
               is
               round
               and
               unsquared
               ,
               to
               take
               
               a
               Line
               ,
               and
               girt
               about
               the
               midst
               of
               th●
               Piece
               ;
               and
               then
               to
               double
               the
               Line
               4
               times
               ,
               and
               account
               that
               4th
               part
               of
               the
               Circumference
               ,
               to
               be
               the
               side
               of
               the
               Square
               ,
               equal
               to
               that
               Circle
               ;
               but
               this
               is
               well
               known
               to
               be
               less
               than
               the
               true
               measure
               ,
               by
               a
               fifth
               part
               of
               the
               true
               Content
               ,
               be
               it
               more
               or
               less
               .
            
             
               Also
               in
               measuring
               Elm
               ,
               and
               Beech
               ,
               and
               Ash
               ,
               whose
               bark
               is
               not
               peeled
               off
               ,
               as
               Oak
               usually
               is
               ;
               to
               cast
               away
               1
               inch
               out
               of
               the
               4th
               part
               of
               the
               Circumference
               ,
               which
               may
               well
               be
               allowed
               when
               the
               Bark
               is
               3
               quarters
               of
               an
               inch
               ,
               or
               more
               in
               thickness
               ,
               and
               the
               Tree
               about
               40
               inches
               about
               ,
               or
               the
               4th
               part
               ,
               10
               inches
               ;
               but
               if
               the
               Bark
               is
               thinner
               ,
               and
               the
               Tree
               less
               ,
               then
               8
               inches-square
               ;
               then
               an
               inch
               is
               too
               much
               to
               be
               allowed
               .
            
             
               Also
               ,
               if
               the
               Tree
               is
               greater
               than
               a
               foot-square
               ,
               and
               the
               Bark
               thick
               ,
               an
               inch
               is
               too
               little
               to
               be
               allowed
               ,
               as
               by
               this
               Rule
               you
               may
               plainly
               see
               ,
               by
               the
               7
               
                 th
                 Problem
              
               of
               Superficial-measure
               in
               the
               7
               
                 th
                 Chapter
              
               .
            
             
               Suppose
               a
               Tree
               be
               48
               inches
               about
               ,
               the
               Diameter
               will
               be
               15
               ¼
               ,
               the
               4th
               of
               48
               ,
               for
               the
               square
               is
               12.
               
            
             
               Now
               if
               I
               take
               away
               1
               inch
               ½
               from
               the
               Diameter
               ,
               then
               the
               Tree
               will
               be
               but
               43
               inches
               and
               1
               /
               ●
               about
               ,
               whose
               4th
               part
               is
               under
               11
               ;
               so
               that
               here
               I
               may
               very
               well
               abate
               1
               inch
               from
               the
               4th
               part
               of
               the
               Line
               ;
               
               So
               consequently
               ,
               if
               the
               Rind
               be
               thinner
               ,
               and
               the
               Tree
               less
               ,
               a
               less
               allowance
               will
               serve
               ;
               and
               if
               the
               Rind
               be
               thicker
               ,
               and
               the
               Tree
               large
               ,
               there
               ought
               to
               be
               more
               ,
               as
               by
               cutting
               the
               Rind
               away
               ,
               and
               then
               taking
               the
               true
               diameter
               ,
               you
               may
               plainly
               see
               .
            
             
               This
               measuring
               by
               the
               4th
               part
               of
               the
               Circumference
               ,
               for
               the
               side
               of
               the
               Square
               ,
               and
               allowance
               for
               the
               Bark
               being
               allowed
               for
               ,
               as
               before
               ,
               I
               say
               will
               prove
               to
               be
               just
               one
               5th
               part
               over-measure
               .
            
             
               Especially
               considering
               this
               ,
               That
               when
               it
               is
               hewed
               ,
               and
               large
               wanes
               left
               ,
               then
               the
               Tree
               is
               marked
               for
               more
               measure
               ,
               sometimes
               by
               10
               foot
               in
               60
               ,
               than
               there
               was
               before
               it
               was
               hewed
               ;
               the
               reason
               is
               ,
               because
               when
               the
               Tree
               is
               round
               and
               unhewn
               ,
               the
               girting
               it
               ,
               and
               counting
               the
               4th
               part
               for
               the
               side
               of
               the
               Square
               ,
               is
               but
               very
               little
               more
               than
               the
               inscribed
               Square
               ;
               and
               then
               being
               hewen
               ,
               and
               that
               scarce
               to
               an
               eight
               Square
               ,
               and
               measuring
               with
               a
               pair
               of
               Callipers
               ,
               to
               the
               extremity
               of
               that
               ,
               doth
               not
               then
               allow
               the
               Square
               equal
               to
               the
               Circle
               for
               the
               side
               of
               the
               Square
               ,
               as
               by
               the
               working
               by
               those
               several
               Squares
               ,
               will
               very
               plainly
               appear
               ,
               which
               being
               foretold
               and
               warned
               of
               ,
               let
               those
               whom
               it
               concerns
               look
               to
               it
               .
            
             
               But
               this
               being
               premised
               ,
               and
               the
               Parties
               agreeing
               ,
               the
               difference
               being
               as
               4
               to
               5
               ,
               the
               
               best
               way
               to
               measure
               round
               Timber
               ,
               I
               conceive
               ,
               is
               by
               the
               Diameter
               taken
               with
               a
               pair
               of
               Callipers
               ,
               and
               the
               length
               ;
               which
               for
               the
               just
               and
               true
               measure
               is
               largely
               handled
               already
               .
            
             
               
                 But
                 if
                 this
                 allowance
                 be
                 agreed
                 on
                 ,
                 then
                 the
                 Proportion
                 for
                 it
                 is
                 thus
              
               ;
            
             
               As
               1-526
               ,
               to
               the
               Diameter
               ;
            
             
               So
               is
               the
               length
               to
               a
               4th
               ,
               and
               so
               is
               that
               4th
               to
               the
               Content
               in
               feet
               .
            
             
               Example
               .
            
             
               The
               Extent
               from
               1-526
               to
               15-26
               ,
               shall
               reach
               ,
               being
               twice
               repeated
               from
               10
               foot
               ,
               the
               length
               ,
               to
               10
               foot
               the
               Content
               required
               ,
               being
               all
               at
               one
               Point
               .
            
             
               
                 Or
                 ,
                 another
                 Example
              
               .
            
             
               The
               Extent
               from
               1-526
               ,
               to
               20
               inches
               the
               Diameter
               ,
               being
               twice
               repeated
               the
               same
               way
               from
               10
               foot
               ,
               the
               length
               ,
               shall
               reach
               to
               17
               foot
               ¼
               the
               Content
               .
            
             
               
                 Or
                 ,
                 if
                 you
                 have
                 the
                 Circumference
                 and
                 length
                 .
              
            
             
               Then
               the
               Extent
               from
               48
               ,
               to
               the
               inches
               about
               ,
               being
               turned
               twice
               the
               same
               way
               from
               the
               length
               in
               feet
               ,
               shall
               reach
               to
               the
               Content
               required
               .
            
             
               The
               Extent
               from
               48
               ,
               to
               62
               ,
               the
               inches
               about
               ,
               being
               turned
               twice
               from
               10
               ,
               the
               same
               
               way
               ,
               shall
               reach
               to
               17
               foot
               ¼
               ,
               the
               Content
               in
               that
               measure
               .
            
             
               Thus
               you
               have
               full
               and
               compleat
               Directions
               for
               the
               measuring
               of
               any
               round
               Timber
               by
               the
               Line
               of
               Numbers
               ,
               by
               having
               the
               Diameter
               and
               length
               given
               ,
               after
               any
               usual
               manner
               ,
               there
               remains
               only
               one
               general
               and
               natural
               way
               ,
               by
               finding
               the
               base
               of
               the
               middle
               ,
               or
               one
               end
               ,
               by
               the
               7
               
                 th
                 Problem
              
               of
               Superficial
               measure
               ;
               and
               then
               to
               multiply
               that
               base
               by
               the
               length
               ,
               will
               give
               the
               true
               Content
               in
               feet
               or
               inches
               .
            
             
               Thus
               ,
            
             
               Having
               found
               the
               Base
               of
               the
               Cillender
               by
               the
               7th
               or
               10th
               Problem
               of
               Superficial-measure
               ;
               then
               if
               you
               multiply
               that
               Base
               being
               found
               in
               square
               inches
               ,
               by
               the
               length
               in
               inches
               you
               shall
               have
               the
               whole
               Content
               in
               Cube
               Inches
               .
            
             
               Example
               .
            
             
               Suppose
               a
               Cillender
               have
               10
               inches
               for
               its
               Diameter
               ,
               then
               by
               the
               7th
               or
               10th
               abovesaid
               ,
               you
               shall
               find
               the
               Base
               to
               be
               78-54
               ;
               then
               if
               you
               multiply
               78-54
               by
               80
               ,
               the
               supposed
               length
               in
               inches
               ,
               you
               shall
               find
               2356-20
               Cube
               Inches
               ,
               which
               divided
               by
               1728
               ,
               the
               inches
               in
               a
               Cube
               Foot
               ,
               sheweth
               how
               many
               feet
               there
               is
               ,
               &c.
               
               And
               as
               to
               the
               number
               of
               figures
               ,
               and
               the
               fractions
               cutting
               off
               ,
               you
               have
               ample
               Directions
               in
               the
               
                 first
                 
                 Problem
              
               ,
               and
               the
               
                 third
                 Section
              
               of
               the
               
                 six●
                 Chapter
              
               .
            
          
           
             
               Problem
               VII
               .
               How
               to
               measure
               a
               Pyramis
               ,
               or
               taper
               Timber
               ,
               or
               the
               Section
               of
               a
               Cone
               .
            
             
               
                 1.
                 
                 First
                 ,
                 get
                 the
                 Perpendiculer
                 length
                 of
                 the
                 Pyramis
                 or
                 Cone
                 ,
                 thus
              
               ;
            
             
               Multiply
               half
               the
               Diameter
               of
               the
               Base
               ,
               AB
               ,
               by
               it self
               ;
               then
               measure
               the
               side
               AD
               ,
               and
               multiply
               that
               by
               it self
               ;
               then
               take
               the
               lesser
               Square
               out
               of
               the
               greater
               ,
               and
               the
               Square
               root
               of
               the
               residue
               is
               the
               Perpendiculer
               Altitude
               required
               ,
               viz.
               DB.
               
            
             
               Example
               .
            
             
               Suppose
               the
               half
               Diameter
               of
               the
               Base
               AC
               ,
               were
               10-25
               ,
               and
               the
               side
               DA
               100
               ,
               AB
               10-25
               ,
               and
               10-25
               multiplied
               together
               ,
               called
               Squaring
               ,
               makes
               105
               ,
               0625
               ;
               DC
               100
               ,
               multiplied
               by
               100
               ,
               called
               Squaring
               ,
               makes
               10000
               ;
               then
               the
               lesser
               Square
               105
               ,
               0625
               ,
               taken
               out
               of
               10000
               ,
               the
               greater
               Square
               ,
               the
               remainder
               is
               9894
               ,
               9375
               ,
               whose
               square
               Root
               found
               by
               the
               8
               
                 th
                 Problem
              
               of
               the
               
                 sixt
                 Chapter
              
               ,
               is
               99-475
               ,
               the
               true
               length
               of
               the
               Line
               DB
               ,
               the
               length
               or
               height
               of
               the
               Cone
               .
            
             
               Then
               if
               you
               multiply
               the
               Area
               or
               Content
               of
               the
               Base
               AC
               20-5
               ,
               which
               by
               the
               
               7th
               or
               10th
               of
               Superficial
               measure
               is
               found
               to
               be
               160-08
               ,
               by
               33-158
               ,
               a
               third
               part
               of
               99-475
               ,
               the
               whole
               height
               makes
               5308
               ,
               cutting
               off
               the
               Fractions
               for
               the
               true
               Content
               of
               the
               Cone
               ,
               whose
               length
               is
               99
               inches
               ,
               and
               near
               a
               half
               ,
               and
               whose
               Base
               is
               20
               inches
               and
               a
               half
               Diameter
               .
            
             
               
                 2.
                 
                 Then
                 Secondly
                 ,
                 for
                 the
                 Segment
                 or
                 Section
                 of
                 a
                 Cone
                 ,
                 the
                 shape
                 or
                 form
                 of
                 all
                 round
                 taper
                 Timber
                 ,
                 the
                 truest
                 way
                 is
                 thus
              
               ;
            
             
               By
               the
               length
               and
               difference
               of
               Diameters
               ,
               find
               the
               whole
               length
               of
               the
               Cone
               ,
               which
               for
               all
               manner
               of
               Timber
               as
               it
               grows
               this
               way
               is
               near
               enough
               .
            
             
               
                 As
                 thus
              
               ;
            
             
               As
               the
               difference
               of
               the
               Diameters
               at
               the
               the
               two
               ends
               ,
               is
               to
               the
               length
               between
               the
               two
               ends
               ;
            
             
               So
               is
               the
               Diameter
               at
               the
               Base
               ,
               to
               the
               whole
               length
               of
               the
               Cone
               .
            
             
               Example
               .
            
             
               The
               difference
               between
               the
               Diameters
               AC
               ,
               and
               EF
               ,
               is
               13-70
               ,
               the
               length
               ,
               AE
               is
               66-32
               .
               then
               the
               Extent
               on
               the
               Line
               of
               Numbers
               from
               13-70
               ,
               the
               difference
               of
               the
               Diameters
               ;
               to
               66-32
               ,
               the
               length
               between
               ,
               shall
               reach
               the
               same
               way
               from
               20-50
               ,
               the
               
               greater
               Diameter
               to
               99
               and
               better
               ,
               the
               length
               that
               makes
               up
               the
               Cone
               ,
               at
               that
               Angle
               o●
               Tapering
               in
               the
               Timber
               ;
               then
               if
               by
               the
               last
               Rule
               you
               measure
               it
               as
               a
               Cone
               of
               that
               length
               ,
               and
               also
               measure
               the
               little
               end
               or
               point
               at
               his
               length
               and
               diameter
               ;
               and
               then
               lastly
               ,
               this
               little
               Cone
               taken
               out
               of
               the
               great
               Cone
               ,
               there
               remains
               the
               true
               Content
               of
               the
               Taper-piece
               that
               was
               to
               be
               measured
               ,
               viz.
               5246-71
               ,
               when
               61-30
               ,
               the
               Content
               of
               the
               small
               Cone
               at
               the
               end
               ,
               is
               taken
               out
               of
               5308
               ,
               the
               Content
               of
               the
               whole
               Pyramid
               .
            
             
               
                 3.
                 
                 If
                 this
                 way
                 seem
                 too
                 troublesome
                 for
                 the
                 common
                 use
                 ,
                 then
                 use
                 this
                 ,
                 being
                 more
                 brief
                 :
              
            
             
               To
               the
               Content
               that
               is
               found
               out
               ,
               by
               the
               Diameter
               in
               the
               midst
               of
               the
               Timber
               ,
               and
               the
               length
               ,
               add
               the
               Content
               of
               a
               Piece
               found
               out
               ,
               by
               half
               the
               difference
               of
               Diameters
               ,
               and
               one
               third
               part
               of
               the
               length
               of
               the
               whole
               Piece
               ,
               and
               the
               sum
               of
               them
               two
               shall
               be
               the
               whole
               Content
               required
               .
            
             
               
                 4.
                 Or
                 else
              
               ;
               
            
             
               Divide
               the
               length
               of
               the
               Tree
               into
               4
               or
               5
               parts
               ,
               and
               measure
               the
               middle
               of
               each
               part
               severally
               ,
               and
               that
               cast
               up
               by
               his
               proper
               length
               ,
               shall
               give
               the
               Content
               of
               each
               Piece
               ;
               then
               the
               sum
               of
               the
               Contents
               of
               all
               the
               Pieces
               put
               together
               ,
               is
               the
               true
               Content
               
               of
               the
               whole
               Taper
               Piece
               ,
               very
               near
               .
            
             
               Note
               ,
               That
               this
               curiosity
               shall
               never
               need
               to
               be
               used
               ,
               but
               when
               you
               meet
               with
               Timber
               much
               Taper
               ,
               and
               Die-square
               ,
               or
               on
               a
               Contest
               or
               Wager
               ;
               for
               according
               to
               the
               usual
               way
               (
               and
               measure
               )
               of
               squaring
               the
               Timber
               ,
               it
               is
               well
               ,
               if
               the
               measure
               of
               the
               Square
               ,
               taken
               with
               Callipers
               from
               side
               to
               side
               ,
               in
               the
               middle
               of
               the
               length
               of
               the
               Piece
               ,
               will
               make
               amends
               for
               half
               the
               Timber
               which
               is
               wanting
               in
               the
               wany
               edges
               of
               your
               squared
               Timber
               ,
               and
               the
               knots
               or
               swellings
               ,
               &
               hollows
               of
               most
               round
               Timber
               ,
               may
               well
               ballance
               this
               over-measure
               found
               by
               the
               Diameter
               taken
               in
               the
               middle
               of
               the
               length
               of
               the
               Piece
               .
               But
               indeed
               for
               Masts
               of
               Ships
               and
               Yards
               ,
               being
               wrought
               true
               and
               smooth
               ,
               where
               the
               price
               of
               a
               Foot
               is
               considerable
               ,
               there
               exactness
               is
               requisite
               ,
               and
               necessary
               to
               be
               used
               ;
               and
               thus
               much
               for
               Solid-measure
               in
               Squares
               and
               Cillenders
               .
            
          
           
             
               Problem
               VIII
               .
               To
               measure
               Globes
               ,
               and
               roundish
               Figures
               .
            
             
               1.
               
               To
               measure
               a
               Sphear
               or
               Globe
               by
               Arithmetick
               ,
               the
               ancient
               way
               ,
               is
               to
               multiply
               the
               Diameter
               by
               it self
               ,
               and
               then
               that
               Product
               ,
               to
               multiply
               by
               the
               Diameter
               again
               ;
               which
               two
               multiplications
               is
               called
               Cubing
               
               of
               the
               Diameter
               ;
               then
               multiply
               this
               Cube
               by
               11
               ,
               and
               then
               divide
               this
               last
               Product
               by
               21
               ,
               and
               the
               Quotient
               shall
               be
               the
               Solid
               Content
               of
               the
               Sphear
               ,
               in
               such
               measure
               as
               the
               Diameter
               was
               .
            
             
               Example
               .
            
             
               Let
               a
               Sphear
               be
               to
               be
               measured
               ,
               whose
               Diameter
               is
               10
               inches
               :
               First
               ,
               10
               times
               10
               ,
               is
               100
               ;
               and
               10
               times
               100
               ,
               is
               1000
               ;
               the
               Cube
               of
               10
               ,
               that
               multiplied
               by
               11
               ,
               makes
               11000
               ;
               which
               being
               divided
               by
               21
               ,
               makes
               523-81
               ,
               for
               the
               Solid
               Content
               .
            
             
               
                 Which
                 by
                 the
                 Line
                 of
                 Numbers
                 ,
                 you
                 may
                 work
                 thus
              
               ;
            
             
               2.
               
               The
               Extent
               from
               1
               ,
               to
               the
               Diameter
               ,
               shall
               reach
               the
               same
               way
               from
               the
               Diameter
               to
               the
               Square
               of
               the
               Diameter
               .
            
             
               
                 Then
                 again
              
               ,
            
             
               The
               Extent
               from
               1
               ,
               to
               the
               Square
               of
               the
               Diameter
               ,
               shall
               reach
               the
               same
               way
               from
               the
               Diameter
               ,
               to
               the
               Cube
               of
               the
               Diameter
               .
            
             
               Then
               ,
            
             
               The
               Extent
               from
               1
               ,
               to
               the
               Cube
               of
               the
               Diameter
               ,
               shall
               reach
               the
               same
               way
               from
               11
               ,
               to
               the
               Product
               of
               the
               Cube
               of
               the
               Diameter
               ,
               multiplied
               by
               11.
               
            
             
               Lastly
               ,
               This
               Extent
               from
               21
               ,
               to
               this
               last
               Product
               ,
               shall
               reach
               the
               same
               way
               from
               1
               ,
               to
               the
               Solid
               Content
               of
               the
               Sphear
               required
               .
            
             
             
               
                 Or
                 else
                 more
                 briefly
                 thus
              
               ;
            
             
               3.
               
               The
               Extent
               from
               1
               ,
               to
               the
               Diameter
               ,
               being
               turned
               three
               times
               the
               same
               way
               from
               0-5238
               ,
               shall
               stay
               at
               the
               Solid
               Content
               of
               the
               Sphear
               ,
               or
               Globe
               ,
               required
               .
            
             
               
                 Example
                 at
                 12
                 Diameter
              
               .
            
             
               The
               Extent
               from
               1
               to
               12
               ,
               being
               turned
               three
               times
               the
               same
               way
               from
               0-5238
               ,
               shall
               reach
               to
               905-143
               ,
               the
               Solid
               Content
               required
               .
            
             
               
                 3.
                 
                 The
                 Diameter
                 given
                 ,
                 to
                 find
                 the
                 Superficial
                 Content
                 .
              
            
             
               Square
               the
               Diameter
               ,
               an●
               multiply
               that
               by
               3-1416
               ,
               and
               the
               Product
               is
               the
               Superficial
               Content
               .
            
             
               
                 Or
                 ,
                 by
                 the
                 Line
                 of
                 Numbers
              
               ;
            
             
               The
               Extent
               from
               1
               ,
               to
               the
               Diameter
               ,
               being
               turned
               twice
               the
               same
               way
               from
               3-141●
               ,
               shall
               reach
               to
               the
               Superficial
               Content
               ,
               of
               the
               out-side
               round
               about
               the
               Gobe
               ,
               viz.
               at
               12
               Diameter
               ,
               452-44
               .
            
             
               
                 4.
                 
                 Having
                 the
                 Superficial
                 Content
                 ,
                 to
                 find
                 the
                 Diameter
                 .
              
            
             
               The
               Extent
               from
               1
               to
               0-3183
               ,
               shall
               reach
               the
               same
               way
               from
               the
               Superficial
               Content
               ,
               to
               the
               Square
               of
               the
               Diameter
               ,
               whose
               
               Square-root
               is
               the
               Diameter
               required
               .
               As
               at
               452-44
               ,
               gives
               144.
               
            
             
               
                 5.
                 
                 Having
                 the
                 Solid
                 Content
                 ,
                 to
                 find
                 the
                 Diameter
                 of
                 a
                 Globe
                 .
              
            
             
               The
               Extent
               from
               1
               to
               1-90986
               ,
               shall
               reach
               from
               the
               Solid
               Content
               to
               the
               Cube
               of
               the
               Diameter
               ,
               as
               ●t
               905-143
               Solidity
               gives
               1728
               ,
               the
               Cube
               of
               12.
               
            
             
               
                 6.
                 
                 Having
                 a
                 Segment
                 of
                 a
                 Sphear
                 ,
                 to
                 find
                 the
                 Superficial
                 Content
                 .
              
            
             
               The
               Extent
               from
               1
               ,
               to
               the
               Chord
               of
               the
               half
               Segment
               ,
               shall
               reach
               ,
               being
               twice
               repeated
               ,
               from
               3-1416
               ,
               to
               the
               Superficial
               Content
               of
               the
               round
               part
               of
               the
               Segment
               ,
               ABC
               .
            
             
               Example
               .
            
             
               Let
               the
               Segment
               be
               the
               half
               Sphear
               ,
               ABC
               ;
               AC
               being
               12
               ,
               then
               BC
               which
               is
               the
               Chord
               of
               the
               Peripheria
               ,
               BC
               is
               8-485
               ,
               whose
               Square
               is
               72.
               
            
             
               Then
               ,
            
             
               The
               Extent
               of
               the
               Compasses
               from
               1
               ,
               to
               8-485
               ,
               being
               turned
               twice
               the
               same
               way
               from
               3-1416
               ,
               shall
               reach
               to
               220-22
               ,
               the
               Superficial
               Content
               of
               the
               round
               part
               of
               the
               Segment
               ,
               or
               half
               Sphear
               or
               Globe
               ,
               to
               which
               if
               you
               add
               the
               Content
               of
               the
               Circle
               or
               Base
               ,
               you
               have
               the
               whole
               Superficies
               round
               about
               .
            
             
             
               
                 7.
                 
                 To
                 find
                 the
                 Solid
                 Content
                 of
                 a
                 Segment
                 of
                 a
                 Globe
                 .
              
            
             
               First
               ,
               you
               must
               find
               the
               Diameter
               of
               tha●
               Sphear
               ,
               of
               which
               the
               given
               Segment
               to
               b●
               measured
               is
               part
               .
            
             
               Thus
               ;
            
             
               Add
               the
               Square
               of
               the
               Altitude
               ,
               and
               the
               Square
               of
               the
               Diameter
               of
               the
               Segment
               together
               ,
               and
               the
               sum
               divide
               by
               the
               Altitude
               of
               the
               Segment
               ,
               the
               Quotient
               shall
               be
               the
               whole
               Sphears
               Diameter
               .
            
             
               Then
               ,
            
             
               Taking
               the
               Altitude
               of
               the
               Segment
               given
               ,
               from
               the
               whole
               Diameter
               ,
               there
               remains
               the
               Altitude
               of
               the
               other
               Segment
               .
            
             
               Then
               ;
            
             
               Extend
               the
               Compasses
               from
               the
               whole
               Diameter
               of
               the
               Sphear
               ,
               to
               1
               ;
               the
               same
               Extent
               applied
               the
               same
               way
               from
               the
               Altitude
               of
               the
               given
               Segment
               ,
               shall
               reach
               to
               a
               4th
               Number
               ,
               on
               a
               Line
               of
               Artificial
               Solid
               Segments
               joyned
               to
               the
               Line
               of
               Numbers
               ,
               which
               4th
               Number
               keep
               .
            
             
               Then
               ,
            
             
             
             
               Example
               .
            
             
               Let
               the
               whole
               Diameter
               of
               a
               Sphear
               be
               14
               ,
               then
               the
               whole
               Solid
               Content
               by
               the
               former
               Rules
               ,
               you
               will
               find
               to
               be
               1437
               ⅓
               ,
               a
               Segment
               of
               that
               Sphear
               whose
               Altitude
               or
               Depth
               is
               4
               ,
               the
               Solidity
               is
               required
               .
            
             
               Extend
               the
               Compasses
               from
               14
               ,
               the
               whole
               Sphears
               Diameter
               ,
               to
               1
               ;
               that
               Extent
               applied
               the
               same
               way
               from
               4
               ,
               the
               Altitude
               of
               the
               Segment
               ,
               shall
               reach
               to
               2-86
               on
               the
               Numbers
               ,
               or
               to
               19-88
               ,
               on
               the
               Line
               of
               Solid
               Segments
               joyn'd
               to
               the
               Line
               of
               Numbers
               ,
               which
               19-88
               ,
               is
               the
               4th
               Number
               to
               be
               kept
               .
            
             
               
                 Then
                 secondly
              
               ,
            
             
               The
               Extent
               from
               1
               to
               1437
               ,
               the
               whole
               Content
               of
               the
               whole
               Sphear
               ,
               shall
               reach
               the
               same
               way
               from
               19-88
               ,
               to
               284
               2
               /
               ●
               ,
               the
               Content
               of
               the
               Segment
               required
               to
               be
               found
               .
            
             
               If
               you
               want
               the
               
                 Line
                 of
                 Segments
              
               ,
               the
               Table
               annexed
               will
               supply
               that
               defect
               :
            
             
               Thus
               ;
            
             
               Look
               for
               the
               4th
               Number
               ,
               found
               on
               the
               
                 Line
                 of
                 Numbers
              
               ,
               among
               the
               figures
               on
               the
               Table
               ,
               and
               the
               number
               answering
               it
               in
               the
               
                 first
                 Column
              
               ,
               is
               the
               Solid
               Segment
               ,
               or
               4th
               to
               be
               kept
               ;
               as
               here
               ,
               on
               the
               Numbers
               ,
               you
               find
               2-86
               ;
               seek
               2-86
               in
               the
               Table
               annexed
               ,
               
               and
               in
               the
               
                 first
                 Column
              
               ,
               you
               find
               near
               20
               fo●
               the
               4th
               in
               Segments
               .
            
             
               
                 8.
                 
                 To
                 perform
                 the
                 same
                 by
                 Arithmetick
                 after
                 the
                 way
                 set
                 forth
                 by
                 Mr.
                 
                   Thomas
                   Diggs
                   ,
                   1574.
                
                 
              
            
             
               
                 To
                 find
                 the
                 Superficial
                 Content
                 of
                 a
                 Globe
                 or
                 Sphear
                 .
              
            
             
               Multiply
               the
               Diameter
               by
               the
               Circumference
               ,
               the
               Product
               shall
               be
               the
               Superficial
               Content
               round
               about
               the
               Globe
               .
            
             
               
                 9.
                 
                 Or
              
               ,
            
             
               Multiply
               the
               Content
               of
               a
               Circle
               ,
               having
               like
               Diameter
               ,
               by
               4
               ,
               the
               Product
               shall
               be
               the
               Superficial
               Content
               .
            
             
               10.
               And
            
             
               If
               you
               multiply
               the
               Superficial
               Content
               ,
               by
               a
               6th
               part
               of
               the
               Diameter
               ,
               the
               Product
               shall
               be
               the
               Solid
               Content
               of
               the
               Sphear
               .
            
             
             
               
                 
                   A
                   Table
                   of
                   Segments
                   .
                
                 
                   
                     
                       Num.
                       Seg
                    
                  
                   
                     
                       Num.
                       Segm.
                    
                     
                  
                   
                     
                       Num.
                       Segm.
                    
                     
                  
                   
                     
                       Num.
                       Segm.
                    
                     
                  
                
                 
                   
                     059
                  
                   
                     335
                  
                   
                     506
                  
                   
                     679
                  
                
                 
                   
                     084
                  
                   
                     342
                  
                   
                     513
                  
                   
                     686
                  
                
                 
                   
                     104
                  
                   
                     349
                  
                   
                     520
                  
                   
                     694
                  
                
                 
                   
                     122
                  
                   
                     356
                  
                   
                     527
                  
                   
                     703
                  
                
                 
                   
                     5
                     137
                  
                   
                     30
                     363
                  
                   
                     55
                     534
                  
                   
                     80
                     712
                  
                
                 
                   
                     152
                  
                   
                     371
                  
                   
                     540
                  
                   
                     720
                  
                
                 
                   
                     164
                  
                   
                     378
                  
                   
                     547
                  
                   
                     728
                  
                
                 
                   
                     176
                  
                   
                     385
                  
                   
                     554
                  
                   
                     737
                  
                
                 
                   
                     188
                  
                   
                     392
                  
                   
                     560
                  
                   
                     746
                  
                
                 
                   
                     10
                     197
                  
                   
                     35
                     399
                  
                   
                     60
                     567
                  
                   
                     85
                     753
                  
                
                 
                   
                     207
                  
                   
                     406
                  
                   
                     574
                  
                   
                     763
                  
                
                 
                   
                     218
                  
                   
                     413
                  
                   
                     580
                  
                   
                     772
                  
                
                 
                   
                     228
                  
                   
                     420
                  
                   
                     587
                  
                   
                     782
                  
                
                 
                   
                     237
                  
                   
                     426
                  
                   
                     594
                  
                   
                     793
                  
                
                 
                   
                     15
                     245
                  
                   
                     40
                     433
                  
                   
                     65
                     601
                  
                   
                     90
                     803
                  
                
                 
                   
                     254
                  
                   
                     440
                  
                   
                     608
                  
                   
                     812
                  
                
                 
                   
                     263
                  
                   
                     447
                  
                   
                     615
                  
                   
                     825
                  
                
                 
                   
                     272
                  
                   
                     453
                  
                   
                     622
                  
                   
                     836
                  
                
                 
                   
                     280
                  
                   
                     460
                  
                   
                     629
                  
                   
                     848
                  
                
                 
                   
                     20
                     288
                  
                   
                     45
                     466
                  
                   
                     70
                     637
                  
                   
                     95
                     865
                  
                
                 
                   
                     297
                  
                   
                     473
                  
                   
                     644
                  
                   
                     878
                  
                
                 
                   
                     306
                  
                   
                     480
                  
                   
                     651
                  
                   
                     896
                  
                
                 
                   
                     314
                  
                   
                     487
                  
                   
                     658
                  
                   
                     916
                  
                
                 
                   
                     321
                  
                   
                     494
                  
                   
                     665
                  
                   
                     941
                  
                
                 
                   
                     25
                     328
                  
                   
                     50
                     500
                  
                   
                     75
                     672
                  
                   
                     100
                     1000
                  
                
              
            
             
             
               
                 11.
                 
                 For
                 the
                 Segment
                 work
                 thus
              
               ;
            
             
               Multiply
               the
               whole
               Superficial
               Content
               of
               the
               whole
               Globe
               ,
               by
               the
               Altitude
               of
               the
               Segment
               ,
               and
               divide
               the
               Product
               by
               the
               Sphears
               whole
               Diameter
               ,
               the
               Quotient
               shall
               be
               the
               Superficial
               Content
               of
               the
               Convexity
               or
               round
               part
               of
               the
               Segment
               .
            
             
               
                 12.
                 
                 But
                 for
                 the
                 Solid
                 Content
                 ,
                 work
                 thus
              
               ;
            
             
               First
               ,
               find
               the
               difference
               between
               the
               height
               of
               the
               Segment
               ,
               and
               the
               half
               Diameter
               of
               the
               Sphear
               ;
               then
               multiply
               this
               difference
               (
               being
               found
               by
               subtracting
               the
               less
               from
               the
               greater
               )
               by
               the
               Superficial
               Content
               of
               the
               Base
               of
               the
               Segment
               ,
               and
               the
               Product
               subtract
               from
               the
               Product
               of
               the
               Sphears
               semi-Diameter
               ,
               and
               the
               Convex
               Superficies
               of
               the
               Segment
               ;
               then
               a
               third
               part
               of
               the
               remainder
               shall
               be
               the
               Solid
               Content
               of
               the
               Segment
               required
               .
            
             
               
                 Example
                 as
                 before
              
               .
            
             
               The
               Sphears
               Diameter
               is
               14
               ,
               the
               Segments
               Altitude
               is
               4
               ,
               the
               Segments
               Altitude
               taken
               from
               7
               ,
               the
               half
               Diameter
               ,
               the
               remainder
               is
               3
               ,
               which
               multiplied
               by
               126
               ,
               the
               Superficial
               Content
               of
               the
               Base
               of
               the
               Segment
               ,
               makes
               378
               ;
               then
               having
               multiplied
               7
               ,
               the
               Sphears
               half
               Diameter
               ,
               by
               the
               Convex
               Superficies
               of
               
               the
               Segment
               176
               ,
               the
               Product
               is
               1232
               ,
               from
               which
               number
               take
               378
               ,
               the
               Product
               last
               found
               ,
               and
               the
               remainder
               is
               854
               ,
               whose
               third
               part
               284
               ⅔
               ,
               is
               the
               Solidity
               of
               the
               Segment
               required
               .
            
             
               There
               are
               other
               fragments
               of
               Sphears
               ,
               as
               Multiformed
               and
               Irregular
               ,
               Cones
               or
               Pyramids
               ,
               and
               Solid
               Angles
               ;
               but
               the
               Mensuration
               of
               these
               I
               shall
               not
               trouble
               my self
               ,
               nor
               the
               Learner
               with
               ,
               for
               whom
               I
               only
               write
               ,
               intending
               the
               Mensuration
               of
               things
               that
               may
               come
               in
               use
               only
               .
            
             
               
                 13.
                 
                 But
                 yet
                 to
                 conclude
                 this
                 Chapter
                 ,
                 take
                 these
                 Observations
                 along
                 with
                 you
                 ,
                 concerning
                 the
                 Proportion
                 of
                 a
                 Cube
                 ,
                 a
                 Prisma
                 ,
                 and
                 a
                 Pyramid
                 ,
                 a
                 Cillender
                 ,
                 Sphear
                 ,
                 and
                 Cone
                 ;
                 whose
                 Shapes
                 and
                 Proportions
                 are
                 as
                 in
                 the
                 Figures
                 .
              
            
             
               If
               a
               Cube
               be
               made
               or
               conceived
               ,
               whose
               side
               is
               12
               inches
               ,
               then
               the
               solidity
               thereof
               is
               1728
               Cube
               inches
               ;
               and
               a
               Prisma
               ,
               having
               the
               same
               Base
               and
               Altitude
               ,
               contains
               864
               Cube
               inches
               ;
               and
               a
               square
               Pyramis
               ,
               of
               the
               same
               Base
               and
               Altitude
               ,
               contains
               576
               Cube
               inches
               ;
               and
               a
               Trianguler
               Pyramid
               ,
               as
               before
               ,
               contains
               249-6
               Cube
               inches
               ;
               A
               Cillender
               contains
               1357
               5
               /
               7
               ,
               being
               the
               same
               Height
               and
               Diameter
               of
               12
               inches
               :
               A
               Sphear
               ,
               whose
               Axis
               is
               12
               inches
               ,
               contains
               
               905
               1
               /
               7
               Cube
               inches
               ;
               and
               a
               Cone
               ,
               of
               the
               same
               Diameter
               and
               Altitude
               ,
               contains
               452
               4
               /
               6.
               
            
             
               The
               Superficies
               of
               the
               Cillender
               about
               ,
               (
               excepting
               the
               top
               and
               bottom
               )
               is
               equal
               to
               the
               Superficial
               Content
               of
               a
               Globe
               .
            
             
               
                 
                   
                     Cube
                  
                   
                     1728
                  
                
                 
                   
                     Cillender
                  
                   
                     1357
                     5
                     /
                     7
                  
                
                 
                   
                     Sphear
                  
                   
                     905
                     2
                     /
                     7
                  
                
                 
                   
                     Prisma
                  
                   
                     864
                  
                
                 
                   
                     Piramis
                  
                   
                     576
                  
                
                 
                   
                     Cone
                  
                   
                     452
                     4
                     /
                     6
                  
                
                 
                   
                     △
                     Tetrahed
                     .
                  
                   
                     2496
                  
                
                 
                   
                     Octahed
                     .
                     whose
                     Triangle
                     Side
                     is
                     12
                  
                   
                     814-6
                  
                
              
            
             
               By
               the
               foregoing
               Proportions
               ,
               it
               is
               evident
               that
               a
               Cube
               is
               double
               the
               Prisma
               ,
               and
               treble
               to
               the
               Square
               Pyramis
               of
               equal
               Base
               and
               Altitude
               ,
               or
               as
               3
               ,
               2
               ,
               1
               ;
               for
               3
               times
               576
               is
               1728
               ,
               and
               2
               times
               864
               is
               1728.
               
            
             
               Also
               a
               Cillender
               is
               11
               /
               14
               of
               a
               Cube
               ;
               and
               a
               Globe
               is
               11
               /
               ●●
               of
               a
               Cube
               ,
               or
               ⅔
               of
               a
               Cillender
               ,
               whose
               Sides
               and
               Diameters
               are
               equal
               ;
               and
               a
               Cone
               is
               ●
               /
               ●
               of
               a
               Cillender
               ;
               so
               that
               the
               Proportion
               between
               the
               Cone
               ,
               Sphear
               ,
               and
               Cillender
               ,
               is
               as
               1
               ,
               2
               ,
               &
               3
               ;
               for
               3
               times
               452
               4
               /
               6
               ,
               makes
               1357
               5
               /
               7
               ;
               and
               two
               thirds
               
               of
               452
               4
               /
               6
               makes
               905
               1
               /
               7
               ;
               the
               Content
               of
               a
               Sphear
               .
               The
               Trianguler
               Pyramid
               is
               little
               more
               than
               1
               /
               7
               of
               a
               Cube
               ;
               so
               that
               if
               any
               one
               have
               frequent
               occasions
               for
               these
               proportions
               ,
               let
               Centers
               be
               put
               in
               the
               Line
               of
               Numbers
               ,
               at
               these
               proportional
               Numbers
               ,
               and
               then
               work
               with
               those
               Points
               from
               the
               Cube
               and
               Cillender
               ,
               as
               is
               directed
               before
               ,
               for
               the
               Circumference
               ,
               and
               Diameter
               ,
               and
               Squares
               ,
               equal
               and
               inscribed
               in
               Chap.
               8.
               
               Prob.
               6.
               
            
             
               So
               much
               for
               the
               measuring
               of
               regular
               ordinary
               Solids
               ;
               for
               the
               extraordinary
               and
               irregular
               ,
               the
               best
               Mechanick
               way
               is
               by
               Weights
               or
               Waters
               to
               measure
               their
               Crassitudes
               or
               Solidities
               ,
               either
               by
               Weight
               or
               Measure
               .
            
             
             
               
                 A
                 further
                 improvement
                 of
                 the
                 
                   Trianguler
                   Quadrant
                
                 ,
                 as
                 I
                 have
                 made
                 it
                 several
                 times
                 ,
                 with
                 a
                 sliding
                 Cover
                 on
                 the
                 in-side
                 ,
                 when
                 made
                 hollow
                 ,
                 to
                 carry
                 Ink
                 ,
                 Pens
                 ,
                 and
                 Compasses
                 ;
                 then
                 on
                 the
                 sliding
                 Cover
                 ,
                 and
                 Edges
                 ,
                 is
                 put
                 the
                 Line
                 of
                 Numbers
                 ,
                 according
                 to
                 Mr.
                 White
                 's
                 first
                 Contrivance
                 for
                 manner
                 of
                 Operation
                 ;
                 but
                 much
                 augmented
                 ,
                 and
                 made
                 easie
                 ,
                 by
                 
                   John
                   Brown.
                
                 
              
            
             
               1.
               
               THe
               description
               thereof
               for
               
                 one
                 side
              
               ▪
               being
               the
               same
               with
               the
               Line
               of
               Numbers
               on
               the
               outter-Edge
               ,
               except
               that
               the
               first
               part
               is
               sometimes
               (
               when
               required
               for
               that
               particular
               purpose
               )
               divided
               into
               12
               parts
               ,
               for
               inches
               ,
               instead
               of
               10
               that
               is
               to
               say
               ,
               The
               space
               between
               the
               first
               1
               ,
               and
               the
               middle
               1
               ,
               on
               the
               Rule
               ;
               (
               the
               space
               I
               say
               )
               between
               every
               Figure
               ,
               on
               the
               first
               half
               part
               ,
               is
               cut
               into
               12
               parts
               ,
               instead
               of
               10
               ,
               to
               answer
               to
               the
               12
               inches
               in
               a
               Foot
               ;
               and
               the
               other
               half
               ,
               as
               the
               Line
               of
               Numbers
               on
               the
               Edge
               .
               And
               in
               the
               same
               manner
               are
               White
               's
               
               sliding
               Rules
               made
               ,
               only
               for
               this
               particular
               purpose
               .
            
             
               2.
               
               On
               the
               
                 other
                 side
              
               ,
               is
               the
               Line
               of
               Numbers
               drawn
               double
               ,
               the
               one
               Line
               to
               the
               other
               ,
               for
               the
               ready
               measuring
               of
               solid
               Measure
               at
               one
               Operation
               ;
               the
               description
               whereof
               in
               brief
               is
               thus
               ;
            
             
               First
               ,
               The
               divisions
               on
               the
               sliding-piece
               in
               hollow-Rules
               ,
               or
               on
               the
               right-side
               in
               sliding-Rules
               ;
               when
               the
               figures
               of
               the
               Timber-side
               stand
               fit
               to
               read
               ,
               I
               call
               the
               right-side
               ,
               or
               single-side
               ,
               being
               alwayes
               toward
               the
               right
               hand
               ,
               and
               a
               single
               Radius
               .
            
             
               The
               divisions
               therefore
               on
               the
               fixed-edge
               of
               the
               Rule
               ,
               must
               needs
               be
               the
               left-side
               ,
               and
               is
               also
               divided
               to
               a
               double
               Radius
               ,
               or
               one
               Radius
               twice
               repeated
               .
            
             
               So
               also
               in
               sliding-Rules
               ,
               the
               double
               Radius
               is
               on
               the
               left-side
               also
               .
               
                 See
                 the
                 Figure
                 thereof
                 ,
                 with
                 right
                 and
                 left-side
                 exprest
                 upon
                 it
                 .
              
            
             
               
                 For
                 the
                 right
                 reading
                 those
                 Lines
                 ,
                 the
                 Method
                 is
                 thus
              
               ;
            
             
               The
               Figures
               on
               the
               right
               or
               single-side
               ,
               do
               usually
               begin
               at
               3
               or
               4
               ,
               and
               so
               proceed
               with
               4
               ,
               5
               ,
               6
               ,
               7
               ,
               8
               ,
               9
               ,
               10
               ,
               11
               ,
               for
               so
               many
               inches
               of
               a
               Foot.
               
            
             
               Then
               1
               ,
               2
               ,
               3
               ,
               4
               ,
               5
               ,
               6
               ,
               7
               ,
               8
               ,
               9
               ,
               10
               ,
               11
               ,
               12
               ,
               13
               ,
               14
               ,
               &c.
               for
               so
               many
               whole
               Feet
               .
            
             
             
               The
               smaller
               Cuts
               between
               the
               first
               Figures
               ,
               from
               3
               inches
               to
               1
               foot
               ,
               being
               quarters
               of
               inches
               ;
               And
               the
               small
               Divisions
               between
               the
               Figures
               ,
               that
               represent
               Feet
               ,
               are
               only
               every
               whole
               Inch
               ;
               The
               halfs
               ,
               and
               quarters
               of
               Feet
               ,
               also
               noted
               by
               a
               longer
               stroke
               ▪
               as
               in
               such
               work
               is
               necessary
               and
               usual
               .
            
             
               3.
               
               On
               the
               same
               right-side
               also
               ,
               for
               more
               ease
               and
               readiness
               in
               the
               use
               ,
               are
               noted
               several
               Gage-points
               (
               as
               it
               were
               )
               ;
               As
               ,
            
             
               First
               ,
               At
               1
               Foot
               is
               the
               word
               square
               .
            
             
               Secondly
               ,
               At
               1
               Foot
               ,
               1
               inch
               and
               ½
               ,
               is
               a
               spot
               ;
               and
               close
               to
               it
               is
               set
               t.d.
               for
               true
               Diameter
               of
               a
               round
               solid
               Cillender
               .
            
             
               Thirdly
               ,
               At
               1
               foot
               3
               inches
               ⅜
               is
               another
               spot
               ,
               and
               near
               to
               it
               is
               set
               D
               ,
               for
               the
               Diameter
               of
               a
               rough
               piece
               of
               Timber
               ,
               according
               to
               the
               usual
               allowance
               for
               unhewed
               Timber
               ,
               according
               to
               the
               fourth
               part
               of
               a
               Line
               girt
               about
               and
               counted
               for
               the
               side
               of
               the
               square
               .
            
             
               Fourthly
               ,
               At
               3
               foot
               6
               inches
               and
               ½
               ,
               and
               near
               to
               it
               is
               set
               t
               :
               r
               :
               for
               the
               true
               Circumference
               of
               a
               round
               Cillender
               .
            
             
               Fiftly
               ,
               At
               4
               foot
               just
               is
               set
               R
               ,
               for
               the
               Circumference
               ,
               according
               to
               the
               former
               allowance
               .
            
             
               Sixtly
               ,
               At
               1
               foot
               5
               inches
               1
               /
               7
               f●rè
               ,
               is
               a
               spot
               ;
               and
               close
               to
               it
               the
               letter
               W
               ,
               as
               the
               Gage-point
               for
               a
               Wine-gallon
               .
            
             
             
               Seventhly
               ,
               At
               near
               19
               inches
               ,
               or
               1
               foot
               7
               inches
               ,
               is
               another
               spot
               ;
               and
               close
               to
               it
               the
               letter
               a
               ,
               as
               the
               Gage-point
               of
               an
               Ale-gallon
               .
            
             
               Eightly
               ,
               At
               2
               foot
               8
               inches
               8
               /
               10
               is
               a
               spot
               ,
               and
               close
               to
               it
               is
               set
               B
               ,
               for
               the
               Gage-point
               of
               a
               Beer
               Barrel
               ;
               and
               at
               2
               foot
               7
               inches
               is
               set
               A
               ,
               for
               an
               Ale
               Barrel
               .
               
                 The
                 Uses
                 whereof
                 in
                 order
                 follow
                 .
              
            
             
               The
               Figures
               on
               the
               left-side
               ,
               or
               fixed-edges
               ,
               are
               read
               and
               counted
               as
               those
               on
               the
               right
               :
               For
               the
               small
               ,
               1
               ,
               2
               ,
               3
               ,
               4
               ,
               5
               ,
               6
               ,
               7
               ,
               8
               ,
               9
               ,
               10
               ,
               11
               ,
               are
               to
               represent
               inches
               ,
               and
               the
               cuts
               between
               ,
               quarters
               of
               inches
               ;
               Then
               the
               1
               ,
               2
               ,
               3
               ,
               4
               ,
               5
               ,
               6
               ,
               7
               ,
               8
               ,
               9
               ,
               10
               Figures
               next
               ,
               somewhat
               bigger
               ,
               as
               to
               represent
               so
               many
               feet
               ,
               and
               the
               cuts
               between
               ,
               are
               whole
               inches
               :
               Then
               20
               ,
               30
               ,
               40
               ,
               50
               ,
               60
               ,
               70
               ,
               80
               ,
               90
               ,
               100
               ,
               150
               ,
               for
               tens
               of
               feet
               ,
               and
               the
               parts
               between
               ,
               for
               single-feet
               ,
               for
               the
               most
               part
               ;
               or
               else
               whole
               and
               half
               feet
               ,
               as
               is
               usual
               .
               The
               Uses
               follow
               .
            
          
           
             
               Use
               I.
               A
               piece
               of
               Timber
               being
               not
               Square
               ,
               (
               or
               having
               its
               breadth
               and
               depth
               unequal
               )
               to
               make
               it
               Square
               ,
               or
               find
               the
               Square
               equal
               ;
            
             
               Set
               the
               breadth
               of
               the
               Piece
               ,
               counted
               on
               the
               right-side
               ,
               to
               the
               same
               breadth
               counted
               
               on
               the
               left-side
               ;
               then
               right
               against
               the
               depth
               found
               on
               the
               left-side
               ,
               on
               the
               right
               or
               single-side
               ,
               is
               the
               inches
               and
               quarters
               square
               required
               .
            
             
               
                 Example
                 ,
                 at
                 15
                 inches
                 broad
                 ,
                 and
                 9
                 inches
                 thick
                 ,
                 or
                 deep
                 .
              
            
             
               Set
               9
               inches
               on
               the
               right-side
               ,
               to
               9
               on
               the
               left
               ;
               then
               right
               against
               15
               inches
               ,
               or
               1
               foot
               3
               inches
               ,
               found
               out
               on
               the
               double
               or
               left-side
               ,
               on
               the
               right
               or
               single-side
               ,
               is
               11
               inches
               and
               ⅝
               the
               Square
               equal
               required
               .
            
             
               Also
               ,
               if
               you
               set
               15
               to
               15
               ,
               then
               right
               against
               9
               ,
               found
               out
               on
               the
               left-side
               ,
               on
               the
               right-side
               is
               11
               inches
               ⅝
               ,
               the
               Square
               equal
               required
               .
            
          
           
             
               Use
               II.
               The
               Side
               of
               the
               Square
               given
               ,
               to
               find
               how
               much
               in
               length
               will
               make
               1
               Foot.
               
            
             
               Set
               the
               inches
               (
               or
               feet
               &
               inches
               )
               found
               out
               on
               the
               right-side
               ,
               to
               1
               foot
               on
               the
               left
               ;
               then
               right
               against
               1
               foot
               on
               the
               right
               ,
               is
               the
               inches
               ,
               or
               the
               feet
               and
               inches
               required
               ,
               to
               make
               a
               Foot
               of
               Timber
               .
            
             
               
                 Example
                 at
                 9
                 inches
                 square
              
               .
            
             
               Set
               9
               inches
               ,
               found
               out
               on
               the
               right-side
               ,
               to
               1
               foot
               on
               the
               left-side
               ;
               then
               right
               against
               1
               ,
               on
               the
               right-side
               ,
               is
               1
               foot
               9
               inches
               2
               /
               ●
               on
               the
               left
               .
            
             
             
               If
               the
               Square
               be
               so
               big
               ,
               that
               the
               1
               on
               the
               right
               falls
               beyond
               the
               End
               at
               the
               beginning
               ,
               then
               right
               against
               10
               foot
               on
               the
               right-side
               ,
               is
               on
               the
               left
               ,
               the
               hundredth
               part
               of
               a
               foot
               ,
               that
               makes
               a
               Foot
               of
               Timber
               .
            
             
               
                 Example
                 ,
                 at
                 4
                 Foot
                 Square
                 .
              
            
             
               Set
               4
               foot
               ,
               found
               out
               on
               the
               right
               ,
               to
               1
               foot
               on
               the
               left
               ;
               then
               right
               against
               10
               foot
               on
               the
               right-side
               ,
               is
               0-063
               on
               the
               left-side
               ,
               or
               against
               12
               foot
               you
               have
               9
               ,
               12
               parts
               of
               1
               inch
               ,
               the
               length
               that
               goes
               to
               make
               1
               foot
               of
               Timber
               required
               .
            
          
           
             
               Use
               III.
               At
               any
               (
               bigness
               or
               )
               Inches
               ,
               or
               feet
               and
               inches
               square
               ,
               to
               find
               how
               much
               is
               in
               1
               Foot
               long
               .
            
             
               Just
               as
               the
               Rule
               stands
               even
               ,
               that
               is
               1
               foot
               on
               the
               right
               ,
               against
               1
               foot
               on
               the
               left
               ,
               seek
               the
               inches
               ,
               or
               feet
               and
               inches
               ,
               the
               Piece
               is
               square
               on
               the
               right
               or
               single-side
               ;
               and
               just
               against
               it
               on
               the
               left
               or
               double-side
               is
               the
               Answer
               required
               ;
               in
               inches
               ,
               or
               feet
               and
               inches
               .
            
             
               
                 Example
                 at
                 19
                 inches
                 square
              
               .
            
             
               Just
               against
               1
               foot
               7
               inches
               ,
               or
               19
               inches
               ,
               (
               which
               is
               all
               one
               )
               found
               out
               on
               the
               right-side
               ,
               
               on
               the
               left-side
               is
               2
               foot
               6
               inches
               ,
               the
               quantity
               of
               Timber
               in
               1
               foot
               long
               ,
               at
               19
               inches
               square
               ;
               which
               Number
               of
               2
               foot
               6
               inches
               ,
               multiplied
               by
               the
               length
               in
               feet
               ,
               gives
               the
               true
               Content
               of
               the
               whole
               Piece
               of
               Timber
               required
               .
            
             
               Note
               ,
               That
               this
               is
               a
               most
               excellent
               way
               for
               great
               Wood
               ,
               and
               very
               exact
               .
            
             
               
                 Also
                 Note
              
               ,
               That
               here
               ,
               by
               inspection
               ,
               you
               may
               square
               a
               small
               Number
               ,
               or
               find
               the
               square-root
               of
               a
               small
               Number
               .
            
             
               
                 As
                 thus
              
               ;
            
             
               The
               square
               of
               8
               &
               ½
               ,
               is
               near
               72
               ;
            
             
               Or
               ,
            
             
               The
               square-root
               of
               72
               ,
               is
               near
               8
               &
               ½
               .
            
          
           
             
               Use
               IV.
               The
               side
               of
               the
               Square
               ,
               and
               the
               length
               of
               any
               Piece
               being
               given
               ,
               to
               find
               the
               Content
               in
               feet
               and
               parts
               .
            
             
               Set
               the
               word
               square
               ,
               or
               1
               foot
               ,
               alwayes
               to
               the
               length
               ,
               found
               out
               on
               the
               left-side
               ;
               then
               right
               against
               the
               inches
               ,
               or
               feet
               and
               inches
               square
               counted
               on
               the
               right
               ,
               on
               the
               left
               is
               the
               Content
               required
               .
            
             
               
                 Example
                 ,
                 at
                 20
                 foot
                 long
                 ,
                 and
                 15
                 inches
                 square
                 .
              
            
             
               Set
               1
               foot
               on
               the
               right
               ,
               to
               20
               on
               the
               left
               ,
               then
               right
               against
               1
               foot
               3
               inches
               on
               the
               
               right
               ,
               is
               31
               foot
               2
               inches
               and
               ½
               ,
               the
               Content
               .
            
             
               Note
               ,
               That
               if
               the
               Piece
               be
               very
               small
               ,
               call
               the
               feet
               on
               the
               left-side
               ,
               inches
               ;
               and
               the
               parts
               between
               12●
               of
               inches
               ;
               then
               the
               Answer
               will
               be
               found
               on
               the
               left-side
               in
               144●
               of
               a
               Foot.
               
            
             
               
                 Example
                 ,
                 at
                 2
                 inches
                 square
                 ,
                 and
                 30
                 foot
                 long
                 ,
                 how
                 much
                 is
                 there
                 ?
              
            
             
               Set
               1
               foot
               on
               the
               right
               ,
               to
               30
               foot
               on
               the
               left
               ;
               then
               right
               against
               2
               foot
               on
               the
               right
               ,
               counted
               as
               2
               inches
               ,
               is
               120
               parts
               of
               a
               foot
               divided
               into
               144
               parts
               ,
               being
               just
               10
               inches
               ,
               for
               10
               times
               12
               is
               120.
               
            
             
               
                 But
                 if
                 it
                 be
                 ●
                 great
                 Piece
                 of
                 Timber
                 ,
                 then
                 work
                 thus
              
               ;
            
             
               Set
               1
               foot
               ,
               or
               the
               word
               square
               ,
               to
               the
               length
               on
               the
               left
               ,
               counting
               the
               single
               feet
               10●
               of
               feet
               ;
               then
               right
               against
               the
               feet
               and
               inches
               square
               are
               the
               100●
               of
               feet
               required
               .
            
             
               
                 Example
                 ,
                 at
                 40
                 foot
                 long
                 ,
                 and
                 4
                 foot
                 square
                 .
              
            
             
               Set
               1
               foot
               to
               4
               foot
               ,
               counted
               as
               40
               ,
               on
               the
               left
               ;
               then
               right
               against
               4
               foot
               on
               the
               right
               ,
               is
               640
               ,
               the
               true
               Content
               ,
               increasing
               the
               10●
               to
               100●
               .
               Thus
               much
               for
               square
               Timber
               .
            
             
             
               Though
               there
               be
               many
               other
               wayes
               and
               manner
               of
               workings
               ,
               some
               whereof
               you
               may
               find
               in
               a
               Book
               set
               forth
               under
               the
               name
               of
               
                 The
                 Carpenters
                 Rule
              
               ,
               1666
               ,
               by
               
                 I.
                 Brown
              
               ,
               and
               well
               known
               abroad
               already
               .
            
          
           
             
               
                 Use
                 V.
              
               For
               Round
               Timber
               .
               The
               middle
               Diameter
               of
               any
               Piece
               given
               ,
               to
               find
               how
               much
               is
               in
               a
               Foot
               long
               ,
               at
               true
               measure
               .
            
             
               Set
               the
               spot
               by
               t.d.
               to
               1
               foot
               on
               the
               left
               ,
               then
               just
               against
               the
               inches
               ,
               or
               feet
               and
               inches
               Diameter
               ,
               found
               on
               the
               right
               ,
               is
               the
               quantity
               of
               Timber
               in
               1
               foot
               long
               on
               the
               left-side
               required
               .
            
             
               
                 Example
                 ,
                 at
                 2
                 foot
                 9
                 inches
                 Diameter
                 .
              
            
             
               Suppose
               a
               piece
               of
               Stone-Pillar
               ,
               or
               Garden-Roul
               ,
               be
               two
               foot
               9
               inches
               Diameter
               ,
               set
               the
               spot
               by
               t.d.
               just
               against
               1
               foot
               ,
               then
               right
               against
               2
               foot
               9
               inches
               ,
               found
               on
               the
               right
               ,
               on
               the
               left
               is
               6
               foot
               ,
               the
               quantity
               of
               solid
               measure
               in
               one
               foot
               long
               ;
               which
               being
               multiplied
               by
               the
               length
               in
               feet
               ,
               gives
               the
               true
               Content
               of
               the
               whole
               Piece
               .
            
             
               Note
               ,
               That
               if
               you
               would
               have
               the
               usual
               allowance
               ,
               set
               D
               to
               1
               ,
               instead
               of
               t.d.
               
            
          
           
             
             
               Use
               VI.
               The
               Diameter
               of
               any
               Piece
               of
               Timber
               given
               ,
               to
               find
               how
               much
               in
               length
               will
               make
               one
               Foot.
               
            
             
               Set
               1
               foot
               on
               the
               left
               ,
               to
               the
               inches
               Diameter
               counted
               on
               the
               right
               ;
               then
               right
               against
               t.d.
               for
               true
               measure
               ;
               or
               D
               for
               the
               usual
               allowance
               ,
               is
               the
               Answer
               required
               ,
               found
               on
               the
               left-side
               .
            
             
               
                 Example
                 ,
                 at
                 9
                 inches
                 Diameter
                 .
              
            
             
               Set
               1
               on
               the
               left
               to
               9
               on
               the
               right
               ;
               then
               just
               against
               t.d.
               is
               2
               foot
               3
               ½
               on
               the
               left
               ;
               and
               right
               against
               D
               ,
               is
               2
               foot
               11
               inches
               on
               the
               left
               ,
               the
               length
               required
               to
               make
               a
               foot
               solid
               at
               true
               measure
               ,
               or
               the
               usual
               allowance
               ,
               when
               the
               4th
               part
               of
               the
               girt
               about
               ,
               is
               counted
               the
               side
               of
               the
               Square
               ,
               equal
               to
               the
               round
               piece
               of
               Timber
               .
            
             
               Note
               ,
               That
               for
               great
               Timber
               ,
               you
               must
               set
               the
               left
               1
               foot
               ,
               to
               the
               feet
               and
               inches
               Diameter
               as
               before
               ;
               but
               count
               the
               t.d.
               or
               D
               ,
               as
               far
               beyond
               12
               foot
               ,
               as
               it
               is
               placed
               beyond
               12
               inches
               ,
               and
               you
               shall
               have
               the
               Answer
               in
               144●
               of
               a
               foot
               .
            
             
               
                 Example
                 ,
                 at
                 5
                 foot
                 Diameter
                 .
              
            
             
               If
               you
               set
               1
               on
               the
               left
               to
               5
               foot
               on
               the
               right
               ,
               and
               count
               so
               much
               beyond
               12
               foot
               
               on
               the
               right
               ,
               as
               t.d.
               is
               beyond
               12
               inches
               ,
               you
               shall
               find
               7
               ¼
               ,
               that
               is
               ,
               7
               144●
               and
               a
               quarter
               ,
               to
               make
               1
               foot
               true
               measure
               ,
               and
               9-12s
               and
               ½
               for
               the
               usual
               allowance
               .
            
             
               But
               for
               small
               Timber
               ,
               set
               1
               foot
               ,
               2
               foot
               ,
               &c.
               on
               the
               right
               ,
               counted
               as
               1
               ,
               2
               ,
               3
               ,
               4
               ,
               and
               inches
               ,
               to
               1
               on
               the
               left
               ;
               then
               right
               against
               t.d.
               or
               D
               ,
               is
               a
               Number
               ,
               that
               multiplied
               by
               12
               ,
               is
               the
               Number
               of
               feet
               required
               .
            
             
               
                 Example
                 ,
                 at
                 2
                 inches
                 Diameter
                 ,
                 how
                 much
                 makes
                 1
                 Foot
                 ?
              
            
             
               Set
               2
               foot
               on
               the
               right
               ,
               counted
               as
               1
               inch
               ,
               to
               1
               on
               the
               left
               ;
               then
               right
               against
               t.d.
               is
               3
               foot
               10
               inches
               ,
               calling
               the
               inches
               feet
               ,
               and
               the
               feet
               10s
               of
               feet
               ;
               which
               3
               foot
               10
               inches
               ,
               multiplied
               by
               12
               ,
               make
               46
               foot
               ,
               for
               the
               length
               of
               1
               foot
               of
               Timber
               at
               2
               inches
               Diameter
               ,
               the
               thing
               required
               for
               true
               measure
               .
            
          
           
             
               Use
               VII
               .
               The
               Diameter
               and
               Length
               given
               ,
               to
               find
               the
               Content
               .
            
             
               Set
               t.d.
               or
               D
               ,
               for
               true
               measure
               ,
               or
               usual
               allowance
               ,
               alwayes
               to
               the
               length
               counted
               on
               the
               left
               ;
               then
               right
               against
               the
               inches
               Diameter
               counted
               on
               the
               right
               ,
               on
               the
               left-side
               is
               the
               Content
               required
               .
            
             
             
               
                 Example
                 ,
                 at
                 6
                 inches
                 Diameter
                 ,
                 an●
                 30
                 Foot
                 long
                 .
              
            
             
               Set
               t.d.
               to
               30
               ;
               then
               right
               against
               6
               inches
               ,
               counted
               on
               the
               right
               ;
               on
               the
               left
               is
               5
               foot
               11
               inches
               ,
               the
               Content
               required
               .
            
             
               Note
               ,
               If
               the
               Piece
               be
               small
               ,
               then
               count
               every
               foot
               on
               the
               right
               as
               inches
               ,
               and
               you
               have
               the
               Answer
               in
               144s
               of
               a
               foot
               ,
               which
               is
               easily
               counted
               by
               having
               1
               set
               at
               12
               ,
               2
               at
               24
               ,
               3
               at
               36
               ,
               4
               at
               48
               ,
               5
               at
               60
               ,
               &c.
               to
               12
               at
               144
               ,
               which
               little
               small
               figures
               are
               counted
               as
               inches
               of
               12s
               of
               a
               foot
               .
            
             
               
                 Example
                 ,
                 at
                 2
                 inches
                 Diameter
                 ,
                 and
                 20
                 foot
                 long
                 .
              
            
             
               You
               shall
               find
               64
               ,
               144s
               ;
               that
               is
               ,
               5
               in
               ⅓
               true
               measure
               .
            
             
               But
               for
               great
               Pieces
               ,
               set
               t.d.
               or
               D
               ,
               to
               the
               length
               ,
               counting
               1
               foot
               ,
               or
               the
               left
               for
               10
               foot
               ,
               then
               have
               you
               the
               Answer
               in
               100s
               of
               feet
               .
            
             
               
                 Example
                 ,
                 at
                 5
                 foot
                 Diameter
                 ,
                 and
                 30
                 foot
                 long
                 .
              
            
             
               Set
               t.d.
               to
               3
               inches
               ,
               counting
               30
               foot
               for
               the
               length
               ;
               then
               right
               against
               5
               foot
               on
               the
               right
               ,
               on
               the
               left
               is
               592
               foot
               ,
               the
               Content
               required
               .
            
          
           
             
             
               Use
               VIII
               .
               To
               measure
               round
               Timber
               ,
               by
               having
               the
               Girt
               ,
               or
               Circumference
               about
               ,
               and
               length
               given
               .
            
             
               This
               being
               the
               same
               in
               operation
               with
               the
               Diameter
               ,
               I
               shall
               pass
               it
               over
               more
               briefly
               ;
               which
               way
               of
               wording
               ,
               may
               serve
               for
               the
               Square
               and
               Diameter
               also
               ;
               only
               I
               labour
               to
               be
               plain
               and
               brief
               .
            
             
               
                 The
                 Circumference
                 given
                 ,
                 to
                 find
                 how
                 much
                 in
                 a
                 Foot
                 long
                 .
              
            
             
               Set
               t.r.
               or
               R
               ,
               for
               true
               round
               ,
               or
               allowance
               to
               1
               foot
               on
               the
               left
               ;
               then
               against
               the
               inches
               about
               ,
               on
               the
               left
               is
               the
               Answer
               required
               .
            
             
               Example
               .
            
             
               At
               2
               foot
               about
               ,
               you
               will
               find
               3
               inches
               and
               10
               /
               12
               in
               a
               foot
               long
               true
               measure
               ;
               or
               just
               3
               inches
               at
               the
               usual
               allowance
               .
            
             
               
                 The
                 Circumference
                 given
                 ,
                 to
                 find
                 how
                 much
                 makes
                 a
                 Foot.
                 
              
            
             
               Set
               the
               inches
               ,
               or
               feet
               and
               inches
               about
               ,
               to
               1
               foot
               on
               the
               left
               ;
               so
               is
               t.r.
               or
               R
               ,
               to
               the
               length
               to
               make
               a
               foot
               .
            
             
             
               
                 Example
                 ,
                 at
                 18
                 inches
                 about
                 .
              
            
             
               As
               single
               18
               ,
               to
               double
               1
               ;
            
             
               So
               is
               t.r.
               to
               5
               foot
               ,
               7
               inches
               ½
               Or
               ,
            
             
               So
               is
               R
               ,
               to
               7
               foot
               2
               inches
               ,
               for
               the
               usual
               allowance
               .
            
             
               
                 The
                 Circumference
                 and
                 length
                 ,
                 to
                 find
                 the
                 Content
                 .
              
            
             
               As
               t.
               r
               ,
               or
               R
               ,
               to
               the
               length
               ;
            
             
               So
               is
               the
               feet
               and
               inches
               about
               ,
               to
               the
               Content
               .
            
             
               
                 Example
                 ,
                 at
                 3
                 foot
                 about
                 ,
                 and
                 30
                 foot
                 long
                 ,
                 true
                 measure
                 .
              
            
             
               As
               t.r.
               to
               30
               ;
               so
               is
               3
               foot
               ,
               to
               21
               foot
               5
               inches
               ,
               the
               Content
               .
            
             
               For
               great
               things
               ,
               call
               1
               foot
               on
               the
               left
               ,
               10
               foot
               ,
               as
               before
               .
            
             
               For
               small
               things
               ,
               call
               1
               foot
               on
               the
               right
               ,
               1
               inch
               ,
               as
               before
               .
            
          
           
             
               Use
               IX
               .
               To
               Gage
               small
               Cask
               by
               the
               mean
               Diameter
               and
               Length
               .
            
             
               Set
               the
               spot
               by
               W
               ,
               for
               Wine-gallons
               ,
               alwayes
               to
               the
               length
               of
               the
               Vessel
               ,
               given
               in
               inches
               ,
               counted
               on
               the
               left-side
               ;
               then
               right
               against
               the
               inches
               ,
               or
               feet
               and
               inches
               Diameter
               ,
               
               counted
               on
               the
               right
               ;
               on
               the
               left
               is
               the
               Content
               in
               Gallons
               required
               .
            
             
               
                 Example
                 ,
                 at
                 24
                 inches
                 ,
                 or
                 2
                 foot
                 mean
                 Diameter
                 ,
                 and
                 30
                 inches
                 long
                 .
              
            
             
               As
               the
               spot
               at
               W
               ,
               to
               30
               inches
               counted
               at
               30
               foot
               ;
            
             
               So
               is
               24
               inches
               ,
               or
               2
               foot
               ,
               to
               58
               Gallons
               3
               quarters
               ,
               the
               Content
               required
               .
            
             
               For
               greater
               Vessels
               ,
               count
               the
               feet
               on
               the
               left
               for
               10s
               s
               of
               inches
               in
               the
               length
               ,
               and
               you
               have
               your
               desire
               .
            
             
               
                 Example
                 ,
                 at
                 60
                 inches
                 long
                 ,
                 and
                 38
                 inches
                 mean
                 Diameter
                 .
              
            
             
               As
               W
               ,
               to
               6
               foot
               on
               the
               left
               for
               60
               inches
               ;
            
             
               So
               is
               3
               foot
               2
               inches
               ,
               or
               38
               inches
               to
               295
               Gallons
               ,
               the
               Answer
               required
               .
            
             
               If
               you
               would
               have
               it
               in
               Ale-gallons
               ,
               use
               the
               mark
               at
               a.
               
            
          
           
             
               Use
               X.
               To
               gage
               Great
               Brewers
               Vessels
               ,
               round
               Tuns
               .
            
             
               
                 The
                 Diameter
                 and
                 length
                 being
                 given
                 in
                 feet
                 and
                 inches
                 ,
                 to
                 find
                 the
                 Content
                 in
                 Beer-Barrels
                 ,
                 at
                 one
                 Operation
                 .
              
            
             
               Set
               the
               spot
               at
               BB
               ,
               to
               the
               depth
               of
               the
               
               Tun
               ,
               counted
               on
               the
               left
               ,
               in
               feet
               and
               inches
               ;
               then
               right
               against
               the
               mean
               Diameter
               ,
               found
               out
               on
               the
               right
               ,
               on
               the
               left
               is
               the
               Content
               in
               Barrels
               required
               .
            
             
               
                 Example
                 ,
                 a
                 Tun
                 4
                 foot
                 deep
                 ,
                 and
                 10
                 foot
                 mean
                 Diameter
                 .
              
            
             
               As
               the
               spot
               at
               BB
               ,
               to
               4
               foot
               ;
            
             
               So
               is
               10
               to
               53
               Barrels
               ,
               and
               2
               third
               parts
               .
            
             
               If
               you
               would
               have
               the
               Content
               in
               Ale-Barrels
               ,
               use
               the
               mark
               at
               AB
               .
            
             
               Thus
               much
               for
               the
               Timber-side
               ,
               the
               use
               of
               the
               other
               ,
               or
               board-side
               ,
               is
               the
               same
               with
               that
               by
               the
               Compasses
               before
               treated
               of
               ,
               and
               therefore
               needs
               here
               no
               repetition
               ,
               unless
               as
               to
               the
               bare
               manner
               of
               working
               with
               it
               .
            
             
               The
               sliding-Rule
               is
               only
               two
               Rules
               ,
               or
               Pieces
               fitted
               together
               ,
               with
               a
               short
               Grove
               ,
               and
               Tenon
               ,
               and
               two
               Braces
               at
               each
               end
               ,
               to
               keep
               it
               from
               falling
               assunder
               ;
               and
               even
               so
               also
               is
               the
               sliding-Cover
               ,
               and
               two
               edges
               on
               the
               inside
               of
               the
               
                 Trianguler
                 Quadrant
              
               ;
               and
               the
               Numbers
               graduated
               thereon
               ,
               are
               cut
               across
               the
               middle
               Joynt
               ,
               having
               the
               same
               divisions
               on
               both
               sides
               ;
               that
               is
               to
               say
               ,
               on
               each
               Rule
               ,
               or
               on
               the
               Cover
               and
               Edge
               of
               the
               inside
               of
               the
               Rule
               .
            
             
               The
               Reading
               and
               description
               is
               the
               same
               with
               that
               in
               Chap.
               I.
               Page
               12
               ,
               13
               ,
               14
               ,
               15
               ,
               
               16
               ;
               and
               the
               general
               Method
               in
               use
               is
               thus
               ;
            
             
               That
               side
               or
               part
               of
               the
               Rule
               ,
               on
               which
               you
               count
               the
               first
               term
               in
               the
               Question
               ,
               is
               called
               alwayes
               the
               first-side
               ;
               then
               the
               other
               must
               needs
               be
               the
               second
               .
            
             
               
                 Then
                 for
                 Multiplication
                 ,
                 thus
              
               ;
            
             
               As
               1
               ,
               on
               the
               first-side
               ,
               to
               the
               Multiplier
               on
               the
               second
               ,
               or
               other-side
               ;
            
             
               So
               is
               the
               Multiplicand
               ,
               on
               the
               first-side
               ,
               where
               1
               was
               ,
               to
               the
               Product
               on
               the
               second
               .
            
             
               
                 For
                 Division
                 ,
                 alwayes
                 thus
              
               ;
            
             
               As
               the
               Divisor
               found
               on
               any
               one
               side
               ,
               is
               to
               1
               on
               the
               second
               ,
               or
               other-side
               ;
            
             
               So
               is
               the
               Dividend
               on
               the
               same
               first-side
               ,
               to
               the
               Quotient
               on
               the
               second
               .
            
             
               
                 For
                 the
                 Rule
                 of
                 Three
              
               .
            
             
               As
               the
               first
               term
               on
               the
               first-side
               ,
               to
               the
               second
               on
               the
               other
               ;
            
             
             
             
               
                 For
                 Superficial
                 Measure
                 ,
                 by
                 Inches
                 and
                 Feet
                 .
              
            
             
               As
               12
               ,
               to
               the
               breadth
               in
               inches
               on
               the
               second
               ;
            
             
               So
               is
               the
               length
               in
               feet
               ,
               to
               the
               Content
               on
               the
               second
               .
            
             
               For
               any
               thing
               else
               ,
               the
               same
               Rules
               and
               Precepts
               you
               find
               in
               Chap.
               VII
               .
               will
               give
               you
               ample
               and
               plain
               directions
               .
            
             
               The
               Lines
               being
               fitted
               ,
               as
               much
               as
               may
               be
               ,
               to
               speak
               out
               the
               Answer
               to
               the
               Question
               ,
               as
               by
               well
               considering
               the
               Figure
               ,
               you
               may
               see
               .
            
          
        
         
           
           
             CHAP.
             XI
             .
             To
             make
             and
             measure
             the
             Five
             Regular
             Platonical
             Bodies
             ,
             with
             their
             Declinations
             and
             Reclinations
             .
          
           
             
               1.
               
               For
               the
               Cube
               ,
               being
               the
               Foundation
               of
               all
               other
               .
            
             
               IT
               is
               a
               Square
               Solid
               Body
               ,
               every
               way
               alike
               ,
               and
               spoken
               of
               largely
               before
               ,
               as
               to
               the
               Mensuration
               thereof
               ,
               and
               obvious
               enough
               to
               every
               indifferent
               Workman
               ,
               as
               to
               the
               making
               thereof
               ,
               and
               needs
               no
               repetition
               in
               this
               place
               .
            
          
           
             
               2.
               
               For
               the
               Tetrahedron
               .
            
             
               It
               is
               a
               Figure
               ,
               comprehended
               of
               4
               equilatteral
               plain
               Triangles
               ,
               or
               a
               Trianguler
               Pyramid
               ,
               last
               mentioned
               ,
               the
               best
               and
               nearest
               way
               ,
               as
               I
               conceive
               of
               making
               ,
               is
               thus
               .
               According
               to
               directions
               of
               Mr.
               
                 Iohn
                 Leak
              
               :
            
             
             
               On
               any
               rough
               Piece
               ,
               make
               one
               side
               plain
               and
               flat
               ,
               so
               large
               ,
               as
               to
               contain
               the
               Triangle
               which
               you
               intend
               shall
               be
               one
               side
               of
               the
               Tetrahedron
               ;
               then
               set
               a
               Bevel
               to
               70
               degrees
               31
               minutes
               and
               42
               seconds
               ;
               and
               plain
               another
               side
               ,
               to
               fit
               the
               former
               side
               ,
               and
               the
               Bevel
               
                 (
                 secundum
                 Artem
              
               )
               ;
               then
               mark
               this
               last
               plained
               side
               ,
               according
               to
               the
               former
               ,
               and
               cut
               away
               the
               residue
               ,
               plaining
               them
               away
               just
               to
               the
               strokes
               ,
               and
               fit
               to
               the
               Bevel
               formerly
               set
               ,
               and
               you
               shall
               constitute
               the
               Tetrahedron
               required
               .
            
             
               The
               Superficial
               Content
               is
               the
               Area
               of
               the
               4
               equilatteral
               Triangles
               mentioned
               before
               ,
               and
               the
               solid
               Content
               is
               found
               by
               multiplying
               the
               Area
               of
               one
               Triangle
               by
               one
               third
               part
               of
               the
               Altitude
               of
               the
               Pyramid
               ,
               or
               Tetrahedron
               ,
               from
               the
               midst
               of
               one
               Plain
               to
               the
               Apex
               ,
               or
               top
               of
               the
               opposite
               Solid
               Angle
               .
            
             
               If
               the
               measuring
               the
               sides
               ,
               Perpendiculer
               ,
               and
               Altitude
               of
               the
               Tetrahedron
               with
               Compasses
               ,
               Callipers
               ,
               and
               Scale
               ,
               serve
               not
               to
               exactness
               ;
               then
               proceed
               thus
               ;
            
             
               
                 First
                 ,
                 for
                 the
                 Perpendiculer
                 ,
                 the
                 Triangle
                 being
                 equilatteral
                 .
              
            
             
               Multiply
               one
               side
               given
               by
               13
               ,
               and
               divide
               the
               Product
               by
               15
               ;
               the
               Quotient
               is
               the
               Perpendiculer
               .
            
             
             
               Example
               .
            
             
               If
               the
               side
               of
               the
               Tetrahedron
               be
               12
               ,
               that
               multiplied
               by
               13
               ,
               gives
               156
               ;
               which
               divided
               by
               15
               ,
               leaves
               10-4
               ,
               for
               the
               length
               of
               the
               Perpendiculer
               in
               the
               equilatteral
               Triangle
               ,
               whose
               side
               is
               12.
               
            
             
               
                 Then
                 for
                 the
                 Perpendiculer
                 Altitude
                 ,
                 work
                 thus
                 ,
                 by
                 the
                 Artificial
                 Numbers
                 and
                 Sines
                 .
              
            
             
               As
               the
               Sine
               of
               90
               ,
               to
               the
               sine
               of
               70
               deg
               .
               31
               min.
               42
               sec
               ;
            
             
               So
               is
               10-4
               ,
               the
               Perpendiculer
               ,
               to
               9-80
               ,
               the
               perpendiculer
               Altitude
               required
               .
            
             
               
                 Or
                 by
                 the
                 Sector
                 ,
                 work
                 thus
              
               ;
            
             
               Take
               12
               ,
               the
               side
               of
               the
               Tetrahedron
               ,
               from
               (
               any
               Scale
               ,
               or
               )
               the
               Line
               of
               Lines
               ,
               and
               set
               the
               Sector
               to
               60
               degrees
               ,
               by
               making
               the
               Latteral
               12
               ,
               a
               Parallel
               12
               ,
               then
               the
               nearest
               distance
               from
               12
               ,
               to
               the
               Line
               of
               Lines
               ,
               is
               the
               true
               Perpendiculer
               ;
               which
               measured
               on
               the
               same
               Line
               of
               Lines
               ,
               will
               be
               found
               to
               be
               10-4
               ,
               as
               before
               ;
               then
               make
               this
               10-4
               a
               Parallel
               Sine
               of
               90
               ,
               and
               90
               the
               Sector
               so
               set
               ,
               take
               out
               the
               Parallel-sine
               of
               70-31-42
               ,
               and
               measure
               it
               on
               the
               same
               Scale
               ,
               and
               it
               shall
               be
               9-8
               ,
               as
               before
               .
            
             
             
               Then
               ,
            
             
               Lastly
               ,
               This
               perpendiculer
               Altitude
               being
               multiplied
               by
               the
               Area
               of
               the
               Base
               ,
               gives
               a
               Number
               ,
               whose
               third
               part
               is
               the
               Solid-Content
               of
               the
               Tetrahedron
               required
               .
            
             
               For
               12
               the
               side
               ,
               and
               5-2
               the
               half
               Perpendiculer
               ,
               makes
               62-4
               ,
               the
               Superficial-Content
               of
               one
               Triangle
               ,
               or
               Base
               ;
               then
               62-4
               ,
               the
               Base
               ,
               multiplied
               by
               9-8
               ,
               the
               perpendiculer
               Altitude
               ,
               gives
               611-52
               ,
               and
               a
               third
               part
               of
               611-52
               is
               203-86
               ,
               the
               solid
               Content
               required
               .
            
             
               The
               three
               Triangles
               recline
               from
               the
               Perpendiculer
               upright
               ,
               19
               degr
               .
               28
               min.
               and
               18
               sec.
               and
               decline
               when
               the
               edge
               is
               South
               60
               ,
               South
               East
               ,
               and
               South
               West
               ,
               and
               the
               opposite
               Plain
               a
               just
               North
               ;
               but
               if
               you
               
               make
               one
               South
               ,
               then
               the
               other
               two
               are
               are
               North-east
               and
               North-west
               60
               deg
               .
            
          
           
             
               3.
               
               For
               the
               Octahedron
               ,
            
             
               Whis
               is
               a
               solid
               body
               ,
               comprehended
               under
               8
               equilatteral
               Triangles
               :
               The
               way
               of
               making
               which
               ,
               
                 is
                 thus
              
               ;
            
             
               Make
               a
               plain
               Parallelepipedon
               ,
               or
               long-Cube
               ,
               if
               the
               breadth
               both
               wayes
               be
               1000
               ,
               let
               the
               length
               be
               1-414
               ;
               or
               if
               the
               length
               be
               500000
               ,
               the
               breadth
               both
               wayes
               must
               be
               3-53553
               ;
               then
               to
               these
               Measures
               square
               it
               exactly
               ;
               then
               divide
               the
               length
               and
               breadth
               just
               in
               the
               midst
               ,
               and
               draw
               Lines
               both
               wayes
               on
               all
               the
               6
               sides
               ;
               then
               draw
               the
               Diagonal-Lines
               from
               the
               midst
               of
               the
               length
               ,
               to
               the
               midst
               of
               the
               breadth
               ;
               and
               cut
               by
               these
               Diagonal-Lines
               ,
               and
               the
               Octahedran
               will
               appear
               to
               be
               truly
               made
               .
            
             
               For
               the
               Mensuration
               thereof
               ,
               it
               is
               the
               same
               as
               in
               the
               Tetrahedron
               ;
               For
               ,
               supposing
               the
               side
               of
               one
               of
               the
               Triangles
               12
               ;
               the
               Base
               is
               144
               in
               Content
               ,
               and
               the
               Triangles
               Perpendiculer
               is
               10-4
               ,
               as
               before
               :
               But
               the
               Perpendiculer
               Altitude
               is
               just
               half
               the
               length
               ,
               viz.
               8-49
               ;
               for
               if
               the
               breadth
               be
               12
               ,
               then
               the
               length
               must
               be
               16-98
               ,
               whose
               halfs
               are
               6
               and
               8-49
               ;
               Then
               if
               you
               multiply
               144
               the
               Base
               ,
               by
               8-49
               the
               perpendiculer
               
               Altitude
               ,
               the
               Product
               will
               be
               1222-56
               ,
               whose
               third
               part
               is
               407-52
               ,
               the
               half
               of
               the
               Tetrahedron
               ,
               and
               815-04
               is
               the
               whole
               solid
               Content
               of
               the
               Tetrahedron
               required
               ,
               as
               near
               as
               we
               can
               see
               by
               Instrumental
               Operation
               ;
               but
               if
               you
               work
               to
               a
               Figure
               more
               ,
               you
               shall
               find
               the
               total
               Area
               to
               be
               but
               814-656
               more
               exact
               .
            
             
               
                 To
                 find
                 this
                 Perpendiculer
                 Altitude
                 by
                 the
                 Sector
                 ,
                 work
                 thus
              
               ;
            
             
               First
               ,
               The
               Triangles
               Perpendiculer
               being
               10-4
               ,
               as
               before
               ;
               Take
               the
               Latteral
               10-4
               ,
               from
               the
               Line
               of
               Lines
               ,
               make
               it
               a
               Parallel
               in
               90
               ,
               lay
               the
               Thred
               exactly
               to
               the
               nearest
               distance
               ,
               and
               there
               keep
               it
               ;
               then
               the
               Parallel-distance
               from
               the
               Sine
               of
               54
               deg
               .
               44
               min.
               45
               sec.
               the
               Reclination
               shall
               be
               8-486
               ,
               the
               true
               Perpendiculer
               Altitude
               required
               .
            
             
               Then
               if
               the
               Octahedron
               stand
               on
               one
               Triangle
               ,
               you
               have
               one
               Horizontal
               Plain
               ,
               and
               one
               South
               and
               North
               Reclining
               and
               Inclining
               19
               deg
               .
               28
               min.
               18
               sec.
               as
               the
               Tetrahedron
               was
               ;
               and
               two
               South
               ,
               and
               two
               North
               ,
               declining
               60
               ,
               and
               reclining
               and
               inclining
               19
               deg
               .
               28
               min.
               18
               sec.
               as
               afore
               .
            
             
               But
               if
               it
               stand
               on
               a
               Point
               ,
               then
               you
               have
               4
               direct
               or
               declining
               45
               ,
               and
               reclining
               54
               ,
               44
               ,
               45
               ;
               and
               4
               incliners
               ,
               inclining
               as
               
               much
               and
               direct
               ,
               or
               declining
               ,
               as
               you
               shall
               please
               to
               set
               them
               .
            
          
           
             
               4.
               
               For
               the
               Dodecahedron
               ,
            
             
               Which
               is
               a
               regular
               solid
               body
               ,
               contained
               under
               (
               or
               made
               up
               of
               )
               12
               Pentagonal
               Pyramids
               ,
               or
               Pyramids
               whose
               Base
               hath
               5
               equal
               sides
               ,
               and
               the
               perpendiculer
               Altitudes
               of
               those
               12
               Pyramids
               equal
               to
               half
               the
               Dodecahedrons
               Altitude
               ,
               standing
               on
               one
               side
               ,
               or
               equal
               to
               the
               semi-diameter
               of
               the
               inscribed
               Sphear
               .
            
             
               To
               cut
               this
               Body
               ,
               take
               any
               round
               Piece
               ,
               and
               if
               the
               Diameter
               be
               100000
               ,
               the
               length
               must
               be
               0-81005
               ,
               or
               as
               4-906
               to
               3-973
               ,
               then
               the
               Piece
               being
               turned
               round
               ,
               and
               the
               two
               Ends
               flat
               to
               the
               former
               measures
               of
               Length
               and
               Diameters
               (
               which
               are
               near
               according
               to
               the
               Sphear
               inscribed
               ,
               and
               to
               the
               Circle
               circumscribing
               )
               being
               measured
               by
               Compasses
               ,
               Callipers
               ,
               and
               Line
               of
               Lines
               very
               carefully
               and
               exactly
               .
               Then
               divide
               the
               Circumference
               of
               the
               two
               Ends
               of
               the
               Cillender
               into
               10
               equal
               parts
               ,
               and
               draw
               Lines
               Perpendiculer
               from
               end
               to
               end
               ,
               and
               plain
               all
               away
               between
               the
               Lines
               flat
               and
               smooth
               ,
               so
               that
               the
               two
               Plains
               on
               both
               ends
               will
               become
               a
               regular
               ten-sided
               Figure
               .
            
             
               Then
               making
               the
               whole
               Diameter
               abovesaid
               ,
               10000
               in
               the
               Line
               of
               Lines
               ,
               take
               out
               
               0-309
               ,
               and
               with
               this
               measure
               (
               as
               a
               Radius
               on
               the
               Center
               )
               at
               both
               ends
               describe
               a
               Circle
               ;
               and
               if
               you
               draw
               Lines
               ,
               from
               every
               opposite
               Line
               of
               the
               10
               first
               drawn
               ,
               you
               shall
               have
               Points
               in
               the
               last
               described
               circle
               ,
               to
               draw
               a
               Pentagon
               by
               ;
               which
               is
               the
               Base
               of
               one
               of
               the
               12
               Pentagonal
               Pyramids
               ,
               contained
               in
               the
               body
               .
               This
               Work
               is
               to
               be
               done
               at
               both
               Ends
               ;
               but
               be
               sure
               that
               the
               Angle
               of
               the
               Pentagon
               at
               one
               end
               ,
               be
               opposite
               to
               a
               side
               of
               the
               Pentagon
               at
               the
               other
               end
               ;
               then
               these
               Lines
               drawn
               ,
               the
               two
               ends
               are
               fully
               marked
               .
            
             
               
                 Then
                 to
                 mark
                 the
                 10
                 Sides
                 ,
                 do
                 thus
              
               ;
            
             
               Count
               the
               first
               length
               1000
               ,
               viz.
               the
               measure
               from
               the
               top
               to
               the
               bottom
               ,
               or
               from
               Center
               to
               Center
               ;
               and
               fit
               this
               length
               in
               10
               and
               10
               ,
               of
               the
               Line
               of
               Lines
               ;
               the
               Sector
               so
               set
               ,
               take
               out
               0-3821
               ,
               and
               lay
               it
               from
               the
               two
               ends
               ,
               and
               either
               draw
               ,
               or
               gage
               Lines
               round
               about
               from
               each
               end
               ;
               and
               in
               the
               midst
               between
               the
               two
               Lines
               will
               remain
               0-2358
               ;
               then
               Lines
               drawn
               Diagonally
               on
               the
               10
               sides
               ,
               will
               guide
               to
               the
               true
               cutting
               of
               the
               Dodecahedron
               .
            
             
               If
               you
               set
               a
               Bevel
               to
               116
               deg
               .
               33
               min.
               54
               sec.
               and
               apply
               it
               from
               the
               two
               ends
               ,
               you
               may
               try
               the
               truth
               of
               your
               Work.
               
            
             
             
               
                 The
                 Declination
                 and
                 Reclination
                 of
                 all
                 the
                 10
                 Pantagonal
                 Plains
                 ,
                 are
                 as
                 followeth
                 .
              
            
             
               First
               ,
               You
               have
               1
               North
               ,
               reclining
               26
               deg
               .
               34
               min
               ;
               and
               1
               South
               ,
               inclining
               as
               much
               .
            
             
               Secondly
               ,
               You
               have
               2
               North
               declining
               72
               ,
               and
               reclining
               26
               ,
               34
               ;
               and
               2
               South
               ,
               declining
               72
               ,
               and
               inclining
               26
               ,
               34.
               
            
             
               Thirdly
               ,
               You
               have
               2
               North
               ,
               declining
               36
               ,
               and
               inclining
               26
               ,
               34
               ;
               and
               2
               South
               ,
               declining
               36
               ,
               and
               reclining
               26
               ,
               34
               ;
               And
               1
               Horizontal
               Plain
               ,
               and
               his
               opposite
               Base
               to
               stand
               on
               .
            
             
               
                 As
                 for
                 the
                 measuring
                 of
                 this
                 Body
                 ,
                 the
                 Plain
                 and
                 Natural
                 way
                 is
                 thus
              
               ;
            
             
               First
               ,
               find
               the
               Superficial
               Content
               of
               the
               Base
               of
               one
               of
               the
               Pentagons
               ,
               by
               multiplying
               the
               measure
               from
               the
               Center
               to
               the
               middle
               of
               one
               of
               the
               Sides
               ,
               (
               which
               is
               the
               contained
               Circles
               semi-diameter
               )
               and
               half
               the
               sum
               of
               the
               measure
               of
               all
               the
               sides
               put
               together
               ;
               and
               then
               to
               multiply
               this
               Product
               by
               one
               third
               part
               of
               half
               the
               Altitude
               of
               the
               body
               ,
               and
               the
               Product
               shall
               be
               the
               Content
               of
               one
               Pentagonal
               Pyramid
               ,
               being
               one
               twelft
               part
               of
               the
               Dodecahedron
               ;
               and
               this
               last
               ,
               multiplied
               by
               12
               ,
               gives
               the
               solid
               
               Content
               of
               the
               Dodecahedron
               ;
               or
               12
               times
               the
               Superficial
               Content
               of
               one
               side
               ,
               is
               the
               Superficial
               Content
               thereof
               .
            
             
               Example
               .
            
             
               Suppose
               the
               side
               of
               a
               Dodecahedron
               be
               6
               ,
               then
               the
               sum
               of
               the
               sides
               measured
               is
               30
               ,
               the
               contained
               Circles
               semi-diameter
               is
               4-12
               ;
               then
               15
               the
               half
               of
               30
               ,
               and
               4-12
               multiplied
               together
               ,
               make
               61-80
               ;
               and
               12
               times
               this
               ,
               makes
               741-60
               ,
               for
               the
               Superficial
               Content
               of
               the
               Dodecahedron
               .
            
             
               Then
               for
               the
               Solid
               Content
               ,
               multiply
               61-80
               ,
               the
               Superficial
               Content
               of
               one
               side
               by
               2-233
               ,
               one
               6th
               part
               of
               13-392
               ,
               the
               whole
               Altitude
               of
               the
               body
               ;
               the
               Product
               is
               137-99940
               :
               Again
               ,
               this
               multiplied
               by
               12
               ,
               the
               number
               of
               Pyramids
               ,
               makes
               16●●
               ▪
               9928
               ,
               the
               Solid
               Content
               ,
               as
               near
               as
               may
               be
               ,
               in
               such
               a
               Decimal
               way
               of
               Computation
               .
            
          
           
             
               5.
               
               For
               the
               Icosahedron
               ,
            
             
               Which
               is
               a
               regular
               solid
               body
               ,
               made
               up
               of
               ,
               or
               contained
               under
               20
               Trianguler
               Pyramids
               ,
               whose
               Base
               (
               or
               one
               of
               whose
               Sides
               )
               is
               an
               equilatteral
               Triangle
               ;
               and
               the
               perpendiculer
               Altitude
               of
               one
               of
               these
               20
               Pyramids
               ,
               is
               equal
               to
               half
               the
               perpendiculer
               Altitude
               of
               the
               Icosahedron
               ,
               from
               any
               one
               side
               ,
               to
               his
               opposite
               side
               ,
               or
               equal
               to
               the
               semi-diameter
               of
               the
               inscribed
               Sphear
               .
            
             
             
               To
               cut
               this
               body
               ,
               take
               any
               round
               Piece
               ,
               and
               if
               the
               Diameter
               thereof
               be
               10000
               ,
               let
               the
               length
               thereof
               be
               turned
               flat
               and
               even
               to
               8075
               ;
               or
               if
               the
               true
               Round
               and
               Cillendrical
               Form
               in
               Diameter
               be
               4910
               ,
               let
               the
               true
               length
               ,
               when
               the
               ends
               are
               plain
               and
               flat
               ,
               be
               3964
               ;
               then
               divide
               the
               Cillendrical
               part
               into
               6
               equal
               parts
               ,
               and
               plain
               away
               all
               to
               the
               Lines
               ,
               so
               that
               the
               two
               ends
               may
               be
               two
               6-sided-figures
               ;
               then
               making
               5000
               ,
               the
               former
               semi-diameter
               ,
               1000
               in
               the
               Line
               of
               Lines
               ,
               take
               out
               616
               ,
               and
               on
               the
               Center
               ,
               at
               each
               end
               ,
               describe
               a
               Circle
               ;
               and
               by
               drawing
               Lines
               to
               each
               opposite
               Point
               ,
               make
               a
               Triangle
               ,
               whose
               circumscribing
               Circle
               may
               be
               the
               Circle
               drawn
               at
               each
               end
               ;
               but
               be
               sure
               to
               mark
               the
               side
               of
               one
               Triangle
               opposite
               to
               the
               Point
               of
               the
               other
               Triangle
               at
               the
               other
               end
               ,
               as
               before
               in
               the
               Dodecahedron
               ;
               thus
               both
               the
               ends
               shall
               be
               fully
               and
               truly
               marked
               .
            
             
               Then
               making
               the
               length
               a
               Parallel
               in
               1000
               ,
               of
               the
               Line
               of
               Lines
               ,
               take
               out
               -379
               ,
               and
               -095
               ,
               and
               prick
               those
               two
               measures
               from
               each
               end
               ,
               and
               by
               those
               Points
               (
               draw
               or
               gage
               )
               Lines
               round
               about
               ,
               on
               the
               6
               sides
               .
               Then
               Diagonal
               Lines
               drawn
               from
               Point
               to
               Line
               ,
               and
               from
               Line
               to
               Point
               round
               about
               ,
               shews
               how
               to
               cut
               the
               Body
               at
               12
               cuts
               .
            
             
             
               Note
               ,
               That
               if
               you
               set
               a
               Bevel
               to
               -
               138-11-23
               ,
               and
               apply
               it
               from
               each
               end
               ,
               it
               will
               guide
               you
               in
               the
               true
               plaining
               of
               the
               sides
               of
               the
               Icosahedron
               .
               And
               a
               Bevel
               set
               to
               100
               degrees
               ,
               will
               fit
               ,
               being
               applied
               from
               the
               midst
               of
               one
               side
               ,
               to
               the
               meeting
               of
               two
               sides
               .
            
             
               The
               Reclination
               of
               the
               three
               Triangles
               ,
               whose
               upper
               sides
               are
               adjacent
               (
               or
               next
               )
               to
               the
               three
               sides
               of
               the
               upper
               Horizontal-Triangle
               is
               48
               11.23
               ,
               from
               the
               Perpendiculer
               ,
               or
               41
               48
               37
               ,
               from
               the
               Horizontal
               ,
               and
               when
               one
               corner
               stands
               South
               ,
               the
               Declination
               of
               one
               of
               these
               3
               ,
               viz.
               that
               opposite
               to
               the
               South-corner
               a
               direct
               North
               ;
               th'
               other
               two
               decline
               60
               degrees
               ,
               one
               South-east
               ,
               the
               other
               South-west
               ;
               the
               other
               6
               ,
               about
               the
               corners
               of
               the
               Horizontal-plain
               ,
               do
               all
               recline
               19
               deg
               .
               28
               min.
               16
               sec.
               the
               two
               that
               behold
               the
               South
               ,
               decline
               22
               deg
               .
               14
               min.
               29
               sec.
               and
               those
               two
               that
               behold
               the
               North
               ,
               decline
               37
               deg
               .
               45
               min.
               51
               sec.
               toward
               the
               East
               and
               West
               ;
               the
               other
               two
               remaining
               ,
               recline
               as
               before
               ,
               and
               decline
               one
               North-east
               ,
               and
               the
               other
               North-west
               82
               deg
               .
               14
               min.
               19
               sec.
               
            
             
               The
               other
               Nine
               under-Plains
               ,
               opposite
               to
               every
               one
               of
               these
               ,
               decline
               and
               incline
               ,
               as
               much
               as
               the
               opposite
               did
               recline
               and
               decline
               ,
               as
               by
               due
               consideration
               will
               plainly
               appear
               .
            
             
             
               For
               the
               measuring
               of
               this
               body
               ,
               do
               as
               you
               did
               by
               the
               Dodecahedron
               ,
               find
               the
               Area
               of
               one
               Triangle
               ,
               and
               multiply
               it
               by
               20
               ,
               gives
               the
               Superficial
               Content
               ;
               and
               the
               Area
               of
               one
               Triangle
               ,
               multiplied
               by
               one
               sixt
               part
               of
               the
               Altitude
               of
               the
               body
               ,
               gives
               the
               solid
               Content
               of
               the
               Trianguler
               Pyramid
               ;
               and
               that
               Product
               multiplied
               by
               20
               ,
               the
               number
               of
               Pyramids
               ,
               gives
               the
               whole
               Solid
               Content
               of
               the
               Icosahedron
               .
            
             
               Example
               .
            
             
               Suppose
               the
               side
               of
               an
               Icosahedron
               be
               12
               ,
               first
               square
               one
               side
               (
               viz.
               12
               ,
               which
               makes
               144
               )
               ;
               then
               multiply
               that
               Square
               by
               13
               ,
               and
               then
               divide
               the
               Product
               by
               30
               ,
               the
               Quotient
               and
               his
               remainder
               is
               the
               Superficial
               Content
               of
               the
               Equilatteral
               Triangle
               ,
               whose
               side
               is
               12
               ;
               namely
               ,
               62-400
               ;
               or
               more
               exactly
               ,
               the
               Square-root
               of
               3888
               ,
               which
               is
               near
               62-354
               ;
               20
               times
               this
               ,
               is
               the
               Superficial
               Content
               ,
               namely
               ,
               1247-08
               .
            
             
               Then
               for
               the
               Solid
               Capacity
               or
               Content
               ,
               multiply
               3-023
               ,
               the
               sixt
               part
               of
               the
               bodies
               Altitude
               ,
               or
               one
               third
               of
               the
               Pyramids
               Altitude
               ,
               by
               62-354
               ,
               the
               Area
               of
               one
               Trianguler
               Base
               ,
               and
               the
               Product
               will
               be
               188-493229
               .
               Lastly
               ,
               this
               multiplied
               by
               20
               ,
               the
               number
               of
               Pyramids
               in
               the
               Body
               ,
               
               the
               Product
               is
               3769-864380
               ,
               the
               true
               solid
               Content
               of
               the
               Icosahedron
               .
            
             
               Thus
               you
               have
               the
               way
               of
               cutting
               ,
               and
               the
               Declinations
               and
               Reclinations
               and
               Measures
               ,
               Superficial
               and
               Solid
               ,
               of
               the
               5
               Regular
               Bodies
               ,
               as
               near
               as
               by
               Decimal
               Accompt
               to
               100
               part
               of
               an
               Integer
               may
               be
               ,
               the
               exact
               measuring
               whereof
               ,
               requires
               the
               help
               of
               Algebra
               ,
               whereof
               I
               am
               ignorant
               .
            
             
               The
               Measures
               of
               the
               Containing
               ,
               and
               Contained
               Sphears
               ,
               Circles
               ,
               and
               Diameters
               ,
               Sides
               and
               Axis's
               ,
               Diagonal-lines
               and
               Altitudes
               of
               the
               five
               Regular
               Bodies
               ,
               gathered
               in
               a
               Table
               to
               a
               Containing
               Sphear
               ,
               whose
               Diameter
               was
               10
               inches
               (
               or
               Integers
               )
               found
               out
               by
               Geometry
               ,
               according
               to
               this
               Scheam
               ,
               taken
               from
               Mr.
               
                 Tho.
                 Diggs
              
               .
            
             
               Let
               the
               Line
               AB
               be
               10
               of
               some
               Diagonal
               Scale
               ,
               representing
               the
               Diameter
               of
               the
               Containing
               Sphear
               .
               Which
               Line
               AB
               ,
               you
               must
               divide
               into
               two
               parts
               at
               C
               ,
               and
               into
               three
               parts
               at
               E
               ;
               AE
               being
               one
               third
               part
               ,
               and
               on
               the
               Points
               C
               and
               E
               ,
               raise
               two
               Lines
               Perpendiculer
               to
               AB
               ;
               and
               with
               5
               of
               your
               Diagonal
               Scale
               ,
               on
               the
               Center
               C
               ,
               describe
               the
               semi-Circle
               AFDB
               ,
               and
               note
               the
               Points
               F
               and
               D
               ,
               in
               the
               semi-Circle
               ,
               with
               F
               and
               D
               ,
               drawing
               Lines
               from
               either
               of
               them
               to
               A
               ,
               and
               from
               F
               to
               B.
               
            
             
             
               Then
               ,
               Divide
               AF
               by
               extream
               and
               mean
               Proportion
               ;
               the
               greater
               Segment
               being
               AG
               ,
               (
               by
               the
               10th
               Problem
               of
               the
               6th
               Chapter
               )
               then
               extend
               the
               Line
               AF
               to
               H
               ,
               making
               FH
               equal
               to
               FG
               ,
               and
               draw
               the
               Line
               HB
               ;
               and
               from
               F
               ,
               draw
               another
               Line
               Parallel
               to
               HB
               ,
               cutting
               the
               Diameter
               in
               I
               ,
               and
               from
               I
               ,
               draw
               a
               Line
               Parallel
               to
               CD
               ,
               as
               IK
               ;
               then
               make
               IL
               a
               third
               part
               of
               IB
               ,
               and
               draw
               ML
               Parallel
               to
               IK
               ;
               also
               ,
               draw
               the
               Line
               MB
               ,
               and
               divide
               it
               into
               two
               parts
               at
               N
               ,
               and
               into
               4
               parts
               at
               5
               ;
               then
               divide
               the
               4th
               part
               ,
               MS
               ,
               by
               extream
               and
               mean
               Proportion
               ,
               whose
               greater
               Segment
               (
               or
               part
               )
               let
               be
               SV
               ;
               then
               divide
               FB
               in
               4
               parts
               ,
               making
               FO
               the
               half
               ,
               and
               FR
               the
               quarter
               ;
               divide
               likewise
               FE
               in
               two
               parts
               ,
               and
               at
               the
               middle
               set
               P
               :
               The
               Figure
               being
               thus
               made
               ,
               then
               with
               your
               Compasses
               ,
               and
               Diagonal
               Scale
               ,
               you
               may
               measure
               all
               the
               Diameters
               ,
               Sides
               ,
               and
               Altitudes
               ,
               of
               all
               the
               5
               Regular
               Bodies
               .
            
             
               
                 As
                 thus
              
               ;
            
             
               AB
               is
               in
               all
               of
               them
               ,
               the
               Contained
               Sphears
               Diameter
               .
            
             
               EC
               ,
               OF
               ,
               RO
               ,
               NC
               ,
               NC
               ,
               the
               Contained
               Sphears
               Semi-Diameter
               .
            
             
               EF
               ,
               OB
               ,
               OF
               ,
               NB
               ,
               MB
               ,
               the
               Containing
               Circles
               Semi-Diameter
               .
            
             
             
               EP
               ,
               OC
               ,
               CO
               ,
               VN
               ,
               MN
               ,
               the
               Contained
               Circles
               Semi-Diameter
               .
            
             
               FB
               ,
               AF
               ,
               AD
               ,
               AG
               ,
               KB
               ,
               the
               Length
               of
               the
               Sides
               of
               each
               Body
               .
            
             
               EB
               ,
               AF
               ,
               FA
               ,
               MA
               ,
               AM
               ,
               the
               Altitude
               of
               the
               Bodies
               .
            
             
               AD
               ,
               FB
               ,
               VB
               ,
               SB
               ,
               the
               Perpendiculer
               Line
               of
               the
               Bases
               .
            
             
               FB
               ,
               AF
               ,
               the
               Diagonal-Line
               of
               the
               Bases
               ,
               as
               in
               the
               Table
               .
            
             
               These
               Measures
               and
               Proportions
               are
               for
               a
               Sphear
               of
               10
               inches
               Diameter
               .
            
             
               
                 If
                 you
                 would
                 have
                 the
                 like
                 for
                 any
                 other
                 ,
                 then
                 say
                 by
                 the
                 Line
                 of
                 Numbers
                 ,
                 or
                 Line
                 of
                 Lines
                 ,
                 or
                 Rule
                 of
                 Three
                 ,
                 thus
              
               ;
            
             
               As
               the
               side
               (
               Diameter
               or
               Altitude
               )
               for
               10
               ,
               as
               in
               the
               Table
               ,
               is
               to
               the
               given
               Side
               ,
               Diameter
               ,
               or
               Altitude
               ;
            
             
               So
               is
               any
               other
               Number
               ,
               in
               the
               Table
               ,
               for
               Diameter
               ,
               Side
               ,
               or
               Altitude
               ,
               to
               his
               Proportional
               Measure
               required
               .
            
             
             
               
                 Example
                 .
                 I
                 have
                 a
                 Dodecahedron
                 ,
                 whose
                 Side
                 is
                 6
                 ,
                 What
                 shall
                 all
                 his
                 other
                 Sphears
                 ,
                 or
                 Circles
                 ,
                 Diameters
                 ,
                 or
                 Altitude
                 be
                 ?
              
            
             
               The
               Extent
               of
               the
               Compasses
               from
               3-570
               ,
               the
               Dodecahedrons-side
               in
               the
               Table
               ,
               to
               6
               the
               side
               given
               ,
               shall
               reach
               from
               10
               ,
               the
               Containing
               Sphears
               Diameter
               in
               the
               Table
               ,
               to
               16
               ,
               the
               Containing
               Sphears
               Diameter
               ,
               for
               a
               Dodecahedron
               ,
               whose
               side
               is
               6
               :
               And
               from
               7-970
               ,
               the
               Contained
               Sphears
               Diameter
               ,
               to
               12-643
               ,
               the
               Contained
               Sphears
               Diameter
               .
               And
               so
               for
               any
               other
               whatsoever
               .
            
             
             
               
                 
                   The
                   Table
                   .
                
                 
                   
                     
                       The
                       Names
                       of
                       the
                       Bodies
                    
                  
                   
                     Tetrahedron
                  
                   
                     Cube
                     .
                  
                   
                     Octahed
                     .
                  
                   
                     Dodecahed
                     .
                  
                   
                     Icosahed
                     .
                  
                
                 
                   
                     Containing
                     Sphear
                     .
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                
                 
                   
                     The
                     containing
                     Sphears
                     Diameter
                     ,
                     that
                     comprehends
                     the
                     body
                     in
                     it
                     ,
                     is
                     for
                     ever
                     one
                     of
                     them
                     .
                  
                   
                     AB
                  
                   
                     AB
                  
                   
                     AB
                  
                   
                     AB
                  
                   
                     AB
                  
                
                 
                   
                     10.000
                  
                   
                     10.000
                  
                   
                     10.000
                  
                   
                     10.000
                  
                   
                     10.000
                  
                
                 
                   
                     Contained
                     Sphear
                     .
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                
                 
                   
                     The
                     contained
                     SphearS
                     Diameter
                     that
                     is
                     contained
                     in
                     the
                     body
                     ,
                     called
                     also
                     Axis
                     ,
                     is
                  
                   
                     2.332
                  
                   
                     8.1648
                  
                   
                     4
                     0824
                  
                   
                     7.970
                  
                   
                     7.970
                  
                
                 
                   
                     EC
                  
                   
                     OF
                  
                   
                     RO
                  
                   
                     NC
                  
                   
                     NC
                  
                
                 
                   
                     The
                     half
                     thereof
                     ,
                     is
                  
                   
                     1.666
                  
                   
                     4.0824
                  
                   
                     2
                     .
                     041●
                  
                   
                     3.983
                  
                   
                     3.985
                  
                
                 
                   
                     Containing
                     Circle
                     .
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                
                 
                   
                     The
                     containing
                     Circles
                     Diameter
                     ,
                     (
                     or
                     the
                     Diameter
                     of
                     that
                     Circle
                     which
                     comprehends
                     one
                     side
                     or
                     base
                     of
                     the
                     body
                     ,
                     is
                  
                   
                     9.420
                  
                   
                     9.648
                  
                   
                     9.644
                  
                   
                     6.070
                  
                   
                     MB
                  
                
                 
                   
                     EF
                  
                   
                     OB
                  
                   
                     OF
                  
                   
                     NB
                  
                   
                     6.070
                  
                
                 
                   
                     The
                     half
                     thereof
                     ,
                     is
                  
                   
                     4.710
                  
                   
                     4.0824
                  
                   
                     4.0824
                  
                   
                     3.035
                  
                   
                     3.035
                  
                
                 
                   
                     Contained
                     Circles
                     .
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                
                 
                   
                     The
                     contained
                     Circles
                     Diameter
                     ,
                     comprehended
                     in
                     the
                     Base
                     of
                     one
                     side
                     ,
                     is
                  
                   
                     4.410
                  
                   
                     5.7840
                  
                   
                     3.7840
                  
                   
                     4.910
                  
                   
                     MN
                  
                
                 
                   
                     EP
                  
                   
                     OC
                  
                   
                     OC
                  
                   
                     VN
                  
                   
                     3.035
                  
                
                 
                   
                     The
                     half
                     thereof
                     ,
                     is
                  
                   
                     2.355
                  
                   
                     2.8920
                  
                   
                     2.8920
                  
                   
                     2.455
                  
                   
                     1.5175
                  
                
                 
                   
                     Sides
                     ,
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                
                 
                   
                     The
                     length
                     of
                     one
                     side
                     of
                     the
                     Triangle
                     Square
                     ,
                     or
                     Pentagon
                     ,
                     being
                     the
                     base
                     of
                     the
                     figure
                     ,
                     is
                  
                   
                     FB
                  
                   
                     AF
                  
                   
                     AD
                  
                   
                     AG
                  
                   
                     KB
                  
                
                 
                   
                     8.1647
                  
                   
                     5.774
                  
                   
                     7.071
                  
                   
                     3.570
                  
                   
                     5.260
                  
                
                 
                   
                     The
                     half
                     therof
                     ,
                  
                   
                     4.08235
                  
                   
                     2.887
                  
                   
                     3.5355
                  
                   
                     1.785
                  
                   
                     2.630
                  
                
                 
                   
                     Altitude
                     .
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                
                 
                   
                     The
                     Altitude
                     from
                     side
                     ,
                     to
                     the
                     side
                     opposite
                     ,
                     or
                     from
                     Side
                     to
                     the
                     Point
                     opposite
                     ,
                  
                   
                     EB
                  
                   
                     AF
                  
                   
                     AF
                  
                   
                     MA
                  
                   
                     MA
                  
                
                 
                   
                     6.666
                  
                   
                     5.774
                  
                   
                     5.774
                  
                   
                     7.960
                  
                   
                     7.960
                  
                
                 
                   
                     The
                     half
                     thereof
                     ,
                  
                   
                     3.3333
                  
                   
                     2.887
                  
                   
                     2.887
                  
                   
                     3.980
                  
                   
                     3.980
                  
                
                 
                   
                     Perpendiculer
                     .
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                
                 
                   
                     The
                     Length
                     of
                     the
                     Perpendiculer
                     Line
                     of
                     any
                     one
                     Side
                     or
                     Base
                     ,
                  
                   
                     AD
                  
                   
                     FB
                  
                   
                      
                  
                   
                     VB
                  
                   
                     SB
                  
                
                 
                   
                     7
                     073
                  
                   
                     8.1647
                  
                   
                     6.123
                  
                   
                     5.485
                  
                   
                     4.556
                  
                
                 
                   
                     The
                     half
                     thereof
                     ,
                  
                   
                     3
                     5365
                  
                   
                     4.0823
                  
                   
                     3.0625
                  
                   
                     2.7425
                  
                   
                     2.278
                  
                
                 
                   
                     Diagonal-Line
                     .
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                
                 
                   
                     The
                     Diagonal-Line
                     ,
                     from
                     Corner
                     to
                     Corner
                     of
                     the
                     same
                     Base
                     ,
                     is
                     ,
                  
                   
                      
                  
                   
                     FB
                  
                   
                      
                  
                   
                     AF
                  
                   
                      
                  
                
                 
                   
                     none
                  
                   
                     8.1647
                  
                   
                     none
                  
                   
                     5.774
                  
                   
                     none
                  
                
                 
                   
                     The
                     half
                     thereof
                     is
                     ,
                  
                   
                      
                  
                   
                     4.0823
                  
                   
                      
                  
                   
                     2.887
                  
                   
                      
                  
                
              
            
             
             
               This
               Table
               was
               gathered
               from
               this
               
                 Geometrical
                 Figure
              
               ,
               drawn
               on
               a
               Slate
               ,
               by
               a
               good
               Diagonal
               Scale
               of
               6
               parts
               in
               a
               Foot
               ,
               whereby
               I
               could
               very
               well
               come
               to
               the
               100th
               part
               of
               an
               Integer
               ;
               and
               is
               true
               enough
               for
               any
               Mechanick
               Operation
               ,
               for
               whose
               use
               I
               only
               do
               it
               ,
               and
               I
               hope
               it
               may
               be
               as
               kindly
               accepted
               ,
               as
               it
               was
               carefully
               Calculated
               ,
               and
               offered
               to
               Publick
               view
               .
            
          
        
         
           
             CHAP.
             XII
             .
             The
             use
             of
             the
             Line
             of
             Numbers
             ,
             in
             Gaging
             of
             Vessels
             ,
             close
             or
             open
             .
          
           
             GAging
             of
             Vessels
             ,
             is
             no
             other
             than
             the
             Measuring
             of
             Solid
             Bodies
             ;
             and
             the
             former
             directions
             for
             solid
             Measure
             ,
             conveniently
             and
             aptly
             applied
             ,
             is
             fully
             sufficient
             ;
             only
             observing
             this
             difference
             ,
             That
             the
             result
             or
             issue
             of
             the
             Question
             is
             to
             be
             rendred
             in
             proper
             terms
             ,
             according
             to
             the
             
             demand
             of
             the
             Question
             ,
             as
             thus
             ;
             in
             measuring
             of
             Timber
             or
             Stone
             ,
             the
             Question
             is
             ,
             
               How
               many
               Feet
               ,
               or
               Inches
               ,
               is
               there
               in
               the
               Solid
               Body
               ?
            
             But
             in
             Gaging
             ,
             the
             Question
             is
             ,
             
               How
               many
               Gallons
               ,
               Kilderkins
               ,
               or
               Barrels
               is
               there
               in
               the
               Vessel
               to
               be
               measured
               ?
            
             For
             which
             purpose
             there
             are
             fit
             Numbers
             ,
             or
             Gage-Points
             ,
             requisite
             to
             be
             known
             ,
             for
             the
             more
             speedy
             attaining
             the
             Answer
             to
             the
             Question
             ,
             of
             which
             in
             their
             order
             ,
             as
             followeth
             ;
          
           
             First
             ,
             You
             are
             to
             remember
             ,
             That
             the
             solid
             capacity
             of
             a
             Wine-Gallon
             ,
             is
             231
             Cube
             Inches
             ;
             a
             Corn-Gallon
             272
             ¼
             Cube
             inches
             ;
             an
             Ale
             or
             Beer-Gallon
             ,
             is
             282
             ¼
             Cube
             inches
             ;
             or
             as
             some
             say
             ,
             288
             Cube
             inches
             ;
             So
             that
             when
             you
             have
             found
             the
             Content
             of
             any
             Vessel
             in
             Cube
             inches
             ,
             if
             you
             divide
             that
             sum
             in
             inches
             ,
             by
             the
             respective
             Number
             for
             the
             Gallons
             you
             would
             have
             ,
             the
             Quotient
             shall
             be
             the
             Content
             in
             Gallons
             required
             .
          
           
             
               Problem
               I.
               To
               measure
               a
               Square
               Vessel
               .
            
             
               From
               hence
               it
               follows
               ,
               That
               to
               measure
               any
               Square
               or
               Oblong
               Vessel
               ,
               you
               must
               multiply
               the
               length
               and
               breadth
               taken
               in
               inches
               ,
               and
               tenth
               parts
               ,
               together
               ;
               that
               is
               to
               
               say
               ,
               The
               one
               by
               the
               other
               ;
               and
               the
               Product
               shall
               be
               the
               Content
               of
               the
               Base
               in
               inches
               ,
               superficially
               :
               Then
               multiply
               this
               Superficial
               Content
               of
               the
               Base
               ,
               by
               the
               inches
               ,
               and
               tenth
               parts
               deep
               ,
               and
               the
               Product
               shall
               be
               the
               solid
               Content
               in
               Cube
               inches
               ;
               then
               divide
               this
               Product
               by
               282
               ,
               gives
               the
               Content
               in
               Ale-Gallons
               in
               the
               Quotient
               ,
               and
               the
               remainder
               ,
               if
               any
               be
               ,
               are
               Cube
               inches
               .
            
             
               But
               if
               you
               divide
               by
               10161
               ,
               the
               Cube
               inches
               in
               a
               Beer
               Barrel
               ;
               or
               ,
               by
               9032
               the
               Cube
               inches
               in
               an
               Ale
               Barrel
               ;
               the
               Quotient
               sheweth
               the
               Number
               of
               Beer
               or
               Ale
               Barrels
               ,
               (
               and
               the
               remainder
               Cube
               inches
               .
               )
            
             
               
                 Example
                 of
                 a
                 Brewers
                 Cooler
              
               .
            
             
               The
               length
               let
               be
               78
               inches
               and
               1
               tenth
               ,
               the
               breadth
               let
               be
               320
               inches
               and
               5
               tenths
               ,
               and
               the
               depth
               9
               inches
               and
               5
               tenths
               ,
               or
               half
               an
               inch
               ;
               by
               multiplying
               and
               dividing
               ,
               as
               above
               ,
               you
               will
               find
               843
               Gallons
               ,
               and
               68
               Cube
               inches
               ,
               to
               be
               the
               solid
               Content
               of
               that
               Cooler
               ;
               which
               work
               is
               very
               readily
               done
               by
               the
               Line
               of
               Numbers
               ,
               in
               this
               manner
               ;
            
             
               Extend
               the
               Compasses
               from
               1
               ,
               to
               the
               breadth
               or
               length
               ;
               and
               the
               same
               Extent
               shall
               reach
               from
               the
               length
               or
               breadth
               to
               a
               4th
               ,
               which
               is
               the
               Superficial
               Content
               of
               the
               Base
               ,
               or
               bottom
               ,
               in
               Superficial
               inches
               .
            
             
             
               Then
               ,
            
             
               The
               Extent
               from
               282
               ¼
               ,
               to
               the
               last
               Number
               found
               ,
               shall
               reach
               the
               same
               way
               from
               the
               inches
               ,
               and
               tenths
               deep
               ,
               to
               the
               Content
               in
               Gallons
               .
            
             
               Example
               .
            
             
               The
               Extent
               from
               1
               ,
               to
               78-1
               ,
               shall
               reach
               the
               same
               way
               from
               320-5
               ,
               to
               25031
               ;
               then
               the
               Extent
               from
               282
               ¼
               ,
               to
               25031
               ,
               shall
               reach
               the
               same
               way
               from
               9-5
               ,
               to
               842-68
               ,
               the
               Solid
               Content
               in
               Gallons
               required
               .
            
             
               Indeed
               ,
               you
               must
               Note
               ,
               You
               cannot
               see
               so
               many
               Figures
               on
               the
               Line
               ,
               as
               the
               Product
               of
               4
               figures
               multiplied
               by
               3
               ;
               yet
               by
               the
               Rules
               (
               in
               Chap.
               6.
               
               Sect.
               3.
               )
               you
               have
               directions
               as
               to
               the
               number
               of
               Figures
               ,
               which
               here
               is
               7
               ;
               the
               two
               last
               (
               next
               the
               right
               hand
               )
               being
               Fractions
               ,
               or
               parts
               of
               an
               Inch
               ,
               and
               is
               therefore
               neglected
               .
            
             
               Again
               ,
            
             
               In
               dividing
               the
               Product
               of
               25031
               ,
               and
               9-5
               ,
               multiplied
               together
               ,
               which
               makes
               6
               Figures
               beside
               the
               Fraction
               by
               282
               ,
               there
               must
               needs
               be
               three
               Figures
               in
               the
               Quotient
               ,
               which
               are
               the
               Gallons
               :
               This
               artificial
               help
               you
               have
               ,
               beside
               the
               present
               view
               of
               the
               Vessel
               ,
               which
               will
               direct
               you
               not
               to
               call
               842
               Gallons
               ,
               only
               84
               ½
               ,
               nor
               8420
               ,
               as
               you
               must
               needs
               do
               ,
               if
               you
               mistake
               as
               to
               the
               denomination
               .
            
             
             
               Again
               ,
            
             
               You
               need
               not
               to
               trouble
               your self
               ,
               to
               know
               what
               the
               4th
               Number
               is
               ;
               but
               having
               found
               the
               Point
               representing
               it
               ,
               keep
               the
               Compass-point
               fixed
               there
               ,
               and
               open
               the
               other
               to
               282
               ¼
               ,
               where
               you
               may
               have
               a
               Brass
               Center-pin
               for
               more
               readiness
               ;
               but
               let
               your
               account
               go
               as
               282
               ¼
               to
               the
               4th
               ,
               for
               methods
               sake
               ,
               and
               not
               as
               the
               4th
               to
               282
               ¼
               ;
               for
               then
               you
               must
               say
               ,
               so
               is
               the
               depth
               the
               contrary
               way
               to
               the
               Content
               in
               Gallons
               .
               All
               this
               is
               hinted
               for
               plainness
               and
               caution
               sake
               ,
               in
               benefit
               to
               young
               Learners
               .
            
             
               
                 Also
                 Note
              
               ,
               That
               if
               you
               would
               have
               had
               the
               Answer
               in
               Ale
               or
               Beer
               Barrels
               ;
               then
               instead
               of
               282
               ¼
               ,
               you
               must
               use
               the
               Point
               at
               9032
               ,
               for
               Ale
               Barrels
               ;
               or
               the
               Point
               at
               10161
               for
               Beer
               Barrels
               ,
               being
               the
               number
               of
               Cube
               inches
               in
               those
               Barrels
               ,
               as
               282
               ¼
               is
               the
               number
               of
               inches
               ,
               in
               a
               Gallon
               of
               Ale
               or
               Beer
               .
            
             
               
                 Example
                 for
                 the
                 same
                 Cooler
              
               .
            
             
               The
               Extent
               from
               1
               to
               78-1
               ,
               shall
               reach
               from
               320-5
               ,
               the
               same
               way
               to
               25031
               ;
               then
               ,
               the
               Extent
               from
               10161
               to
               25031
               ,
               shall
               reach
               the
               same
               way
               from
               9-5
               to
               23
               ⅓
               ,
               the
               true
               number
               of
               Beer
               Barrels
               required
               .
               Or
               ,
            
             
               The
               Extent
               from
               9032
               ,
               to
               25031
               ,
               shall
               reach
               to
               26
               Barrels
               ,
               and
               near
               1
               third
               :
               which
               
               is
               as
               quick
               and
               ready
               a
               way
               as
               can
               be
               for
               Square
               Vessels
               .
            
          
           
             
               Problem
               II.
               To
               Gage
               or
               Measure
               any
               round
               Tunn
               or
               Vessel
               .
            
             
               The
               plain
               and
               natural
               way
               for
               measuring
               of
               a
               round
               Tun
               ,
               is
               this
               ;
               Measure
               the
               Diameter
               in
               inches
               and
               tenths
               ,
               and
               set
               down
               half
               thereof
               ;
               Measure
               also
               ,
               the
               Compass
               round
               about
               the
               inside
               ,
               and
               set
               down
               the
               half
               of
               that
               also
               ,
               in
               inches
               and
               tenth
               parts
               ;
               and
               multiply
               those
               two
               Numbers
               together
               ,
               the
               Product
               shall
               be
               the
               Content
               of
               the
               Base
               ,
               or
               bottom
               ,
               in
               Superficial
               inches
               ;
               then
               this
               Product
               multiplied
               by
               the
               depth
               in
               inches
               ,
               gives
               the
               solid
               Content
               in
               inches
               ;
               then
               lastly
               ,
               this
               Product
               divided
               by
               282
               ,
               or
               by
               10161
               ,
               or
               by
               9032
               ,
               gives
               the
               solid
               Content
               in
               Gallons
               ,
               or
               Beer
               ,
               or
               Ale
               Barrels
               ,
               as
               before
               .
            
             
               For
               ,
               half
               the
               Diameter
               ,
               and
               half
               the
               Circumference
               ,
               doth
               reduce
               the
               round
               Vessel
               to
               an
               Oblong
               Vessel
               ,
               equal
               to
               that
               round
               Vessel
               .
            
             
               Which
               Vessel
               ,
               when
               it
               is
               brought
               to
               a
               Square
               ,
               by
               taking
               of
               half
               the
               Diameter
               ,
               and
               half
               the
               Circumference
               ;
               then
               the
               Rule
               
               last
               mentioned
               ,
               for
               Square
               Vessels
               ,
               performs
               the
               work
               exactly
               ,
               to
               Gallons
               ,
               or
               Barrels
               ,
               as
               you
               please
               .
            
             
               But
               when
               the
               Vessel
               is
               Taper
               ,
               that
               is
               to
               say
               ,
               the
               bottom
               and
               top
               of
               different
               Diameters
               ,
               as
               generally
               they
               all
               are
               ;
               then
               the
               chief
               care
               is
               to
               come
               by
               the
               true
               Diameters
               ,
               which
               is
               best
               done
               by
               a
               sliding
               Rule
               applied
               to
               the
               inside
               ,
               whose
               regular
               equal
               computation
               is
               thus
               to
               be
               ordered
               ;
            
             
               When
               the
               Vessel
               is
               taper
               ,
               and
               the
               Sides
               go
               streight
               ,
               like
               the
               Segment
               of
               a
               Con●
               ;
               then
               you
               may
               add
               the
               Diameters
               at
               top
               and
               bottom
               together
               ,
               and
               count
               the
               half
               sum
               for
               the
               mean
               Diameter
               of
               that
               taper
               Vessel
               ,
               and
               multiply
               half
               the
               Diameter
               ,
               and
               half
               his
               proportional
               Circumference
               ,
               as
               before
               ;
               and
               multiply
               and
               divide
               ,
               to
               get
               the
               solid
               Content
               in
               Gallons
               ,
               or
               Barrels
               .
            
             
               But
               when
               the
               Staves
               are
               bending
               ,
               as
               most
               of
               your
               close
               Cask
               are
               ,
               then
               the
               readiest
               way
               to
               come
               to
               a
               mean
               Diameter
               ,
               is
               thus
               ;
            
             
               Say
               ,
            
             
               As
               10
               to
               7
               ,
               or
               as
               10
               to
               6
               &
               3
               /
               10
               ,
               4
               /
               10
               ,
               5
               /
               10
               ,
               6
               /
               10
               ,
               7
               /
               10
               ,
               8
               /
               10
               ,
               9
               /
               10
               ;
               according
               as
               you
               shall
               find
               most
               true
               for
               several
               Cask
               :
            
             
               So
               is
               the
               difference
               of
               Diameters
               to
               a
               4th
               Number
               ,
               which
               is
               to
               be
               added
               to
               the
               least
               of
               the
               Diameters
               ,
               to
               make
               up
               a
               mean
               Diameter
               .
            
             
             
               
                 As
                 for
                 Example
              
               .
            
             
               If
               the
               Sides
               be
               round
               or
               arching
               ,
               and
               the
               less
               Diameter
               be
               30
               inches
               ,
               and
               the
               greater
               40
               inches
               ;
               then
               ,
               As
               10
               to
               7
               ;
               So
               is
               10
               ,
               the
               difference
               to
               7
               inches
               ;
               which
               makes
               (
               being
               added
               to
               30
               ,
               the
               least
               Diameter
               )
               37
               ,
               for
               a
               mean
               Diameter
               .
            
             
               
                 But
                 Note
              
               ,
               It
               is
               hinted
               by
               Mr.
               Dary
               ,
               That
               Vessels
               ,
               usually
               ,
               are
               between
               a
               Spheroid
               and
               a
               Parabolick
               Spindle
               ;
               then
               ,
               if
               as
               10
               to
               7
               ,
               be
               too
               much
               to
               add
               to
               the
               least
               Diameter
               ;
               
                 You
                 may
                 say
              
               ,
            
             
               As
               10
               ,
               to
               6
               ½
               ;
               Or
               ,
               As
               10
               ,
               to
               6
               6
               /
               10
               ,
               7
               /
               10
               ,
               8
               /
               10
               ,
               9
               /
               10
               ;
            
             
               So
               is
               the
               difference
               of
               Diameters
               to
               a
               4th
               Number
               ,
               which
               you
               must
               add
               to
               the
               least
               Diameter
               ,
               to
               make
               a
               mean
               Diameter
               .
            
             
               Having
               thus
               gained
               a
               mean
               Diameter
               ,
               you
               may
               work
               as
               before
               ;
               or
               rather
               thus
               more
               readily
               and
               easily
               ,
               by
               the
               Line
               of
               Numbers
               ,
               thus
               ;
            
             
               As
               the
               Gage-point
               is
               to
               the
               mean
               Diameter
               ;
            
             
               So
               is
               the
               Length
               to
               a
               4th
               ,
               and
               that
               4th
               to
               the
               Content
               required
               .
            
             
               
               
                 The
                 Gage-point
                 for
                 Wine
                 ,
                 and
                 Oyl-gallons
                 ,
                 at
                 231
                 Cube
                 inches
                 in
                 a
                 Gallon
                 ,
                 is
              
               
                 17
                 —
                 15
              
               
                 The
                 Gage-point
                 for
                 Ale-gallons
                 ,
                 at
                 282
                 ¼
                 ,
                 is
                 ,
              
               
                 18
                 —
                 95
              
               
                 The
                 Gage-point
                 for
                 Ale
                 ,
                 or
                 Beer-gallons
                 ,
                 at
                 288
                 ,
                 is
                 ,
              
               
                 19
                 —
                 15
              
               
                 The
                 Gage-point
                 for
                 a
                 Corn-gallon
                 ,
                 at
                 272
                 ¼
                 ,
                 is
                 ,
              
               
                 18
                 —
                 62
              
               
                 The
                 Gage-point
                 for
                 a
                 Beer
                 Barrel
                 ,
                 at
                 10161
                 ,
                 is
                 ,
              
               
                 35
                 —
                 96
              
               
                 The
                 Gage-point
                 for
                 an
                 Ale
                 Barrel
                 ,
                 at
                 9032
                 ,
                 is
                 ,
              
               
                 33
                 —
                 91
              
            
             
               The
               Extent
               of
               the
               Compasses
               ,
               on
               the
               Line
               of
               Numbers
               ,
               from
               the
               Gage-point
               to
               the
               mean
               Diameter
               of
               a
               Vessel
               ;
               being
               turned
               two
               times
               the
               same
               way
               from
               the
               length
               of
               a
               Vessel
               ,
               shall
               reach
               to
               the
               Content
               of
               the
               Vessel
               ,
               in
               Gallons
               or
               Barrels
               ,
               according
               to
               the
               nature
               of
               the
               Gage-point
               .
            
             
               Example
               .
            
             
               A
               mean
               Diameter
               being
               30
               ,
               and
               the
               Length
               40
               ,
               the
               Content
               is
               in
               Wine-gallons
               123
               ,
               near
               .
            
             
               In
               the
               lesser
               Ale
               or
               Beer-gallons
               ,
               100-½
               .
            
             
               In
               the
               greater
               Ale-gallons
               ,
               at
               288-098
               gallons
               and
               a
               half
               .
            
             
             
               In
               Corn-gallons
               ,
               at
               272-¼
               ,
               —
               104
               Gallons
               .
            
             
               In
               Beer
               Barrels
               ,
               by
               his
               Gage-point
               you
               will
               find
               2-78
               ,
               or
               2
               three
               quarters
               :
               2
               —
               78.
               
            
             
               In
               Ale
               Barrels
               ,
               you
               will
               find
               3
               and
               11
               of
               a
               hundred
               :
               3
               —
               11.
               
            
             
               And
               the
               like
               for
               any
               other
               Measure
               ,
               whose
               Gage-point
               is
               known
               .
            
          
           
             
               Problem
               III.
               To
               find
               the
               Gage-point
               of
               any
               Measure
               .
            
             
               The
               Gage-point
               of
               any
               Solid
               Measure
               ,
               is
               only
               the
               Diameter
               of
               a
               Circle
               ,
               whose
               Superficial
               Content
               is
               equal
               to
               the
               Solid
               Content
               of
               the
               same
               Measure
               .
            
             
               
                 As
                 thus
                 more
                 plainly
              
               ;
            
             
               The
               Solid
               Content
               of
               a
               Wine
               Gallon
               is
               231
               Cube
               inches
               :
               Now
               if
               you
               have
               a
               Circle
               that
               contains
               231
               Superficial
               inches
               ,
               the
               Diameter
               thereof
               will
               be
               found
               to
               be
               17
               inches
               ,
               and
               15
               of
               a
               hundred
               ;
               as
               by
               the
               7th
               Problem
               of
               the
               7th
               Chapter
               ,
               is
               well
               seen
               .
            
             
               These
               Directions
               may
               serve
               for
               any
               round
               Vessel
               ,
               either
               close
               or
               open
               ;
               yet
               Mr.
               Oughtred
               ,
               a
               very
               able
               Mathematition
               ,
               hath
               a
               
               way
               accounted
               somewhat
               more
               exact
               ,
               and
               consequently
               more
               tedious
               and
               troublesome
               to
               use
               either
               by
               the
               Pen
               or
               Compasses
               ,
               which
               is
               this
               ;
            
             
               You
               must
               measure
               the
               Diameters
               at
               head
               and
               bung
               ,
               or
               the
               top
               and
               bottom
               in
               inches
               and
               10ths
               ,
               the
               length
               also
               by
               the
               same
               measure
               ;
               then
               find
               out
               the
               Superficial
               Content
               of
               the
               Circles
               ,
               answerable
               to
               those
               two
               Diameters
               ,
               and
               take
               two
               thirds
               of
               the
               greatest
               ,
               and
               one
               third
               of
               the
               least
               ,
               and
               add
               them
               together
               in
               one
               sum
               ;
               which
               sum
               you
               must
               multiply
               by
               the
               length
               in
               inches
               and
               tens
               ,
               and
               the
               Product
               shall
               be
               the
               Content
               in
               Cube
               inches
               ;
               which
               Product
               divided
               by
               282
               ,
               gives
               Ale
               Gallons
               ;
               or
               by
               231
               ,
               gives
               Wine
               Gallons
               ,
               as
               before
               .
            
             
               
                 By
                 the
                 Line
                 of
                 Numbers
                 ,
                 this
                 way
                 is
                 more
                 easie
                 and
                 ready
                 thus
              
               ;
            
             
               The
               Extent
               from
               1
               ,
               to
               0-5236
               ,
               a
               Number
               fit
               for
               2
               thirds
               ,
               of
               the
               Circle
               at
               the
               bung
               ;
            
             
               So
               is
               the
               Square
               of
               the
               Diameter
               at
               the
               bung
               to
               a
               4th
               .
            
             
             
               
                 Then
                 again
              
               ;
            
             
               As
               1
               ,
               to
               0-2618
               ,
               the
               half
               of
               the
               former
               Number
               ,
               and
               fit
               for
               one
               third
               of
               the
               Circle
               at
               head
               ;
            
             
               So
               is
               the
               Square
               of
               the
               Diameter
               at
               head
               to
               a
               4th
               .
            
             
               
                 These
                 two
                 4ths
                 add
                 together
                 ,
                 then
                 say
              
               ;
            
             
               As
               231
               (
               for
               Wine
               ,
               or
               282
               ¼
               for
               Ale-Gallons
               )
               ,
               is
               to
               the
               sum
               of
               the
               two
               4ths
               added
               together
               ;
            
             
               So
               is
               the
               length
               to
               the
               Content
               in
               Wine-Gallons
               .
            
             
               
                 Example
                 ,
                 at
                 18
                 inches
                 at
                 head
                 ,
                 and
                 32
                 at
                 bung
                 ,
                 that
                 old
                 Example
                 .
              
            
             
               The
               Square
               of
               32
               ,
               is
               1024
               ;
            
             
               The
               Square
               of
               18
               ,
               is
               324
               :
            
             
               Then
               ,
            
             
               The
               Extent
               of
               the
               Compasses
               from
               1
               ,
               to
               0-5236
               ,
               shall
               reach
               from
               1024
               ,
               the
               Square
               of
               32
               ,
               to
               536-4
               ,
               two
               thirds
               of
               the
               bung-Circle
               .
            
             
               Again
               ,
            
             
               The
               Extent
               from
               1
               ,
               to
               0-2618
               ,
               shall
               reach
               from
               324
               ,
               the
               Square
               of
               18
               ,
               the
               Diameter
               at
               head
               ,
               to
               84-9
               ,
               the
               sum
               of
               536-4
               ,
               and
               84-9
               ,
               is
               621-3
               .
            
             
               
                 Then
                 lastly
              
               ,
            
             
               The
               Extent
               from
               231
               ,
               the
               621-3
               ,
               shall
               
               reach
               from
               40
               the
               length
               ,
               to
               107-58
               ,
               or
               107
               Gallons
               and
               a
               half
               ,
               and
               better
               ,
               the
               Content
               in
               Wine-gallons
               ,
               as
               briefly
               as
               can
               be
               done
               this
               way
               .
            
             
               But
               if
               you
               take
               the
               Diameters
               at
               head
               and
               bung
               ,
               with
               a
               Line
               called
               Oughtred's
               Gage-line
               ;
               and
               set
               the
               measure
               found
               at
               the
               bung
               by
               that
               Line
               ,
               down
               twice
               ;
               and
               the
               measure
               found
               at
               the
               head
               ,
               found
               by
               the
               same
               Line
               ,
               once
               ,
               and
               bring
               them
               into
               one
               sum
               ;
               then
               multiply
               that
               sum
               by
               the
               length
               of
               the
               Vessel
               in
               inches
               ,
               and
               10
               parts
               ,
               and
               then
               the
               Product
               shall
               be
               the
               Content
               in
               Wine-gallons
               required
               .
            
             
               As
               if
               I
               should
               measure
               a
               Cask
               of
               18
               ,
               and
               32
               ,
               as
               before
               :
               right
               against
               18
               inches
               on
               Oughtreds-Line
               ,
               you
               find
               0-367
               ;
               and
               right
               against
               32
               ,
               you
               shall
               find
               1-161
               ;
               this
               last
               set
               down
               twice
               ,
               and
               0-367
               once
               ;
               added
               ,
               makes
               2-689
               ;
               and
               then
               this
               sum
               multiplied
               by
               40
               ,
               makes
               107-56
               ,
               being
               very
               near
               to
               the
               former
               operation
               ,
               but
               differing
               about
               2
               Gallons
               ,
               from
               the
               way
               set
               before
               by
               the
               mean
               Diameter
               and
               Gage-point
               ,
               by
               reason
               of
               the
               extream
               swelling
               of
               the
               Cask
               ;
               But
               if
               this
               way
               should
               prove
               the
               truest
               in
               the
               Book
               of
               
                 the
                 Carpenters
                 Rule
              
               ,
               you
               have
               a
               Table
               to
               rectifie
               this
               difference
               ,
               which
               you
               will
               very
               seldom
               have
               occasion
               to
               use
               .
            
             
             
               
                 Note
                 also
              
               ,
               That
               this
               Line
               ,
               called
               Oughtred's
               Gage-Line
               ,
               is
               very
               excellently
               improved
               to
               find
               the
               Content
               of
               Great
               Vessels
               ,
               either
               in
               the
               whole
               ,
               or
               inch
               by
               inch
               ;
               which
               you
               will
               find
               at
               large
               in
               the
               Book
               before
               mentioned
               .
            
             
               Also
               ,
               The
               use
               of
               the
               Lines
               called
               Diagonal-Lines
               ,
               and
               Lines
               to
               find
               the
               emptiness
               of
               Cask
               ,
               and
               to
               measure
               Corn-measures
               by
               ,
               to
               which
               I
               shall
               ,
               for
               the
               present
               ,
               refer
               you
               .
            
          
           
             
               Problem
               IV.
               The
               Diameter
               and
               content
               of
               a
               Vessel
               being
               given
               ,
               to
               find
               the
               length
               of
               the
               Vessel
               .
            
             
               Extend
               the
               Compasses
               from
               the
               Diameter
               to
               the
               Gage-point
               ,
               the
               same
               Extent
               twice
               repeated
               from
               the
               Content
               ,
               shall
               give
               the
               length
               required
               .
            
             
               Example
               .
            
             
               If
               the
               Content
               be
               60
               ,
               and
               the
               Diameter
               24
               ,
               then
               extend
               the
               Compasses
               from
               24
               ,
               to
               17-15
               ,
               the
               Gage-point
               for
               Wine
               ;
               this
               Extent
               turned
               twice
               the
               same
               way
               ,
               from
               60
               the
               Content
               ,
               shall
               reach
               to
               30
               inches
               ,
               and
               6
               tenths
               ,
               and
               a
               half
               ,
               the
               length
               required
               .
            
          
           
             
             
               Problem
               V.
               The
               Length
               and
               Content
               of
               a
               Vessel
               given
               ,
               to
               find
               the
               Diameter
               .
            
             
               Divide
               the
               space
               on
               the
               Line
               of
               Numbers
               ,
               between
               the
               Length
               and
               the
               Content
               ,
               into
               two
               equal
               parts
               ;
               the
               Compasses
               so
               set
               ,
               shall
               reach
               the
               same
               way
               from
               the
               Gage-point
               to
               the
               Diameter
               of
               the
               Vessel
               .
            
             
               Example
               .
            
             
               The
               half
               distance
               between
               31-65
               the
               length
               ,
               and
               60
               the
               Content
               ,
               shall
               reach
               the
               same
               way
               from
               17-15
               the
               Gage-point
               ,
               to
               24
               the
               Diameter
               required
               .
            
             
               These
               two
               last
               Problems
               may
               be
               useful
               for
               Coopers
               ,
               to
               make
               Cask
               of
               any
               length
               ,
               diameter
               ,
               and
               quantity
               .
            
          
           
             
               Problem
               VI.
               To
               find
               what
               is
               wanting
               in
               any
               close
               Cask
               ,
               at
               any
               number
               of
               inches
               and
               parts
               ,
               (
               the
               Cask
               lying
               after
               the
               usual
               manner
               ,
               with
               the
               bung-hole
               uppermost
               )
               from
               the
               bung-hole
               to
               the
               superficies
               of
               the
               Liquor
               given
               ,
               
                 two
                 wayes
              
               .
            
             
               This
               Problem
               I
               shall
               resolve
               
                 two
                 wayes
              
               ,
               either
               of
               which
               is
               experimented
               to
               come
               
               near
               the
               truth
               ,
               and
               will
               very
               well
               serve
               ,
               till
               a
               better
               comes
               to
               light
               .
            
             
               The
               One
               ,
               by
               a
               Line
               of
               Segments
               ,
               joyned
               to
               the
               Line
               of
               Numbers
               ,
               as
               before
               in
               the
               measuring
               the
               Fragments
               of
               a
               Globe
               ;
               But
               ,
            
             
               The
               Other
               ,
               is
               by
               a
               way
               found
               out
               by
               Mr.
               Bennit
               ,
               a
               Cooper
               ,
               that
               hath
               long
               exercised
               the
               way
               of
               Gaging
               ,
               which
               is
               by
               comparing
               a
               Cask
               known
               ,
               and
               its
               quantity
               of
               emptiness
               ,
               to
               a
               Cask
               unknown
               ,
               and
               its
               inches
               of
               emptiness
               ,
               as
               followeth
               .
            
             
               
                 First
                 ,
                 by
                 the
                 Line
                 of
                 Numbers
                 ,
                 and
                 Artificial
                 Line
                 of
                 Segments
                 ,
                 to
                 find
                 the
                 quantity
                 of
                 Gallons
                 that
                 any
                 Vessel
                 wants
                 of
                 being
                 full
                 ,
                 at
                 any
                 number
                 of
                 Inches
                 ,
                 from
                 the
                 inside
                 of
                 the
                 bung-hole
                 ,
                 to
                 the
                 superficies
                 of
                 the
                 Liquor
                 ,
                 which
                 is
                 usually
                 called
                 Inches
                 dry
                 .
              
            
             
               Extend
               the
               Compasses
               ,
               on
               the
               Line
               of
               Numbers
               ,
               from
               the
               inches
               and
               tenths
               diameter
               at
               the
               bung
               ,
               to
               100
               on
               the
               Line
               of
               Segments
               ,
               the
               same
               extent
               applied
               the
               same
               way
               from
               the
               inches
               and
               parts
               dry
               ,
               shall
               reach
               to
               a
               4th
               Number
               ,
               on
               the
               Line
               of
               Artificial
               Segments
               ;
               which
               4th
               Number
               you
               must
               keep
               .
               (
               Or
               ,
               if
               you
               will
               ,
               you
               may
               use
               the
               inches
               wet
               ,
               laying
               the
               same
               extent
               from
               the
               inches
               wet
               ,
               and
               that
               also
               will
               on
               the
               Segments
               
               give
               a
               4th
               Number
               ,
               which
               you
               must
               likewise
               keep
               .
               )
            
             
               
                 Then
                 secondly
              
               ,
            
             
               As
               the
               Extent
               from
               1
               ,
               to
               the
               whole
               Content
               of
               the
               Vessel
               in
               Wine
               or
               Ale-gallons
               ;
               So
               is
               the
               4th
               Number
               kept
               to
               the
               Gallons
               of
               emptiness
               ,
               or
               fullness
               ,
               that
               it
               wants
               of
               being
               full
               ,
               or
               the
               quantity
               of
               Gallons
               in
               the
               Vessel
               .
            
             
               
                 Example
                 ,
                 of
                 a
                 Canary-Pipe
                 ,
                 whose
                 Diameter
                 at
                 bung
                 ,
                 is
                 28
                 inches
                 and
                 7
                 ,
                 and
                 full
                 Content
                 is
                 Gallons
                 
                   116
                   ½
                
                 ,
                 at
                 12
                 inches
                 dry
                 ,
                 or
                 16
                 inches
                 ,
                 and
                 7
                 tenths
                 wet
                 .
              
            
             
               The
               Extent
               of
               the
               Compasses
               from
               28-7
               ,
               to
               100
               ,
               (
               at
               the
               end
               of
               the
               Line
               of
               Segments
               )
               shall
               reach
               the
               same
               way
               from
               12
               ,
               the
               inches
               dry
               ,
               to
               39
               ½
               on
               the
               Line
               of
               Segments
               for
               a
               4th
               ;
               or
               from
               16-7
               wet
               ,
               to
               60
               2
               /
               10
               on
               the
               Segments
               ,
               for
               his
               4th
               also
               ,
               which
               two
               4ths
               keep
               .
            
             
               
                 Then
                 secondly
              
               ,
            
             
               The
               Extent
               from
               1
               ,
               to
               116
               ½
               ,
               the
               whole
               Content
               in
               Gallons
               ,
               shall
               rea●h
               from
               39
               ,
               the
               dry
               4th
               ,
               on
               the
               Line
               of
               Numbers
               ,
               to
               46
               3
               /
               10
               ,
               for
               the
               gallons
               dry
               or
               wanting
               :
               or
               the
               same
               extent
               shall
               reach
               the
               same
               way
               ,
               on
               the
               Line
               of
               Numbers
               ,
               from
               60
               2
               /
               10
               ,
               the
               4th
               Number
               for
               wet
               ,
               to
               70
               gallons
               ,
               and
               2
               tenths
               
               in
               the
               Vessel
               ,
               at
               16
               inches
               and
               7
               tenths
               wet
               ;
               which
               two
               Numbers
               put
               together
               ,
               makes
               up
               116
               gallons
               and
               a
               half
               ,
               the
               full
               Content
               .
            
             
               The
               like
               manner
               of
               working
               serves
               for
               any
               Cask
               whatsoever
               ,
               and
               the
               nearer
               the
               Vessel
               wants
               of
               being
               half
               empty
               ,
               the
               more
               near
               to
               the
               truth
               will
               your
               work
               be
               ,
               and
               the
               most
               errour
               in
               very
               round
               and
               swelling
               Cask
               ,
               when
               the
               emptiness
               is
               not
               above
               one
               or
               two
               inches
               ;
               but
               in
               Vessels
               near
               to
               Cillenders
               ,
               it
               will
               give
               the
               Answer
               very
               true
               ,
               and
               as
               readily
               as
               any
               way
               whatsoever
               .
            
             
               
                 Observe
                 also
              
               ,
               That
               if
               you
               use
               the
               Segments
               in
               taking
               the
               wants
               ,
               you
               must
               abate
               of
               the
               gallons
               found
               ,
               till
               you
               come
               to
               the
               2
               thirds
               of
               the
               half
               diameter
               ;
               that
               is
               to
               say
               ,
               the
               Rule
               sayes
               ,
               there
               is
               more
               wanting
               than
               indeed
               there
               is
               ;
               and
               that
               somewhat
               considerable
               about
               the
               first
               6
               inches
               in
               a
               vessel
               of
               30
               inches
               diameter
               :
               So
               that
               I
               find
               a
               Table
               made
               as
               a
               mean
               between
               the
               Superficial
               and
               solid
               Segments
               ,
               would
               do
               the
               work
               the
               truest
               and
               best
               of
               any
               other
               ;
               Or
               else
               ,
               use
               the
               mean
               diameter
               and
               mean
               parts
               of
               emptiness
               ;
               found
               thus
               .
            
             
               Take
               the
               equaded
               diameter
               ,
               from
               the
               diameter
               at
               the
               bung
               ;
               and
               note
               the
               difference
               :
               then
               half
               this
               difference
               taken
               from
               the
               inches
               and
               parts
               empty
               gives
               the
               mean
               
               emptiness
               ;
               then
               use
               the
               mean
               diameter
               ,
               and
               mean
               emptiness
               ,
               instead
               of
               the
               other
               ,
               and
               the
               work
               is
               more
               exact
               .
            
             
               
                 The
                 other
                 way
                 of
                 Mr.
                 Bennits
                 invention
                 is
                 thus
              
               ;
            
             
               First
               ,
               you
               are
               to
               fill
               an
               ordinary
               Cask
               ,
               of
               a
               competent
               magnitude
               ,
               as
               60
               or
               100
               gallons
               ,
               of
               a
               mean
               form
               ,
               between
               a
               Spheriord
               (
               or
               roundish
               form
               )
               and
               a
               Cillenderical
               form
               ;
               or
               else
               fill
               two
               Casks
               of
               each
               form
               ,
               and
               learn
               the
               true
               Content
               ,
               and
               Diameter
               of
               that
               mean
               Vessel
               ,
               or
               rather
               of
               both
               those
               Vessels
               ;
               and
               the
               Vessel
               being
               full
               ,
               draw
               off
               with
               a
               true
               gallon-measure
               ,
               and
               on
               the
               drawing
               off
               every
               gallon
               ,
               take
               the
               exact
               quantity
               of
               inches
               and
               10th
               parts
               ,
               that
               the
               drawing
               off
               of
               every
               gallon
               makes
               in
               the
               emptiness
               or
               driness
               of
               that
               mean
               Vessel
               ,
               or
               rather
               both
               those
               Vessels
               ,
               at
               least
               until
               you
               have
               drawn
               off
               the
               half
               quantity
               of
               the
               Vessel
               ,
               which
               number
               of
               gallons
               drawn
               off
               ,
               and
               the
               inches
               and
               tenth
               parts
               of
               emptiness
               ,
               or
               fulness
               ,
               or
               driness
               or
               wetness
               ,
               you
               must
               draw
               into
               a
               Table
               ,
               or
               insert
               them
               on
               a
               Rule
               ,
               making
               the
               inches
               as
               equal
               parts
               ,
               and
               the
               gallons
               ,
               and
               his
               proportional
               part
               of
               a
               gallon
               ,
               the
               unequal
               parts
               ;
               then
               with
               the
               Line
               of
               Numbers
               ,
               and
               this
               mean
               Table
               ,
               or
               rather
               two
               Tables
               or
               Scales
               ,
               which
               you
               may
               
               put
               on
               a
               Rule
               ,
               as
               Mr.
               Bennit
               hath
               done
               ,
               you
               may
               find
               out
               the
               wants
               of
               any
               Cask
               whatsoever
               ;
               either
               Spherioid
               ,
               or
               Cillender-like
               ,
               as
               followeth
               .
            
             
               This
               measured
               Cask
               on
               the
               Scale
               ,
               or
               Table
               ,
               for
               methods
               sake
               ,
               and
               avoiding
               tautologie
               ,
               I
               shall
               call
               the
               first
               Cask
               ,
               and
               the
               Vessel
               or
               Cask
               ,
               whose
               wants
               you
               would
               know
               ,
               I
               shall
               call
               the
               second
               Cask
               ;
               then
               the
               proportion
               is
               thus
               .
            
             
               As
               the
               Diameter
               at
               the
               bung
               of
               the
               second
               Cask
               ,
               is
               to
               the
               bung
               diameter
               of
               the
               first
               Cask
               (
               which
               is
               always
               fixed
               )
               ;
            
             
               So
               is
               the
               inches
               dry
               of
               the
               second
               Cask
               to
               a
               4th
               (
               on
               the
               Line
               of
               Numbers
               )
               which
               4th
               Number
               sought
               on
               the
               inches
               of
               your
               Table
               ,
               or
               Scale
               ,
               on
               the
               opposite-part
               of
               your
               Scale
               or
               Table
               ,
               gives
               a
               5th
               Number
               ,
               which
               you
               must
               keep
               .
            
             
               Then
               ,
            
             
               As
               the
               whole
               Content
               of
               the
               first
               Cask
               ,
               is
               to
               the
               whole
               Content
               of
               the
               second
               Cask
               ;
            
             
               So
               is
               the
               first
               Number
               kept
               ,
               to
               the
               Number
               of
               Gallons
               the
               Vessel
               wants
               of
               being
               full
               ,
               at
               so
               many
               inches
               dry
               .
            
             
               Example
               .
            
             
               There
               is
               such
               a
               Scale
               made
               on
               purpose
               for
               Victuallers
               use
               ,
               to
               measure
               what
               they
               want
               of
               a
               Barrel
               of
               Ale
               ,
               being
               put
               into
               a
               Beer-barrel
               ,
               
               which
               Scale
               I
               shall
               here
               use
               ,
               to
               try
               this
               former
               Example
               by
               .
            
             
               Suppose
               ,
               as
               before
               ,
               a
               Canary
               Pipe
               want
               12
               inches
               of
               being
               full
               ,
               and
               the
               Content
               116
               ½
               gallons
               ,
               and
               28
               inches
               and
               7
               tenths
               diameter
               at
               bung
               ;
               The
               Extent
               on
               the
               Line
               of
               Numbers
               from
               28-7
               ,
               to
               22-5
               ,
               shall
               reach
               from
               12
               ,
               to
               9-4
               ;
               then
               just
               against
               9
               inches
               and
               4
               tenths
               ,
               on
               that
               Barrel
               Scale
               ,
               I
               find
               14
               gallons
               of
               Beer
               ,
               which
               is
               17
               gallons
               and
               a
               half
               of
               Wine
               ,
               being
               the
               5th
               Number
               to
               be
               kept
               .
            
             
               Then
               the
               Extent
               from
               44
               ,
               the
               Content
               of
               a
               Barrel
               in
               Wine-gallons
               ,
               to
               116
               ½
               ,
               the
               Content
               of
               a
               Canary-Pipe
               in
               the
               same
               gallons
               ,
               shall
               reach
               the
               same
               way
               from
               17
               ½
               the
               Number
               kept
               ,
               to
               46
               ,
               and
               near
               a
               half
               ,
               the
               gallons
               wanting
               at
               12
               inches
               dry
               ,
               in
               the
               Canary
               Pipe
               ,
               and
               46
               gallons
               ,
               and
               3
               quarts
               ,
               is
               the
               Number
               Mr.
               Bennit
               finds
               in
               a
               Canary-Pipe
               ,
               by
               measuring
               at
               12
               inches
               dry
               .
            
             
               Thus
               you
               have
               an
               account
               of
               the
               two
               easie
               Mechanick
               wayes
               ,
               to
               discover
               the
               wants
               of
               Cask
               ,
               very
               applicable
               ,
               and
               ready
               ,
               and
               experimented
               to
               be
               
                 Propè
                 verum
              
               .
            
             
             
               
                 
                   The
                   Gallons
                   wanting
                   in
                   a
                   Barrel
                   ,
                   at
                   every
                   inch
                   and
                   quarter
                   .
                
                 
                   
                      
                  
                   
                     
                       Beer
                       Gall.
                    
                     
                  
                   
                     
                       Wine
                       Gall
                    
                  
                   
                      
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     gal
                     .
                     pi
                     .
                     100
                  
                   
                     gal
                     .
                     pi
                     .
                     100
                  
                   
                     gal
                     .
                     1000
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     0
                     0
                     40
                  
                   
                     0
                     0
                     49
                  
                   
                     0
                     0612
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     0
                     1
                     20
                  
                   
                     0
                     1
                     47
                  
                   
                     0
                     184
                  
                   
                     22
                  
                
                 
                   
                      
                  
                   
                     0
                     2
                     10
                  
                   
                     0
                     2
                     57
                  
                   
                     0
                     321
                  
                   
                      
                  
                
                 
                   
                     1
                  
                   
                     0
                     3
                     10
                  
                   
                     0
                     3
                     80
                  
                   
                     0
                     475
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     0
                     4
                     33
                  
                   
                     0
                     5
                     30
                  
                   
                     0
                     663
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     0
                     6
                     00
                  
                   
                     0
                     7
                     35
                  
                   
                     0
                     920
                  
                   
                     21
                  
                
                 
                   
                      
                  
                   
                     0
                     7
                     60
                  
                   
                     1
                     1
                     29
                  
                   
                     1
                     161
                  
                   
                      
                  
                
                 
                   
                     2
                  
                   
                     1
                     1
                     80
                  
                   
                     1
                     4
                     00
                  
                   
                     1
                     500
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     1
                     3
                     90
                  
                   
                     1
                     6
                     56
                  
                   
                     1
                     821
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     1
                     6
                     10
                  
                   
                     2
                     1
                     22
                  
                   
                     2
                     153
                  
                   
                     20
                  
                
                 
                   
                      
                  
                   
                     2
                     0
                     66
                  
                   
                     2
                     4
                     34
                  
                   
                     2
                     543
                  
                   
                      
                  
                
                 
                   
                     3
                  
                   
                     2
                     3
                     50
                  
                   
                     2
                     7
                     98
                  
                   
                     2
                     998
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     2
                     6
                     16
                  
                   
                     3
                     3
                     10
                  
                   
                     3
                     388
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     3
                     0
                     70
                  
                   
                     3
                     6
                     20
                  
                   
                     3
                     772
                  
                   
                     19
                  
                
                 
                   
                      
                  
                   
                     3
                     3
                     80
                  
                   
                     4
                     2
                     00
                  
                   
                     4
                     250
                  
                   
                      
                  
                
                 
                   
                     4
                  
                   
                     3
                     6
                     50
                  
                   
                     4
                     5
                     30
                  
                   
                     4
                     663
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     4
                     1
                     80
                  
                   
                     5
                     1
                     35
                  
                   
                     5
                     169
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     4
                     5
                     25
                  
                   
                     5
                     5
                     60
                  
                   
                     5
                     700
                  
                   
                     18
                  
                
                 
                   
                      
                  
                   
                     5
                     0
                     42
                  
                   
                     6
                     1
                     45
                  
                   
                     6
                     182
                  
                   
                      
                  
                
                 
                   
                     5
                  
                   
                     5
                     3
                     90
                  
                   
                     6
                     5
                     70
                  
                   
                     6
                     713
                  
                   
                      
                  
                
                 
                   
                     
                  
                   
                     5
                     7
                     20
                  
                   
                     7
                     1
                     70
                  
                   
                     7
                     213
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     6
                     2
                     80
                  
                   
                     7
                     6
                     20
                  
                   
                     7
                     777
                  
                   
                     17
                  
                
                 
                   
                      
                  
                   
                     6
                     6
                     50
                  
                   
                     8
                     2
                     65
                  
                   
                     8
                     333
                  
                   
                      
                  
                
                 
                   
                     6
                  
                   
                     7
                     2
                     20
                  
                   
                     8
                     7
                     20
                  
                   
                     8
                     900
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     7
                     5
                     50
                  
                   
                     9
                     3
                     20
                  
                   
                     9
                     400
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     8
                     1
                     10
                  
                   
                     9
                     7
                     70
                  
                   
                     9
                     960
                  
                   
                     16
                  
                
                 
                   
                      
                  
                   
                     8
                     4
                     80
                  
                   
                     10
                     4
                     20
                  
                   
                     10
                     525
                  
                   
                      
                  
                
                 
                   
                     7
                  
                   
                     9
                     0
                     70
                  
                   
                     11
                     1
                     00
                  
                   
                     11
                     125
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     9
                     4
                     50
                  
                   
                     11
                     5
                     40
                  
                   
                     11
                     806
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     10
                     0
                     40
                  
                   
                     12
                     2
                     20
                  
                   
                     12
                     275
                  
                   
                     15
                  
                
                 
                   
                      
                  
                   
                     10
                     4
                     30
                  
                   
                     12
                     7
                     00
                  
                   
                     12
                     876
                  
                   
                      
                  
                
                 
                   
                     8
                  
                   
                     11
                     0
                     50
                  
                   
                     13
                     4
                     10
                  
                   
                     13
                     513
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     11
                     4
                     30
                  
                   
                     14
                     0
                     80
                  
                   
                     14
                     110
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     12
                     0
                     30
                  
                   
                     14
                     5
                     80
                  
                   
                     14
                     725
                  
                   
                     14
                  
                
                 
                   
                      
                  
                   
                     12
                     4
                     29
                  
                   
                     15
                     2
                     80
                  
                   
                     15
                     350
                  
                   
                      
                  
                
                 
                   
                     9
                  
                   
                     13
                     0
                     30
                  
                   
                     15
                     7
                     70
                  
                   
                     15
                     926
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     13
                     4
                     30
                  
                   
                     16
                     4
                     60
                  
                   
                     16
                     577
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     14
                     0
                     40
                  
                   
                     17
                     1
                     60
                  
                   
                     17
                     200
                  
                   
                     13
                  
                
                 
                   
                      
                  
                   
                     14
                     4
                     60
                  
                   
                     17
                     6
                     60
                  
                   
                     17
                     827
                  
                   
                      
                  
                
                 
                   
                     10
                  
                   
                     15
                     0
                     50
                  
                   
                     18
                     3
                     40
                  
                   
                     18
                     425
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     15
                     4
                     48
                  
                   
                     19
                     0
                     30
                  
                   
                     19
                     037
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     16
                     0
                     80
                  
                   
                     19
                     6
                     50
                  
                   
                     19
                     815
                  
                   
                     12
                  
                
                 
                   
                      
                  
                   
                     16
                     5
                     50
                  
                   
                     20
                     3
                     25
                  
                   
                     20
                     40●
                  
                   
                      
                  
                
                 
                   
                     11
                  
                   
                     17
                     2
                     20
                  
                   
                     21
                     1
                     00
                  
                   
                     21
                     225
                  
                   
                      
                  
                
                 
                   
                     
                  
                   
                     17
                     7
                     90
                  
                   
                     22
                     0
                     00
                  
                   
                     21
                     000
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     18
                     5
                     49
                  
                   
                     22
                     6
                     98
                  
                   
                     22
                     644
                  
                   
                     11
                  
                
                 
                   
                      
                  
                   
                     19
                     2
                     00
                  
                   
                     23
                     4
                     31
                  
                   
                     23
                     391
                  
                   
                      
                  
                
                 
                   
                     12
                  
                   
                     19
                     6
                     16
                  
                   
                     24
                     1
                     48
                  
                   
                     24
                     184
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     20
                     3
                     00
                  
                   
                     24
                     7
                     30
                  
                   
                     24
                     961
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     20
                     7
                     40
                  
                   
                     25
                     4
                     60
                  
                   
                     25
                     575
                  
                   
                     10
                  
                
                 
                   
                      
                  
                   
                     21
                     3
                     10
                  
                   
                     26
                     1
                     36
                  
                   
                     26
                     170
                  
                   
                      
                  
                
                 
                   
                     13
                  
                   
                     21
                     7
                     40
                  
                   
                     26
                     6
                     38
                  
                   
                     26
                     799
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     22
                     3
                     00
                  
                   
                     27
                     3
                     36
                  
                   
                     27
                     130
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     22
                     7
                     00
                  
                   
                     28
                     2
                     18
                  
                   
                     28
                     174
                  
                   
                     9
                  
                
                 
                   
                      
                  
                   
                     23
                     3
                     00
                  
                   
                     28
                     5
                     18
                  
                   
                     28
                     648
                  
                   
                      
                  
                
                 
                   
                     14
                  
                   
                     23
                     7
                     30
                  
                   
                     29
                     2
                     19
                  
                   
                     29
                     275
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     24
                     3
                     70
                  
                   
                     29
                     7
                     40
                  
                   
                     29
                     926
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     24
                     7
                     40
                  
                   
                     30
                     3
                     90
                  
                   
                     30
                     488
                  
                   
                     8
                  
                
                 
                   
                      
                  
                   
                     25
                     3
                     60
                  
                   
                     31
                     1
                     00
                  
                   
                     31
                     125
                  
                   
                      
                  
                
                 
                   
                     15
                  
                   
                     25
                     7
                     50
                  
                   
                     31
                     5
                     80
                  
                   
                     31
                     726
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     26
                     3
                     30
                  
                   
                     32
                     2
                     60
                  
                   
                     32
                     325
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     26
                     7
                     00
                  
                   
                     32
                     7
                     00
                  
                   
                     32
                     875
                  
                   
                     7
                  
                
                 
                   
                      
                  
                   
                     27
                     3
                     00
                  
                   
                     33
                     3
                     80
                  
                   
                     33
                     475
                  
                   
                      
                  
                
                 
                   
                     16
                  
                   
                     27
                     6
                     40
                  
                   
                     34
                     0
                     30
                  
                   
                     34
                     037
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     28
                     2
                     20
                  
                   
                     34
                     4
                     80
                  
                   
                     34
                     600
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     28
                     5
                     80
                  
                   
                     35
                     0
                     80
                  
                   
                     35
                     100
                  
                   
                     6
                  
                
                 
                   
                      
                  
                   
                     29
                     1
                     40
                  
                   
                     35
                     5
                     34
                  
                   
                     35
                     668
                  
                   
                      
                  
                
                 
                   
                     17
                  
                   
                     29
                     4
                     80
                  
                   
                     36
                     1
                     80
                  
                   
                     36
                     225
                  
                   
                      
                  
                
                 
                   
                     
                  
                   
                     30
                     0
                     40
                  
                   
                     36
                     6
                     29
                  
                   
                     36
                     788
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     30
                     4
                     10
                  
                   
                     37
                     2
                     29
                  
                   
                     37
                     287
                  
                   
                     5
                  
                
                 
                   
                      
                  
                   
                     30
                     7
                     50
                  
                   
                     37
                     6
                     54
                  
                   
                     37
                     820
                  
                   
                      
                  
                
                 
                   
                     18
                  
                   
                     31
                     3
                     00
                  
                   
                     38
                     2
                     39
                  
                   
                     38
                     299
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     31
                     6
                     10
                  
                   
                     38
                     6
                     64
                  
                   
                     38
                     833
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     32
                     1
                     80
                  
                   
                     39
                     2
                     70
                  
                   
                     39
                     338
                  
                   
                     4
                  
                
                 
                   
                      
                  
                   
                     32
                     5
                     00
                  
                   
                     39
                     6
                     00
                  
                   
                     39
                     752
                  
                   
                      
                  
                
                 
                   
                     19
                  
                   
                     32
                     7
                     80
                  
                   
                     40
                     1
                     80
                  
                   
                     40
                     225
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     33
                     2
                     10
                  
                   
                     40
                     4
                     90
                  
                   
                     40
                     614
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     33
                     4
                     80
                  
                   
                     41
                     0
                     10
                  
                   
                     41
                     012
                  
                   
                     3
                  
                
                 
                   
                      
                  
                   
                     33
                     7
                     40
                  
                   
                     41
                     3
                     65
                  
                   
                     41
                     457
                  
                   
                      
                  
                
                 
                   
                     20
                  
                   
                     34
                     2
                     00
                  
                   
                     41
                     6
                     77
                  
                   
                     41
                     848
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     34
                     4
                     30
                  
                   
                     42
                     1
                     43
                  
                   
                     42
                     180
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     34
                     6
                     20
                  
                   
                     42
                     4
                     00
                  
                   
                     42
                     500
                  
                   
                     2
                  
                
                 
                   
                      
                  
                   
                     35
                     0
                     10
                  
                   
                     42
                     6
                     70
                  
                   
                     42
                     840
                  
                   
                      
                  
                
                 
                   
                     21
                  
                   
                     35
                     2
                     00
                  
                   
                     43
                     0
                     64
                  
                   
                     43
                     055
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     35
                     3
                     60
                  
                   
                     43
                     2
                     70
                  
                   
                     43
                     338
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     35
                     4
                     80
                  
                   
                     43
                     4
                     19
                  
                   
                     43
                     524
                  
                   
                     1
                  
                
                 
                   
                      
                  
                   
                     35
                     6
                     00
                  
                   
                     43
                     5
                     42
                  
                   
                     43
                     678
                  
                   
                      
                  
                
                 
                   
                     22
                  
                   
                     35
                     6
                     80
                  
                   
                     43
                     6
                     51
                  
                   
                     43
                     816
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     35
                     7
                     40
                  
                   
                     43
                     7
                     50
                  
                   
                     43
                     938
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     36
                     0
                     00
                  
                   
                     44
                     0
                     00
                  
                   
                     44
                     000
                  
                   
                      
                  
                
              
            
          
        
         
           
           
             CHAP.
             XIII
             .
             The
             use
             of
             the
             Line
             of
             Numbers
             ,
             in
             Questions
             of
             Interest
             and
             Annuities
             .
          
           
             
               Problem
               I.
               At
               any
               rate
               of
               Interest
               
                 per
                 annum
              
               for
               a
               hundred
               pounds
               ,
               to
               find
               what
               the
               Interest
               of
               any
               greater
               or
               lesser
               sum
               comes
               to
               in
               one
               year
               .
            
             
               EXtend
               the
               Compasses
               from
               100
               to
               the
               increase
               of
               100
               l.
               in
               one
               year
               ,
               the
               same
               Extent
               shall
               reach
               from
               the
               sum
               propounded
               ,
               to
               its
               increase
               for
               one
               year
               ,
               at
               that
               rate
               propounded
               .
            
             
               
                 Example
                 .
                 What
                 is
                 the
                 increase
                 or
                 profit
                 of
                 124
                 l.
                 10
                 s.
                 for
                 one
                 year
                 ,
                 at
                 6
                 per
                 cent
                 .
                 per
                 annum
                 ?
              
            
             
               The
               Extent
               of
               the
               Compasses
               ,
               from
               100
               to
               6
               ,
               being
               laid
               the
               same
               way
               from
               124
               l.
               
               10
               s.
               (
               which
               is
               at
               124-5
               )
               shall
               reach
               to
               7-47
               ,
               which
               is
               7
               l.
               —
               9
               s.
               —
               4
               d.
               the
               profit
               of
               124·
               10
               s
               in
               one
               year
               .
            
          
           
             
               Problem
               II.
               Any
               sum
               of
               Money
               ,
               and
               the
               rate
               of
               Interest
               propounded
               ,
               to
               find
               what
               it
               will
               increase
               to
               ,
               at
               any
               number
               of
               years
               ,
               counting
               Interest
               upon
               Interest
               .
            
             
               The
               Extent
               of
               the
               Compasses
               from
               100
               ,
               to
               the
               increase
               of
               100
               ,
               being
               turned
               as
               many
               times
               from
               the
               sum
               propounded
               the
               same
               way
               ,
               as
               there
               be
               years
               propounded
               ,
               shall
               at
               last
               stay
               at
               the
               Principal
               and
               Interest
               required
               .
            
             
               
                 Example
                 .
                 To
                 what
                 sum
                 shall
                 143
                 pounds
                 10
                 shillings
                 ,
                 amount
                 to
                 in
                 10
                 years
                 ,
                 counting
                 Interest
                 upon
                 Interest
                 ,
                 at
                 6
                 per
                 cent
                 ?
              
            
             
               The
               Extent
               of
               the
               Compasses
               from
               100
               ,
               to
               106
               ,
               being
               turned
               10
               times
               from
               143
               ½
               ,
               shall
               reach
               to
               257
               l.
               0
               s.
               the
               sum
               of
               Principal
               and
               Interest
               at
               10
               years
               end
               .
            
             
               Note
               ,
               That
               in
               doing
               this
               ,
               you
               ought
               to
               be
               very
               precise
               ,
               in
               taking
               the
               first
               Extent
               from
               100
               ,
               to
               106
               ;
               but
               to
               cure
               the
               uncertainty
               thereof
               ,
               you
               have
               this
               very
               good
               remedy
               :
               If
               you
               have
               a
               Diagonal
               Scale
               ,
               equal
               
               to
               the
               Radius
               of
               the
               Line
               of
               Numbers
               ,
               then
               use
               that
               ;
               if
               not
               ,
               use
               the
               Line
               of
               Lines
               on
               the
               Sector-side
               ,
               which
               should
               be
               made
               fit
               to
               (
               or
               the
               double
               ,
               or
               the
               half
               of
               )
               the
               Radius
               of
               the
               Line
               of
               Numbers
               .
            
             
               
                 As
                 thus
              
               ;
            
             
               Take
               the
               Extent
               from
               the
               Line
               of
               Numbers
               ,
               between
               100
               ,
               and
               106
               ;
               this
               Extent
               measured
               on
               the
               Line
               of
               Lines
               ,
               will
               be
               0253058
               ,
               could
               you
               see
               so
               many
               Figures
               ,
               but
               02531
               ,
               will
               serve
               your
               turn
               very
               well
               ;
               which
               Number
               you
               must
               note
               ,
               is
               the
               Logarithm
               of
               106
               ,
               neglecting
               the
               Caracteristick
               ;
               then
               this
               Number
               multiplied
               by
               10
               ,
               the
               Number
               of
               years
               ,
               is
               25310
               ;
               this
               Extent
               taken
               from
               the
               Center
               ,
               on
               the
               Line
               of
               Lines
               ,
               and
               laid
               increasing
               from
               143
               ½
               ,
               shall
               reach
               to
               257
               l.
               0
               s.
               0
               d.
               the
               true
               Number
               of
               the
               Use
               and
               Principal
               of
               143
               l.
               10
               s.
               put
               out
               ,
               or
               forborn
               for
               ten
               years
               .
            
          
           
             
               Problem
               III.
               A
               sum
               of
               Money
               being
               due
               at
               any
               time
               to
               come
               ,
               to
               find
               what
               it
               is
               worth
               in
               ready
               Money
               to
               be
               paid
               presently
               ,
               at
               any
               rate
               propounded
               .
            
             
               This
               Problem
               is
               the
               contrary
               to
               the
               last
               ,
               for
               if
               you
               shall
               turn
               the
               Extent
               between
               100
               and
               106
               ,
               ten
               times
               backward
               from
               
               257
               ,
               it
               will
               stay
               at
               143
               ½
               ,
               the
               worth
               in
               ready
               Money
               .
            
             
               
                 Or
                 ,
                 to
                 make
                 use
                 of
                 the
                 former
                 remedy
              
               ;
            
             
               Multiply
               0253058
               ,
               the
               Logarithm
               of
               106
               by
               10
               ;
               then
               this
               Extent
               taken
               and
               laid
               the
               decreasing
               way
               from
               257
               ,
               shall
               reach
               to
               143
               ½
               .
            
             
               
                 For
                 Note
              
               ,
               That
               the
               Line
               of
               Lines
               is
               the
               Scale
               of
               equal
               parts
               ,
               that
               makes
               the
               Line
               of
               Numbers
               ,
               and
               10
               ,
               or
               7
               ,
               or
               15
               ,
               or
               any
               other
               Number
               multiplied
               by
               the
               Logarithm
               of
               106
               ,
               taken
               from
               that
               Scale
               of
               Lines
               all
               at
               once
               ,
               is
               equal
               to
               so
               many
               repetitions
               ;
               and
               consequently
               more
               exact
               ,
               because
               of
               the
               difficulty
               of
               taking
               the
               10
               ,
               12
               ,
               or
               15th
               part
               of
               any
               Number
               whatsoever
               ;
               and
               observe
               ,
               That
               so
               much
               as
               you
               err
               in
               the
               first
               ,
               it
               will
               be
               10
               ,
               12
               ,
               or
               15
               ,
               or
               20
               times
               so
               much
               at
               last
               ,
               which
               may
               be
               considerable
               in
               this
               .
            
          
           
             
               Problem
               IV.
               A
               yearly
               Rent
               ,
               or
               Annuity
               being
               forborn
               a
               certain
               number
               of
               years
               ,
               to
               find
               what
               the
               Arrears
               thereof
               will
               amount
               unto
               ,
               according
               to
               any
               rate
               propounded
               .
            
             
               First
               ,
               you
               must
               find
               out
               the
               Principal-Money
               ,
               that
               answers
               to
               the
               Rent
               ,
               or
               Annuity
               in
               question
               ;
               then
               find
               the
               sum
               of
               
               that
               Principal
               and
               Use
               ,
               at
               the
               end
               of
               the
               term
               given
               ,
               at
               the
               rate
               propounded
               ;
               then
               the
               Principal
               taken
               out
               of
               this
               sum
               ,
               both
               of
               Arrears
               and
               Principal
               ,
               the
               Arrears
               do
               remain
               ,
               which
               is
               the
               sum
               you
               look
               for
               .
            
             
               Example
               .
            
             
               Suppose
               a
               Landlord
               live
               far
               from
               his
               Tennant
               ,
               and
               yet
               judging
               his
               Tennant
               honest
               ,
               and
               able
               ,
               is
               content
               to
               take
               his
               Rent
               once
               in
               every
               fourth
               year
               ,
               which
               should
               be
               paid
               every
               year
               ,
               or
               every
               quarter
               of
               the
               year
               ;
               and
               suppose
               the
               Rent
               be
               10
               
                 l.
                 per
                 annum
              
               ,
               and
               the
               rate
               of
               profit
               ,
               for
               the
               forbearance
               ,
               be
               8
               
                 per
                 cent
              
               .
            
             
               First
               ,
               to
               find
               the
               Principal
               for
               10
               
                 l.
                 per
                 annum
              
               ,
               at
               the
               rate
               of
               8
               
                 l.
                 per
                 cent
                 .
                 Say
              
               ,
               If
               8
               l.
               have
               100
               for
               his
               Principal
               ,
               what
               shall
               10
               l.
               have
               ?
               The
               Answer
               will
               be
               125
               ;
               for
               the
               Extent
               from
               8
               to
               100
               ,
               shall
               reach
               from
               10
               ,
               the
               same
               way
               ,
               to
               125
               ;
               then
               by
               the
               2d
               Problem
               of
               this
               Chapter
               ,
               125
               l.
               forborn
               for
               four
               years
               ,
               will
               come
               to
               170
               l.
               which
               is
               170
               l.
               0
               s.
               0
               d.
               from
               which
               sum
               ,
               if
               you
               substract
               125
               l.
               there
               remains
               45
               l.
               the
               Arrears
               for
               10
               
                 l.
                 per
                 annum
              
               forborn
               four
               years
               ,
               at
               the
               rate
               of
               8
               
                 per
                 cent
              
               .
            
             
               But
               if
               you
               would
               have
               the
               profit
               of
               these
               Arrearages
               ,
               supposing
               2
               l.
               —
               10
               s.
               the
               4th
               part
               of
               10
               
                 l.
                 per
                 annum
              
               to
               be
               paid
               quarterly
               ,
               and
               to
               count
               Use
               upon
               Use
               at
               the
               rate
               abovesaid
               ,
               
               then
               you
               will
               find
               the
               Principal
               and
               Arrears
               to
               be
               171
               l.
               10
               s
               ,
               For
               if
               you
               multiply
               0086
               ,
               the
               log
               .
               of
               102
               l.
               the
               Interest
               and
               Principal
               of
               100
               l.
               for
               a
               quarter
               of
               a
               year
               by
               16
               ,
               the
               quarters
               in
               four
               years
               ,
               it
               will
               be
               1376
               ,
               which
               Number
               taken
               from
               the
               Line
               of
               Lines
               ,
               and
               laid
               from
               120
               ,
               on
               the
               Line
               of
               Numbers
               ,
               shall
               reach
               to
               171
               ½
               ,
               or
               171
               l.
               10
               s.
               being
               30
               s.
               more
               than
               the
               former
               sum
               ,
               when
               150
               l.
               the
               Principal
               is
               taken
               away
               ,
               the
               residue
               Arreares
               is
               46
               l.
               10
               s.
               
            
             
               Or
               ,
            
             
               If
               you
               turn
               the
               distance
               on
               the
               Numbers
               between
               100
               and
               102
               ,
               16
               times
               from
               125
               ,
               which
               you
               may
               help
               thus
               ;
               turn
               first
               4
               times
               ,
               then
               take
               them
               4
               times
               in
               one
               Extent
               ,
               and
               turn
               3
               times
               more
               ,
               and
               you
               will
               stay
               at
               271
               ½
               ,
               the
               Answer
               required
               .
            
          
           
             
               Problem
               V.
               A
               yearly
               Rent
               ,
               or
               Annuity
               propounded
               ,
               to
               find
               the
               worth
               thereof
               in
               ready
               Money
               ,
               at
               any
               rate
               whatsoever
               .
            
             
               First
               ,
               by
               the
               4th
               Problem
               ,
               find
               the
               Arrears
               that
               shall
               be
               due
               at
               the
               end
               of
               the
               term
               ,
               and
               at
               the
               rate
               propounded
               ;
               then
               by
               the
               3d
               Problem
               ,
               find
               what
               those
               Arrears
               are
               worth
               in
               ready
               money
               ,
               which
               shall
               be
               the
               worth
               of
               the
               Annuity
               ,
               or
               Rent
               required
               .
            
             
             
               
                 Example
                 .
                 There
                 is
                 a
                 Lease
                 of
                 a
                 House
                 or
                 Land
                 worth
                 12
                 l.
                 per
                 annum
                 ,
                 and
                 there
                 is
                 16
                 years
                 yet
                 to
                 come
                 ;
                 which
                 Lease
                 a
                 man
                 would
                 buy
                 ,
                 provided
                 he
                 may
                 lay
                 out
                 his
                 money
                 to
                 gain
                 after
                 the
                 rate
                 of
                 10
                 l.
                 per
                 cent
                 :
                 the
                 question
                 is
                 ,
                 What
                 is
                 it
                 worth
                 ?
              
            
             
               First
               ,
               by
               the
               last
               ,
               if
               10
               l.
               have
               100
               for
               his
               Principal
               ,
               What
               shall
               12
               ?
               the
               Answer
               is
               120
               ;
               Then
               by
               the
               second
               part
               of
               the
               second
               ,
               120
               l.
               forborn
               16
               years
               ,
               comes
               to
               551
               l.
               the
               Principal
               and
               Interest
               :
               from
               which
               sum
               ,
               taking
               120
               l.
               the
               Principal
               ,
               there
               remains
               431
               the
               Arrears
               .
               Then
               by
               the
               third
               Problem
               find
               what
               431
               due
               16
               years
               to
               come
               ,
               is
               worth
               in
               ready
               money
               ;
               and
               the
               Answer
               will
               be
               at
               10
               in
               the
               100
               ,
               93
               l.
               14
               s.
               
            
             
               
                 Also
                 herein
                 observe
              
               ,
               That
               if
               there
               be
               any
               Reversion
               of
               a
               Lease
               to
               be
               expired
               ,
               before
               it
               may
               be
               injoyed
               ;
               then
               you
               are
               to
               find
               the
               worth
               of
               431
               l.
               after
               so
               many
               years
               more
               ;
               as
               suppose
               it
               be
               5
               years
               before
               the
               Annuity
               begin
               ;
               then
               find
               the
               worth
               of
               431
               ,
               forborn
               21
               years
               ,
               which
               will
               be
               58
               l.
               4
               s.
               
            
          
           
             
             
               Problem
               VI.
               A
               sum
               of
               Money
               is
               propounded
               ,
               and
               the
               rate
               whereby
               a
               man
               intends
               to
               Purchase
               ,
               to
               find
               what
               Annuity
               ,
               and
               how
               many
               years
               to
               continue
               ,
               that
               sum
               of
               money
               will
               buy
               .
            
             
               Take
               any
               known
               Annuity
               at
               pleasure
               ,
               and
               find
               by
               the
               last
               ,
               the
               value
               of
               that
               in
               ready
               money
               ,
               then
               this
               proportion
               holds
               ;
            
             
               As
               the
               value
               found
               ,
               is
               to
               the
               Annuity
               supposed
               ;
            
             
               So
               is
               the
               sum
               of
               money
               to
               be
               improved
               ,
               to
               the
               Annuity
               required
               .
            
             
               
                 Example
                 .
                 What
                 Annuity
                 ,
                 to
                 continue
                 16
                 years
                 ,
                 will
                 500
                 l.
                 Purchase
                 ,
                 whereby
                 a
                 man
                 may
                 gain
                 after
                 the
                 rate
                 of
                 10
                 l.
                 per
                 cent
                 ?
              
            
             
               By
               the
               last
               Problem
               I
               find
               ,
               That
               93
               l.
               14
               s.
               will
               purchase
               12
               l.
               a
               year
               ,
               for
               16
               years
               ,
               at
               10
               per
               cent
               .
            
             
               Therefore
               ,
            
             
               The
               Extent
               of
               the
               Compasses
               from
               93
               l.
               7
               ,
               to
               12
               l.
               
                 per
                 annum
              
               ,
               shall
               reach
               the
               same
               from
               500
               ,
               to
               64
               l.
               
                 per
                 annum
              
               .
               For
               such
               an
               Annuity
               ,
               to
               continue
               16
               years
               ,
               will
               500
               l.
               purchase
               ,
               to
               gain
               10
               l.
               
                 per
                 annum
                 ,
                 per
                 cent
              
               .
               for
               your
               Money
               .
            
          
           
             
             
               Problem
               VII
               .
               Or
               ,
               first
               rather
               ;
               Lands
               or
               Houses
               ,
               sold
               at
               any
               certain
               number
               of
               years
               Purchase
               ;
               to
               find
               what
               the
               value
               of
               the
               whole
               will
               be
               ?
            
             
               The
               usual
               way
               of
               valuing
               Land
               or
               Houses
               ,
               is
               by
               the
               years
               Purchase
               ,
               and
               Land
               Fee-simple
               is
               usually
               vallued
               at
               20
               years
               Purchase
               ;
               Coppy-hold-Land
               ,
               at
               15
               or
               16
               years
               Purchase
               ;
               and
               good
               ,
               strong
               ,
               and
               new
               Houses
               ,
               at
               12
               ,
               13
               ,
               or
               14
               years
               Purchase
               for
               Fee-simple
               .
            
             
               But
               a
               Lease
               of
               〈…〉
               of
               21
               years
               about
               7
               years
               Purchase
               ;
               and
               a
               Lease
               of
               31
               years
               ,
               about
               8
               years
               Purchase
               ,
               rather
               less
               than
               more
               ;
               and
               a
               Lease
               of
               60
               ,
               or
               100
               ,
               not
               worth
               above
               8
               ½
               years
               Purchase
               .
            
             
               Again
               ,
            
             
               The
               usual
               profit
               allowed
               for
               Land
               in
               Fee-simple
               ,
               is
               not
               above
               5
               l.
               in
               the
               100
               
                 per
                 annum
              
               ,
               because
               of
               the
               certainty
               thereof
               ;
               for
               Coppy-hold
               Land
               ,
               full
               6
               l.
               in
               the
               100
               
                 per
                 annum
              
               ;
               for
               the
               best
               Houses
               ,
               7
               and
               8
               l.
               in
               the
               100
               Fee-simple
               .
            
             
               But
               in
               laying
               out
               Money
               on
               Leases
               ,
               either
               of
               Land
               or
               Houses
               ,
               Men
               shall
               hardly
               be
               savers
               ,
               if
               they
               gain
               not
               8
               ,
               9
               ,
               or
               10
               in
               the
               100
               
                 per
                 annum
              
               ,
               for
               their
               Money
               ;
               The
               reason
               
               and
               demonstration
               whereof
               ,
               you
               may
               read
               at
               large
               in
               Mr.
               Phillips
               his
               
                 Purchasers
                 Pattern
              
               .
            
             
               Thus
               the
               number
               of
               years
               Purchase
               agreed
               on
               ,
               (
               which
               ought
               to
               be
               cleer
               ,
               from
               Quit-rent
               ,
               and
               Taxes
               ,
               and
               the
               like
               ;
               the
               Rent
               is
               usually
               various
               ,
               according
               to
               the
               place
               ,
               and
               time
               where
               ,
               and
               wherein
               ,
               the
               Purchase
               shall
               happen
               to
               be
               )
               then
               to
               find
               the
               quantity
               of
               the
               whole
               Purchase
               ,
            
             
               Say
               ,
            
             
               As
               1
               ,
               to
               20
               ,
               18
               ,
               15
               ,
               14
               ,
               12
               ,
               10
               ,
               or
               8
               ,
               the
               number
               of
               years
               Purchase
               ,
               for
               Fee-simple
               ,
               or
               Coppy-hold
               Land
               ,
               or
               Houses
               Fee-simple
               ,
               or
               Coppy-hold
               ;
               For
               Leases
               of
               60
               ,
               50
               ,
               40
               ,
               or
               30
               years
               ,
               or
               21
               years
               ;
            
             
               So
               is
               the
               yearly
               Rent
               to
               the
               whole
               value
               .
            
             
               Example
               .
            
             
               A
               Parcel
               of
               Land
               worth
               10
               
                 l.
                 per
                 annum
              
               Fee-simple
               ,
               valued
               at
               20
               years
               Purchase
               ,
               will
               amount
               to
               200
               l.
               
            
             
               For
               ,
            
             
               The
               Extent
               from
               1
               ,
               to
               20
               ,
               will
               reach
               the
               same
               way
               from
               10
               to
               200
               ,
               the
               whole
               price
               of
               20
               years
               Purchase
               ,
               at
               10
               
                 l.
                 per
                 annum
              
               .
            
          
        
         
           
           
             CHAP.
             XIV
             .
             The
             Use
             of
             the
             Line
             of
             Numbers
             IN
             Military
             Questions
             .
          
           
             
               Problem
               I.
               Any
               Number
               of
               Souldiers
               being
               propounded
               ,
               to
               order
               them
               into
               a
               Square
               Battel
               of
               Men
               ;
               that
               is
               ,
               as
               many
               in
               Rank
               as
               in
               File
               .
            
             
               FInd
               the
               Square-root
               of
               the
               Number
               of
               Souldiers
               ,
               and
               that
               shall
               be
               the
               Number
               of
               Men
               in
               Rank
               and
               File
               required
               .
            
             
               As
               suppose
               it
               were
               required
               to
               order
               1770
               Men
               ,
               in
               the
               order
               abovesaid
               ,
               you
               shall
               ,
               by
               the
               8th
               Probl.
               of
               the
               6th
               Chapt.
               find
               ,
               that
               the
               Square-root
               of
               1770
               is
               42
               ,
               and
               6
               over
               ,
               which
               here
               is
               not
               considerable
               .
            
          
           
             
             
               Problem
               II.
               Any
               number
               of
               Souldiers
               propounded
               ,
               to
               order
               them
               into
               a
               double
               Battel
               of
               Men
               ;
               that
               is
               to
               say
               ,
               twice
               as
               many
               in
               Rank
               as
               File
               .
            
             
               Find
               the
               Square-root
               of
               half
               the
               Number
               of
               Men
               ,
               and
               that
               is
               the
               Number
               of
               Men
               in
               File
               ,
               and
               the
               double
               the
               Number
               in
               Rank
               .
            
             
               
                 As
                 for
                 Example
              
               .
            
             
               If
               2603
               ,
               were
               so
               to
               be
               placed
               ,
               the
               half
               of
               2603
               ,
               is
               1301
               ;
               whose
               Square-root
               by
               the
               8th
               of
               6th
               ,
               is
               36
               ;
               the
               number
               of
               Men
               in
               File
               :
               and
               72
               ,
               the
               double
               thereof
               ,
               is
               the
               number
               in
               Rank
               .
               For
               if
               you
               shall
               multiply
               72
               by
               36
               ,
               the
               Product
               is
               2592
               ,
               almost
               the
               number
               of
               Men
               propounded
               .
            
          
           
             
               Problem
               III.
               Any
               Number
               of
               Souldiers
               being
               propounded
               ,
               to
               order
               them
               into
               a
               Quadruple
               Battel
               of
               Men
               ;
               
                 viz.
                 4
              
               times
               as
               many
               in
               Rank
               as
               File
               .
            
             
               Find
               the
               Square-root
               of
               a
               4th
               part
               of
               the
               Number
               of
               Men
               ,
               and
               that
               shall
               be
               the
               Number
               in
               File
               ;
               and
               4
               times
               so
               many
               the
               Number
               in
               Rank
               .
            
             
             
               So
               the
               4th
               part
               of
               2603
               ,
               is
               650
               ;
               whose
               Square-root
               is
               25
               ½
               ,
               and
               4
               times
               25
               is
               100
               ,
               the
               Number
               in
               Rank
               .
            
          
           
             
               Problem
               IV.
               Any
               Number
               of
               Souldiers
               being
               given
               ,
               together
               with
               their
               Distance
               one
               from
               another
               in
               Rank
               and
               File
               ,
               to
               order
               them
               into
               a
               Square
               Battel
               of
               Ground
               .
            
             
               As
               suppose
               I
               would
               order
               3000
               Men
               so
               ,
               that
               being
               7
               foot
               asunder
               in
               File
               ,
               and
               3
               foot
               apart
               in
               Rank
               ,
               the
               Ground
               whereon
               they
               stood
               should
               be
               Square
               .
            
             
               Extend
               the
               Compasses
               from
               7
               foot
               ,
               the
               distance
               in
               File
               ;
               to
               3
               foot
               ,
               the
               distance
               in
               Rank
               ;
               then
               that
               Extent
               applied
               the
               same
               way
               from
               3000
               ,
               the
               Number
               of
               Souldiers
               ,
               reaches
               to
               1286
               ,
               whose
               greatest
               Square-root
               is
               35-7
               ;
               that
               is
               ,
               35
               ,
               the
               Number
               of
               Men
               to
               be
               placed
               in
               File
               .
            
             
               Then
               ,
            
             
               If
               you
               divide
               3000
               ,
               the
               whole
               Number
               ,
               by
               35-7
               ,
               the
               Quotient
               is
               84
               ,
               the
               Number
               in
               Rank
               ,
               to
               use
               and
               imploy
               a
               Square
               plat
               of
               ground
               to
               stand
               in
               .
            
             
               As
               7
               ,
               to
               3
               ;
               so
               is
               3000
               ,
               to
               1286
               ,
               whose
               Square-root
               is
               35-7
               .
               Then
               ,
            
             
               As
               ●5-7
               ,
               to
               1
               ;
               so
               is
               3000
               ,
               to
               84.
               
            
          
           
             
             
               Problem
               V.
               Any
               Number
               of
               Souldiers
               propounded
               ,
               to
               order
               them
               into
               Rank
               and
               File
               ,
               according
               to
               the
               ratio
               of
               any
               two
               Numbers
               given
               .
            
             
               This
               Question
               is
               all
               one
               with
               the
               former
               ;
            
             
               For
               ,
            
             
               As
               the
               Number
               given
               for
               the
               distance
               in
               File
               ,
               is
               to
               that
               for
               the
               distance
               in
               Rank
               ;
            
             
               So
               is
               the
               whole
               Number
               of
               Souldiers
               to
               a
               4th
               ,
               whose
               Square-root
               is
               the
               Number
               of
               Men
               in
               Rank
               .
            
             
               Then.
               
            
             
               The
               whole
               Number
               divided
               by
               the
               Number
               in
               Rank
               ,
               the
               Quotient
               is
               the
               Number
               to
               be
               placed
               in
               File
               .
            
             
               Example
               .
            
             
               Suppose
               3000
               Souldiers
               were
               to
               be
               ordered
               in
               Rank
               and
               File
               :
               As
               5
               is
               to
               10
               ,
               or
               as
               5
               is
               to
               9
               ;
               that
               is
               to
               say
               ,
               that
               the
               Men
               in
               Rank
               ,
               might
               be
               in
               Proportion
               to
               them
               in
               File
               ,
               as
               9
               is
               to
               5.
               
            
             
               
                 Say
                 thus
              
               ;
            
             
               As
               the
               Extent
               from
               5
               ,
               to
               9
               ;
            
             
               So
               is
               3000
               ,
               to
               5400
               ,
               whose
               Square-root
               is
               73
               ½
               ,
               the
               Number
               of
               Men
               in
               Rank
               .
            
             
             
               Then
               ,
            
             
               As
               73
               ½
               ,
               to
               1
               ;
            
             
               So
               is
               3000
               ,
               to
               near
               41
               ,
               the
               Number
               in
               File
               .
            
          
           
             
               Problem
               VI.
               There
               are
               8100
               Men
               to
               be
               ordered
               into
               a
               Square
               Body
               of
               Men
               ,
               and
               to
               have
               so
               many
               Pikes
               ,
               as
               to
               arm
               the
               main
               Square-Body
               round
               about
               ,
               with
               6
               Ranks
               of
               Pikes
               ;
               the
               Question
               is
               ,
               How
               many
               Ranks
               must
               be
               in
               the
               whole
               Square
               Battel
               ?
               And
               ,
               How
               many
               Pikes
               and
               Musquets
               ?
            
             
               First
               ,
               the
               Square-root
               of
               8100
               ,
               is
               90
               ,
               the
               Number
               of
               Men
               in
               ,
               and
               Number
               of
               Ranks
               and
               Files
               ;
               now
               in
               regard
               that
               there
               must
               be
               6
               Ranks
               of
               Pikes
               round
               about
               the
               Musquetiers
               ,
               there
               will
               be
               12
               Ranks
               less
               of
               them
               ,
               both
               in
               Front
               and
               Flank
               ,
               than
               in
               the
               whole
               Body
               ;
               therefore
               substracting
               of
               12
               from
               90
               ,
               rest
               78
               ,
               whose
               Square
               is
               6084
               ,
               the
               number
               of
               Musquetiers
               ;
               which
               taken
               from
               8100
               ,
               there
               remains
               2061
               ,
               the
               number
               of
               Pikes
               .
            
          
           
             
             
               Problem
               VII
               .
               To
               three
               Numbers
               given
               ,
               to
               find
               a
               fourth
               in
               a
               doubled
               Proportion
               .
            
             
               For
               as
               much
               as
               like
               Squar●●
               ,
               are
               in
               double
               the
               Proportion
               of
               their
               answerable
               sides
               ;
               therefore
               you
               must
               work
               by
               their
               Squares
               ,
               and
               Square-root
               .
            
             
               
                 But
                 by
                 the
                 Line
                 of
                 Numbers
                 ,
                 in
                 this
                 manner
                 .
              
            
             
               If
               a
               Fathom
               of
               Rope
               ,
               of
               6
               inches
               compass
               about
               ,
               weigh
               6
               pound
               ,
               2
               ounces
               ,
               (
               or
               6-●25
               /
               1000
               )
               what
               shall
               a
               Fathom
               of
               Rope
               of
               12
               inches
               compass
               weigh
               ?
            
             
               
                 Here
                 Note
                 alwayes
              
               ,
               That
               when
               the
               two
               Numbers
               of
               like
               denomination
               ,
               which
               are
               given
               ,
               are
               of
               Lines
               ,
               or
               sides
               of
               Squares
               ,
               or
               Diameters
               of
               Circles
               ;
               then
               the
               Extent
               of
               the
               Compasses
               upon
               the
               Line
               of
               Numbers
               ,
               from
               one
               Line
               to
               the
               other
               ,
               or
               from
               one
               side
               to
               the
               other
               side
               ;
               that
               Extent
               turned
               twice
               the
               same
               way
               from
               the
               given
               Area
               ,
               or
               Content
               ,
               shall
               reach
               to
               the
               other
               required
               :
            
             
               So
               here
               ,
               the
               Extent
               of
               the
               Compasses
               from
               6
               to
               12
               ,
               being
               turned
               two
               times
               the
               same
               way
               from
               6-125
               ,
               shall
               reach
               to
               24-50
               ,
               for
               24
               pound
               and
               a
               half
               ,
               the
               weight
               required
               .
            
             
             
               But
               if
               the
               two
               terms
               given
               of
               one
               denomination
               ,
               are
               of
               Squares
               ,
               or
               Superficies
               ,
               or
               Areas
               ;
               then
               the
               half
               distance
               ,
               on
               the
               Line
               of
               Numbers
               ,
               between
               one
               Area
               and
               the
               other
               ,
               being
               turned
               the
               same
               way
               on
               the
               Line
               ,
               from
               the
               given
               Line
               or
               Side
               ,
               it
               shall
               reach
               to
               the
               Side
               ,
               or
               Line
               ,
               required
               .
            
             
               For
               the
               half-distance
               ,
               between
               24-50
               ,
               and
               6-125
               ,
               shall
               reach
               from
               12
               to
               6
               ;
               or
               the
               contrary
               ,
               from
               6
               to
               12.
               
            
             
               An
               Example
               whereof
               ,
               you
               have
               in
               the
               4th
               and
               5th
               Problems
               of
               the
               12th
               Chapter
               :
               Also
               ,
               in
               the
               6th
               and
               7th
               Problems
               of
               the
               8th
               Chapter
               ,
               which
               treates
               of
               Superficial-measure
               ,
               in
               measuring
               of
               Land.
               
            
             
               
                 Note
                 also
              
               ,
               That
               if
               you
               have
               three
               Lines
               of
               Numbers
               ,
               viz.
               a
               Great
               ,
               a
               Mean
               ,
               and
               a
               Less
               ;
               after
               Mr.
               Windgates
               way
               ;
               then
               these
               Questions
               are
               wrought
               without
               doubling
               ,
               or
               halving
               ,
               and
               very
               neatly
               and
               speedily
               .
            
             
               
                 As
                 thus
              
               ;
            
             
               The
               Extent
               on
               the
               mean
               Line
               ,
               from
               24-50
               ,
               to
               6-125
               ,
               the
               weight
               of
               the
               two
               Ropes
               ,
               shall
               reach
               on
               the
               great
               Line
               ,
               from
               12
               to
               6
               ;
               or
               from
               6
               to
               12
               ,
               the
               inches
               in
               compass
               about
               of
               each
               Rope
               .
            
          
           
             
             
               Problem
               VIII
               .
               To
               three
               Numbers
               given
               ,
               to
               find
               a
               fourth
               in
               a
               tripled
               Proportion
               .
            
             
               For
               as
               much
               as
               like
               Solids
               ,
               are
               in
               a
               tripled
               Proportion
               to
               their
               answerable
               side
               ;
               the
               Cubes
               of
               their
               sides
               are
               proportional
               one
               to
               another
               ;
               therefore
               ,
               to
               work
               these
               Questions
               by
               the
               Line
               of
               Numbers
               ,
               
                 do
                 thus
              
               ;
            
             
               When
               the
               two
               given
               terms
               ,
               of
               like
               denomination
               in
               the
               Question
               ,
               are
               of
               Sides
               ,
               Lines
               ,
               or
               Diameters
               ;
               then
               the
               Extent
               of
               the
               Compasses
               ,
               on
               the
               Line
               of
               Numbers
               ,
               from
               one
               side
               to
               the
               other
               ,
               that
               is
               ,
               from
               the
               side
               ,
               whose
               Cube
               or
               Solidity
               is
               also
               given
               ,
               to
               the
               other
               ;
               the
               same
               Extent
               ,
               turned
               three
               times
               from
               the
               given
               Cube
               ,
               or
               Solidity
               ,
               shall
               reach
               to
               the
               inquired
               Cube
               ,
               or
               Solidity
               .
            
             
               
                 As
                 for
                 Example
              
               .
            
             
               If
               a
               side
               of
               a
               Cube
               ,
               being
               12
               inches
               ,
               contain
               in
               Solidity
               1728
               cube
               inches
               ;
               How
               many
               inches
               is
               there
               in
               a
               Cube
               ,
               whose
               side
               is
               8
               inches
               ?
               The
               Extent
               from
               12
               to
               8
               ,
               being
               turned
               three
               times
               from
               1728
               ,
               shall
               reach
               to
               512
               ,
               the
               Solidity
               required
               of
               the
               Cube
               ,
               whose
               side
               is
               8
               inches
               every
               way
               .
            
             
             
               
                 Again
                 ,
                 on
                 the
                 contrary
              
               .
            
             
               When
               the
               two
               terms
               of
               the
               same
               denomination
               ,
               are
               Cubes
               or
               Solidities
               ,
               then
               divide
               the
               space
               on
               the
               Line
               of
               Numbers
               ,
               between
               the
               two
               Solidities
               ,
               into
               three
               equal
               parts
               ,
               and
               lay
               that
               Extent
               the
               same
               way
               ,
               as
               the
               reason
               of
               the
               Question
               doth
               require
               ,
               either
               increasing
               or
               diminishing
               ,
               from
               the
               given
               Side
               or
               Line
               ,
               and
               it
               shall
               reach
               to
               the
               inquired
               Side
               ,
               or
               Line
               .
            
             
               Example
               .
            
             
               If
               1728
               ,
               be
               the
               Cube
               of
               12
               ,
               the
               Root
               or
               side
               ;
               what
               shall
               be
               the
               Root
               or
               side
               of
               864
               ,
               the
               half
               of
               1728.
               being
               half
               a
               foot
               of
               Timber
               ?
               The
               Extent
               between
               1728
               ,
               and
               864
               ,
               being
               divided
               into
               three
               parts
               ,
               and
               that
               third
               part
               ,
               laid
               decreasing
               from
               12
               ,
               shall
               reach
               to
               9-525
               ,
               the
               side
               or
               root
               required
               of
               half
               a
               foot
               of
               Timber
               ,
               though
               not
               exactly
               ,
               yet
               very
               near
               .
            
             
               
                 Again
                 for
                 another
                 Example
              
               .
            
             
               If
               an
               Iron
               Bullet
               ,
               of
               6
               inches
               Diameter
               ,
               weigh
               30
               pound
               ;
               what
               shall
               a
               Bullet
               of
               7
               inches
               Diameter
               weigh
               ?
               The
               Extent
               from
               6
               to
               7
               ,
               shall
               reach
               ,
               being
               turned
               three
               times
               ,
               to
               47-7
               .
            
             
               Again
               ,
            
             
               If
               a
               Ship
               ,
               whose
               Burthen
               is
               300
               Tun
               ,
               be
               
               75
               Foot
               by
               the
               Keel
               ;
               what
               shall
               that
               Ship
               be
               ,
               whose
               Keel
               is
               100
               Feet
               ?
               The
               Extent
               between
               75
               and
               100
               ,
               turned
               three
               times
               from
               300
               ,
               shall
               reach
               to
               713
               Tun
               Burthen
               .
            
             
               Again
               ,
            
             
               If
               a
               Ship
               of
               29
               Foot
               and
               a
               half
               at
               the
               beam
               ,
               be
               300
               Tun
               Burthen
               ,
               what
               shall
               a
               Ship
               of
               713
               Tun
               burthen
               be
               ?
               The
               third
               part
               of
               the
               distance
               between
               300
               and
               713
               ,
               shall
               reach
               from
               29
               ½
               to
               39-35
               ,
               its
               measure
               at
               the
               beam
               .
            
             
               Again
               ,
            
             
               If
               a
               Ship
               of
               300
               Tun
               be
               13
               Foot
               in
               hold
               ,
               what
               shall
               a
               Ship
               of
               713
               Tun
               be
               in
               hold
               ?
               The
               third
               part
               between
               300
               and
               713
               ,
               shall
               reach
               from
               13
               foot
               ,
               to
               17-35
               ,
               the
               Feet
               in
               the
               Hold
               of
               a
               Ship
               of
               713
               Tun.
               
            
             
               If
               you
               have
               a
               treble
               Line
               ,
               then
               you
               may
               save
               the
               dividing
               ,
               by
               taking
               from
               the
               little-Line
               ,
               and
               measuring
               on
               the
               great-Line
               ,
               and
               the
               contrary
               ,
               as
               the
               nature
               of
               the
               Question
               doth
               require
               .
            
             
               Lastly
               ,
            
             
               Know
               ,
               that
               by
               adding
               of
               twelve
               Centers
               and
               Points
               ,
               the
               Line
               may
               be
               made
               to
               speak
               ,
               as
               it
               were
               ,
               and
               so
               made
               more
               fit
               for
               any
               mans
               more
               particular
               occasions
               .
            
          
           
             
             
               A
               Brief
               Touch
               of
               the
               Use
               of
               the
               Logarithms
               ,
               or
               Tables
               ,
               of
               the
               Artificial
               Numbers
               ,
               Sines
               ,
               and
               Tangents
               .
            
             
               See
               more
               in
               Gunter's
               Works
               .
            
             
               IT
               may
               happen
               ,
               that
               some
               may
               meet
               with
               this
               Book
               ,
               that
               had
               rather
               use
               the
               
                 Tables
                 of
                 Logarithms
              
               ,
               from
               whence
               these
               Lines
               are
               framed
               ,
               than
               the
               Lines
               on
               the
               Rule
               ;
               or
               out
               of
               curiosity
               to
               prove
               the
               truth
               of
               their
               work
               ,
               for
               whose
               sakes
               I
               have
               added
               these
               following
               plain
               Precepts
               ,
               without
               Examples
               .
            
             
               1.
               
               To
               multiply
               one
               Number
               by
               another
               .
            
             
               Set
               the
               Logarithm
               of
               the
               Multiplicator
               ,
               and
               Multiplicand
               ,
               right
               under
               one
               another
               ,
               and
               add
               them
               together
               ,
               and
               the
               sum
               is
               the
               Logarithm
               of
               the
               Product
               .
            
             
               2.
               
               To
               divide
               one
               Number
               by
               another
               .
            
             
               Set
               down
               first
               the
               Logarithm
               of
               the
               Dividend
               ,
               and
               then
               right
               under
               it
               the
               Logarithm
               of
               the
               Divisor
               ,
               and
               then
               substract
               the
               log
               ▪
               of
               the
               Divisor
               ,
               from
               the
               log
               .
               of
               the
               
               Dividend
               ,
               and
               the
               remainder
               is
               the
               Log.
               of
               the
               Quotient
               required
               .
            
             
               3.
               
               To
               find
               the
               Square-root
               of
               a
               Number
               .
            
             
               Half
               the
               Logarithm
               of
               the
               Number
               given
               ,
               is
               the
               whole
               Logarithm
               of
               the
               Square-root
               of
               it
               .
            
             
               4.
               
               To
               find
               the
               Cubick-root
               of
               a
               Number
               .
            
             
               One
               third
               part
               of
               the
               Logarithm
               of
               the
               given
               Number
               ,
               is
               the
               full
               Logarithm
               of
               the
               Cubick-root
               of
               the
               given
               Number
               ,
               as
               a
               third
               of
               14313637
               ;
               the
               logarithm
               of
               27
               is
               0-4771212
               ,
               the
               Log.
               of
               3
               ,
               the
               Cube-root
               of
               27
               ,
               required
               .
            
             
               5.
               
               To
               work
               the
               Rule
               of
               Three
               direct
               ,
               or
               three
               Numbers
               given
               ,
               to
               find
               a
               4th
               by
               the
               Logarithms
               .
            
             
               5.
               
               To
               work
               the
               Rule
               of
               Three
               direct
               ,
               or
               three
               Numbers
               given
               ,
               to
               find
               a
               4th
               by
               the
               Logarithms
               .
            
             
               Set
               down
               the
               Logarithms
               of
               the
               1
               ,
               2
               ,
               &
               3
               Numbers
               ,
               one
               right
               over
               another
               ;
               then
               add
               the
               logarithms
               of
               the
               second
               and
               third
               together
               ;
               and
               from
               the
               sum
               ,
               substract
               the
               logarithms
               of
               the
               first
               ,
               and
               the
               remainders
               is
               the
               logarithms
               of
               the
               4th
               required
               .
            
             
             
               6.
               
               When
               in
               common
               Arithmetick
               the
               second
               term
               is
               divided
               by
               the
               first
               ,
               and
               the
               Quotient
               multiplied
               by
               the
               third
               .
            
             
               
                 Then
                 by
                 Logarithms
              
               ,
            
             
               Take
               the
               Logarithm
               of
               the
               first
               term
               ,
               from
               the
               Logarithm
               of
               the
               second
               ;
               and
               add
               the
               difference
               to
               the
               log
               .
               of
               the
               third
               ,
               and
               the
               sum
               is
               the
               log
               .
               of
               the
               4th
               .
            
             
               7.
               
               When
               in
               common
               Arithmetick
               the
               second
               term
               is
               divided
               by
               the
               first
               ,
               and
               the
               third
               by
               the
               Quotient
               .
            
             
               Then
               take
               the
               log
               .
               of
               the
               second
               ,
               from
               the
               log
               .
               of
               the
               first
               term
               ;
               and
               take
               the
               difference
               out
               of
               the
               log
               .
               of
               the
               third
               ,
               and
               the
               remainder
               is
               the
               log
               .
               of
               the
               4th
               term
               required
               .
            
             
               8.
               
               Between
               two
               extream
               Numbers
               ,
               to
               find
               a
               mean
               Proportional
               .
            
             
               Add
               the
               logarithms
               of
               the
               two
               extream
               Numbers
               together
               〈…〉
               sum
               is
               the
               〈…〉
               .
            
             
             
               9.
               
               To
               work
               the
               Rule
               of
               Three
               in
               the
               Logarithms
               of
               Artificial
               Numbers
               ,
               Sines
               ,
               and
               Tangents
               .
            
             
               1.
               
               When
               Radius
               is
               the
               first
               term
               .
            
             
               Add
               the
               Logarithms
               of
               the
               second
               and
               third
               terms
               together
               ,
               and
               Radius
               ,
               or
               a
               unite
               ,
               in
               the
               first
               place
               ,
               taken
               from
               the
               sum
               ,
               there
               shall
               remain
               the
               logarithm
               of
               the
               4th
               term
               required
               ;
               according
               to
               the
               5th
               Precept
               .
            
             
               2.
               
               When
               Radius
               is
               in
               the
               second
               place
               ,
               or
               term
               .
            
             
               Then
               the
               first
               term
               (
               and
               second
               virtually
               )
               taken
               from
               the
               third
               ,
               cutting
               off
               a
               unite
               in
               the
               first
               place
               for
               Radius
               ,
               is
               the
               4th
               term
               .
            
             
               3.
               
               When
               Radius
               is
               in
               the
               third
               place
               .
            
             
               Then
               substract
               the
               logarithm
               of
               the
               second
               term
               ,
               from
               the
               log
               .
               of
               the
               first
               term
               ,
               cutting
               off
               a
               unite
               for
               Radius
               ,
               and
               the
               remainder
               is
               the
               4th
               term
               .
            
             
               4.
               
               If
               Radius
               be
               none
               of
               the
               three
               terms
               .
            
             
               Then
               add
               the
               Logarithms
               of
               the
               second
               and
               third
               terms
               ;
               and
               from
               the
               sum
               ,
               
               substract
               the
               logarithm
               of
               the
               first
               term
               ,
               and
               the
               remainder
               is
               the
               logarithm
               of
               the
               4th
               term
               .
            
             
               5.
               
               Or
               else
               .
            
             
               Set
               down
               the
               Arithmetical
               complement
               of
               the
               first
               term
               ,
               and
               the
               logarithms
               of
               the
               second
               and
               third
               term
               ,
               and
               add
               all
               together
               ,
               and
               the
               sum
               cutting
               off
               Radius
               ,
               is
               the
               4th
               term
               .
            
             
               10.
               
               When
               the
               Number
               is
               not
               to
               be
               found
               in
               the
               Canon
               of
               Logarithms
               of
               Numbers
               ,
               Sines
               ,
               or
               Tangents
               ;
               take
               the
               next
               nearest
               ,
               or
               for
               more
               exactness
               use
               the
               part
               proportional
               .
            
             
               11.
               
               Though
               Numbers
               and
               Sines
               ,
               or
               Numbers
               and
               Tangents
               are
               used
               together
               ,
               the
               work
               is
               all
               one
               ,
               as
               with
               Sines
               and
               Tangents
               ,
               as
               to
               the
               Precept
               in
               working
               .
            
             
               12.
               
               In
               using
               the
               Logarithms
               ,
               great
               regard
               is
               to
               be
               had
               to
               the
               Index
               ,
               or
               Charracteristick
               ,
               to
               rule
               in
               the
               Number
               of
               places
               ;
               the
               Characteristick
               being
               one
               unite
               less
               than
               the
               Number
               of
               places
               to
               express
               that
               Number
               ;
               thus
               the
               Characteristick
               of
               3-5932861
               ,
               the
               logarithm
               of
               3920
               is
               3
               ,
               being
               one
               less
               than
               the
               Number
               of
               places
               in
               3920
               ,
               which
               consists
               of
               4
               figures
               .
            
          
        
         
           
           
             CHAP.
             XV.
             The
             use
             of
             the
             Trianguler
             Quadrant
             ,
             in
             Geometry
             ,
             and
             Astronomy
             .
          
           
             
               Use
               I.
               The
               Radius
               of
               a
               Circle
               ,
               or
               Line
               being
               given
               ,
               to
               find
               readily
               ,
               any
               required
               Sine
               ,
               Tangent
               ,
               or
               Secant
               ,
               or
               Chord
               ,
               to
               that
               Radius
               .
            
             
               
                 And
                 first
                 ,
                 
                 to
                 do
                 it
                 by
                 the
                 Quadrantal-side
                 .
              
            
             
               FIrst
               ,
               If
               your
               Radius
               happen
               to
               be
               equal
               to
               the
               greater
               Scale
               of
               (
               Altitudes
               or
               )
               Sines
               ,
               issuing
               from
               the
               Center
               ,
               
               then
               the
               measure
               of
               any
               degree
               or
               minuite
               ,
               from
               the
               Center
               toward
               the
               head
               ,
               shall
               be
               the
               Sine
               ,
               the
               measure
               from
               the
               Center-point
               at
               600
               ,
               on
               the
               degrees
               ,
               to
               any
               degree
               and
               minuit
               required
               ,
               shall
               be
               
               the
               Tangent
               to
               the
               same
               Radius
               ;
               
               and
               the
               measure
               from
               the
               Tangent
               to
               the
               Center
               ,
               shall
               be
               the
               Secant
               ,
               
               to
               the
               same
               Radius
               .
               And
               if
               you
               have
               an
               Index
               ,
               or
               a
               Bead
               upon
               your
               Thred
               ,
               and
               set
               the
               Bead
               ,
               when
               the
               Thred
               is
               drawn
               streight
               ,
               to
               the
               Center
               at
               600
               on
               the
               degrees
               or
               Tangents
               ,
               or
               to
               the
               Sine
               of
               90
               ;
               
               then
               if
               you
               lay
               the
               Thred
               to
               any
               Number
               of
               degrees
               and
               minuits
               ,
               counted
               from
               90
               ,
               and
               there
               keep
               it
               ;
               then
               the
               extent
               from
               the
               Sine
               of
               90
               ,
               to
               the
               Bead
               ,
               shall
               be
               the
               Chord
               of
               the
               Angle
               the
               Thred
               is
               laid
               to
               ,
               to
               the
               Radius
               of
               the
               greater
               Scale
               of
               Sines
               ,
               issuing
               from
               the
               Center
               .
            
             
               But
               if
               this
               happen
               to
               be
               too
               large
               ,
               then
               the
               other
               lesser
               Line
               of
               Sines
               ,
               issuing
               upwards
               from
               the
               Center
               ,
               being
               about
               one
               third
               part
               of
               the
               other
               ,
               hath
               first
               it self
               for
               Sines
               ;
               
               secondly
               ,
               the
               degrees
               on
               the
               loose-piece
               for
               Tangents
               ,
               counting
               from
               the
               Center
               at
               60
               ;
               
               thirdly
               ,
               the
               measure
               from
               the
               Tangent
               ,
               
               to
               the
               Center
               ,
               for
               a
               Secant
               ;
               fourthly
               ,
               the
               Bead
               and
               Thred
               ,
               for
               a
               Chord
               ,
               
               as
               before
               ;
               all
               at
               once
               to
               one
               Radius
               ,
               clearly
               and
               distinctly
               ,
               without
               any
               interruption
               ,
               to
               75
               degrees
               of
               the
               Tangent
               ,
               or
               Secant
               .
            
             
               But
               if
               any
               other
               Radius
               be
               given
               ,
               then
               they
               will
               not
               be
               had
               so
               readily
               altogether
               ,
               but
               thus
               in
               order
               one
               after
               another
               ,
               and
               
               first
               for
               the
               Sine
               ,
               by
               the
               
                 Trianguler
                 Quadrant
              
               .
            
             
               Take
               the
               Radius
               between
               your
               Compasses
               ,
               set
               one
               foot
               in
               90
               ,
               with
               the
               other
               lay
               the
               Thred
               to
               the
               nearest
               distance
               ,
               and
               there
               keep
               it
               ;
               then
               take
               the
               nearest
               distance
               from
               the
               Sine
               of
               any
               Ark
               or
               Angle
               you
               would
               have
               ,
               and
               that
               shall
               be
               the
               Sine
               of
               the
               Ark
               or
               Angle
               required
               to
               the
               given
               Radius
               .
            
             
               1.
               
               But
               by
               the
               Sector-side
               work
               thus
               ,
               being
               near
               alike
               ,
               
               fit
               the
               given
               Radius
               in
               the
               Parallel-sine
               of
               90
               &
               90
               ;
               then
               take
               out
               the
               Parallel-sine
               ,
               of
               the
               Ark
               or
               Angle
               required
               ,
               and
               you
               have
               your
               desire
               .
               
            
             
               2.
               
               Also
               the
               Sector
               being
               so
               set
               ,
               if
               you
               take
               out
               any
               Parallel
               Tangent
               under
               45
               ,
               
               you
               have
               that
               also
               to
               the
               same
               Radius
               .
            
             
               3.
               
               Also
               ,
               if
               you
               would
               have
               any
               Tangent
               under
               76
               ,
               as
               the
               Sector
               stands
               ,
               
               take
               out
               the
               parallel
               Tangent
               thereof
               ,
               and
               that
               shall
               be
               the
               4th
               part
               of
               the
               Tangent
               required
               to
               the
               same
               Radius
               ,
               and
               is
               to
               be
               turned
               4
               times
               for
               that
               greater
               Radius
               .
            
             
               4.
               
               Also
               ,
               
               if
               you
               want
               a
               Secant
               under
               60
               degrees
               ;
               at
               the
               same
               Radius
               take
               out
               the
               parallel
               Secant
               of
               the
               Ark
               or
               Angle
               required
               ;
               and
               that
               shall
               be
               the
               half
               of
               the
               Secant
               required
               ;
               for
               note
               ,
               the
               Secant
               of
               one
               degree
               is
               more
               than
               Radius
               ;
               and
               why
               
               I
               use
               a
               half
               ,
               rather
               than
               a
               4th
               part
               ,
               in
               time
               you
               may
               well
               see
               .
            
             
               
               By
               the
               Artificial
               Numbers
               ,
               Sines
               ,
               and
               Tangents
               ,
               this
               cannot
               properly
               be
               done
               ;
               only
               thus
               you
               may
               do
               by
               them
               ,
               counting
               your
               given
               Radius
               (
               be
               it
               great
               or
               little
               )
               10000
               parts
               ,
               you
               may
               by
               them
               find
               out
               readily
               how
               many
               of
               them
               parts
               will
               go
               to
               make
               the
               Sine
               ,
               Tangent
               ,
               or
               Secant
               ,
               to
               any
               Number
               of
               degrees
               and
               minuts
               .
               
                 As
                 thus
              
               ;
            
             
               
               Take
               the
               distance
               from
               the
               Sine
               of
               90
               ,
               on
               the
               Artificial
               Sines
               ,
               to
               the
               Sine
               of
               any
               degree
               and
               minuit
               required
               ;
               and
               set
               the
               same
               distance
               ,
               the
               same
               way
               ,
               from
               10
               on
               the
               Line
               of
               Numbers
               ,
               reading
               it
               as
               a
               Scale
               of
               equal
               parts
               ,
               and
               that
               shall
               be
               the
               Natural
               Sine
               of
               the
               degree
               and
               minuit
               required
               .
            
             
               Or
               ,
            
             
               If
               you
               lay
               a
               Square
               to
               the
               Sine
               given
               ,
               on
               the
               Numbers
               ,
               it
               cuts
               the
               Natural
               Sine
               required
               .
            
             
               Example
               .
            
             
               Right
               against
               the
               Artificial
               Sine
               of
               30
               ,
               on
               the
               Line
               of
               Numbers
               ,
               you
               find
               5000
               ,
               which
               is
               the
               Natural
               Number
               thereof
               .
            
             
               But
               ,
               if
               you
               measure
               this
               distance
               from
               10
               ,
               in
               the
               Line
               of
               Lines
               ,
               it
               will
               give
               the
               Logarithmal
               Sine
               thereof
               ,
               viz.
               69897.
               
            
             
               
               And
               the
               like
               for
               the
               Tangent
               also
               ,
               under
               45
               ,
               in
               the
               same
               manner
               .
            
             
             
               But
               for
               the
               Secants
               ,
               and
               ,
               
               for
               the
               Tangents
               above
               45
               ,
               you
               must
               count
               thus
               ;
            
             
               Measure
               ,
               as
               before
               from
               90
               ,
               to
               the
               Co-sine
               of
               the
               Angle
               ,
               required
               ,
               for
               a
               Secant
               ;
               and
               from
               45
               ,
               to
               the
               co-Tangent
               of
               45
               ,
               for
               a
               Tangent
               ;
               This
               extent
               laid
               the
               contrary
               way
               from
               1
               ,
               in
               the
               Numbers
               ,
               shews
               how
               many
               Radiusses
               ,
               and
               also
               how
               much
               above
               Radius
               ,
               you
               must
               have
               to
               make
               up
               the
               Natural
               Tangent
               ,
               or
               Secant
               required
               ,
               in
               Numbers
               .
            
             
               Example
               .
            
             
               The
               Secant
               of
               50
               degrees
               ,
               and
               the
               Tangent
               of
               57
               degrees
               ,
               16
               minuts
               ,
               
               being
               near
               alike
               ,
               is
               1
               Radius
               5556
               ;
               for
               the
               Natural
               Number
               thereof
               :
               and
               this
               distance
               measured
               on
               the
               Line
               of
               Lines
               ,
               gives
               Radius
               because
               above
               45
               ,
               and
               1919
               ;
               more
               for
               the
               Artificial
               Tangent
               of
               57-16
               ,
               or
               the
               Secant
               of
               50
               degrees
               .
            
             
               This
               have
               I
               hinted
               ,
               in
               the
               first
               place
               ,
               that
               thereby
               you
               might
               see
               the
               nature
               of
               the
               Lines
               ,
               and
               the
               making
               of
               the
               Instrument
               ,
               with
               its
               great
               convenience
               in
               the
               Contrivance
               of
               the
               Work
               on
               both
               sides
               ,
               and
               the
               harmony
               ,
               and
               proportion
               ,
               the
               Natural
               way
               hath
               to
               the
               Artificial
               ;
               also
               hereby
               you
               may
               readily
               prove
               the
               truth
               of
               your
               Instrument
               ,
               being
               an
               equilatteral
               Triangle
               ,
               whether
               you
               use
               the
               greater
               or
               the
               lesser
               Sines
               ;
               For
               the
               
               measure
               from
               the
               Center
               ,
               
               where
               the
               Thred
               is
               fastened
               ,
               to
               the
               Center-point
               of
               Brass
               on
               the
               moveable-leg
               ,
               and
               loose-piece
               at
               60
               on
               the
               degrees
               ,
               ought
               to
               be
               equal
               to
               each
               Line
               of
               Sines
               ;
               and
               also
               to
               the
               Tangent
               of
               45
               on
               the
               Tangent
               Line
               :
               The
               measure
               from
               the
               Center
               to
               the
               rectifying-point
               on
               the
               Head
               ,
               at
               the
               meeting
               of
               the
               Lines
               for
               the
               Hour
               and
               Azimuth
               ,
               and
               the
               Lines
               for
               the
               Sines
               and
               Lines
               ,
               is
               equal
               to
               the
               Tangent
               of
               twice
               30
               on
               that
               piece
               .
            
             
               Again
               ,
            
             
               The
               measure
               from
               the
               Center
               ,
               to
               the
               rectifying-point
               on
               the
               end
               of
               the
               Head-leg
               ,
               shall
               reach
               from
               thence
               to
               30
               on
               the
               loose-piece
               ;
               and
               being
               turned
               twice
               ,
               reaches
               to
               060
               on
               the
               loose-piece
               :
               Also
               ,
               the
               Radius
               ,
               or
               Tangent
               of
               45
               ,
               turned
               twice
               from
               060
               on
               the
               loose-piece
               ,
               shall
               reach
               to
               75
               ,
               as
               by
               comparing
               the
               Natural
               Numbers
               together
               ,
               will
               most
               exactly
               appear
               ;
               Though
               perhaps
               without
               this
               hint
               ,
               it
               might
               not
               have
               beeen
               observed
               by
               an
               ordinary
               eye
               .
            
             
               Having
               been
               so
               large
               ,
               and
               plain
               ,
               in
               this
               first
               Use
               ,
               I
               shall
               be
               ,
               I
               hope
               ,
               as
               plain
               ,
               though
               far
               more
               brief
               in
               all
               the
               rest
               ;
               for
               if
               you
               look
               back
               to
               Chapt.
               VI.
               Probl.
               I.
               Sect.
               3.
               you
               shall
               there
               see
               the
               full
               explaining
               of
               Latteral
               and
               Parallel
               ,
               and
               Nearest-distance
               ,
               
               and
               how
               to
               take
               them
               ;
               the
               mark
               for
               Latteral
               being
               thus
               —
               ;
               The
               mark
               for
               Parallel
               thus
               =
               ;
               Nearest-distance
               thus
               ND
               ,
               &c.
               
            
          
           
             
               Use
               II.
               The
               Sine
               of
               any
               Ark
               or
               Angle
               given
               ,
               to
               find
               the
               Radius
               to
               it
               .
            
             
               Take
               the
               Sine
               between
               your
               Compasses
               ,
               
               and
               setting
               one
               foot
               of
               the
               Compasses
               in
               the
               given
               Sine
               ;
               and
               with
               the
               other
               Point
               lay
               the
               Thred
               to
               the
               nearest-distance
               ,
               and
               there
               keep
               it
               ;
               then
               the
               nearest-distance
               from
               the
               Sine
               of
               90
               to
               the
               Thred
               ,
               shall
               be
               the
               Radius
               required
               .
            
             
               Make
               the
               given
               Sine
               a
               Parallel
               Sine
               ,
               
               and
               then
               take
               out
               the
               Parallel
               Radius
               ,
               and
               you
               have
               your
               desire
               .
            
             
               The
               Artificial
               Sines
               and
               Tangents
               ,
               are
               not
               proper
               for
               this
               work
               ,
               further
               then
               to
               give
               the
               Natural
               Number
               thereof
               ,
               as
               before
               ;
               therefore
               I
               shall
               only
               add
               the
               use
               of
               them
               when
               it
               is
               convenient
               in
               the
               fit
               place
               .
            
          
           
             
               Use
               III.
               The
               Radius
               ,
               or
               any
               known
               Sine
               being
               given
               ,
               to
               find
               the
               quantity
               of
               any
               other
               unknown
               Sine
               ,
               to
               the
               same
               Radius
               .
            
             
               Take
               the
               Radius
               ,
               or
               known
               Sine
               given
               ,
               
               
               and
               make
               it
               a
               Parallel
               in
               the
               Sine
               of
               90
               for
               Radius
               ,
               or
               in
               the
               Sine
               of
               the
               known
               Angle
               given
               ,
               and
               lay
               the
               Thred
               to
               ND
               .
               Then
               ,
               take
               the
               unknown
               Sine
               between
               your
               Compasses
               ,
               and
               carry
               one
               Point
               along
               the
               Line
               of
               Sines
               ,
               till
               the
               other
               foot
               being
               turned
               about
               ,
               will
               but
               just
               touch
               the
               Thred
               ;
               then
               the
               place
               where
               the
               Compasses
               stayes
               ,
               shall
               be
               the
               Sine
               of
               the
               unknown
               Angle
               required
               ,
               to
               that
               Radius
               or
               known
               Sine
               .
            
             
               
               Make
               the
               given
               Radius
               a
               Parallel
               Radius
               ,
               or
               the
               given
               Sine
               a
               =
               Sine
               ,
               in
               the
               answerable
               Sine
               thereof
               :
               Then
               ,
               taking
               the
               unknown
               Sine
               ,
               carry
               it
               parallelly
               along
               the
               Line
               of
               Sines
               till
               it
               stay
               in
               like
               parts
               ,
               which
               parts
               shall
               be
               the
               Numerator
               to
               the
               Sine
               required
               .
            
          
           
             
               Use
               IV.
               The
               Radius
               being
               given
               ,
               by
               the
               Sines
               alone
               to
               find
               any
               Tangent
               or
               Secant
               to
               that
               Radius
               .
            
             
               
               Take
               the
               Radius
               between
               your
               Compasses
               ,
               and
               set
               one
               Point
               in
               the
               Sine
               complement
               of
               the
               Tangent
               required
               ,
               and
               lay
               the
               Thred
               to
               the
               ND
               ;
               then
               the
               ND
               from
               the
               Sine
               of
               the
               Tangent
               required
               ,
               to
               the
               Thred
               ,
               shall
               be
               the
               Tangent
               required
               :
               And
               the
               ND
               from
               90
               ,
               to
               the
               Thred
               ,
               shall
               be
               the
               Secant
               required
               .
            
             
             
               Make
               the
               given
               Radius
               a
               =
               in
               the
               co-Sine
               of
               the
               Tangent
               required
               ;
               
               then
               the
               =
               Sine
               (
               of
               the
               inquired
               Ark
               or
               Angle
               )
               shall
               be
               the
               Tangent
               required
               ;
               and
               =
               90
               shall
               be
               the
               Secant
               required
               to
               that
               Radius
               .
            
          
           
             
               Use
               V.
               Any
               Tangent
               or
               Secant
               being
               given
               ,
               to
               find
               the
               answerable
               Radius
               ;
               and
               then
               any
               other
               proportionable
               Tangent
               ,
               or
               Secant
               ,
               by
               Sines
               only
               .
            
             
               First
               ,
               if
               it
               be
               a
               Tangent
               that
               is
               given
               ,
               
               take
               it
               between
               your
               Compasses
               ,
               and
               setting
               one
               foot
               in
               the
               Sine
               thereof
               ,
               lay
               the
               Thred
               to
               ND
               ,
               then
               the
               =
               Co-sine
               thereof
               shall
               be
               Radius
               ;
            
             
               But
               ,
               if
               it
               be
               a
               Secant
               ,
               take
               it
               between
               your
               Compasses
               ,
               and
               set
               one
               foot
               alwayes
               in
               90
               ,
               lay
               the
               Thred
               to
               the
               ND
               ,
               then
               the
               nearest
               distance
               from
               the
               Co-sine
               to
               the
               Thred
               (
               or
               the
               =
               Co-sine
               )
               shall
               be
               the
               Radius
               required
               .
            
             
               Take
               the
               given
               Tangent
               ,
               
               make
               it
               a
               =
               in
               the
               Sine
               thereof
               ;
               then
               the
               =
               Co-sine
               thereof
               shall
               be
               Radius
               .
            
             
               
                 Or
                 ,
                 if
                 it
                 be
                 a
                 Secant
                 given
                 ,
                 then
              
            
             
               Take
               the
               given
               Secant
               ,
               make
               it
               a
               =
               in
               90
               ,
               
               then
               the
               =
               Co-sine
               thereof
               ,
               shall
               be
               the
               Radius
               required
               .
            
             
               Then
               having
               gotten
               Radius
               ,
               the
               4th
               Use
               shewes
               how
               to
               come
               by
               any
               Tangent
               ,
               or
               Secant
               ,
               by
               the
               Sines
               only
               .
            
          
           
             
               Use
               VI.
               To
               lay
               down
               any
               Chord
               ,
               to
               any
               Radius
               ;
               less
               then
               the
               Sine
               of
               30
               degrees
               .
            
             
               Take
               the
               given
               Radius
               ▪
               set
               one
               Point
               in
               the
               Sine
               of
               30
               ,
               
               lay
               the
               Thred
               to
               the
               ND
               (
               and
               for
               your
               more
               ready
               setting
               it
               again
               ,
               note
               ,
               what
               degree
               and
               minuit
               the
               Thred
               doth
               stay
               at
               ,
               on
               the
               degrees
               )
               and
               there
               keep
               it
               .
               Then
               the
               ND
               from
               the
               Sine
               of
               half
               the
               Angle
               you
               would
               have
               ,
               shall
               be
               the
               Chord
               of
               the
               Angle
               required
               .
            
             
               
               Take
               the
               given
               Radius
               ,
               and
               make
               it
               alwayes
               a
               =
               in
               30
               ,
               and
               30
               of
               Sines
               ;
               the
               =
               Sine
               of
               half
               the
               Chord
               ,
               shall
               be
               the
               Chord
               required
               .
            
          
           
             
               Use
               VII
               .
               To
               lay
               down
               any
               Chord
               to
               the
               Radius
               of
               the
               whole
               Line
               of
               Sines
               .
            
             
               
               Take
               the
               Radius
               between
               your
               Compasses
               ,
               and
               setting
               one
               Point
               in
               90
               of
               the
               Sines
               ,
               
               lay
               the
               Thred
               to
               the
               ND
               ,
               observing
               the
               place
               ,
               there
               keep
               it
               .
            
             
               Then
               taking
               the
               =
               Sine
               of
               the
               Angle
               required
               ,
               with
               it
               set
               one
               Point
               in
               the
               Line
               to
               which
               you
               would
               draw
               the
               Angle
               ,
               as
               far
               from
               the
               Center
               as
               the
               Radius
               is
               ;
               then
               draw
               the
               Convexity
               of
               an
               Ark
               ,
               and
               by
               that
               Convexity
               ,
               and
               the
               Center
               ,
               draw
               the
               Line
               for
               the
               Angle
               required
               .
            
             
               Example
               .
            
             
               Let
               AB
               be
               a
               Radius
               of
               any
               length
               ,
               under
               or
               equal
               to
               the
               whole
               Line
               of
               Sines
               :
               Take
               AB
               between
               your
               Compasses
               ,
               and
               setting
               one
               Point
               in
               90
               ;
               lay
               the
               Thred
               to
               ND
               ,
               then
               take
               out
               the
               =
               Sine
               of
               38
               ,
               or
               any
               other
               Number
               you
               please
               ,
               and
               setting
               one
               Point
               in
               B
               ,
               the
               end
               of
               the
               Radius
               from
               A
               the
               Center
               ,
               and
               trace
               the
               Ark
               DC
               ,
               by
               the
               Convexity
               of
               which
               Ark
               ,
               draw
               the
               Line
               AC
               for
               the
               Angle
               required
               .
            
             
               Take
               the
               given
               Radius
               AB
               ,
               
               make
               it
               a
               =
               in
               90
               ,
               and
               90
               of
               Sines
               ;
               then
               take
               out
               =
               38
               ,
               and
               setting
               one
               foot
               in
               B
               ,
               draw
               the
               Ark
               DC
               ,
               and
               draw
               AC
               for
               the
               Angle
               required
               .
            
             
               
                 Or
                 else
                 work
                 thus
              
               ;
            
             
               Take
               AB
               ,
               the
               given
               Radius
               ,
               (
               having
               drawn
               the
               Ark
               BE
               )
               and
               make
               it
               a
               =
               in
               
               the
               Co-sine
               of
               half
               the
               Angle
               required
               ;
               and
               lay
               the
               Thred
               to
               ND
               ,
               (
               or
               set
               the
               Sector
               )
               .
            
             
               Then
               ,
            
             
               Take
               the
               =
               ND
               ,
               from
               the
               right-sine
               of
               the
               Angle
               required
               ,
               and
               it
               shall
               be
               BE
               ,
               the
               Chord
               required
               to
               be
               found
               .
            
             
               Note
               ,
               That
               the
               contrary
               work
               finds
               Radius
               .
            
          
           
             
               Use
               VIII
               .
               To
               lay
               off
               any
               Angle
               by
               the
               Line
               of
               Tangents
               ,
               or
               Secants
               ,
               to
               prove
               it
               .
            
             
               
               Having
               drawn
               the
               Ground-Line
               ,
               AB
               ,
               at
               the
               Point
               B
               ,
               raise
               a
               Perpendiculer
               ,
               as
               the
               Line
               BC
               extended
               at
               length
               ,
               then
               make
               AB
               ,
               the
               Radius
               ,
               a
               =
               Tangent
               in
               45
               and
               45
               ;
               then
               take
               out
               the
               =
               Tangent
               of
               the
               Angle
               required
               ,
               and
               lay
               it
               from
               B
               to
               C
               in
               the
               Perpendiculer
               ,
               and
               draw
               the
               Line
               AC
               for
               the
               Angle
               required
               .
            
             
               Also
               ,
               If
               you
               take
               out
               the
               Secant
               of
               the
               Angle
               ,
               as
               the
               Sector
               stands
               ,
               and
               lay
               it
               twice
               in
               the
               Line
               AE
               ,
               it
               will
               reach
               just
               to
               C
               ,
               the
               Point
               required
               .
            
             
               
                 Also
                 Note
              
               ,
               That
               if
               you
               want
               an
               Angle
               above
               45
               degrees
               ,
               as
               the
               Sector
               stands
               ,
               take
               the
               same
               from
               the
               small
               Tangent
               that
               proceeds
               to
               75
               ,
               and
               turn
               that
               Extent
               4
               times
               from
               B
               ,
               and
               it
               shall
               give
               the
               Point
               required
               in
               the
               Line
               BC.
               
            
          
           
             
             
               Use
               IX
               .
               To
               lay
               down
               ,
               or
               protract
               any
               Angle
               by
               the
               Tangent
               of
               45
               only
               .
            
             
               First
               ,
               make
               a
               Geometrical
               Square
               ,
               a
               ABCD
               ,
               and
               let
               A
               be
               the
               Anguler
               Point
               ;
               then
               making
               AB
               Radius
               ,
               make
               AB
               a
               =
               Co-sine
               of
               the
               Angle
               you
               would
               have
               ,
               and
               lay
               the
               Thred
               to
               the
               nearest
               distance
               ,
               then
               the
               ND
               from
               the
               right
               Sine
               of
               the
               Angle
               to
               the
               Thred
               ,
               shall
               be
               the
               Tangent
               required
               .
            
             
               Example
               .
            
             
               I
               make
               AB
               Radius
               a
               =
               in
               50
               ,
               the
               co-Sine
               of
               40
               ,
               then
               the
               =
               Sine
               of
               40
               shall
               be
               BE.
               
            
             
               Again
               ,
            
             
               If
               I
               make
               AD
               equal
               to
               AB
               ,
               the
               =
               co-Sine
               of
               30
               ,
               viz.
               60
               ;
               and
               then
               take
               out
               the
               =
               Sine
               of
               30
               ,
               and
               lay
               it
               from
               D
               to
               F
               ,
               it
               shall
               be
               an
               Angle
               of
               60
               from
               AB
               ,
               or
               30
               from
               D
               to
               F.
               
            
             
               
                 But
                 by
                 the
                 Sector
                 this
                 is
                 more
                 easie
              
               ;
            
             
          
           
             
             
               Use
               X.
               To
               take
               out
               readily
               ,
               any
               Tangent
               above
               45
               ,
               by
               the
               Tangent
               to
               45
               on
               the
               Sector-side
               .
            
             
               
               Take
               the
               given
               Radius
               ,
               make
               it
               a
               =
               in
               the
               co-Tangent
               of
               the
               Tangent
               required
               ;
               then
               the
               =
               Tangent
               of
               45
               ,
               shall
               be
               the
               Tangent
               required
               .
            
             
               Example
               .
            
             
               I
               would
               have
               a
               Tangent
               to
               80
               degrees
               ;
               take
               the
               given
               Radius
               ,
               make
               it
               a
               =
               in
               10
               ,
               the
               complement
               of
               80
               ;
               then
               the
               =
               Tangent
               of
               45
               ,
               shall
               be
               the
               Tangent
               of
               80
               required
               .
            
             
               But
               if
               your
               Radius
               be
               so
               big
               ,
               that
               you
               cannot
               enter
               it
               ,
               then
               take
               the
               half
               ,
               or
               a
               quarter
               of
               your
               Radius
               ,
               and
               then
               =
               45
               will
               be
               the
               half
               ,
               or
               the
               quarter
               of
               the
               Tangent
               required
               .
            
          
           
             
               Use
               XI
               .
               How
               to
               work
               Proportions
               ,
               in
               Sines
               alone
               ,
               by
               the
               Natural
               Sines
               .
            
             
               There
               are
               4
               Varieties
               in
               this
               Work
               ,
               that
               include
               all
               Proportions
               ,
               viz.
               
            
             
             
               
                 1.
                 
                 When
                 the
                 Sine
                 of
                 90
                 is
                 the
                 first
                 term
                 ,
                 then
                 the
                 work
                 is
                 thus
              
               ;
            
             
               Lay
               the
               Thred
               to
               the
               second
               term
               ,
               counted
               on
               the
               degrees
               from
               the
               Head
               ,
               
               toward
               the
               loose-piece
               ;
               and
               count
               the
               third
               term
               on
               the
               Line
               of
               Sines
               ,
               from
               the
               Center
               downwards
               ;
               and
               taking
               the
               nearest
               distance
               from
               thence
               to
               the
               Thred
               ,
               and
               that
               distance
               measured
               from
               the
               Center
               downwards
               ,
               on
               the
               Line
               of
               Sines
               ,
               gives
               the
               4th
               term
               required
               .
            
             
               Example
               .
            
             
               As
               Sine
               90
               ,
               to
               Sine
               23-30
               ;
            
             
               So
               is
               30
               ,
               to
               11-31
               .
            
             
               Take
               the
               Latteral
               second
               term
               ,
               
               make
               it
               a
               =
               Sine
               of
               90
               ;
               then
               take
               out
               the
               =
               third
               term
               ,
               and
               measuring
               it
               from
               the
               Center
               ,
               it
               gives
               the
               4th
               term
               required
               .
            
             
               
                 2.
                 
                 When
                 the
                 Sine
                 of
                 90
                 is
                 the
                 third
                 term
                 ,
                 then
                 work
                 thus
              
               ;
            
             
               Take
               the
               —
               Sine
               ,
               of
               the
               second
               term
               ,
               from
               the
               Center
               downwards
               ,
               
               and
               make
               it
               a
               =
               Sine
               in
               the
               first
               term
               ,
               laying
               the
               Thred
               to
               ND
               ;
               then
               on
               the
               degrees
               ,
               the
               Thred
               shall
               give
               the
               4th
               term
               required
               .
            
             
             
               Example
               .
            
             
               As
               the
               Sine
               of
               30
               ,
               to
               23-31
               ;
            
             
               So
               is
               the
               Sine
               of
               90
               ,
               to
               Sine
               of
               52-56
               .
            
             
               
                 But
                 by
                 the
                 Sector
              
               ,
            
             
               
               Take
               the
               —
               Sine
               of
               the
               second
               ,
               make
               it
               a
               =
               Sine
               of
               the
               first
               term
               ;
               then
               take
               out
               =
               90
               ,
               and
               measure
               it
               from
               the
               Center
               ,
               and
               it
               shall
               give
               the
               4th
               term
               required
               .
               Example
               as
               before
               .
            
             
               
                 3.
                 
                 When
                 the
                 Radius
                 ,
                 or
                 Sine
                 of
                 90
                 ,
                 is
                 in
                 the
                 second
                 place
                 ,
                 work
                 thus
              
               ;
            
             
               
               Take
               —
               90
               from
               a
               lesser
               Scale
               ,
               as
               the
               uppermost
               Sine
               above
               the
               Center
               ,
               or
               the
               Line
               of
               Right-Ascentions
               ,
               or
               the
               Azimuth-Scale
               ,
               or
               the
               like
               ;
               and
               make
               it
               a
               =
               in
               the
               Sine
               of
               the
               first
               ,
               laying
               the
               Thred
               to
               ND
               ,
               then
               the
               =
               third
               term
               ,
               taken
               and
               measured
               on
               the
               same
               Scale
               that
               90
               was
               taken
               from
               ,
               shall
               give
               the
               4th
               term
               required
               .
            
             
               Example
               .
            
             
               As
               —
               90
               ,
               on
               the
               Line
               of
               Right-Ascention
               ,
               is
               to
               =
               30
               ;
            
             
               So
               is
               =
               20
               ,
               to
               43-12
               ,
               measured
               on
               the
               same
               Line
               that
               90
               was
               taken
               from
               .
            
             
             
               
                 Or
                 else
                 secondly
                 ,
                 work
                 thus
              
               ;
            
             
               As
               —
               30
               ,
               to
               =
               90
               ;
            
             
               So
               is
               —
               20
               ,
               to
               =
               43-12
               .
            
             
               By
               carrying
               the
               Compasses
               till
               it
               so
               stayes
               ,
               as
               that
               the
               foot
               turned
               about
               ,
               will
               but
               just
               touch
               the
               Thred
               ,
               at
               the
               nearest
               distance
               .
            
             
               
                 Or
                 else
                 thus
                 ,
                 thirdly
              
               ;
            
             
               By
               transposing
               the
               terms
               ,
               when
               the
               third
               is
               not
               greater
               than
               the
               first
               :
               thus
               ;
            
             
               As
               the
               first
               ,
               to
               the
               third
               ;
            
             
               So
               is
               the
               second
               term
               ,
               to
               the
               4th
               :
            
             
               Where
               the
               Radius
               being
               brought
               to
               the
               third
               place
               ,
               it
               is
               wrought
               by
               the
               second
               Rule
               ,
               as
               before
               .
            
             
               
                 By
                 the
                 Sector
              
               .
            
             
               Take
               a
               smaller
               —
               Sine
               of
               90
               ,
               make
               it
               a
               =
               in
               30
               ;
               then
               the
               =
               Sine
               of
               20
               ,
               taken
               and
               measured
               on
               the
               small
               Sine
               ,
               gives
               43-12
               ,
               as
               before
               .
            
             
               Again
               ,
            
             
               As
               —
               90
               ,
               to
               =
               30
               ;
            
             
               So
               is
               =
               20
               ,
               to
               —
               43-12
               .
            
             
               Again
               ,
            
             
               As
               —
               30
               ,
               to
               =
               90
               ;
            
             
               So
               is
               =
               20
               ,
               to
               =
               43-12
               :
            
             
             
               
                 Lastly
                 ,
                 by
                 transposing
              
               .
            
             
               As
               —
               20
               ,
               to
               =
               30
               ;
            
             
               So
               is
               =
               Radius
               ,
               to
               —
               43-12
               ;
               as
               before
               .
            
             
               
                 4.
                 
                 When
                 Radius
                 is
                 none
                 of
                 the
                 given
                 terms
                 .
              
            
             
               Then
               when
               the
               first
               term
               is
               greater
               than
               the
               second
               and
               third
               ,
               
               
                 work
                 thus
              
               ;
            
             
               Take
               the
               —
               second
               term
               ,
               make
               it
               a
               =
               in
               the
               first
               ,
               laying
               the
               Thred
               to
               the
               ND
               ;
               then
               the
               nearest
               distance
               ,
               from
               the
               third
               term
               ,
               to
               the
               Thred
               measured
               from
               the
               Center
               downward
               ,
               give
               the
               4th
               Sine
               required
               .
            
             
               Example
               .
            
             
               As
               20
               ,
               to
               12
               ;
               so
               is
               18
               ,
               to
               10-50
               .
            
             
               
                 By
                 the
                 Quadrant
              
               .
            
             
               As
               —
               12
               ,
               to
               =
               20
               ;
            
             
               So
               is
               =
               18
               ,
               to
               —
               10-50
               .
            
             
               When
               only
               the
               second
               term
               is
               greater
               than
               the
               first
               ,
               then
               transpose
               the
               terms
               ,
               and
               work
               as
               before
               :
               Or
               else
               use
               a
               double
               Radius
               ,
               which
               is
               on
               this
               Instrument
               very
               easily
               done
               ,
               having
               several
               Radiusses
               .
            
             
             
               Or
               ,
            
             
               Lastly
               ,
               use
               a
               Parallel
               entrance
               ,
               or
               answer
               rather
               ,
               as
               before
               ,
               which
               being
               carefully
               wrought
               ,
               will
               do
               very
               well
               .
            
             
               
                 By
                 the
                 Sector
              
               .
            
             
               The
               same
               manner
               of
               work
               ,
               is
               as
               before
               by
               the
               Quadrant
               ,
               and
               the
               setting
               the
               Sector
               ,
               is
               all
               one
               to
               the
               laying
               the
               Thred
               ,
               as
               will
               be
               largely
               seen
               in
               all
               the
               following
               Propositions
               ,
               wrought
               both
               by
               the
               Artificial
               and
               Natural
               Lines
               ,
               of
               Numbers
               ,
               Sines
               ,
               and
               Tangents
               ,
               as
               followeth
               .
            
          
           
             
               Use
               XII
               .
               Having
               the
               day
               of
               the
               Month
               ,
               or
               Suns
               place
               given
               ,
               to
               find
               his
               Declination
               .
            
             
               Lay
               the
               Thred
               on
               the
               day
               of
               the
               Month
               in
               the
               Kalender
               ,
               
               and
               in
               the
               Line
               of
               degrees
               ,
               on
               the
               Moving-leg
               ,
               you
               have
               his
               Declination
               ,
               either
               Northward
               ,
               or
               Southward
               ,
               according
               to
               the
               time
               of
               the
               year
               ,
               counting
               from
               600
               ,
               toward
               the
               Head
               ,
               for
               North-declination
               ;
               or
               toward
               the
               End
               ,
               for
               South-declination
               .
            
             
             
               
                 By
                 the
                 Artificial
                 Sines
                 and
                 Tangents
                 on
                 the
                 Edge
                 of
                 the
                 Instrument
                 .
              
            
             
               Extend
               the
               Compasses
               from
               the
               Sine
               of
               90
               ,
               to
               the
               Sine
               of
               23
               degrees
               31
               minuts
               ,
               the
               Suns
               greatest
               Declination
               :
               The
               same
               Extent
               applied
               the
               same
               way
               ,
               from
               the
               Sine
               of
               the
               Suns
               place
               ,
               or
               the
               Suns
               distance
               from
               the
               next
               Equinoctial-point
               ,
               shall
               cause
               the
               Moving-point
               to
               fall
               ,
               on
               the
               sine
               of
               the
               Suns
               declination
               ;
               This
               being
               the
               general
               way
               of
               working
               .
            
             
               Example
               .
            
             
               The
               Extent
               from
               the
               sine
               of
               90
               ,
               to
               the
               sine
               of
               23-31
               ,
               shall
               reach
               from
               the
               sine
               of
               30
               ,
               to
               11
               deg
               .
               31
               min.
               the
               Suns
               declination
               ,
               in
               ♉
               Taurus
               30
               degrees
               from
               ♈
               Aries
               ,
               the
               next
               Equinoctial-point
               ,
               and
               from
               60
               degrees
               ,
               the
               Suns
               distance
               in
               ♊
               Gemini
               60
               degrees
               ,
               from
               ♈
               20
               deg
               .
               12
               min.
               the
               Suns
               declination
               then
               .
               This
               being
               the
               manner
               of
               working
               by
               these
               Lines
               ,
               by
               extending
               the
               Compasses
               from
               the
               first
               to
               the
               second
               term
               :
               I
               shall
               for
               the
               rest
               wave
               this
               large
               repetition
               of
               extending
               the
               Compasses
               ,
               and
               render
               it
               only
               thus
               by
               the
               words
               of
               the
               Cannon-general
               in
               all
               Books
               ;
            
             
               As
               Sine
               90
               ,
               to
               Sine
               23-31
               ;
            
             
               So
               is
               the
               Sine
               of
               30
               ,
               to
               Sine
               11-31
               .
            
             
             
               Lay
               the
               Thred
               to
               23-31
               ,
               
               on
               the
               degrees
               on
               the
               Moveable-piece
               ,
               counted
               from
               the
               Head
               toward
               the
               End
               ;
               then
               count
               the
               Suns
               place
               from
               the
               next
               Equinoctial-point
               ,
               on
               the
               Line
               of
               Sines
               from
               the
               Center
               downwards
               ,
               and
               take
               the
               ND
               from
               thence
               to
               the
               Thred
               ;
               then
               this
               distance
               being
               measured
               from
               the
               Center
               downwards
               ,
               shall
               be
               the
               sine
               of
               the
               Suns
               declination
               ,
               required
               for
               that
               distance
               ,
               from
               the
               next
               Equinoctial-point
               ;
               (
               
                 by
                 the
              
               1
               
                 st
                 Rule
                 abovesaid
              
               )
               .
            
             
               
                 By
                 the
                 Sector
              
               .
            
             
               Take
               —
               23-31
               ,
               from
               the
               Sines
               ,
               make
               it
               a
               =
               in
               the
               sine
               of
               90
               ;
               then
               the
               =
               sine
               of
               the
               Suns
               distance
               from
               the
               next
               Equinoctial-point
               ,
               shall
               be
               the
               —
               sine
               of
               the
               Suns
               declination
               ;
               Example
               as
               before
               (
               
                 Rule
                 the
              
               1
               st
               )
               .
            
          
           
             
               Use
               XIII
               .
               The
               Suns
               Declination
               being
               given
               ,
               to
               find
               his
               true
               place
               or
               distance
               from
               ♈
               or
               ♎
               ,
               the
               two
               Equinoctial
               Points
               .
            
             
               Lay
               the
               Thred
               to
               the
               Declination
               counted
               in
               the
               degrees
               from
               600
               ,
               
               and
               in
               the
               Line
               of
               the
               Suns
               place
               ,
               is
               his
               true
               place
               required
               .
            
             
             
               Example
               .
            
             
               When
               the
               Suns
               declination
               is
               12
               degrees
               Northward
               ,
               the
               dayes
               increasing
               ,
               then
               the
               Sun
               will
               be
               31
               deg
               .
               and
               23
               min.
               from
               ♈
               ,
               or
               1
               deg
               .
               23
               min.
               in
               ♉
               ,
               his
               true
               place
               required
               .
            
             
               As
               Sine
               of
               23-31
               ,
               the
               Suns
               greatest
               declination
               ,
               
               to
               Sine
               of
               90
               ;
            
             
               So
               Sine
               of
               12-00
               ,
               the
               Suns
               present
               declination
               ,
               to
               Sine
               of
               Suns
               distance
               from
               ♈
               or
               ♎
               31-23
               .
            
             
               Which
               ,
               by
               considering
               the
               time
               of
               the
               year
               ,
               gives
               his
               true
               place
               ,
               by
               looking
               on
               the
               Months
               and
               Line
               of
               Suns
               place
               on
               the
               Quadrantal-side
               .
            
             
               Take
               the
               —
               Sine
               of
               the
               present
               declination
               ,
               
               make
               it
               a
               =
               Sine
               in
               the
               greatest
               declination
               ,
               laying
               the
               Thred
               to
               ND
               ;
               and
               on
               the
               degrees
               the
               Thred
               shall
               give
               the
               Suns
               distance
               from
               ♈
               ,
               or
               ♎
               ,
               required
               .
               Example
               as
               before
               .
            
             
               Make
               —
               Sine
               of
               the
               given
               Suns
               declination
               ,
               
               a
               =
               Sine
               in
               the
               Suns
               greatest
               declination
               ,
               then
               =
               Sine
               of
               90
               ,
               measured
               from
               the
               Center
               ,
               is
               the
               =
               Sine
               of
               the
               Suns
               distance
               ,
               from
               ♈
               or
               ♎
               ,
               required
               ;
               or
               count
               30
               deg
               .
               for
               one
               sign
               ,
               and
               the
               Center
               for
               the
               next
               Equinoctial-point
               ,
               
               and
               90
               for
               the
               two
               Tropicks
               of
               Cancer
               ,
               and
               Capricorn
               .
               ♋
               .
               ♑
               .
            
          
           
             
               Use
               XIV
               .
               The
               Suns
               place
               ,
               or
               Day
               of
               the
               Month
               ,
               and
               greatest
               Declination
               given
               ;
               to
               find
               his
               Right
               Ascention
               from
               the
               same
               Equinoctial
               .
            
             
               Lay
               the
               Thred
               to
               the
               day
               of
               the
               Month
               ,
               
               or
               place
               given
               ,
               and
               in
               the
               Line
               of
               the
               Suns
               Right
               Ascention
               ,
               you
               have
               his
               Right
               Ascention
               in
               degrees
               ,
               or
               hours
               and
               minutes
               ,
               counting
               4
               minuts
               for
               every
               degree
               .
            
             
               Example
               .
            
             
               On
               the
               9th
               of
               April
               ,
               near
               night
               ,
               the
               Sun
               being
               then
               entring
               ♉
               ,
               the
               Suns
               Right
               Ascention
               will
               be
               1
               hour
               52
               min.
               or
               28
               degrees
               of
               Right
               Ascention
               ,
               distant
               from
               ♈
               .
            
             
               As
               the
               Sine
               of
               90
               ,
               
               to
               the
               Sine
               complement
               of
               the
               Suns
               greatest
               declination
               (
               or
               C.S.
               )
               of
               23-31
               ,
               counting
               backwards
               from
               90
               ,
               which
               will
               be
               at
               the
               Sine
               of
               66-29′
               .
               )
            
             
               So
               is
               the
               Tangent
               of
               the
               Suns
               distance
               from
               the
               next
               Equinoctial-point
               ;
               to
               the
               Tangent
               of
               the
               Suns
               Right
               Ascention
               from
               the
               same
               Equinoctial-point
               .
            
             
             
               
               Take
               the
               —
               co-sine
               of
               the
               greatest
               declination
               from
               the
               Center
               downwards
               ,
               being
               the
               —
               sine
               of
               66-29′
               .
               make
               it
               a
               =
               sine
               of
               90
               ,
               laying
               the
               Thred
               to
               ND
               ;
               and
               note
               what
               degree
               and
               minuit
               it
               cuts
               ,
               for
               this
               is
               fixed
               to
               this
               Proportion
               :
               Then
               take
               the
               Tangent
               of
               the
               Suns
               distance
               from
               the
               next
               Equinoctial-point
               ,
               from
               the
               Center
               at
               600
               ,
               on
               the
               degrees
               toward
               the
               End
               ,
               and
               lay
               it
               on
               the
               sines
               ,
               from
               the
               Center
               downwards
               ,
               and
               note
               the
               Point
               where
               it
               stayeth
               ,
               for
               the
               ND
               from
               thence
               to
               the
               Thred
               ,
               shall
               be
               the
               Tangent
               of
               the
               Suns
               Right
               Ascention
               required
               .
            
             
               Note
               ,
               That
               if
               the
               Suns
               distance
               from
               ♈
               ,
               or
               ♎
               ,
               be
               above
               45
               degrees
               ,
               then
               the
               Tangents
               on
               the
               loose-piece
               ,
               are
               to
               be
               used
               instead
               of
               the
               Tangents
               on
               the
               moveable-leg
               .
            
             
               
                 Or
                 ,
                 by
                 Sines
                 only
                 thus
              
               ;
            
             
               Or
               ,
               Take
               —
               Sine
               of
               the
               present
               Suns
               declination
               ,
               make
               it
               a
               =
               in
               the
               Sine
               of
               the
               Suns
               greatest
               declination
               ,
               and
               lay
               the
               Thred
               to
               ND
               ;
               then
               take
               =
               Co-sine
               of
               the
               Suns
               greatest
               declination
               ,
               and
               make
               it
               a
               =
               in
               Co-sine
               of
               the
               Suns
               present
               declination
               ,
               and
               lay
               the
               Thred
               to
               ND
               ,
               and
               in
               the
               degrees
               it
               cuts
               the
               Suns
               Right
               Ascention
               ,
               required
               .
            
             
             
               Make
               —
               Co-sine
               of
               23-31
               ,
               
               viz.
               the
               right
               Sine
               of
               66-29
               ,
               a
               =
               sine
               of
               90
               ,
               then
               the
               =
               Tangent
               of
               the
               Suns
               distance
               from
               ♈
               ,
               or
               ♎
               ,
               is
               the
               =
               Tangent
               of
               the
               Suns
               Right
               Ascention
               from
               the
               same
               Point
               of
               ♈
               ,
               or
               ♎
               ;
               as
               at
               30
               from
               ♈
               ,
               it
               is
               28
               degrees
               ,
               or
               1
               hour
               and
               52
               minuts
               from
               ♈
               ,
               (
               neer
               )
               .
            
          
           
             
               Use
               XV.
               Having
               the
               Suns
               Right
               Ascention
               ,
               and
               greatest
               Declination
               ,
               to
               find
               the
               Angle
               of
               the
               Ecliptick
               and
               Meridian
               .
            
             
               As
               Sine
               90
               ,
               to
               Sine
               23-31
               ;
               
            
             
               So
               is
               the
               Co-sine
               of
               the
               Suns
               Right
               Ascention
               ,
               to
               the
               Co-sine
               of
               the
               Angle
               of
               the
               Ecliptick
               and
               Meridian
               .
            
             
               Lay
               the
               Thred
               to
               23-31
               ,
               
               counted
               on
               the
               degrees
               from
               the
               Head
               ;
               then
               count
               the
               Co-sine
               of
               the
               Right
               Ascention
               ,
               from
               the
               Center
               downward
               ,
               or
               the
               Sine
               from
               90
               upwards
               ,
               and
               take
               the
               ND
               from
               thence
               to
               the
               Thred
               ,
               and
               measure
               it
               from
               the
               Center
               ,
               and
               it
               shall
               reach
               to
               the
               Co-sine
               of
               the
               Angle
               required
               .
            
             
               Example
               .
            
             
               The
               Right
               Ascention
               being
               30
               degrees
               ,
               or
               2
               hours
               ,
               the
               Angle
               shall
               be
               69-50
               .
            
             
             
               
               Make
               the
               —
               right
               sine
               of
               23-31
               ,
               a
               =
               sine
               of
               90
               ;
               then
               the
               =
               co-sine
               of
               30
               ,
               viz.
               =
               60
               ,
               shall
               make
               the
               —
               sine
               of
               69-50
               ,
               the
               Angle
               of
               the
               Ecliptick
               and
               Meridian
               .
            
          
           
             
               Use
               XVI
               .
               Having
               the
               Latitude
               ,
               and
               Declination
               of
               the
               Sun
               or
               Stars
               ,
               to
               find
               the
               Suns
               or
               Stars
               Amplitude
               ,
               at
               rising
               or
               Setting
               .
            
             
               
               Take
               the
               Suns
               declination
               ,
               from
               the
               particular
               Scale
               of
               Sines
               ,
               and
               lay
               it
               from
               6
               ,
               in
               the
               hour
               or
               Azimuth-line
               ,
               and
               it
               shall
               give
               the
               Amplitude
               from
               South
               ,
               as
               it
               is
               figured
               ;
               or
               from
               East
               ,
               or
               West
               ,
               counting
               from
               90
               ;
               observing
               to
               turn
               the
               Compasses
               the
               same
               way
               from
               90
               or
               6
               ,
               as
               the
               declination
               is
               Northward
               ,
               or
               Southwards
               .
            
             
               Example
               .
            
             
               The
               Suns
               declination
               being
               10
               degrees
               Northward
               ,
               the
               Suns
               Amplitude
               ,
               or
               Line
               ,
               is
               106-12
               ,
               from
               the
               South
               ,
               or
               16-12
               from
               the
               East-point
               .
            
             
               
               As
               co-sine
               of
               the
               Latitude
               ,
               to
               S.
               90
               ;
            
             
               So
               is
               S.
               of
               the
               Suns
               declination
               ,
               to
               S.
               of
               the
               Amplitude
               .
            
             
               
               Take
               the
               —
               Sine
               of
               the
               Suns
               declination
               ,
               make
               it
               a
               =
               in
               the
               co-sine
               of
               the
               Latitude
               ,
               
               and
               lay
               the
               Thred
               to
               the
               nearest
               distance
               ,
               
               and
               on
               the
               degrees
               the
               Thred
               shall
               shew
               the
               true
               Amplitude
               required
               .
            
             
               Make
               the
               —
               right
               Sine
               of
               the
               Suns
               declination
               ,
               a
               =
               in
               co-sine
               latitude
               ,
               then
               =
               90
               ,
               taken
               and
               measured
               from
               the
               Center
               ,
               gives
               the
               Amplitude
               or
               Line
               .
            
          
           
             
               Use
               XVII
               .
               Having
               the
               same
               Amplitude
               ,
               and
               Declination
               ,
               to
               find
               the
               Latitude
               .
            
             
               As
               S.
               of
               the
               Suns
               Amplitude
               ,
               
               to
               S.
               the
               Suns
               Declination
               ;
            
             
               So
               is
               S
               90
               ,
               to
               Co-sine
               Latitude
               .
            
             
               Take
               the
               —
               sine
               of
               the
               Suns
               declination
               ;
               
               set
               one
               Point
               in
               the
               Sine
               of
               the
               Suns
               Amplitude
               ,
               lay
               the
               Thred
               to
               ND
               ,
               and
               on
               the
               degrees
               it
               sheweth
               the
               complement
               of
               the
               Latitude
               required
               .
            
             
               Example
               .
            
             
               The
               Declination
               being
               20
               degrees
               ,
               and
               the
               Amplitude
               33-15
               ,
               the
               complement
               of
               the
               Latitude
               will
               be
               38-28
               —
               ,
               counting
               from
               the
               Head
               ,
               toward
               the
               End.
               
            
             
               Make
               the
               right
               Sine
               of
               the
               Suns
               Declination
               ,
               
               a
               =
               sine
               in
               the
               Suns
               Amplitude
               ;
               then
               the
               =
               sine
               of
               90
               ,
               shall
               be
               the
               —
               co-sine
               of
               the
               Latitude
               required
               .
            
          
           
             
             
               Use
               XVIII
               .
               Having
               the
               Latitude
               ,
               and
               Suns
               Declination
               ,
               to
               find
               his
               Altitude
               at
               East
               or
               West
               ,
               commonly
               called
               the
               Vertical-Circle
               ;
               or
               Azimuth
               of
               East
               or
               West
               .
            
             
               Take
               the
               Suns
               Declination
               from
               the
               particular
               Line
               of
               Sines
               ,
               
               set
               one
               Point
               in
               90
               on
               the
               Azimuth-line
               ,
               and
               lay
               the
               Thred
               to
               the
               ND
               ,
               and
               on
               the
               degrees
               it
               sheweth
               the
               Altitude
               required
               ;
               counting
               from
               600
               toward
               the
               End.
               
            
             
               As
               S.
               latitude
               S.
               of
               90
               ;
            
             
               
               So
               S.
               of
               Suns
               declination
               ,
               to
               S.
               Suns
               height
               ,
               at
               East
               or
               West
               .
            
             
               
               Take
               the
               —
               sine
               of
               the
               Suns
               declination
               ,
               make
               it
               a
               =
               in
               the
               sine
               of
               the
               latitude
               ,
               and
               lay
               the
               Thred
               to
               ND
               ,
               and
               on
               the
               degrees
               it
               shall
               shew
               the
               Suns
               Altitude
               ,
               at
               East
               and
               West
               required
               .
            
             
               Example
               .
            
             
               Declination
               10.
               
               Latitude
               51-32
               ;
               the
               Altitude
               is
               12
               degrees
               ,
               and
               50
               minuts
               .
            
             
               As
               —
               S.
               of
               the
               Suns
               Declination
               ,
               to
               =
               S.
               of
               Latitude
               ;
               
            
             
               So
               is
               the
               =
               S.
               of
               90
               ,
               to
               —
               S.
               of
               Vertical
               Altitude
               .
            
          
           
             
             
               Use
               XIX
               .
               Having
               the
               Latitude
               ,
               and
               Suns
               Declination
               ,
               to
               find
               the
               time
               when
               the
               Sun
               will
               be
               due
               East
               or
               West
               .
            
             
               Having
               gotten
               the
               Altitude
               by
               the
               last
               Rule
               ,
               take
               it
               from
               the
               particular
               Sine
               ;
               
               then
               lay
               the
               Thred
               to
               the
               Suns
               declination
               ,
               counted
               on
               the
               degrees
               ;
               then
               setting
               one
               Point
               in
               the
               Hour-line
               ,
               so
               as
               the
               other
               turned
               about
               ,
               shall
               but
               just
               touch
               the
               Thred
               ,
               and
               the
               Compass-point
               shall
               stay
               at
               the
               hour
               and
               minuit
               of
               time
               required
               .
            
             
               As
               Tangent
               latitude
               ,
               to
               Sine
               90
               ;
               
            
             
               So
               is
               the
               Tangent
               of
               the
               Suns
               declination
               ,
               to
               Co-sine
               of
               the
               hour
               .
            
             
               Or
               ,
            
             
               As
               sine
               90
               ,
               to
               Tangent
               Suns
               declination
               ;
            
             
               So
               is
               Co-tangent-latitude
               ,
               to
               Co-sine
               of
               the
               hour
               from
               noon
               .
            
             
               Example
               .
            
             
               Latitude
               51-32
               ,
               declination
               10
               ,
               the
               Sun
               will
               be
               due
               East
               at
               6-32
               ,
               and
               West
               at
               5-28
               .
            
             
               Take
               the
               —
               Tangent
               of
               the
               Latitude
               (
               on
               the
               loose-piece
               ,
               
               counting
               from
               60
               towards
               the
               moveable-leg
               ;
               or
               else
               from
               600
               ,
               on
               the
               
               moving-leg
               ,
               or
               degrees
               ,
               according
               as
               the
               Latitude
               is
               above
               or
               under
               45
               degrees
               )
               and
               lay
               it
               from
               the
               Center
               downwards
               ,
               and
               note
               the
               Point
               where
               it
               ends
               .
               Then
               take
               from
               the
               same
               Tangent
               ,
               the
               Tangent
               of
               the
               Suns
               declination
               ,
               and
               setting
               one
               foot
               in
               the
               Point
               last
               noted
               ,
               lay
               the
               Thred
               to
               ND
               ,
               then
               the
               =
               sine
               of
               90
               ,
               shall
               be
               the
               —
               sine
               of
               the
               hour
               from
               6.
               
            
             
               
                 Or
                 by
                 the
                 Sines
                 only
                 work
                 thus
              
               ;
            
             
               Take
               the
               —
               sine
               of
               the
               Suns
               declination
               ,
               make
               it
               a
               =
               in
               sine
               of
               the
               latitude
               ;
               lay
               the
               Thred
               to
               ND
               ,
               then
               take
               ND
               from
               the
               Co-sine
               latitude
               to
               the
               Thred
               ;
               then
               set
               one
               foot
               in
               the
               Co-sine
               of
               the
               Suns
               declination
               ,
               lay
               the
               Thred
               to
               ND
               ,
               and
               on
               the
               degrees
               it
               gives
               the
               hour
               from
               noon
               ,
               as
               it
               is
               figured
               ,
               or
               the
               hour
               from
               6
               ,
               counting
               from
               the
               head
               ,
               counting
               4
               minuts
               for
               every
               degree
               .
            
             
               Make
               the
               small
               Tangent
               of
               the
               Latitude
               ,
               if
               above
               45
               ,
               
               taken
               from
               the
               Center
               ,
               a
               =
               sine
               of
               90
               ;
               then
               the
               —
               Tangent
               of
               the
               Suns
               declination
               ,
               taken
               from
               the
               same
               small
               Tangent
               ,
               and
               carried
               Parallely
               till
               it
               stay
               in
               like
               Sines
               ,
               shall
               be
               the
               Sine
               of
               the
               hour
               from
               6.
               
            
             
             
               
                 Or
                 ,
                 as
                 before
                 ,
                 by
                 Sines
                 only
                 .
              
            
             
               Make
               —
               sine
               Declination
               ,
               a
               =
               sine
               Latitude
               ;
               then
               take
               =
               Co-sine
               Latitude
               ,
               and
               make
               it
               a
               =
               Co-sine
               of
               the
               Suns
               Declination
               ;
               then
               take
               =
               90
               ,
               and
               lay
               it
               from
               the
               Center
               ,
               it
               gives
               the
               Sine
               of
               the
               hour
               from
               6.
               
            
          
           
             
               Use
               XX.
               Having
               the
               Latitude
               ,
               and
               Suns
               Declination
               ,
               to
               find
               the
               Ascentional
               Difference
               ,
               or
               the
               Suns
               Rising
               and
               Setting
               ,
               and
               Oblique
               Ascention
               .
            
             
               Lay
               the
               Thred
               to
               the
               Day
               of
               the
               Month
               ,
               (
               or
               to
               the
               Suns
               Declination
               ,
               or
               true
               Place
               ,
               
               or
               to
               his
               Right
               Ascention
               ;
               for
               the
               Thred
               being
               laid
               to
               any
               one
               of
               them
               ,
               is
               then
               also
               laid
               to
               all
               the
               rest
               )
               then
               in
               the
               Azimuth-line
               ,
               it
               cuts
               the
               Ascentional
               difference
               ,
               if
               it
               you
               count
               from
               90
               ,
               or
               the
               Suns
               Rising
               ,
               as
               you
               count
               the
               morning
               hours
               ;
               or
               his
               Setting
               ,
               counting
               the
               afternoon
               hours
               .
            
             
               The
               Oblique
               Ascention
               is
               found
               out
               for
               the
               six
               Northern
               signs
               ,
               
               or
               Summer
               half-year
               ,
               by
               substracting
               the
               Suns
               difference
               of
               Ascentions
               ,
               out
               of
               the
               Suns
               Right
               Ascention
               .
               But
               for
               the
               other
               Winter-half
               year
               ,
               or
               six
               Southern
               signs
               ,
               it
               is
               found
               by
               adding
               the
               
               Suns
               difference
               of
               Ascentions
               to
               his
               Right
               Ascention
               ;
               this
               sum
               in
               Winter
               ,
               and
               the
               remainder
               as
               above-said
               in
               Summer
               ,
               shall
               be
               the
               Suns
               Oblique
               Ascention
               required
               .
            
             
               
               As
               Co-tangent
               Lat.
               to
               sine
               90
               ;
            
             
               To
               is
               the
               Tangent
               of
               the
               Suns
               declination
               ,
               to
               the
               sine
               of
               the
               Suns
               Ascentional
               difference
               .
            
             
               
               Take
               the
               —
               co-tangent
               latitude
               ,
               from
               the
               loose
               or
               moveable-piece
               ,
               as
               it
               is
               above
               or
               under
               45
               degrees
               ,
               make
               it
               a
               =
               in
               sine
               90
               ,
               lay
               the
               Thred
               to
               ND
               ,
               then
               take
               the
               —
               Tangent
               of
               the
               Suns
               declination
               from
               the
               same
               Tangents
               ,
               and
               carry
               it
               =
               till
               it
               stay
               in
               the
               parts
               ,
               that
               the
               other
               foot
               ,
               turned
               about
               ,
               will
               but
               just
               touch
               the
               Thred
               ,
               which
               parts
               shall
               be
               the
               Sine
               of
               the
               Suns
               Ascentional
               difference
               required
               .
            
             
               
                 Or
                 ●hus
                 ,
                 by
                 Sines
                 only
              
               ;
            
             
               Make
               the
               —
               sine
               of
               Declination
               ,
               a
               =
               Co-sine
               of
               the
               Latitude
               ;
               lay
               the
               Thred
               to
               ND
               ,
               then
               take
               the
               =
               sine
               of
               Latitude
               ,
               make
               it
               a
               =
               in
               Co-sine
               of
               the
               declination
               ,
               and
               lay
               the
               Thred
               to
               ND
               ,
               and
               on
               the
               degrees
               it
               shall
               cut
               the
               Suns
               Ascentional-difference
               required
               ;
               which
               being
               turned
               into
               time
               ,
               by
               counting
               4
               minuts
               for
               every
               degree
               ,
               
               and
               added
               to
               ,
               or
               taken
               from
               6
               ,
               gives
               the
               Suns
               Rising
               in
               Summer
               ,
               or
               Winter
               .
            
             
               Make
               the
               —
               Co-tangent
               Latitude
               ,
               
               a
               =
               sine
               of
               90
               ;
               then
               take
               —
               Tangent
               of
               the
               Suns
               declination
               ,
               and
               carry
               it
               =
               till
               it
               stay
               in
               like
               parts
               ,
               viz.
               the
               Sine
               of
               the
               Suns
               Ascentional
               difference
               required
               .
            
             
               
                 Example
                 otherwise
              
               ;
            
             
               As
               —
               sine
               90
               ,
               to
               =
               Tangent
               38-28
               ;
            
             
               So
               is
               =
               Tangent
               of
               23-31
               ,
               the
               Suns
               greatest
               declination
               ,
               to
               the
               —
               sine
               of
               the
               Suns
               greatest
               Ascentional
               difference
               ,
               33
               deg
               .
               and
               12
               min.
               
            
          
           
             
               Use
               XXI
               .
               The
               Latitude
               and
               Suns
               Declination
               given
               ,
               to
               find
               the
               Suns
               Meridian
               Altitude
               .
            
             
               When
               the
               Latitude
               and
               Declination
               is
               both
               alike
               ,
               viz.
               both
               North
               ,
               or
               both
               South
               ;
               then
               substract
               the
               Declination
               out
               of
               the
               Latitude
               ,
               or
               the
               less
               from
               the
               greater
               ,
               and
               the
               remainder
               shall
               be
               the
               complement
               of
               the
               Suns
               Meridian
               Altitude
               .
            
             
               But
               when
               they
               be
               unlike
               ,
               then
               add
               them
               together
               ,
               and
               the
               sum
               shall
               be
               the
               complement
               of
               the
               Meridian
               Altitude
               :
               The
               contrary
               work
               serves
               when
               the
               complement
               of
               the
               Latitude
               and
               Declination
               is
               given
               ,
               to
               find
               the
               Meridian
               Altitude
               .
            
             
             
               Lay
               the
               Thred
               to
               the
               Declination
               ,
               counted
               on
               the
               degrees
               from
               600
               ,
               the
               right
               way
               ,
               toward
               the
               Head
               for
               North
               ,
               and
               toward
               the
               End
               for
               South
               declination
               .
            
             
               Then
               ,
            
             
               Take
               the
               nearest
               distance
               ,
               from
               the
               Center-prick
               at
               12
               ,
               in
               the
               Hour-line
               ,
               to
               the
               Thred
               ;
               this
               distance
               measured
               on
               the
               Particular-line
               of
               Sines
               ,
               shall
               shew
               the
               Suns
               Meridian-Altitude
               required
               .
            
          
           
             
               Use
               XXI
               .
               The
               Latitude
               ,
               and
               Hour
               from
               the
               midnight
               Meridian
               given
               ,
               to
               find
               the
               Angle
               of
               the
               Suns
               Position
               ,
               viz.
               the
               Angle
               between
               the
               Hour
               and
               Azimuth-lines
               in
               the
               Center
               of
               the
               Sun.
               
            
             
               
               As
               Sine
               90
               ,
               to
               Co-sine
               of
               the
               Latitude
               ;
            
             
               So
               is
               the
               Sine
               of
               the
               Hour
               from
               Midnight
               ,
               to
               the
               sine
               of
               the
               Angle
               of
               Position
               .
            
             
               Example
               .
            
             
               As
               Sine
               90
               ,
               to
               Co-sine
               Latitude
               38-28
               ;
            
             
               So
               is
               the
               Co-sine
               of
               the
               Hour
               from
               midnight
               ,
               120
               ,
               for
               which
               you
               must
               use
               60
               ,
               to
               32-34
               ,
               the
               Angle
               of
               Position
               .
            
             
               
               Take
               the
               distance
               from
               the
               Hour
               to
               the
               90
               Azimuth
               on
               the
               Hour-line
               ,
               and
               measure
               it
               in
               the
               particular
               sines
               ,
               and
               it
               shall
               shew
               
               the
               Angle
               of
               Position
               required
               .
               This
               holds
               in
               the
               Equinoctial
               .
            
             
               Take
               —
               Co-sine
               Latitude
               ,
               
               make
               it
               a
               =
               in
               sine
               90
               ;
               then
               take
               out
               the
               =
               Co-sine
               of
               the
               Hour
               from
               the
               Meridian
               ,
               and
               it
               shall
               be
               the
               —
               sine
               of
               the
               Suns
               Position
               .
            
             
               Make
               —
               Co-sine
               Latitude
               a
               =
               sine
               90
               ;
               
               then
               =
               Co-sine
               of
               the
               Hour
               ,
               shall
               be
               —
               sine
               of
               the
               Suns
               Position
               .
            
             
               Note
               ,
            
             
               The
               Angle
               of
               the
               Suns
               Position
               may
               be
               varied
               ,
               and
               it
               is
               generally
               the
               Angle
               made
               in
               the
               Center
               of
               the
               Sun
               ,
               by
               his
               Meridian
               or
               Hour-circle
               ,
               being
               a
               Circle
               passing
               thorow
               the
               Pole
               of
               the
               World
               ,
               and
               the
               Center
               of
               the
               Sun
               ;
               and
               any
               other
               principal
               Circle
               ,
               as
               the
               Meridian
               ,
               the
               Horizon
               ,
               or
               any
               Azimuth
               ,
               the
               Anguler-Point
               being
               alwayes
               the
               Center
               of
               the
               Sun.
               
            
          
           
             
               Use
               XXII
               .
               The
               Suns
               Declination
               given
               ,
               to
               find
               the
               beginning
               and
               end
               of
               Twi-light
               ,
               or
               Day-break
               ▪
            
             
               Lay
               the
               Thred
               to
               the
               Declination
               on
               the
               degrees
               ,
               but
               counted
               the
               contrary
               way
               ,
               
               viz.
               South-declination
               toward
               the
               〈…〉
               North-declination
               toward
               the
               〈◊〉
               ;
               then
               take
               18
               degrees
               from
               the
               particul●●
               〈◊〉
               
               Sines
               for
               Twi-light
               ,
               or
               13
               degrees
               for
               Day-break
               ,
               or
               clear
               light
               ;
               Then
               carry
               this
               distance
               of
               18
               for
               Twi-light
               ,
               or
               13
               for
               Day-break
               ,
               along
               the
               Line
               of
               Hours
               on
               that
               side
               of
               the
               Thred
               next
               the
               End
               ,
               till
               the
               other
               Foot
               ,
               turned
               about
               ,
               will
               but
               just
               touch
               the
               Thred
               ,
               then
               shall
               the
               Point
               shew
               the
               time
               of
               Twi-light
               ,
               or
               Day-break
               ,
               required
               .
            
             
               Example
               .
            
             
               The
               Suns
               Declination
               being
               12
               degrees
               North
               ,
               the
               Twi-light
               continues
               ,
               till
               9
               hours
               24
               minuts
               ;
               or
               it
               begins
               in
               the
               morning
               at
               38
               minuts
               after
               2
               ;
               but
               the
               Day-break
               is
               not
               till
               22
               minuts
               after
               3
               in
               the
               morning
               ,
               or
               38
               minuts
               after
               8
               at
               night
               ,
               and
               last
               no
               longer
               .
            
             
               
                 To
                 work
                 this
                 for
                 any
                 other
                 place
                 ,
                 where
                 the
                 Latitude
                 doth
                 vary
                 ,
                 do
                 thus
              
               ;
            
             
               Find
               the
               Hour
               that
               answers
               to
               18
               degrees
               of
               Altitude
               ,
               in
               as
               much
               Declination
               the
               contrary
               way
               ,
               and
               that
               shall
               be
               the
               time
               of
               Twi-light
               ;
               or
               at
               13
               degrees
               for
               Day-break
               ,
               according
               to
               the
               Rules
               in
               the
               26th
               Use
               ,
               where
               the
               way
               how
               is
               largely
               handled
               to
               the
               33d
               Use
               ,
               both
               wayes
               generally
               .
            
          
           
             
             
               Use
               XXIII
               .
               To
               find
               for
               what
               Latitude
               your
               Instrument
               is
               particularly
               made
               for
               ;
            
             
               Take
               the
               nearest
               distance
               from
               the
               Center
               on
               the
               Head-leg
               ,
               
               to
               the
               Azimuth-line
               on
               the
               moveable-leg
               ;
               this
               distance
               measured
               on
               the
               particular
               Scale
               of
               Sines
               ,
               shall
               shew
               the
               Latitude
               required
               ;
               or
               the
               Extent
               from
               0
               to
               90
               ,
               on
               the
               Azimuth-line
               ,
               shall
               shew
               the
               complement
               of
               the
               Latitude
               ,
               being
               measured
               as
               before
               .
            
          
           
             
               Use
               XXIV
               .
               Having
               the
               Meridian
               Altitude
               given
               ,
               to
               find
               the
               time
               of
               Sun
               Rising
               or
               Setting
               ,
               true
               Place
               ,
               or
               Declination
               .
            
             
               Take
               the
               Suns
               Meridian
               Altitude
               from
               the
               particular
               Scale
               ,
               
               and
               setting
               on
               Point
               in
               ☉
               on
               the
               Azimuth-line
               ;
               lay
               the
               Thred
               to
               the
               ND
               ,
               and
               on
               the
               Hour-line
               it
               sheweth
               the
               time
               of
               Rising
               or
               Setting
               ;
               and
               on
               the
               degrees
               ,
               the
               Declination
               ;
               and
               the
               rest
               in
               their
               respective
               Lines
               .
            
             
               Example
               .
            
             
               The
               Meridian
               Altitude
               being
               50
               ,
               the
               Sun
               riseth
               at
               5
               ,
               and
               sets
               at
               7.
               
            
          
           
             
             
               Use
               XXV
               .
               The
               Latitude
               and
               Declination
               given
               ,
               to
               find
               the
               Suns
               height
               at
               6.
               
            
             
               Lay
               the
               Thred
               to
               the
               Day
               of
               the
               Month
               ,
               or
               Declination
               ,
               
               then
               take
               the
               ND
               from
               the
               Hour-point
               of
               06
               ,
               and
               6
               to
               the
               Thred
               ,
               and
               that
               distance
               measured
               on
               the
               particular
               Scale
               of
               Sines
               ,
               shall
               be
               the
               Suns
               Altitude
               at
               6
               in
               Summer
               time
               ,
               or
               his
               depression
               under
               the
               Horizon
               in
               the
               Winter
               time
               .
            
             
               
               As
               sine
               of
               90
               ,
               to
               sine
               of
               the
               Suns
               Declination
               ;
            
             
               So
               is
               sine
               Latitude
               ,
               to
               sine
               of
               the
               Suns
               Altitude
               at
               6.
               
            
             
               
               Count
               the
               Suns
               declination
               on
               the
               degrees
               from
               90
               ,
               toward
               the
               End
               ,
               and
               there
               lay
               the
               Thred
               ;
               then
               the
               least
               distance
               from
               the
               sine
               of
               the
               Latitude
               to
               the
               Thred
               ,
               measured
               from
               the
               Center
               downwards
               ,
               shall
               be
               the
               sine
               of
               the
               Suns
               Altitude
               at
               6.
               
            
             
               Make
               the
               —
               sine
               of
               the
               Declination
               a
               =
               sine
               of
               90
               ;
               
               then
               the
               =
               sine
               of
               the
               Latitude
               ,
               shall
               be
               the
               —
               sine
               of
               the
               Suns
               height
               at
               6.
               
            
             
               Example
               .
            
             
               Latitude
               51-32
               ,
               Declination
               23-31
               ,
               the
               height
               at
               6
               ,
               is
               18
               deg
               .
               13
               min.
               
            
          
           
             
             
               Use
               XXVI
               .
               Having
               the
               Latitude
               ,
               the
               Suns
               Declination
               and
               Altitude
               ,
               to
               find
               the
               Hour
               of
               the
               Day
               .
            
             
               Take
               the
               Suns
               Altitude
               ,
               
               from
               the
               particular
               Scale
               of
               Sines
               ,
               between
               the
               Compasses
               ;
               then
               lay
               the
               Thred
               to
               the
               Day
               of
               the
               Month
               ,
               or
               Declination
               ;
               then
               carry
               the
               Compasses
               along
               the
               Line
               of
               Hours
               ,
               between
               the
               Thred
               and
               the
               End
               ,
               till
               the
               other
               Point
               (
               being
               turned
               about
               )
               will
               but
               just
               touch
               the
               Thred
               ,
               and
               then
               the
               fixed
               Point
               shall
               shew
               the
               true
               hour
               and
               min.
               required
               ,
               in
               the
               Fore
               ,
               or
               After-noon
               ;
               if
               you
               be
               in
               doubt
               which
               it
               is
               ,
               then
               another
               Observation
               presently
               after
               ,
               will
               determine
               it
               .
            
             
               Example
               .
            
             
               May
               10th
               ,
               at
               30
               degrees
               of
               Altitude
               ,
               the
               hour
               will
               be
               32
               minuts
               after
               7
               in
               the
               Morning
               ,
               or
               28
               minuts
               after
               4
               in
               the
               Afternoon
               .
            
             
               This
               Work
               being
               somewhat
               more
               difficult
               than
               the
               former
               ,
               I
               shall
               part
               it
               thus
               ;
            
             
               
                 1.
                 
                 First
                 ,
                 to
                 find
                 the
                 Hour
                 the
                 Sun
                 being
                 in
                 the
                 Equinoctial
                 .
              
            
             
               Take
               the
               —
               sine
               of
               the
               Suns
               Altitude
               ,
               
               
               make
               it
               a
               =
               Co-sine
               of
               the
               Latitude
               ;
               lay
               the
               Thred
               to
               ND
               ,
               and
               on
               the
               degrees
               it
               shall
               give
               the
               Hour
               from
               12
               ,
               as
               it
               is
               figured
               ,
               counting
               15
               degrees
               for
               an
               hour
               ,
               or
               from
               6
               ,
               counting
               from
               the
               Head
               at
               90.
               
            
             
               Example
               .
            
             
               Latitude
               51-30
               ,
               Altitude
               20
               ,
               the
               hour
               is
               8
               &
               12′
               in
               the
               forenoon
               ,
               or
               3-48′
               in
               the
               afternoon
               .
            
             
               
                 The
                 same
                 by
                 Artificial
                 Sines
                 &
                 Tangents
                 .
              
            
             
               As
               Co-sine
               Latitude
               ,
               to
               sine
               90
               ;
            
             
               So
               is
               the
               sine
               of
               the
               Suns
               Altitude
               ,
               to
               sine
               of
               the
               hour
               from
               6.
               
            
             
               
               Make
               —
               S.
               ☉
               Altitude
               ,
               a
               =
               S.
               in
               ☉
               Latitude
               ;
               then
               take
               out
               =
               S.
               90
               ,
               and
               it
               shall
               be
               the
               —
               sine
               of
               the
               hour
               from
               6.
               
            
             
               
                 2.
                 
                 The
                 Latitude
                 ,
                 Declination
                 ,
                 and
                 Altitude
                 given
                 ,
                 to
                 find
                 the
                 Hour
                 at
                 any
                 time
                 .
              
            
             
               
               First
               by
               the
               25th
               Use
               ,
               find
               the
               Suns
               Altitude
               or
               depression
               at
               6
               ;
               then
               in
               Summer-time
               ,
               lay
               this
               distance
               from
               the
               Center
               downwards
               ;
               and
               in
               Winter-time
               ,
               lay
               it
               upwards
               from
               the
               Center
               toward
               the
               End
               of
               the
               Head-leg
               ;
               and
               note
               that
               Point
               for
               that
               day
               ,
               or
               degree
               of
               Declination
               ;
               for
               by
               taking
               the
               distance
               from
               thence
               to
               the
               Suns
               Altitude
               ,
               on
               the
               General
               Scale
               ,
               you
               have
               
               added
               ,
               or
               substracted
               the
               Altitude
               at
               6
               ,
               to
               ,
               or
               from
               the
               present
               Altitude
               .
            
             
               (
               For
               by
               taking
               the
               distance
               from
               that
               noted
               Point
               ,
               over
               ,
               or
               under
               the
               Center
               ,
               to
               the
               Suns
               present
               Altitude
               ,
               you
               have
               in
               Summer
               the
               difference
               between
               the
               Suns
               present
               Altitude
               ,
               and
               his
               Altitude
               at
               6.
               
               And
               in
               Winter
               you
               have
               the
               sum
               of
               the
               present
               Altitude
               ,
               and
               the
               Altitude
               at
               6.
               )
            
             
               This
               Operation
               is
               plainly
               hinted
               at
               ,
               in
               the
               4th
               Chapter
               ,
               and
               9th
               and
               10th
               Section
               ,
               which
               being
               understood
               ,
               take
               the
               whole
               Operation
               in
               shorter
               terms
               ,
               thus
               ;
            
             
               Count
               the
               Suns
               Declination
               from
               90
               ,
               toward
               the
               end
               ,
               and
               thereunto
               lay
               the
               Thred
               ;
               the
               nearest
               distance
               from
               the
               sine
               of
               the
               Latitude
               to
               the
               Thred
               ,
               is
               the
               Suns
               height
               ,
               or
               depression
               at
               6
               :
               In
               Winter
               use
               the
               sum
               of
               ,
               in
               Summer
               the
               difference
               between
               ,
               the
               Suns
               Altitude
               at
               6
               ,
               and
               his
               present
               Altitude
               ;
               with
               this
               distance
               between
               your
               Compasses
               ,
               set
               one
               Point
               in
               the
               co-sine
               of
               the
               Latitude
               ;
               lay
               the
               Thred
               to
               ND
               ,
               then
               take
               the
               ND
               from
               90
               ,
               to
               the
               Thred
               ;
               then
               set
               one
               foot
               in
               the
               Co-sine
               of
               the
               Suns
               declination
               ,
               and
               lay
               the
               Thred
               to
               ND
               ,
               and
               on
               the
               degrees
               it
               gives
               the
               hour
               required
               ;
               from
               6
               counting
               from
               90
               ,
               or
               from
               12
               ,
               as
               it
               is
               figured
               .
            
             
             
               Example
               .
            
             
               On
               April
               20
               ,
               at
               30
               deg
               .
               20
               min.
               of
               Altitude
               ,
               Latitude
               51-32
               ,
               the
               hour
               will
               be
               found
               to
               be
               just
               2
               hours
               from
               6
               ,
               or
               just
               8.
               
            
             
               Again
               ,
            
             
               On
               the
               10th
               of
               November
               ,
               at
               8
               deg
               .
               25
               min.
               high
               ,
               it
               is
               just
               3
               hours
               from
               6
               ,
               or
               9
               a
               clock
               in
               the
               forenoon
               ,
               or
               3
               afternoon
               .
            
             
               
                 Or
                 somewhat
                 differing
                 thus
              
               ;
            
             
               Take
               the
               —
               sine
               of
               the
               sum
               ,
               or
               difference
               ,
               of
               the
               Suns
               present
               Altitude
               ,
               and
               Altitude
               at
               6
               ,
               and
               make
               it
               a
               =
               in
               the
               co-sine
               of
               the
               Latitude
               ,
               and
               lay
               the
               Thred
               to
               the
               nearest
               distance
               ;
               then
               take
               out
               the
               =
               Secant
               of
               the
               declination
               beyond
               90d
               ,
               and
               make
               it
               a
               =
               sine
               of
               90
               ;
               and
               laying
               the
               Thred
               to
               the
               nearest
               distance
               ,
               on
               the
               degrees
               it
               shall
               shew
               the
               hour
               from
               6
               required
               .
            
             
               
               First
               ,
               by
               Use
               25
               ,
               find
               the
               Suns
               height
               at
               6
               ,
               or
               depression
               in
               Winter
               ;
               then
               by
               the
               former
               2d
               ,
               find
               the
               sum
               or
               difference
               between
               the
               Altitude
               at
               6
               ,
               and
               the
               Suns
               present
               Altitude
               ;
               but
               if
               you
               have
               Tables
               of
               Natural
               Sines
               and
               Tangents
               ;
               then
               in
               Winter
               ,
               add
               the
               Natural
               Sines
               of
               the
               two
               Altitudes
               together
               ;
               and
               in
               Summer
               ,
               substract
               the
               lesser
               out
               of
               the
               greater
               ,
               and
               find
               the
               Ark
               of
               difference
               more
               exactly
               .
            
             
             
               Then
               ,
            
             
               As
               the
               Co-sine
               of
               the
               Latitude
               ,
               to
               the
               Secant
               of
               the
               Declination
               (
               counted
               beyond
               90
               ,
               as
               much
               forward
               as
               from
               90
               to
               the
               Co-sine
               of
               the
               Suns
               Declination
               )
               ;
            
             
               So
               is
               the
               Sine
               of
               the
               sum
               ,
               or
               difference
               ,
               to
               the
               hour
               from
               6
               ▪
               required
               .
            
             
               
                 Or
                 else
                 ●hus
              
               ;
            
             
               As
               the
               Co-sine
               of
               the
               Latitude
               ,
               to
               the
               Sine
               of
               the
               sum
               ,
               or
               difference
               ;
            
             
               So
               is
               sine
               of
               90
               ,
               to
               a
               4th
               .
            
             
               Then
               ,
            
             
               As
               the
               Co-sine
               of
               the
               Suns
               declination
               ,
               to
               that
               4th
               ;
            
             
               So
               is
               sine
               90
               ,
               to
               the
               hour
               from
               6.
               
            
             
               
                 By
                 the
                 Sector
              
               .
            
             
               Take
               the
               —
               secant
               of
               the
               Suns
               declination
               ,
               make
               it
               a
               =
               in
               the
               co-sine
               of
               the
               Latitude
               ;
               then
               take
               out
               the
               =
               sine
               of
               the
               sum
               or
               difference
               ,
               and
               turn
               it
               twice
               from
               the
               Center
               lattera●ly
               ,
               and
               it
               shall
               be
               the
               sine
               of
               the
               hour
               from
               6
               ,
               required
               .
            
             
             
               Example
               .
            
             
               April
               20
               ,
               the
               Suns
               Declination
               is
               15
               degrees
               ;
               and
               the
               Suns
               Height
               at
               6
               ,
               then
               is
               ,
               11
               deg
               .
               42
               min.
               now
               the
               Natural
               sine
               of
               11-42
               ,
               20278
               ,
               taken
               from
               the
               Natural
               sine
               of
               30
               deg
               .
               20
               min.
               50502
               ,
               the
               Suns
               present
               Altitude
               ,
               the
               residue
               is
               30224
               ,
               the
               sine
               of
               17
               deg
               .
               35
               min.
               and
               a
               half
               .
            
             
               Then
               ,
            
             
               The
               —
               Secant
               of
               15
               made
               a
               =
               sine
               of
               38-28
               ,
               and
               the
               Sector
               so
               set
               ,
               the
               =
               sine
               of
               17-35
               ½
               ,
               turned
               latterally
               twice
               from
               the
               Center
               ,
               shall
               reach
               to
               30
               ,
               the
               sine
               of
               2
               hours
               from
               6
               ,
               the
               hours
               required
               .
            
          
           
             
               Use
               XXVII
               .
               Having
               the
               Latitude
               ,
               the
               Suns
               Declination
               ,
               and
               Altitude
               ,
               to
               find
               the
               Suns
               Azimuth
               .
            
             
               
               Take
               the
               Declination
               from
               the
               particular
               Scale
               of
               Sines
               ,
               for
               the
               particular
               Latitude
               the
               Instrument
               is
               made
               for
               ;
               Then
               ,
               count
               the
               given
               Altitude
               on
               the
               degrees
               from
               600
               toward
               the
               loose-piece
               ,
               and
               sometimes
               on
               the
               loose-piece
               also
               ;
               and
               thereunto
               lay
               the
               Thred
               ,
               then
               carry
               the
               Compasses
               ,
               so
               set
               ,
               along
               the
               Azimuth-line
               on
               the
               right-side
               of
               the
               Thred
               in
               Northern-declinations
               ,
               and
               
               on
               the
               left-side
               in
               Southern-declinations
               ,
               till
               the
               other
               foot
               ,
               turned
               about
               ,
               will
               but
               just
               touch
               the
               Thred
               ;
               then
               the
               fixed-point
               shall
               stay
               at
               the
               Suns
               true
               Azimuth
               required
               .
            
             
               
                 Take
                 two
                 or
                 three
                 Examples
              
               .
            
             
               1.
               
               First
               ,
               When
               the
               Sun
               is
               in
               the
               Equinoctial
               and
               hath
               no
               Declination
               ,
               then
               there
               is
               nothing
               to
               take
               between
               your
               Compasses
               ,
               but
               just
               to
               lay
               the
               Thred
               to
               the
               Suns
               Altitude
               ,
               counted
               from
               600
               on
               the
               loose-piece
               toward
               the
               End
               :
               then
               〈◊〉
               the
               Azimuth-line
               ,
               it
               cuts
               the
               Azimuth
               from
               the
               South
               required
               .
            
             
               Example
               .
            
             
               At
               00
               degrees
               high
               ,
               the
               Azimuth
               is
               90
               from
               South
               ;
               and
               at
               10
               degrees
               high
               ,
               it
               is
               77-5
               ;
               at
               20
               high
               ,
               the
               Azimuth
               is
               62-45
               ;
               at
               30
               degrees
               high
               ,
               it
               is
               43-30
               ;
               at
               34
               degrees
               high
               ,
               it
               is
               32
               degrees
               of
               Azimuth
               from
               South
               ;
               and
               at
               38-28
               degrees
               high
               ,
               it
               is
               just
               South
               .
            
             
               2.
               
               Secondly
               ,
               at
               10
               degrees
               of
               Declination
               Northward
               ,
               and
               20
               degrees
               of
               Altitude
               ,
               take
               10
               degrees
               from
               the
               particular
               Scale
               ,
               and
               lay
               the
               Thred
               to
               the
               Suns
               present
               Altitude
               ,
               as
               before
               ,
               and
               carry
               the
               Compasses
               on
               the
               right-side
               of
               the
               Thred
               on
               the
               Azimuth-line
               ,
               till
               the
               other
               foot
               ,
               being
               turned
               about
               ,
               will
               but
               just
               touch
               it
               ;
               then
               
               shall
               the
               Point
               rest
               at
               80
               degrees
               ,
               42
               min.
               of
               Azimuth
               from
               the
               South
               .
            
             
               3.
               
               But
               if
               the
               Declimation
               be
               the
               same
               to
               the
               Southwards
               ,
               and
               the
               Altitude
               also
               the
               same
               ;
               then
               carry
               the
               Compasses
               on
               the
               left-side
               of
               the
               Thred
               ,
               on
               the
               Azimuth-line
               ,
               till
               the
               other
               foot
               ,
               turned
               about
               ,
               will
               but
               just
               touch
               it
               ,
               and
               you
               shall
               find
               the
               Point
               to
               stay
               at
               41
               deg
               .
               10
               min.
               the
               true
               Azimuth
               from
               the
               South
               required
               .
            
             
               Note
               ,
               That
               any
               thing
               ,
               as
               thick
               as
               the
               Rule
               ,
               laid
               by
               the
               Rule
               ,
               and
               the
               Thred
               drawn
               over
               ,
               it
               will
               keep
               the
               Thred
               steady
               ,
               till
               you
               get
               the
               nearest
               distance
               more
               truly
               .
            
             
               
                 First
                 ,
                 by
                 the
                 18th
                 Use
                 ,
                 find
                 the
                 Suns
                 Altitude
                 in
                 the
                 Vertical
                 Circle
                 ,
                 or
                 Circle
                 of
                 East
                 and
                 West
                 ,
                 thus
              
               ;
            
             
               
               Take
               the
               sine
               of
               the
               Suns
               Declination
               ,
               and
               set
               one
               foot
               in
               the
               sine
               of
               Latitude
               ,
               lay
               the
               Thred
               to
               ND
               ,
               and
               in
               the
               degrees
               you
               shall
               have
               the
               Altitude
               at
               East
               and
               West
               required
               .
            
             
               Which
               Vertical
               Altitude
               in
               Summer
               or
               Northern
               Declinations
               ,
               you
               must
               substract
               out
               of
               the
               Suns
               present
               Altitude
               ;
               or
               take
               the
               lesser
               from
               the
               greater
               ,
               to
               find
               a
               difference
               ;
               but
               in
               Winter
               ,
               you
               must
               add
               this
               depression
               in
               the
               Vertical
               Circle
               ,
               to
               the
               Suns
               
               present
               Altitude
               to
               get
               a
               sum
               ,
               which
               must
               be
               done
               on
               a
               Line
               of
               Natural
               Sines
               ,
               or
               by
               the
               TABLE
               of
               Natural
               Sines
               ,
               as
               before
               ,
               in
               the
               Hour
               ,
               by
               laying
               it
               over
               or
               under
               the
               Center
               ,
               and
               taking
               from
               that
               noted
               Point
               to
               the
               Suns
               present
               Altitude
               all
               that
               day
               .
               Then
               take
               the
               distance
               from
               the
               Center
               to
               the
               Tangent
               of
               the
               Suns
               present
               Altitude
               on
               the
               loose-piece
               ,
               which
               is
               the
               Secant
               of
               the
               Suns
               present
               Altitude
               ,
               and
               lay
               it
               from
               the
               Center
               on
               the
               Line
               of
               Sines
               ,
               and
               note
               the
               place
               ;
               then
               take
               the
               distance
               from
               60
               ,
               on
               the
               loose-piece
               ,
               to
               the
               co-tangent
               of
               the
               Latitude
               (
               by
               counting
               10
               ,
               20
               ,
               30
               ,
               &c.
               from
               60
               ,
               toward
               the
               moveable-leg
               )
               between
               your
               Compasses
               ;
               then
               setting
               one
               Point
               on
               the
               Secant
               of
               the
               Suns
               Altitude
               last
               found
               ,
               and
               noted
               on
               the
               Line
               of
               Sines
               ;
               and
               with
               the
               other
               ,
               lay
               the
               Thred
               to
               the
               nearest
               distance
               ,
               and
               there
               keep
               it
               ,
               (
               by
               noting
               what
               degree
               ,
               day
               of
               the
               month
               ,
               or
               hour
               &
               minut
               ,
               or
               Azimuth
               it
               cuts
               )
               .
            
             
               Then
               take
               the
               —
               distance
               on
               the
               Sines
               ,
               from
               the
               sine
               of
               the
               Suns
               Vertical
               Altitude
               ,
               to
               his
               present
               Altitude
               ,
               for
               a
               difference
               in
               Summer
               ;
               Or
               ,
            
             
               The
               distance
               from
               a
               Point
               made
               beyond
               the
               Center
               ,
               (
               equal
               to
               the
               sine
               of
               the
               Suns
               Vertical
               depression
               )
               to
               the
               Suns
               present
               Altitude
               ,
               for
               a
               sum
               in
               Winter
               .
            
             
             
               Then
               having
               this
               —
               distance
               of
               sum
               or
               difference
               ,
               for
               Winter
               or
               Summer
               ,
               between
               your
               Compasses
               ;
               carry
               one
               Point
               parallelly
               on
               the
               Line
               of
               Sines
               ,
               till
               the
               other
               ,
               being
               turned
               about
               ,
               shall
               just
               touch
               the
               Thred
               at
               the
               ND
               ,
               the
               place
               where
               the
               Point
               stayeth
               ,
               shall
               be
               the
               Azimuth
               from
               East
               or
               West
               ,
               as
               it
               is
               figured
               from
               the
               Center
               ;
               or
               from
               North
               or
               South
               ,
               counting
               from
               90.
               
            
             
               
                 Which
                 work
                 in
                 brief
                 ,
                 may
                 be
                 sufficiently
                 worded
                 thus
              
               ;
            
             
               As
               —
               co-tangent
               of
               the
               Latitude
               ,
               to
               the
               =
               secant
               of
               the
               Suns
               present
               Altitude
               ,
               laying
               the
               Thred
               to
               ND
               ;
            
             
               So
               is
               the
               —
               sine
               of
               the
               sum
               ,
               or
               difference
               ,
               of
               the
               Suns
               present
               Altitude
               ,
               &
               Vertical
               depression
               in
               Winter
               ,
               or
               the
               difference
               between
               his
               Vertical
               and
               present
               Altitude
               in
               Summer
               ;
               to
               the
               =
               sine
               of
               the
               Suns
               Azimuth
               ,
               at
               that
               Altititude
               and
               Declination
               .
            
             
               
                 Yet
                 again
                 ,
                 more
                 short
              
               .
            
             
               As
               —
               C.T.
               Lat.
               to
               =
               Sec.
               ☉
               Alt.
               
            
             
               So
               —
               S.
               of
               sum
               or
               difference
               ,
               to
               =
               S.
               ☉
               Azimuth
               .
            
             
             
               
                 But
                 note
                 ,
                 That
                 in
                 Latitudes
                 under
                 45
                 ,
                 when
                 the
                 complements
                 of
                 the
                 Latitude
                 are
                 too
                 large
                 ,
                 then
                 work
                 thus
              
               ;
            
             
               As
               the
               —
               co-sine
               of
               the
               Suns
               Altitude
               ,
               to
               =
               Tangent
               of
               the
               Latitude
               ,
               taken
               from
               the
               degrees
               on
               the
               moveable-leg
               ,
               laying
               the
               Thred
               to
               ND
               ,
               then
               the
               —
               sine
               of
               the
               sum
               or
               difference
               carried
               parallelly
               ,
               shall
               stay
               at
               the
               Suns
               Azimuth
               required
               .
            
             
               If
               the
               Tangents
               are
               too
               small
               ,
               on
               the
               Sector-side
               is
               a
               larger
               ;
               and
               if
               the
               Sines
               are
               too
               great
               ,
               on
               the
               Head-leg
               there
               is
               a
               less
               .
            
             
               Find
               the
               Vertical
               Altitude
               by
               Use
               18
               ,
               and
               the
               sum
               or
               difference
               of
               the
               present
               and
               Vertical
               Altitude
               by
               the
               Table
               ,
               
               or
               Line
               of
               Natural
               Sines
               ,
               as
               before
               shewed
               ;
               then
               the
               Canon
               or
               Proportion
               runs
               thus
               ;
            
             
               As
               the
               Co-sine
               of
               the
               Suns
               Altitude
               ,
               to
               the
               Tangent
               of
               the
               Latitude
               ;
            
             
               So
               is
               the
               sine
               of
               the
               sum
               or
               difference
               ,
               to
               the
               sine
               of
               the
               Azimuth
               ,
               from
               East
               or
               West
               .
            
             
               Or
               ,
            
             
               As
               Co-tangent
               Latitude
               ,
               to
               Secant
               of
               the
               Suns
               Altitude
               ;
            
             
               So
               is
               the
               sine
               of
               the
               sum
               or
               difference
               ,
               to
               the
               sine
               of
               the
               Azimuth
               .
            
             
             
               
               Make
               the
               —
               Secant
               of
               the
               Suns
               Altitude
               ,
               a
               =
               Co-tangent
               of
               Latitude
               ;
               then
               the
               —
               sine
               of
               the
               sum
               or
               difference
               ,
               shall
               be
               half
               the
               —
               sine
               of
               the
               Azimuth
               ;
               or
               being
               turned
               twice
               from
               the
               Center
               ,
               the
               whole
               sine
               .
            
             
               
                 Or
                 else
                 thus
              
               ;
            
             
               Make
               the
               —
               Tangent
               of
               the
               Latitude
               ,
               a
               =
               Co-sine
               of
               the
               Suns
               Altitude
               ;
               then
               the
               =
               sine
               of
               the
               sum
               or
               difference
               ,
               shall
               be
               the
               —
               sine
               of
               the
               Azimuth
               ,
               measured
               on
               the
               Sine
               ,
               equal
               to
               the
               Radius
               of
               the
               Tangents
               first
               taken
               .
            
             
               Example
               .
            
             
               In
               Latitude
               51-32
               ,
               Declination
               North
               and
               South
               13-15
               ,
               the
               Vertical
               Altitude
               or
               Depression
               being
               17-01
               ,
               and
               the
               present
               Altitude
               20
               ;
               the
               Azimuth
               for
               South-declination
               will
               be
               found
               to
               be
               31-45
               ,
               from
               South
               ,
               the
               Depression
               at
               East
               and
               West
               being
               17-01
               ;
               and
               the
               sum
               of
               the
               present
               Altitude
               and
               Depression
               39-25
               .
            
             
               Again
               ,
            
             
               For
               North-declination
               ,
               or
               Summer-time
               ,
               the
               difference
               between
               the
               Vertical
               and
               present
               Altitude
               ,
               is
               2-54
               ;
               and
               the
               Azimuth
               from
               South
               ,
               will
               be
               found
               to
               be
               86
               degrees
               and
               15
               minuts
               .
            
          
           
             
             
               Use
               XXVIII
               .
               To
               make
               a
               Scale
               ,
               whereby
               to
               perform
               all
               th●se
               Propositions
               ,
               by
               the
               former
               Rules
               ,
               agreeable
               to
               the
               Trianguler
               Quadrant
               ,
               being
               added
               chiefly
               as
               a
               Demonstration
               of
               the
               Instrument
               ,
               and
               former
               Operations
               .
            
             
             
               Then
               making
               DE
               Radius
               ,
               describe
               the
               Circle
               90
               EI
               ,
               and
               divide
               it
               into
               180
               equal
               degrees
               ;
               Also
               ▪
               draw
               the
               lesser
               Circle
               90
               F
               to
               the
               Radius
               DF
               ,
               then
               a
               Rule
               laid
               to
               the
               Center
               D
               ,
               and
               every
               one
               of
               the
               180
               degrees
               ,
               shall
               divide
               the
               Tangent
               Lines
               AC
               ,
               and
               BC
               ,
               into
               180
               degrees
               ;
               and
               if
               you
               work
               right
               ,
               you
               will
               meet
               with
               all
               the
               former
               Points
               ,
               F
               ,
               G
               ,
               45
               ,
               I
               ,
               69-54
               ,
               60-45
               ,
               H
               ,
               and
               E
               ,
               in
               their
               true
               places
               ,
               as
               first
               drawn
               .
            
             
               Also
               ,
               Perpendiculers
               let
               fall
               from
               every
               degree
               in
               the
               Circle
               90
               EI
               to
               the
               Line
               DB
               ,
               shall
               divide
               the
               Line
               of
               Sines
               ,
               D
               90
               ,
               to
               the
               the
               greater
               Radius
               ;
               and
               the
               like
               Perpendiculers
               from
               the
               degrees
               in
               the
               lesser
               Circle
               ,
               to
               the
               Line
               DA
               ,
               shall
               divide
               the
               lesser
               Line
               of
               Sines
               ;
               Also
               ,
               the
               Extent
               from
               the
               Center
               D
               ,
               to
               the
               Tangent
               of
               any
               Ark
               or
               Angle
               in
               
               the
               Line
               AC
               ,
               counting
               from
               F
               ,
               shall
               be
               the
               Secant
               to
               that
               Ark
               or
               Angle
               ,
               to
               the
               lesser
               Radius
               ;
               and
               the
               measure
               from
               the
               Center
               D
               ,
               to
               the
               Tangent
               of
               any
               Ark
               or
               Angle
               in
               the
               Line
               CB
               (
               but
               counted
               from
               E
               )
               shall
               be
               the
               Secant
               to
               that
               Ark
               or
               Angle
               ,
               to
               the
               greater
               Radius
               .
            
             
               This
               little
               Instrument
               thus
               made
               ,
               and
               a
               Thred
               fastened
               at
               D
               ,
               will
               perform
               any
               Proposition
               by
               the
               Rules
               here
               inserted
               ,
               and
               is
               the
               very
               making
               of
               the
               
                 Trianguler
                 Quadrant
              
               ;
               or
               you
               may
               put
               these
               Lines
               on
               a
               Rule
               as
               a
               plain
               Scale
               ,
               and
               use
               them
               thus
               :
            
             
               
                 As
                 for
                 Example
                 ,
                 for
                 the
                 Azimuth
                 last
                 treated
                 on
                 .
              
            
             
               First
               draw
               a
               streight
               Line
               ,
               as
               AB
               ,
               representing
               the
               Line
               AB
               in
               the
               Trianguler-Quadrant
               ;
               then
               appoint
               in
               that
               Line
               any
               Point
               for
               a
               Center
               ,
               as
               C
               ;
               then
               for
               this
               Proposition
               of
               finding
               the
               Azimuth
               ,
               the
               Sines
               and
               Tangents
               being
               on
               a
               streight
               Scale
               ,
               work
               thus
               ;
            
             
               
                 First
                 ,
                 to
                 find
                 the
                 Suns
                 Altitude
                 ,
                 or
                 Depression
                 in
                 the
                 Vertical-Circle
                 .
              
            
             
               Take
               the
               Sine
               of
               the
               Latitude
               ,
               and
               lay
               it
               from
               C
               to
               51-30
               ;
               then
               take
               out
               the
               Sine
               
               of
               13-15
               ,
               between
               your
               Compasses
               ,
               and
               setting
               one
               Point
               in
               the
               Point
               51-30
               ,
               last
               made
               in
               the
               Line
               AB
               ,
               and
               strike
               the
               touch
               of
               the
               Arch
               at
               D
               ,
               and
               draw
               the
               Line
               CD
               ;
               also
               ,
               on
               the
               Line
               CB
               ,
               lay
               down
               from
               C
               the
               sine
               of
               90
               out
               of
               the
               Scale
               ,
               then
               the
               nearest
               distance
               from
               the
               Point
               for
               90
               in
               CB
               ,
               to
               the
               Line
               CD
               ,
               shall
               be
               the
               Sine
               of
               the
               Suns
               Altitude
               in
               the
               Vertical
               ,
               in
               Summer
               or
               Northern
               declination
               ,
               or
               his
               depression
               in
               Winter
               ,
               viz.
               17-01
               .
            
             
               Then
               ,
               as
               before
               ,
               on
               the
               Line
               of
               Sines
               ,
               find
               a
               sum
               for
               Winter
               ,
               or
               a
               difference
               in
               Summer
               ,
               between
               the
               Vertical
               and
               present
               Altitude
               ;
               Now
               supposing
               the
               Altitude
               15
               ,
               the
               sum
               is
               33-30
               ,
               or
               the
               difference
               is
               1-58
               ,
               which
               you
               must
               remember
               .
            
             
               Then
               take
               the
               Secant
               of
               15
               ,
               the
               Suns
               present
               Altitude
               from
               the
               Scale
               ,
               lay
               it
               from
               C
               to
               E
               ;
               then
               take
               out
               the
               Co-tangent
               of
               the
               Latitude
               between
               your
               Compasses
               ,
               set
               one
               Point
               in
               E
               ,
               and
               strike
               the
               touch
               of
               an
               Ark
               ,
               as
               at
               F
               ,
               and
               draw
               the
               Line
               CF
               ;
               then
               take
               the
               sine
               of
               33-30
               ,
               the
               sum
               ,
               if
               it
               be
               Winter
               ,
               or
               1-58
               ,
               if
               it
               be
               Summer
               ,
               between
               your
               Compasses
               ,
               carry
               one
               Point
               in
               the
               Line
               CB
               ,
               higher
               or
               lower
               ,
               till
               the
               other
               foot
               ,
               turned
               about
               ,
               will
               but
               just
               touch
               the
               Line
               CF
               ;
               then
               the
               measure
               from
               thence
               to
               the
               Point
               C
               ,
               shall
               be
               the
               Sine
               of
               the
               Azimuth
               
               required
               ,
               viz.
               in
               Winter
               43-50
               ;
               and
               in
               Summer
               92-30
               ,
               from
               the
               South
               ,
               because
               the
               present
               Altitude
               is
               less
               than
               the
               Vertical
               ,
               or
               East
               and
               West
               .
            
             
               But
               when
               the
               Co-tangent
               of
               the
               Latitude
               is
               too
               large
               for
               a
               Parallel
               entrance
               ,
               then
               prick
               off
               first
               the
               Tangent
               of
               the
               Latitude
               ,
               and
               take
               the
               Co-sine
               of
               the
               Suns
               Altitude
               to
               work
               in
               a
               Parallel
               way
               which
               will
               remedy
               the
               inconveniences
               ;
               Thus
               you
               see
               that
               by
               drawing
               three
               Lines
               only
               this
               work
               is
               done
               ;
               yet
               not
               so
               soon
               by
               far
               ,
               as
               by
               the
               Instrument
               with
               the
               Thred
               ,
               which
               represents
               those
               Lines
               more
               certainly
               and
               exactly
               ,
               after
               the
               same
               way
               of
               Operation
               .
            
             
               
                 To
                 find
                 the
                 Suns
                 Azimuth
                 in
                 Southern
                 Declinations
                 .
              
            
             
               As
               the
               Co-sine
               of
               the
               Latitude
               ,
               to
               the
               Sine
               of
               the
               Suns
               present
               Altitude
               ;
               
            
             
               So
               is
               the
               Sine
               of
               the
               Latitude
               to
               a
               4th
               sine
               ;
               which
               4th
               sine
               is
               to
               be
               added
               to
               the
               Suns
               Amplitude
               ,
               for
               that
               time
               ,
               on
               a
               Line
               of
               Natural
               sines
               ,
               and
               the
               sum
               observed
               ,
               as
               a
               5th
               .
            
             
               Then
               ,
            
             
               As
               the
               Co-sine
               of
               the
               present
               Altitude
               ,
               is
               to
               the
               sine
               of
               the
               sum
               last
               found
               ;
            
             
             
               So
               is
               the
               sine
               of
               90
               ,
               to
               the
               sine
               of
               the
               Suns
               Azimuth
               ,
               from
               East
               or
               West
               ,
               required
               .
            
             
               
                 For
                 the
                 Amplitude
                 ,
                 work
                 thus
              
               ;
            
             
               As
               Co-sine
               Lat.
               to
               S.
               Suns
               declination
               ;
            
             
               So
               is
               S.
               90
               ,
               to
               the
               sine
               of
               Amplitude
               .
            
          
           
             
               Use
               XXIX
               .
               Having
               the
               Latitude
               ,
               Suns
               Altitude
               ,
               and
               Vertical
               Altitude
               ,
               to
               find
               the
               Azimuth
               .
            
             
               
                 And
                 first
                 for
                 Northern-Declinations
              
               .
            
             
               
               First
               ,
               find
               the
               Vertical
               Altitude
               by
               the
               former
               Rule
               ,
               and
               find
               the
               difference
               between
               it
               and
               the
               present
               Altitude
               ,
               by
               the
               Line
               of
               Sines
               :
               then
               take
               this
               difference
               from
               the
               general
               Sines
               between
               your
               Compasses
               ,
               and
               setting
               one
               foot
               in
               the
               Co-sine
               of
               the
               Latitude
               ,
               lay
               the
               Thred
               to
               the
               ND
               ,
               then
               take
               the
               ND
               from
               the
               sine
               of
               the
               Latitude
               to
               the
               Thred
               ;
               having
               this
               distance
               ,
               set
               one
               one
               foot
               in
               the
               Co-sine
               of
               the
               Suns
               Altitude
               ,
               and
               lay
               the
               Thred
               to
               ND
               ,
               and
               on
               the
               degrees
               it
               shall
               shew
               the
               Suns
               true
               Azimuth
               at
               that
               Altitude
               and
               Declination
               required
               .
            
             
             
               Example
               .
            
             
               The
               Suns
               Declination
               being
               7
               ,
               the
               Virtical
               Altitude
               is
               8-57
               ;
               the
               Suns
               present
               Altitude
               being
               30
               ,
               the
               difference
               or
               residue
               in
               Sines
               will
               be
               20-13
               ,
               and
               the
               Suns
               Azimuth
               found
               thereby
               will
               be
               60-12′
               .
            
             
               
                 The
                 same
                 by
                 Artificial
                 Sines
                 and
                 Tangents
                 ,
                 in
                 Summer
                 .
              
            
             
               As
               Co-S.
               Lat.
               to
               S.
               of
               residue
               ;
            
             
               So
               is
               S.
               90
               ,
               to
               a
               4th
               sine
               .
            
             
               Then
               ,
            
             
               As
               Co-S.
               ☉
               Alt.
               to
               the
               4th
               sine
               ;
            
             
               So
               is
               S.
               90
               ,
               to
               S.
               of
               ☉
               Azimuth
               ,
               from
               East
               or
               West
               .
            
             
               
                 Secondly
                 ,
                 in
                 Southern-Declinations
                 ,
                 work
                 thus
              
               ;
            
             
               First
               ,
               find
               the
               Suns
               Amplitude
               for
               that
               Declination
               ,
               thus
               ;
               Take
               the
               —
               sine
               of
               the
               Declination
               ,
               make
               it
               a
               =
               in
               the
               C●-sine
               of
               the
               Latitude
               ;
               lay
               the
               Thred
               to
               ND
               ,
               and
               on
               the
               degrees
               it
               gives
               the
               Suns
               Am●litude
               for
               that
               Declination
               ,
               which
               you
               must
               remember
               .
            
             
               Then
               ,
            
             
               Take
               the
               —
               sine
               of
               the
               Suns
               present
               Altitude
               ,
               make
               it
               a
               =
               in
               the
               Co-sine
               of
               
               Latitude
               ,
               lay
               the
               Thred
               to
               the
               ND
               ,
               then
               take
               the
               ND
               from
               the
               sine
               of
               the
               Latitude
               to
               the
               Thred
               ,
               and
               as
               the
               Compasses
               so
               stand
               ,
               set
               one
               foot
               in
               the
               sine
               of
               the
               Suns
               Amplitude
               first
               found
               ,
               and
               turn
               the
               other
               foot
               onward
               toward
               90
               ;
               then
               take
               from
               thence
               to
               the
               Center
               .
               Thus
               have
               you
               added
               the
               Amplitude
               ,
               and
               last
               found
               distance
               together
               on
               Sines
               ,
               then
               this
               added
               Latteral-distance
               ,
               must
               be
               made
               a
               Parallel
               in
               the
               Suns
               Co-altitude
               ,
               and
               the
               Thred
               laid
               to
               the
               nearest
               distance
               in
               the
               degrees
               ,
               gives
               the
               Azimuth
               required
               .
            
             
               Example
               .
            
             
               At
               15
               degrees
               of
               Declination
               ,
               and
               10
               degrees
               of
               Altitude
               ,
               the
               Azimuth
               will
               be
               found
               to
               be
               49
               degrees
               46
               minuts
               from
               the
               South
               ,
               and
               the
               Amplitude
               24-30
               in
               51-32
               of
               Latitude
               .
            
             
               
                 The
                 same
                 ,
                 work
                 by
                 Artificial
                 Sines
                 and
                 Tangents
                 ,
                 in
                 Winter
                 .
              
            
             
               As
               co-S.
               of
               Lat.
               to
               S.
               of
               ☉
               present
               Alt.
               
            
             
               So
               is
               S.
               of
               Lat.
               to
               a
               4th
               ;
               which
               you
               must
               add
               to
               the
               Suns
               Amplitude
               on
               Natural
               Sines
               ,
               and
               keep
               it
               as
               a
               sum
               ;
            
             
               Then
               ,
            
             
               As
               co-S.
               of
               ☉
               Alt.
               to
               S.
               of
               the
               sum
               ;
            
             
               So
               is
               S.
               90
               ,
               to
               S.
               Azim
               .
               from
               East
               or
               West
               .
            
          
           
             
             
               Use
               XXX
               .
               Having
               the
               Latitude
               ,
               the
               Suns
               Declination
               ,
               his
               Meridian
               and
               present
               Altitude
               given
               ,
               to
               find
               the
               Hour
               .
            
             
               Make
               the
               —
               Secant
               of
               the
               Latitude
               ,
               
               a
               =
               in
               the
               Co-sine
               of
               the
               Suns
               declination
               ,
               laying
               the
               Thred
               to
               ND
               (
               and
               note
               the
               place
               )
               ;
               then
               take
               the
               —
               distance
               on
               the
               sines
               ,
               between
               the
               Suns
               Meridian
               and
               present
               Altitude
               ,
               and
               lay
               it
               from
               the
               Center
               toward
               90
               ;
               then
               the
               ND
               from
               that
               Point
               to
               the
               Thred
               (
               as
               before
               laid
               )
               shall
               be
               the
               versed
               Sine
               of
               the
               Hour
               ,
               measured
               on
               a
               Line
               of
               versed
               Sines
               ,
               equal
               in
               Radius
               to
               the
               Line
               of
               Secants
               first
               taken
               ,
               as
               the
               Sines
               above
               90
               are
               .
            
             
               Make
               the
               —
               Secant
               Latitude
               ,
               a
               =
               Sine
               of
               Co-declination
               ;
               
               then
               the
               —
               distance
               between
               the
               Suns
               Meridian
               and
               present
               Altitude
               ,
               laid
               on
               both
               Legs
               from
               the
               Center
               latterally
               ,
               and
               the
               =
               distance
               between
               ,
               measured
               on
               versed
               Sines
               ,
               equal
               to
               the
               Secants
               ,
               shall
               give
               the
               hour
               required
               ;
               as
               the
               great
               Line
               of
               Sines
               on
               the
               Sector
               are
               ,
               by
               turning
               the
               Compasses
               twice
               ,
               because
               the
               Line
               of
               Secants
               is
               half
               the
               Radius
               of
               those
               Sines
               ,
               as
               at
               first
               was
               hinted
               .
            
             
             
               Example
               .
            
             
               Latitude
               51-32
               ,
               Meridian
               Altitude
               50
               ,
               present
               Altitude
               40
               ,
               Declination
               North
               11-30
               ,
               the
               Hour
               will
               be
               found
               to
               be
               9-32
               in
               the
               forenoon
               ,
               or
               2
               h
               28′
               in
               the
               afternoon
               .
            
             
               Note
               ,
               That
               if
               the
               Suns
               Meridian
               Altitude
               be
               above
               90
               ,
               count
               the
               excess
               from
               90
               toward
               the
               Center
               ,
               and
               take
               from
               thence
               to
               the
               present
               Altitude
               .
            
          
           
             
               Use
               XXXI
               .
               To
               find
               the
               Suns
               Azimuth
               ,
               by
               having
               the
               Latitude
               ,
               Declination
               ,
               and
               Suns
               present
               Altitude
               .
            
             
               
               First
               ,
               make
               the
               —
               Secant
               of
               the
               Latitude
               ,
               a
               =
               Sine
               in
               the
               Suns
               Co-altitude
               ,
               and
               lay
               the
               Thred
               to
               ND
               ,
               and
               note
               the
               exact
               place
               where
               it
               is
               laid
               ;
               then
               find
               by
               Addition
               the
               sum
               of
               the
               complements
               of
               the
               Latitude
               ,
               and
               Suns
               Altitude
               ;
               and
               observe
               whether
               the
               sum
               of
               them
               ,
               be
               above
               or
               under
               90.
               
            
             
               For
               when
               under
               90
               ,
               count
               it
               from
               90
               toward
               the
               Center
               ,
               as
               is
               usual
               ,
               and
               take
               the
               distance
               from
               thence
               to
               the
               Center
               ,
               and
               substract
               this
               on
               the
               Line
               of
               Sines
               ,
               out
               of
               the
               declination
               ,
               or
               take
               the
               Sine
               of
               the
               declination
               ,
               
               out
               of
               this
               ,
               (
               and
               lay
               the
               residue
               from
               the
               Center
               )
               ;
               and
               note
               the
               place
               .
            
             
               
                 Or
                 shorter
                 thus
              
               ;
            
             
               If
               the
               sum
               of
               the
               complements
               of
               the
               Co-latitude
               and
               Co-altitude
               be
               under
               90
               ,
               then
               count
               the
               same
               on
               the
               Line
               of
               Sines
               from
               90
               ,
               toward
               the
               Center
               ,
               and
               take
               the
               distance
               from
               thence
               to
               the
               Sine
               of
               the
               Suns
               Declination
               ,
               and
               lay
               this
               from
               the
               Center
               ;
               for
               the
               =
               distance
               from
               thence
               to
               the
               Thred
               (
               as
               first
               laid
               )
               shall
               be
               the
               versed
               Sine
               of
               the
               Azimuth
               from
               the
               South
               required
               .
            
             
               Example
               .
            
             
               
                 
                   
                     Latitude
                  
                   
                     51
                  
                   
                     32
                  
                
                 
                   
                     Declination
                  
                   
                     15
                  
                   
                     00
                  
                
                 
                   
                     Altitude
                  
                   
                     50
                  
                   
                     00
                  
                
                 
                   
                     Co-altitude
                  
                   
                     40
                  
                   
                     0
                  
                
                 
                   
                     Co-latitude
                  
                   
                     38
                  
                   
                     28
                  
                
                 
                   
                     Sum
                  
                   
                     78
                  
                   
                     28
                  
                
                 
                   
                     Azimuth
                     is
                     ,
                  
                   
                     33
                  
                   
                     0
                  
                
              
            
             
               But
               if
               the
               sum
               of
               the
               complements
               of
               the
               Latitude
               and
               Suns
               Altitude
               be
               above
               90
               ,
               then
               count
               from
               90
               to
               the
               Center
               as
               90
               ,
               
               and
               reckon
               the
               excess
               above
               90
               ,
               from
               the
               Center
               toward
               90
               ,
               and
               take
               from
               thence
               to
               the
               Center
               ,
               and
               add
               this
               distance
               on
               the
               Sines
               to
               the
               Suns
               declination
               toward
               90
               ,
               and
               take
               from
               thence
               the
               nearest
               distance
               to
               the
               Thred
               ,
               and
               that
               shall
               be
               the
               versed
               Sine
               of
               the
               Suns
               Azimuth
               from
               noon
               .
            
             
               But
               when
               the
               complements
               are
               under
               90
               ,
               then
               the
               ND
               from
               the
               noted
               place
               to
               the
               Thred
               ,
               shall
               be
               the
               versed
               Sine
               of
               the
               Azimuth
               required
               .
            
             
               But
               in
               Winter
               ,
               when
               the
               sum
               of
               the
               complements
               are
               above
               90
               ,
               and
               are
               counted
               backwards
               ,
               from
               the
               Center
               ,
               towards
               90
               ;
               take
               the
               —
               distance
               from
               thence
               to
               the
               Sine
               of
               the
               Suns
               declination
               ,
               the
               lesser
               from
               the
               greater
               ,
               and
               set
               this
               distance
               ,
               or
               residue
               ,
               from
               the
               Center
               downwards
               ;
               then
               the
               nearest
               distance
               from
               thence
               to
               the
               Thred
               ,
               shall
               be
               the
               versed
               Sine
               of
               the
               Azimuth
               .
            
             
               But
               when
               the
               Latitude
               is
               less
               than
               the
               Suns
               Declination
               ,
               and
               the
               same
               way
               ;
               then
               take
               the
               —
               distance
               (
               on
               the
               Sines
               )
               from
               the
               sum
               of
               the
               Suns
               Altitude
               ,
               and
               Co-latitude
               ,
               found
               by
               Addition
               ,
               when
               under
               90
               ,
               and
               counted
               from
               the
               Center
               to
               the
               declination
               ,
               and
               lay
               that
               from
               the
               Center
               ,
               as
               before
               is
               shewed
               .
            
             
               But
               if
               the
               sum
               of
               the
               Suns
               Altitude
               ,
               and
               the
               complement
               of
               the
               Latitude
               ,
               be
               above
               
               90
               ;
               then
               ,
               having
               counted
               forwards
               from
               the
               Center
               to
               90
               ,
               count
               the
               excess
               from
               90
               toward
               the
               Center
               ,
               and
               take
               the
               —
               distance
               from
               thence
               ,
               to
               the
               Sine
               of
               the
               Suns
               declination
               ,
               and
               lay
               it
               from
               the
               Center
               ,
               as
               before
               ;
               then
               the
               ND
               from
               thence
               to
               the
               Thred
               ,
               shall
               give
               the
               versed
               Sine
               of
               the
               Suns
               Azimuth
               on
               the
               small
               Sines
               beyond
               the
               Center
               .
            
             
               The
               very
               same
               manner
               of
               Operation
               that
               serves
               for
               the
               General-Quadrant
               ,
               
               serves
               also
               for
               the
               Sector
               ,
               and
               this
               way
               being
               more
               troublesome
               than
               the
               rest
               ,
               I
               shall
               say
               no
               more
               to
               it
               ,
               but
               proceed
               to
               others
               .
            
          
           
             
               Use
               XXXII
               .
               The
               Suns
               Altitude
               ,
               the
               Latitude
               ,
               and
               Declination
               given
               ,
               to
               find
               the
               Hour
               .
            
             
               Add
               the
               Co-latitude
               ,
               Co-altitude
               ,
               and
               Suns
               distance
               from
               the
               Elevated
               Pole
               together
               ,
               for
               a
               sum
               ;
               and
               find
               the
               half
               sum
               ,
               
               and
               the
               difference
               between
               the
               half
               sum
               and
               the
               Co-altitude
               .
            
             
               
                 Then
                 say
              
               ;
            
             
               As
               Sine
               90
               ,
               to
               Co-sine
               Latitude
               ;
            
             
               So
               is
               the
               Sine
               of
               Suns
               distance
               from
               the
               Pole
               ,
               to
               a
               4th
               Sine
               .
            
             
             
               Again
               ,
            
             
               As
               the
               4th
               Sine
               ,
               to
               the
               Sine
               of
               the
               half
               -
               sum
               ;
            
             
               So
               is
               the
               Sine
               of
               the
               difference
               ,
               to
               the
               versed
               Sine
               of
               the
               Hour
               ,
               if
               you
               have
               them
               on
               the
               Rule
               ;
               if
               not
               ,
               to
               a
               7th
               Sine
               ,
               whose
               half-distance
               on
               the
               Sines
               towards
               90
               ,
               gives
               a
               Sine
               ,
               whose
               complement
               doubled
               ,
               and
               turned
               into
               time
               ,
               is
               the
               Hour
               from
               South
               required
               .
            
             
               
                 Example
                 ,
                 at
                 36
                 deg
                 .
                 42
                 min.
                 Altitude
                 ,
                 and
                 23
                 deg
                 .
                 31
                 min.
                 Declination
                 ,
                 Latitude
                 51-32
                 North.
                 
              
            
             
               53-18
               ,
               the
               Co-altitude
               ;
               38-28
               ,
               the
               Co-latitude
               ;
               and
               66-29
               ,
               added
               together
               ,
               makes
               158-15
               for
               a
               sum
               ;
               then
               the
               half
               -
               sum
               is
               79-07
               ,
               and
               the
               difference
               between
               79-07
               and
               53-18
               ,
               is
               25-49
               for
               a
               difference
               .
            
             
               Then
               ,
            
             
               The
               Extent
               from
               sine
               90
               ,
               to
               the
               Sine
               of
               38-28
               ,
               will
               reach
               the
               same
               way
               from
               the
               sine
               of
               66-29
               ,
               to
               the
               sine
               of
               34-47
               ,
               for
               a
               4th
               Sine
               .
            
             
               Again
               ,
            
             
               The
               Extent
               from
               Sine
               34-47
               ,
               to
               Sine
               79-7
               ,
               shall
               reach
               the
               same
               way
               from
               the
               
               Sine
               of
               25-49
               ,
               the
               difference
               ,
               to
               the
               Sine
               of
               48-34
               ,
               a
               7th
               Sine
               ,
               right
               against
               which
               ,
               on
               the
               versed
               Sines
               ,
               is
               60
               ,
               viz.
               4
               hours
               from
               noon
               .
            
             
               
                 Or
                 else
              
               ,
            
             
               The
               half-distance
               ,
               between
               Sine
               48-34
               ,
               and
               the
               Sine
               of
               90
               ,
               is
               the
               Sign
               of
               60
               degrees
               ,
               whose
               complement
               ,
               viz.
               30
               doubled
               is
               60
               degrees
               ,
               or
               4
               hours
               in
               time
               ,
               from
               noon
               .
            
          
           
             
               Use
               XXXIII
               .
               To
               find
               the
               Suns
               Azimuth
               ,
               having
               the
               same
               things
               given
               ,
               viz.
               Co-latitude
               ,
               Co-altitude
               ,
               and
               Suns
               distance
               from
               the
               Pole.
               
            
             
               Add
               ,
               as
               before
               ,
               
               the
               three
               Nunbers
               together
               ,
               and
               thereby
               find
               the
               sum
               ,
               and
               half
               -
               sum
               ,
               and
               the
               difference
               between
               the
               half
               -
               sum
               ,
               and
               the
               Suns
               distance
               from
               the
               Elevated
               Pole.
               
            
             
               
                 Then
                 say
              
               ,
            
             
               As
               the
               sine
               of
               90
               ,
               to
               the
               Co-sine
               of
               the
               Latitude
               ;
            
             
               So
               is
               the
               Co-sine
               of
               the
               Altitude
               ,
               to
               a
               4th
               sine
               .
            
             
             
               Again
               ,
            
             
               As
               the
               Sine
               of
               the
               4th
               ,
               to
               the
               Sine
               of
               the
               half
               -
               sum
               ;
            
             
               So
               is
               the
               Sine
               of
               the
               difference
               ,
               to
               the
               versed
               Sine
               of
               the
               Suns
               Azimuth
               ,
               from
               South
               ,
               (
               or
               to
               a
               7th
               sine
               ,
               whose
               half-distance
               ,
               toward
               90
               ,
               gives
               a
               sine
               ,
               whose
               complement
               doubled
               ,
               is
               the
               Azimuth
               from
               South
               )
               .
            
             
               
                 Example
                 ,
                 Latitude
                 51-32
                 ,
                 Altitude
                 41-53
                 ,
                 Declination
                 North
                 13.
                 
              
            
             
               The
               3
               Numbers
               ,
               viz.
               38-28
               ,
               49-7
               ,
               and
               77-0
               ,
               added
               together
               ,
               makes
               163-35
               ;
               whose
               half
               is
               81-47
               ½
               ,
               and
               the
               difference
               between
               the
               half
               -
               sum
               ,
               and
               the
               Suns
               distance
               from
               the
               Pole
               ,
               is
               4-47
               ½
               .
            
             
               Then
               ,
            
             
               As
               sine
               90
               ,
               to
               sine
               38-28
               ;
            
             
               So
               is
               sine
               48-7
               ,
               to
               sine
               27-36
               .
            
             
               Then
               ,
            
             
               As
               sine
               27-36
               ,
               to
               sine
               81-47
               ;
            
             
               So
               sine
               4-47
               ½
               ,
               to
               (
               V.S.
               of
               130
               ,
               the
               Azimuth
               from
               the
               North
               :
               )
               the
               sine
               of
               10-15
               ,
               a
               7th
               sine
               ,
               whose
               half-distance
               toward
               90
               ,
               is
               25
               ,
               whose
               complement
               65
               doubled
               ,
               is
               130
               ,
               the
               Azimuth
               from
               the
               North
               ,
               whose
               complement
               to
               180
               ,
               viz.
               50
               ,
               is
               the
               Azimuth
               from
               South
               .
            
             
             
               
                 Having
                 the
                 same
                 complements
                 ,
                 to
                 find
                 the
                 Hour
                 ,
                 and
                 Azimuth
                 ,
                 by
                 the
                 General-Quadrant
                 and
                 Sector
                 ;
                 and
                 first
                 for
                 the
                 Azimuth
                 .
              
            
             
               First
               ,
               
               of
               the
               complements
               of
               the
               Latitude
               ,
               and
               Suns
               present
               Altitude
               ,
               by
               substraction
               find
               the
               difference
               .
            
             
               Secondly
               ,
               Count
               this
               difference
               on
               the
               Line
               of
               Natural
               Sines
               from
               90
               ,
               toward
               the
               Center
               ,
               as
               the
               smaller
               figures
               are
               counted
               .
            
             
               Thirdly
               ,
               Take
               the
               distance
               on
               the
               Sines
               ,
               from
               thence
               to
               the
               —
               sine
               of
               the
               Suns
               declination
               .
               But
               note
               ,
               That
               when
               the
               Latitude
               and
               Declination
               differ
               ,
               viz.
               one
               North
               ,
               and
               the
               other
               South
               ,
               as
               it
               is
               with
               us
               in
               Winter
               ;
               you
               must
               count
               the
               Suns
               Declination
               beyond
               the
               Center
               ,
               and
               call
               it
               the
               Suns
               distance
               from
               the
               Elevated
               Pole
               ,
               and
               take
               from
               thence
               .
            
             
               Fourthly
               ,
               Make
               this
               —
               distance
               ,
               a
               =
               in
               the
               Co-sine
               of
               the
               Latitude
               ,
               laying
               the
               Thred
               to
               ND
               ,
               or
               keeping
               the
               Sector
               at
               that
               opening
               .
               Then
               ,
            
             
               Fiftly
               ,
               Take
               out
               the
               =
               sine
               of
               90
               ,
               And
            
             
               Sixtly
               ,
               Make
               it
               a
               =
               sine
               in
               the
               Suns
               Co-altitude
               ,
               setting
               the
               Sector
               ,
               or
               laying
               the
               Thred
               to
               the
               (
               nearest
               distance
               )
               ND
               .
            
             
               Seventhly
               ,
               Take
               out
               the
               =
               sine
               of
               90.
               
            
             
             
               And
               ,
            
             
               Eightly
               ,
               Measure
               it
               from
               the
               sine
               of
               90
               ,
               towards
               (
               and
               if
               need
               be
               beyond
               )
               the
               Center
               ,
               and
               it
               shall
               reach
               to
               the
               versed
               sine
               of
               the
               Suns
               Azimuth
               from
               North
               or
               South
               ,
               when
               you
               count
               from
               90
               ;
               or
               from
               East
               or
               West
               ,
               if
               you
               count
               from
               the
               Center
               ,
               on
               a
               Line
               of
               Sines
               ,
               or
               middle
               of
               the
               Line
               of
               versed
               Sines
               .
            
             
               Note
               ,
               That
               if
               the
               general
               Sines
               are
               too
               big
               ,
               you
               have
               a
               less
               adjoyning
               ,
               whereon
               to
               begin
               and
               end
               the
               Work
               ;
               as
               sometime
               the
               Hour-Scale
               ,
               and
               sometimes
               the
               Line
               of
               Right
               Ascentions
               .
            
             
               Example
               .
            
             
               In
               the
               Latitude
               of
               51-32
               ,
               the
               Suns
               Declination
               18-30
               ,
               the
               Suns
               Altitude
               48-12
               ,
               you
               shall
               find
               the
               Suns
               Azimuth
               to
               be
               130
               from
               the
               North
               ,
               or
               50
               from
               South
               .
            
             
               
                 Secondly
                 ,
                 for
                 the
                 Hour
                 ,
                 by
                 the
                 same
                 data
                 ,
                 or
                 things
                 given
                 .
              
            
             
               1.
               
               First
               of
               the
               complement
               of
               the
               Latitude
               ,
               and
               the
               Suns
               distance
               from
               the
               Elevated
               Pole
               ,
               find
               the
               difference
               by
               Substraction
               .
            
             
               2.
               
               Count
               it
               on
               the
               Line
               of
               Sines
               from
               90
               
               toward
               the
               Center
               ,
               (
               or
               beginning
               of
               the
               Sines
               )
               .
            
             
               3.
               
               Take
               the
               —
               distance
               from
               thence
               ,
               to
               to
               the
               sine
               of
               the
               Suns
               present
               Altitude
               .
            
             
               4.
               
               Make
               this
               —
               distance
               ,
               a
               =
               in
               the
               Co-sine
               of
               the
               Latitude
               ,
               setting
               the
               Sector
               ,
               or
               laying
               the
               Thred
               to
               the
               ND
               ,
               and
               there
               keep
               it
               .
            
             
               5.
               
               Then
               take
               out
               the
               =
               sine
               of
               90
               ;
            
             
               And
               ,
            
             
               6.
               
               Make
               that
               a
               =
               in
               the
               Co-sine
               of
               the
               Suns
               Declination
               ,
               laying
               the
               Thred
               to
               ND
               .
            
             
               7.
               
               Then
               take
               out
               the
               =
               sine
               of
               90
               again
               .
            
             
               And
               ,
            
             
               8.
               
               Measure
               it
               from
               90
               ,
               toward
               the
               Center
               ,
               and
               it
               shall
               shew
               the
               versed
               sine
               of
               the
               Hour
               from
               Mid-night
               ,
               or
               the
               contrary
               from
               noon
               ;
               or
               from
               6
               ,
               if
               you
               count
               from
               the
               Center
               of
               the
               Sines
               ,
               or
               the
               middle
               on
               versed
               Sines
               .
            
             
               Example
               .
            
             
               Latitude
               51-32
               ,
               Declination
               North
               20-14
               ,
               Altitude
               50-55
               ,
               you
               shall
               find
               the
               Hour
               to
               be
               150
               from
               North
               ,
               viz.
               10
               in
               the
               fore-noon
               ,
               or
               30
               degrees
               short
               of
               South
               .
            
          
           
             
             
               Use
               XXXIV
               .
               Having
               the
               Latitude
               ,
               Suns
               Altitude
               ,
               and
               distance
               from
               the
               Elevated
               Pole
               ,
               to
               find
               the
               Hour
               ,
               by
               the
               Line
               of
               versed
               Sines
               ,
               on
               the
               Sector
               .
            
             
               First
               ,
               By
               Addition
               ,
               find
               the
               sum
               of
               ,
               and
               by
               Substraction
               ,
               the
               difference
               between
               the
               complement
               of
               the
               Latitude
               ,
               and
               the
               Suns
               distance
               from
               the
               Elevated
               Pole.
               
            
             
               Secondly
               ,
               Count
               this
               sum
               and
               difference
               from
               the
               Center
               ,
               or
               the
               versed
               Sines
               on
               the
               Sector
               ,
               (
               or
               the
               beginning
               of
               the
               Azimuth-Line
               ,
               if
               you
               use
               that
               ,
               or
               any
               other
               ,
               which
               is
               not
               drawn
               from
               a
               Center
               )
               and
               with
               Compasses
               take
               the
               —
               distance
               between
               them
               .
            
             
               Thirdly
               ,
               Make
               this
               —
               distance
               ,
               a
               =
               versed
               Sine
               of
               180.
               
            
             
               Fourthly
               ,
               Take
               the
               —
               distance
               between
               the
               versed
               sine
               of
               the
               sum
               ,
               and
               the
               complement
               of
               the
               Suns
               Altitude
               ,
               and
               carry
               parallelly
               till
               it
               stay
               in
               like
               versed
               Sines
               ,
               which
               shall
               be
               the
               versed
               Sine
               of
               the
               Hour
               from
               the
               North
               Meridian
               ,
               or
               mid-night
               .
            
             
               Or
               ,
            
             
               If
               you
               take
               the
               —
               distance
               from
               the
               difference
               to
               the
               Co-altitude
               ,
               and
               carry
               that
               =
               till
               it
               stay
               in
               like
               sines
               ,
               it
               shall
               be
               the
               hour
               from
               noon
               ;
               counting
               the
               Center
               12
               
               at
               noon
               ,
               the
               middle
               at
               90
               ,
               the
               two
               sixes
               and
               180
               at
               the
               end
               ,
               for
               12
               at
               night
               .
            
          
           
             
               Use
               XXXV
               .
               Having
               the
               Latitude
               ,
               the
               Suns
               Altitude
               ,
               and
               Distance
               from
               the
               Elevated
               Pole
               ,
               to
               find
               his
               true
               Azimuth
               from
               South
               or
               North
               ,
               by
               Natural
               versed
               Sines
               .
            
             
               First
               ,
               Of
               the
               Co-altitude
               ,
               and
               Co-latitude
               ,
               find
               the
               sum
               and
               difference
               ,
               by
               Addition
               and
               Substraction
               .
            
             
               Secondly
               ,
               Count
               the
               sum
               and
               difference
               from
               the
               Center
               ,
               and
               take
               the
               —
               distance
               between
               them
               with
               Compasses
               on
               the
               versed
               Sines
               .
            
             
               Thirdly
               ,
               Make
               it
               a
               =
               versed
               sine
               of
               180
               ,
               and
               so
               keep
               the
               Sector
               .
            
             
               Fourthly
               ,
               Take
               the
               —
               distance
               ,
               between
               the
               sum
               ,
               and
               the
               Suns
               distance
               from
               the
               Pole
               ,
               (
               counting
               the
               Center
               the
               Elevated
               Pole
               ,
               and
               90
               the
               Equinoctial
               )
               and
               carry
               it
               =
               till
               it
               stay
               in
               like
               parts
               ,
               which
               shall
               be
               the
               Azimuth
               from
               South
               .
            
             
               Or
               ,
            
             
               If
               you
               take
               the
               —
               distance
               from
               the
               difference
               ,
               to
               the
               Suns
               distance
               from
               the
               Pole
               ,
               and
               carry
               it
               as
               before
               ,
               it
               shall
               stay
               at
               the
               versed
               sine
               of
               the
               Azimuth
               ,
               from
               the
               North
               part
               of
               the
               Horizon
               .
            
             
             
               These
               five
               general
               wayes
               of
               finding
               the
               Hour
               and
               Azimuth
               ,
               are
               not
               all
               needful
               to
               be
               learned
               by
               every
               one
               ,
               but
               to
               delight
               the
               ingenious
               ,
               and
               to
               hold
               forth
               the
               usefulness
               of
               the
               Instrument
               ,
               and
               to
               supply
               defects
               that
               at
               some
               times
               may
               happen
               by
               Excursions
               ,
               and
               as
               a
               four-fold
               Testimony
               ,
               to
               shew
               the
               harmony
               in
               several
               wayes
               of
               Operation
               ;
               the
               first
               particular
               way
               ,
               and
               this
               last
               by
               versed
               Sines
               ,
               being
               most
               easie
               and
               comprehensive
               of
               any
               other
               .
            
          
           
             
               Use
               XXXVI
               .
               To
               work
               the
               last
               without
               the
               Line
               of
               versed
               Sines
               .
            
             
               Note
               ,
               That
               if
               for
               want
               of
               room
               ,
               the
               versed
               Sines
               be
               set
               but
               on
               one
               Leg
               ,
               then
               it
               is
               to
               be
               laid
               at
               the
               nearest
               distance
               instead
               of
               like
               parts
               ,
               after
               the
               manner
               of
               using
               the
               Thred
               on
               the
               
                 General
                 Quadrant
              
               .
            
             
               Also
               ,
               If
               you
               have
               it
               not
               at
               all
               ,
               then
               the
               Azimuth-line
               for
               the
               particular
               Latitude
               ;
               and
               if
               that
               be
               too
               large
               ,
               the
               little
               Line
               of
               Sines
               beyond
               the
               Center
               ,
               will
               supply
               this
               defect
               very
               well
               thus
               ;
            
             
               First
               ,
               Turn
               the
               Radius
               ,
               or
               whole
               length
               of
               that
               Line
               of
               Sines
               ,
               two
               times
               from
               the
               Center
               downwards
               ,
               (
               which
               in
               Sea-Instruments
               ,
               will
               most
               conveniently
               stay
               at
               30
               on
               
               the
               large
               Line
               of
               Sines
               ,
               or
               general
               Scale
               ,
               as
               was
               hinted
               in
               the
               28th
               Use
               ,
               being
               just
               4
               times
               as
               much
               one
               as
               the
               other
               )
               .
               For
               a
               Point
               representing
               180
               of
               versed
               Sines
               ,
               to
               set
               the
               Compasses
               in
               ,
               when
               you
               lay
               the
               Thred
               to
               ND
               ,
               and
               to
               take
               any
               versed
               sine
               above
               90
               degrees
               ;
               this
               being
               premised
               ,
               the
               Operation
               is
               thus
               :
            
             
               Example
               .
            
             
               Lat.
               51-32
               ,
               ☉
               Dist.
               from
               Pole
               80
               ,
               ☉
               Alt.
               25
               ,
               to
               find
               the
               Hour
               ;
               
            
             
               The
               sum
               of
               Co-lat
               .
               38-28
               ,
               and
               80
               ,
               is
               118-28
               ;
            
             
               And
               the
               difference
               is
               41-32
               .
            
             
               Now
               in
               regard
               the
               sum
               is
               above
               90
               ,
               count
               the
               Center
               90
               ,
               10
               on
               the
               smaller
               Sines
               100
               ,
               and
               20
               on
               the
               same
               Sines
               ,
               110
               ,
               and
               28
               deg
               .
               28
               min
               ;
               118
               deg
               .
               28
               min.
               turn
               this
               distance
               the
               other
               way
               from
               the
               Center
               downwards
               ,
               and
               note
               that
               place
               ,
               for
               the
               Point
               ,
               representing
               the
               sum
               on
               the
               versed
               sines
               .
            
             
               Then
               ,
            
             
               The
               —
               Extent
               between
               this
               sum
               ,
               and
               the
               difference
               41-32
               ,
               as
               the
               smaller
               figures
               reckon
               it
               ,
               being
               taken
               between
               your
               Compasses
               ,
               set
               one
               Point
               in
               180
               ,
               the
               Point
               first
               found
               ,
               and
               lay
               the
               Thred
               to
               ND
               ,
               and
               there
               keep
               it
               (
               or
               observe
               where
               it
               cuts
               )
               ,
               then
               taking
               
               the
               —
               distance
               between
               the
               versed
               sine
               of
               the
               difference
               ,
               counted
               as
               the
               small
               figures
               are
               reckoned
               ,
               and
               the
               sine
               of
               the
               Suns
               Altitude
               25
               ,
               as
               the
               greater
               figures
               are
               reckoned
               from
               the
               Center
               toward
               the
               End
               ;
               and
               carrying
               this
               Extent
               parallelly
               along
               the
               greater
               Line
               of
               sines
               ,
               till
               the
               other
               Point
               will
               but
               just
               touch
               the
               Thred
               at
               ND
               ;
               Then
               ,
               I
               say
               ,
               the
               measure
               from
               that
               Point
               to
               the
               Center
               ,
               measured
               on
               the
               small
               sines
               ,
               as
               versed
               sines
               ,
               shall
               be
               the
               versed
               sine
               of
               the
               Hour
               required
               ,
               viz.
               62
               from
               South
               ,
               or
               7
               hours
               52
               minuts
               from
               mid-night
               .
            
             
               This
               Rule
               ,
               or
               Use
               ,
               is
               longer
               far
               in
               wording
               ,
               than
               the
               Operation
               need
               be
               in
               working
               ;
               for
               if
               you
               shall
               approve
               of
               this
               way
               ,
               the
               adding
               of
               two
               brass
               Center-pins
               will
               shew
               you
               the
               two
               Points
               most
               used
               very
               readily
               ,
               and
               the
               Thred
               is
               sooner
               laid
               ,
               than
               the
               Legs
               can
               be
               opened
               or
               shut
               ,
               and
               the
               Instrument
               keeps
               its
               Trianguler
               form
               as
               it
               is
               in
               ,
               during
               the
               time
               of
               Observation
               .
            
          
           
             
               Use
               XXXVII
               .
               Having
               the
               Latitude
               ,
               Suns
               Declination
               ,
               and
               Hour
               ,
               to
               find
               his
               Altitude
               .
            
             
               This
               Problem
               being
               not
               of
               such
               use
               as
               the
               contrary
               ,
               viz.
               having
               the
               Altitude
               ,
               to
               
               〈◊〉
               the
               hour
               ,
               it
               shall
               suffice
               to
               hint
               only
               
                 two
                 ●ayes
              
               ,
               the
               most
               convenient
               .
               And
               ,
            
             
               
                 First
                 by
                 the
                 Particular
                 Quadrant
              
               .
            
             
               Lay
               the
               Thred
               to
               the
               Day
               ,
               or
               Declination
               ,
               then
               the
               ND
               from
               the
               Hour
               to
               the
               Thred
               ,
               measured
               in
               the
               particular
               Scale
               of
               Altitudes
               ,
               shall
               shew
               the
               Suns
               Altitude
               required
               .
            
             
               
                 Secondly
                 ,
                 by
                 the
                 versed
                 Sines
                 .
              
            
             
               1.
               
               First
               ,
               of
               the
               Co-latitude
               ,
               and
               Suns
               distance
               from
               the
               Pole
               ,
               find
               the
               sum
               and
               difference
               .
            
             
               2.
               
               Take
               the
               —
               distance
               between
               them
               ,
               and
               make
               it
               a
               =
               versed
               sine
               of
               180
               ,
               by
               setting
               the
               Sector
               ,
               or
               laying
               the
               Thred
               to
               ND
               .
            
             
               3.
               
               Then
               take
               the
               =
               versed
               sine
               of
               the
               Hour
               ,
               and
               lay
               it
               latterally
               from
               the
               sum
               ,
               and
               it
               shall
               give
               the
               complement
               of
               the
               Altitude
               required
               .
            
             
               This
               work
               is
               the
               same
               ,
               both
               by
               the
               Sector
               ,
               or
               General
               Quadrant
               ,
               as
               is
               shewed
               in
               Use
               ●he
               36th
               ,
               and
               is
               nothing
               else
               but
               a
               backward
               working
               ;
               but
               the
               Altitude
               at
               any
               Azimuth
               ,
               is
               not
               so
               to
               be
               done
               .
            
             
             
               
                 To
                 do
                 the
                 same
                 by
                 the
                 Natural-Sines
                 .
              
            
             
               First
               ,
               having
               the
               Latitude
               ,
               and
               the
               Suns
               Declination
               ,
               find
               the
               Suns
               Altitude
               ,
               or
               Depression
               at
               6
               ;
               and
               note
               the
               Point
               ,
               either
               below
               ,
               or
               above
               ,
               or
               in
               the
               Center
               ,
               as
               is
               largely
               shewed
               in
               Use
               the
               26th
               ,
               where
               the
               Altitude
               is
               given
               ,
               to
               find
               the
               hour
               in
               any
               Latitude
               .
            
             
               Then
               ,
            
             
               Lay
               the
               Thred
               to
               the
               Hour
               ,
               counted
               in
               the
               degrees
               either
               from
               12
               ,
               or
               6
               ;
            
             
               Then
               ,
            
             
               Take
               the
               ND
               from
               the
               Co-sine
               of
               the
               Suns
               Declination
               ,
               and
               make
               it
               a
               =
               in
               the
               sine
               of
               90
               ,
               laying
               the
               Thred
               to
               the
               ND
               ;
               then
               the
               ND
               from
               the
               sine
               complement
               of
               the
               Latitude
               to
               the
               Thred
               ,
               shall
               reach
               from
               the
               noted
               Point
               ,
               for
               the
               Suns
               Altitude
               or
               Depression
               at
               6
               ,
               to
               the
               Suns
               Altitude
               required
               .
            
             
               Example
               .
            
             
               Latitude
               51-32
               ,
               Declination
               23-71
               ,
               a
               8
               or
               4
               ,
               viz.
               2
               hours
               from
               6
               Southwards
               the
               Altitude
               will
               be
               found
               to
               be
               36-42′
               .
            
             
             
               
                 1.
                 
                 For
                 the
                 Altitude
                 at
                 6
                 ,
                 at
                 any
                 time
                 of
                 the
                 year
                 ,
                 say
              
               ;
            
             
               As
               the
               sine
               of
               90
               ;
               
               to
               sine
               of
               the
               Latitude
               ;
            
             
               So
               is
               the
               sine
               of
               the
               Suns
               Declination
               ,
               to
               the
               sine
               of
               the
               Suns
               Altitude
               at
               6.
               
            
             
               
                 2.
                 
                 For
                 the
                 Suns
                 Altitude
                 ,
                 at
                 any
                 hour
                 or
                 quarter
                 ,
                 in
                 Aries
                 or
                 Libra
                 ,
                 (
                 the
                 Equinoctial
                 )
                 .
              
            
             
               As
               sine
               90
               ,
               to
               Co-sine
               Latitude
               ;
            
             
               So
               is
               the
               sine
               of
               the
               Suns
               distance
               from
               6
               in
               degrees
               ,
               to
               sine
               of
               the
               Suns
               Altitude
               .
            
             
               
                 3.
                 
                 For
                 the
                 Suns
                 Altitude
                 at
                 all
                 other
                 hours
                 ,
                 or
                 times
                 of
                 the
                 year
                 .
              
            
             
               As
               sine
               90
               ,
               to
               Co-tangent
               Latitude
               ;
            
             
               So
               is
               sine
               of
               the
               Suns
               distance
               from
               6
               ,
               to
               the
               Tangent
               of
               a
               4th
               Ark
               ,
               in
               the
               Tangents
               .
            
             
               Which
               4th
               Ark
               being
               taken
               from
               the
               Suns
               distance
               from
               the
               Elevated
               Pole
               ,
               then
               the
               residue
               is
               the
               5th
               Ark
               ;
               but
               for
               hours
               before
               and
               after
               6
               ,
               add
               the
               4th
               Ark
               ,
               and
               the
               Suns
               distance
               from
               the
               Pole
               together
               ,
               to
               make
               a
               5th
               Ark.
               
            
             
             
               
                 Then
                 say
              
               ,
            
             
               As
               the
               Co-sine
               of
               the
               4th
               Ark
               ,
               to
               the
               sine
               of
               the
               Latitude
               ;
            
             
               So
               is
               the
               Co-sine
               of
               the
               residue
               (
               or
               sum
               )
               being
               the
               5th
               Ark
               ,
               to
               the
               sine
               of
               the
               Suns
               Altitude
               at
               that
               hour
               .
            
          
           
             
               Use
               XXXVIII
               .
               The
               Latitude
               ,
               Suns
               Azimuth
               and
               Declination
               given
               ,
               to
               find
               the
               Altitude
               ,
               or
               height
               thereof
               .
            
             
               
                 First
                 ,
                 to
                 find
                 the
                 Suns
                 Altitude
                 at
                 all
                 Azimuths
                 in
                 the
                 Equinoctial
                 .
              
            
             
               
               As
               sine
               90
               ,
               to
               Co-tangent
               Latitude
               ;
            
             
               So
               is
               the
               Co-sine
               Azimuth
               from
               South
               ,
               to
               the
               Tangent
               of
               the
               Suns
               Altitude
               in
               Aries
               .
            
             
               Or
               ,
            
             
               As
               sine
               90
               ,
               to
               the
               Co-sine
               of
               the
               Azimuth
               from
               South
               ;
            
             
               So
               is
               Co-tangent
               Lat.
               to
               the
               Tangent
               of
               the
               Suns
               Altitude
               ,
               at
               that
               Azimuth
               in
               the
               Equinoctial
               ,
               which
               you
               must
               gather
               into
               a
               Table
               for
               every
               single
               degree
               .
            
             
             
               Then
               ,
            
             
               As
               the
               sine
               Lat.
               to
               the
               sine
               of
               the
               Suns
               Declination
               ;
            
             
               So
               is
               the
               Co-sine
               of
               the
               Suns
               Altitude
               in
               Equinoctial
               ,
               to
               the
               sine
               of
               a
               4th
               Ark.
               
            
             
               Then
               ,
            
             
               When
               the
               Latitude
               and
               Declination
               are
               alike
               ,
               as
               both
               North
               ,
               or
               South
               ;
               then
               add
               the
               4th
               Ark
               and
               the
               Altitude
               (
               in
               the
               Equator
               )
               together
               ,
               and
               the
               sum
               is
               the
               Altitude
               required
               .
            
             
               But
               in
               Winter-time
               ,
               when
               the
               Latitude
               and
               Declination
               is
               unlike
               ,
               take
               the
               4th
               Ark
               out
               of
               the
               reciprocal
               Altitude
               in
               the
               Equator
               ,
               and
               the
               residue
               is
               the
               Suns
               Altitude
               required
               .
            
             
               Also
               ,
               in
               all
               Azimuths
               from
               East
               and
               West
               Northwards
               ,
               in
               Summer-time
               also
               ,
               you
               must
               use
               Substraction
               also
               ,
               and
               not
               Addition
               ;
               as
               the
               Rule
               before-going
               suggests
               .
            
             
               
                 By
                 the
                 Particular
                 Quadrant
                 ,
                 work
                 thus
              
               ;
            
             
               Take
               the
               Sun
               or
               Stars
               Declination
               from
               the
               particular
               Scale
               ,
               and
               setting
               one
               Point
               in
               the
               Suns
               Azimuth
               ,
               on
               the
               Azimuth
               Line
               ,
               and
               with
               the
               other
               lay
               the
               Thred
               to
               the
               ND
               ,
               the
               right
               way
               ,
               and
               on
               the
               degrees
               the
               Thred
               cuts
               the
               Altitude
               required
               .
            
             
             
               
                 By
                 the
                 General
                 Quadrant
              
               .
            
             
               As
               the
               —
               Co-tangent
               Latitude
               ,
               taken
               from
               the
               Moving-leg
               ,
               or
               Loose-piece
               ,
               to
               =
               sine
               of
               90
               ,
               laying
               the
               Thred
               to
               ND
               ;
            
             
               So
               is
               the
               =
               Co-sine
               of
               the
               Suns
               Azimuth
               from
               South
               ,
               to
               the
               —
               Tangent
               of
               the
               Suns
               Altitude
               in
               the
               Equator
               ,
               at
               that
               reciprocal
               Azimuth
               .
            
             
               
                 Which
                 being
                 remembred
                 ,
                 or
                 gathered
                 into
                 a
                 Table
                 together
                 ,
                 then
                 say
              
               ;
            
             
               As
               the
               —
               Co-sine
               of
               the
               Suns
               Altitude
               in
               the
               Equator
               ,
               to
               the
               =
               sine
               of
               the
               Latitude
               ,
               laying
               the
               Thred
               to
               the
               ND
               ;
            
             
               So
               is
               the
               =
               sine
               of
               the
               Suns
               Declination
               ,
               to
               the
               —
               sine
               of
               the
               4th
               Ark.
               
            
             
               Which
               4th
               Ark
               is
               to
               be
               added
               ,
               or
               substracted
               ,
               as
               immediately
               before
               is
               directed
               ,
               and
               the
               sum
               or
               residue
               ,
               shall
               be
               the
               true
               Altitude
               required
               .
            
             
               Example
               .
            
             
               At
               60
               degrees
               of
               Azimuth
               from
               South
               ,
               the
               Equinoctial
               Altitude
               will
               be
               found
               to
               be
               21-40
               ,
               for
               London
               latitude
               of
               51-32
               ;
               and
               the
               4th
               Ark
               in
               ♋
               or
               ♑
               is
               28-16
               .
            
             
             
               Then
               ,
            
             
               21-40
               ,
               the
               Suns
               Altitude
               at
               60
               in
               ♈
               ,
               and
               28-16
               ,
               the
               reciprocal
               4th
               Ark
               in
               ♋
               added
               ,
               makes
               49-56
               ,
               the
               Suns
               Altitude
               at
               60
               degrees
               from
               the
               South
               in
               ♋
               .
            
             
               The
               same
               way
               of
               working
               serves
               for
               the
               Sector
               ,
               as
               is
               used
               for
               the
               
                 General
                 Quadrant
              
               ,
               
               only
               observing
               to
               set
               the
               Sector
               ,
               instead
               of
               laying
               the
               Thred
               to
               the
               nearest
               distance
               ,
               as
               the
               Ingenious
               will
               soon
               perceive
               .
            
          
           
             
               Use
               XXXIX
               .
               Having
               the
               Latitude
               ,
               Declination
               ,
               Azimuth
               ,
               and
               Altitude
               ,
               to
               find
               the
               Hour
               .
            
             
               As
               the
               —
               Co-sine
               of
               the
               Suns
               Altitude
               ,
               
               to
               =
               Co-sine
               of
               the
               Suns
               Declination
               ;
            
             
               So
               is
               the
               =
               sine
               of
               the
               Suns
               Azimuth
               ,
               to
               —
               sine
               of
               the
               Hour
               .
            
             
               
                 Or
                 else
                 thus
              
               ;
            
             
               First
               find
               the
               Altitude
               ,
               at
               that
               Azimuth
               ;
               and
               then
               at
               that
               Altitude
               ,
               and
               Declination
               ,
               
               the
               Hour
               .
            
             
               As
               —
               Co-sine
               of
               Declination
               ,
               to
               =
               sine
               of
               the
               Azimuth
               ;
            
             
               So
               is
               the
               —
               Co-sine
               Altitude
               ,
               to
               =
               sine
               of
               the
               Hour
               .
            
             
             
               
               As
               Co-sine
               Declination
               ,
               to
               the
               Sine
               of
               the
               Azimuth
               ;
            
             
               So
               is
               Co-sine
               Altitude
               ,
               to
               Sine
               of
               the
               Hour
               .
            
          
           
             
               Use
               XL.
               Having
               the
               Latitude
               ,
               Declination
               ,
               Hour
               ,
               and
               Altitude
               ,
               to
               find
               the
               Azimuth
               .
            
             
               
               As
               —
               Co-sine
               of
               Declination
               ,
               to
               =
               Co-sine
               of
               the
               Suns
               Altitude
               ;
            
             
               So
               is
               =
               sine
               of
               the
               Hour
               ,
               to
               —
               sine
               of
               the
               Azimuth
               .
            
             
               
               First
               ,
               find
               the
               Altitude
               at
               that
               Hour
               ,
               and
               then
               the
               Azimuth
               at
               that
               Altitude
               ,
               as
               before
               .
            
             
               
               As
               Co-sine
               of
               Altitude
               ,
               to
               sine
               of
               the
               Hour
               ;
            
             
               So
               is
               Co-sine
               Declination
               ,
               to
               sine
               of
               the
               Azimuth
               from
               South
               ,
               or
               North
               ,
               as
               the
               Hour
               is
               counted
               ;
               
                 that
                 is
                 to
                 say
              
               ,
               from
               South
               ,
               if
               the
               Hour
               is
               between
               6
               at
               morning
               ,
               and
               6
               at
               night
               ;
               and
               from
               the
               North
               if
               the
               contrary
               ;
               
                 that
                 is
                 to
                 say
              
               ,
               between
               6
               at
               night
               ,
               and
               6
               next
               morning
               ,
               or
               next
               to
               midnight
               .
            
          
           
             
             
               Use
               XLI
               .
               Having
               the
               Latitude
               ,
               and
               the
               Suns
               Declination
               ,
               to
               find
               the
               Suns
               Azimuth
               at
               6.
               
            
             
               As
               the
               sine
               of
               90
               ,
               
               to
               the
               Co-sine
               of
               the
               Latitude
               ;
            
             
               So
               is
               the
               Tangent
               of
               the
               Suns
               Declination
               ,
               to
               Co-tangent
               of
               the
               Suns
               Azimuth
               from
               the
               North
               ,
               at
               the
               hour
               of
               6.
               
            
             
               First
               find
               the
               Suns
               height
               at
               6
               ,
               and
               then
               the
               Suns
               Azimuth
               at
               that
               Altitude
               .
               
            
             
               Make
               the
               —
               Tangent
               of
               the
               Declination
               ,
               a
               =
               sine
               of
               90
               ,
               
               laying
               the
               Thred
               to
               ND
               ,
               then
               the
               =
               Co-sine
               Latitude
               shall
               be
               the
               —
               Co-tangent
               of
               the
               Suns
               Azimuth
               from
               the
               North
               at
               6.
               
            
          
           
             
               Use
               XLII
               .
               To
               find
               the
               Amplitude
               ,
               Azimuth
               ,
               Rising
               ,
               Setting
               ,
               and
               Southing
               of
               the
               fixed
               Stars
               ,
               having
               the
               Latitude
               ,
               Altitude
               ,
               and
               Declination
               ,
               or
               time
               of
               the
               year
               given
               .
            
             
               First
               for
               the
               Amplitude
               ,
               
               Take
               the
               Stars
               Declination
               ,
               out
               of
               the
               particular
               Scale
               of
               Altitudes
               ,
               and
               measure
               it
               from
               90
               in
               the
               
               Azimuth-line
               ;
               and
               count
               the
               same
               way
               ,
               and
               the
               other
               Point
               shall
               shew
               the
               Stars
               Amplitude
               required
               .
            
             
               Example
               .
            
             
               The
               Declination
               of
               the
               
                 Bulls
                 Eye
              
               ,
               being
               15-48
               ;
               if
               you
               take
               15-48
               from
               the
               particular
               Scale
               ,
               and
               lay
               it
               from
               90
               in
               the
               Azimuth-line
               ,
               it
               will
               reach
               to
               26
               degrees
               ,
               counting
               from
               90
               towards
               either
               end
               ,
               the
               same
               as
               for
               the
               Sun
               in
               Use
               16.
               
               But
               in
               other
               Latitudes
               ,
               work
               as
               you
               do
               for
               the
               Sun
               by
               the
               Rules
               in
               the
               16th
               Use
               abovesaid
               .
            
             
               
                 For
                 a
                 Stars
                 Azimuth
              
               .
            
             
               The
               work
               here
               is
               the
               same
               as
               for
               the
               
                 Sun
                 ,
                 thus
              
               ;
               Take
               the
               Stars
               Declination
               from
               the
               particular
               Scale
               of
               Altitudes
               ,
               or
               Sines
               ,
               between
               your
               Compasses
               ,
               and
               lay
               the
               Thred
               to
               the
               Stars
               Altitude
               ,
               counted
               from
               600
               toward
               the
               Loose-piece
               ;
               then
               carry
               the
               Compasses
               ,
               or
               the
               right-side
               of
               the
               Thred
               ,
               for
               Northern-stars
               ;
               and
               on
               the
               left-side
               for
               Southern-stars
               ,
               along
               the
               Azimuth-line
               ,
               till
               the
               other
               foot
               ,
               being
               turned
               about
               ,
               will
               but
               just
               touch
               the
               Thred
               ;
               then
               the
               fixed
               Point
               on
               the
               Azimuth-line
               shall
               shew
               the
               Stars
               Azimuth
               ,
               from
               the
               South
               ,
               required
               .
            
             
             
               Example
               .
            
             
               The
               
                 Bulls
                 Eye
              
               being
               30
               degrees
               high
               ,
               shall
               have
               77
               degrees
               and
               10
               minuts
               of
               Azimuth
               from
               the
               South
               ▪
            
             
               If
               you
               be
               in
               other
               Latitudes
               ,
               use
               the
               general
               wayes
               ,
               as
               for
               the
               Sun
               in
               all
               respects
               ,
               having
               the
               same
               Declination
               that
               the
               Star
               hath
               North
               or
               South
               .
            
             
               
                 To
                 find
                 the
                 Stars
                 Rising
                 ,
                 or
                 Setting
                 .
              
            
             
               Count
               the
               Stars
               Declination
               on
               the
               degrees
               ,
               as
               you
               count
               the
               Suns
               ,
               North
               ,
               or
               South
               ,
               and
               there
               lay
               the
               Thred
               ;
               and
               in
               the
               Line
               of
               Hours
               is
               the
               Stars
               Rising
               ,
               or
               Setting
               ,
               when
               the
               Stars
               Right
               Ascention
               and
               Declination
               are
               equal
               .
            
             
               
                 But
                 at
                 other
                 times
                 ,
                 you
                 must
                 reckon
                 thus
              
               ;
            
             
               First
               ,
               find
               the
               Suns
               Right
               Ascention
               ,
               by
               Use
               14
               ,
               and
               set
               down
               the
               complement
               thereof
               to
               12
               Hours
               ,
               and
               the
               Stars
               Right-Ascention
               ,
               and
               the
               hour
               of
               Rising
               the
               Thred
               cuts
               ,
               and
               add
               them
               into
               one
               sum
               ,
               and
               the
               sum
               ,
               if
               under
               12
               ,
               is
               the
               time
               of
               his
               Rising
               in
               common
               hours
               ;
               or
               if
               you
               add
               the
               hour
               of
               Setting
               that
               the
               Thred
               sheweth
               ,
               it
               shall
               give
               his
               setting
               .
            
             
             
               Example
               .
            
             
               If
               you
               lay
               the
               Thred
               to
               15-48
               ,
               the
               Declination
               of
               the
               
                 Bulls
                 Eye
              
               ,
               in
               the
               Hour-line
               it
               cuts
               4
               hours
               36
               min.
               for
               Rising
               ;
               or
               7-24
               ,
               for
               his
               Setting
               ;
               then
               if
               you
               work
               ,
               for
               April
               the
               23d
               ,
               the
               Suns
               Right
               Ascention
               ,
               then
               is
               2-44
               ,
               and
               the
               complement
               thereof
               to
               12
               ,
               is
               9-16
               ;
               and
               the
               Stars
               Right-Ascention
               is
               4
               hours
               and
               16
               minutes
               ;
               and
               the
               Hour
               cut
               ,
               is
               4-36
               for
               Rising
               ;
               and
               the
               three
               Numbers
               ,
               viz.
               9-16
               ,
               the
               complement
               of
               the
               Suns
               Right
               Ascention
               ,
               and
               4-16
               ,
               the
               Stars
               Right
               Ascention
               ,
               and
               4-36
               ,
               the
               Hour
               of
               Rising
               the
               Thred
               cuts
               ,
               being
               added
               ,
               makes
               18-8
               ;
               from
               which
               ,
               taking
               12
               ,
               rest
               6-8
               ,
               the
               time
               that
               the
               
                 Bulls
                 Eye
              
               Riseth
               on
               April
               23
               ;
               and
               if
               you
               add
               7-24
               ,
               the
               time
               of
               Setting
               that
               the
               thred
               cuts
               ,
               there
               comes
               forth
               8-56
               ,
               viz.
               one
               hour
               and
               32
               min.
               after
               the
               Sun.
               
            
             
               
                 To
                 find
                 the
                 time
                 of
                 a
                 Stars
                 coming
                 to
                 South
                 .
              
            
             
               Substract
               the
               Right
               Ascention
               of
               the
               Sun
               ,
               from
               the
               Right
               Ascention
               of
               the
               Star
               ,
               increased
               by
               24
               ,
               when
               you
               cannot
               do
               without
               ,
               and
               the
               remainder
               ,
               if
               less
               than
               12
               ,
               is
               the
               time
               between
               12
               at
               noon
               ,
               and
               12
               at
               
               night
               ;
               but
               if
               the
               remainder
               be
               more
               than
               12
               ,
               it
               is
               the
               time
               between
               mid-night
               ,
               and
               mid-day
               ,
               following
               .
            
             
               Example
               .
            
             
               The
               Lyons-Heart
               ,
               whose
               Right
               Ascention
               is
               9-50
               ,
               will
               come
               to
               the
               South
               on
               March
               10
               ,
               at
               9-48
               ,
               the
               Suns
               Right
               Ascention
               ,
               being
               then
               only
               2
               minuts
               .
            
             
               
                 By
                 the
                 Line
                 of
                 24
                 hours
                 (
                 say
                 ,
                 or
                 )
                 work
                 thus
              
               ;
            
             
               Extend
               the
               Compasses
               from
               the
               Suns
               Right
               Ascention
               ,
               to
               the
               Stars
               Right
               Ascention
               ;
               that
               distance
               laid
               the
               same
               way
               from
               12
               at
               the
               middle
               ,
               or
               at
               the
               beginning
               ,
               shall
               reach
               to
               the
               time
               of
               the
               Stars
               coming
               to
               South
               .
            
             
               
                 To
                 find
                 the
                 time
                 of
                 the
                 Stars
                 continuance
                 above
                 the
                 Horizon
                 .
              
            
             
               First
               ,
               find
               what
               the
               Suns
               semi-diurnal-Ark
               is
               ,
               having
               the
               same
               declination
               ,
               and
               that
               doubled
               ,
               is
               the
               whole
               time
               of
               continuance
               ;
               Or
               ,
               if
               you
               shall
               add
               and
               substract
               it
               to
               ,
               or
               from
               the
               time
               of
               the
               Stars
               coming
               to
               South
               ,
               you
               shall
               find
               the
               time
               of
               Setting
               or
               Rising
               .
            
             
               
                 Or
                 else
              
               ,
            
             
               By
               laying
               the
               Thred
               to
               the
               Stars
               Declination
               ,
               it
               sheweth
               the
               Ascentional
               difference
               
               in
               this
               Latitude
               ,
               which
               added
               in
               those
               Stars
               that
               have
               North
               declination
               ,
               or
               substracted
               in
               Southern
               to
               6
               hours
               ,
               gives
               the
               semi-diurnal
               Ark
               of
               the
               Star
               above
               the
               Horizon
               .
            
             
               Example
               .
            
             
               The
               Eye
               's
               Ascentional-difference
               ,
               is
               one
               hour
               and
               24
               minuts
               ;
               which
               added
               to
               6
               hours
               ,
               because
               of
               Northern
               declination
               ,
               makes
               7-24
               ,
               for
               the
               semi-diurnal-Ark
               ,
               or
               14●
               48′
               ,
               for
               the
               whole
               time
               of
               being
               above
               the
               Horizon
               .
            
             
               Note
               ,
               That
               to
               work
               this
               for
               other
               Latitudes
               ,
               the
               Suns
               Ascentional-difference
               is
               to
               be
               found
               for
               that
               Latitude
               you
               are
               in
               ,
               and
               the
               Operation
               is
               general
               for
               all
               places
               .
            
             
               
                 To
                 find
                 a
                 Meridian
                 Line
                 by
                 the
                 Sun.
                 
              
            
             
               On
               any
               flat
               Horizontal-Plain
               ,
               set
               up
               a
               streight
               Wyre
               in
               the
               Center
               of
               a
               Circle
               ;
               or
               hold
               up
               a
               Thred
               or
               Plummet
               ,
               till
               the
               shadow
               of
               the
               Thred
               cut
               the
               Center
               ,
               and
               any
               where
               in
               the
               Circumference
               ,
               which
               two
               Points
               you
               must
               note
               ;
               then
               immediately
               take
               the
               Suns
               Altitude
               ,
               and
               find
               the
               Suns
               Azimuth
               ,
               and
               count
               so
               many
               degrees
               in
               the
               Circle
               the
               right
               way
               ,
               as
               the
               Suns
               Azimuth
               comes
               to
               ,
               from
               the
               Points
               of
               the
               shadow
               marked
               in
               the
               Circumference
               ,
               and
               draw
               that
               Line
               for
               a
               true
               Meridian-line
               .
               This
               Work
               is
               best
               done
               before
               10
               in
               the
               
               morning
               ,
               and
               after
               two
               afternoon
               ;
               or
               in
               the
               night
               ,
               by
               two
               Plumb-lines
               ,
               set
               in
               a
               right-Line
               with
               the
               North-Star
               ,
               at
               a
               right
               scituation
               .
            
          
           
             
               Use
               XLIII
               .
               To
               find
               the
               Hour
               of
               the
               Night
               by
               the
               Fixed
               Stars
               .
            
             
               First
               ,
               find
               the
               Stars
               Altitude
               ,
               by
               looking
               along
               the
               Fixed
               or
               Moveable-leg
               ,
               to
               the
               middle
               of
               the
               Star
               ,
               letting
               the
               Thred
               ,
               with
               a
               weighty
               Plummet
               ,
               play
               evenly
               by
               the
               degrees
               ,
               between
               your
               Thumb
               and
               Fore-finger
               ,
               to
               the
               end
               you
               may
               command
               the
               Thred
               ,
               and
               know
               whether
               it
               playeth
               well
               or
               no
               by
               feeling
               .
            
             
               Then
               ,
            
             
               Take
               the
               Altitude
               found
               ,
               from
               the
               particular
               Scale
               of
               Sines
               ,
               and
               laying
               the
               Thred
               over
               the
               Stars
               declination
               ,
               which
               for
               readiness
               sake
               is
               marked
               with
               1
               ,
               2
               ,
               3
               ,
               4
               ,
               5
               ,
               6
               ,
               7
               ,
               8
               ,
               9
               ,
               10
               ,
               11
               ,
               12
               ,
               according
               to
               the
               Figures
               set
               to
               the
               12
               Names
               of
               the
               12
               Stars
               on
               the
               Rule
               ;
               and
               then
               carrying
               the
               Compasses
               as
               you
               do
               in
               finding
               the
               hour
               by
               the
               Sun
               ,
               you
               shall
               find
               how
               much
               the
               Star
               wants
               ,
               or
               is
               past
               the
               Meridian
               ,
               which
               is
               called
               the
               Stars-Hour
               ;
               And
               note
               ,
               That
               if
               the
               Star
               be
               past
               the
               South
               ,
               it
               is
               an
               aft●rnoon
               hour
               ;
               if
               not
               
               come
               to
               the
               South
               a
               morning
               hour
               ,
               which
               you
               must
               remember
               .
            
             
               Also
               ,
               knowing
               the
               Suns
               Right
               Ascention
               ,
               set
               one
               Point
               of
               the
               Compasses
               in
               the
               Suns
               Right
               Ascention
               ,
               (
               counted
               in
               the
               Line
               of
               twice
               12
               ,
               or
               24
               hours
               ,
               on
               the
               outward-leg
               of
               the
               fixed-piece
               ,
               next
               to
               the
               particular
               Scale
               of
               Sines
               )
               and
               open
               the
               other
               to
               the
               Stars
               Right
               Ascention
               ,
               noting
               which
               way
               you
               turn
               the
               Compasses
               ;
               for
               the
               same
               Extent
               ,
               applied
               the
               same
               way
               ,
               from
               the
               Stars
               hour
               last
               found
               ,
               shall
               shew
               the
               true
               hour
               of
               the
               night
               required
               .
            
             
               Example
               .
            
             
               Suppose
               on
               the
               10th
               of
               Ianuary
               ,
               I
               should
               observe
               the
               Altitude
               of
               the
               
                 Bulls
                 Eye
              
               to
               be
               20
               degrees
               ;
               if
               you
               take
               20
               degrees
               ,
               the
               Altitude
               ,
               from
               the
               particular
               Scale
               ,
               and
               lay
               the
               Thred
               on
               15-48
               ,
               the
               Stars
               declination
               Northward
               ,
               and
               measure
               from
               the
               Hour-scale
               the
               nearest
               distance
               to
               the
               Thred
               ,
               you
               shall
               find
               the
               Compass-point
               to
               stay
               at
               6-49
               on
               the
               East-side
               of
               the
               Meridian
               ;
               (
               suppose
               )
            
             
               Also
               ,
               The
               Suns
               Right
               Ascention
               ,
               the
               same
               day
               ,
               is
               8
               hours
               and
               12
               minuts
               .
            
             
               Then
               ,
            
             
               The
               Extent
               from
               8
               hours
               12
               minuts
               (
               on
               the
               Line
               of
               twice
               12
               hours
               )
               the
               Suns
               Right
               Ascention
               ,
               to
               4-16
               ,
               the
               Stars
               Right
               Ascention
               ,
               
               shall
               reach
               the
               same
               way
               from
               6-49
               ,
               the
               Stars
               hour
               ,
               to
               2-53
               ,
               the
               true
               hour
               .
            
          
           
             
               Use
               XLIV
               .
               To
               find
               the
               Hour
               of
               the
               Night
               by
               the
               Moon
               .
            
             
               First
               ,
               by
               an
               Almanack
               ,
               or
               Ephemerides
               ,
               find
               the
               Moons
               Age
               ,
               and
               true
               Place
               for
               the
               present
               time
               ;
               then
               ,
               by
               laying
               the
               Thred
               on
               the
               Moons
               place
               ,
               you
               may
               have
               her
               Right
               Ascention
               ,
               and
               also
               the
               Suns
               Right
               Ascention
               ;
               and
               by
               the
               Moons
               Altitude
               ,
               taken
               from
               the
               particular
               Scale
               ,
               and
               the
               Thred
               laid
               over
               the
               Moons
               place
               ,
               you
               find
               what
               the
               Moon
               wants
               ,
               or
               is
               past
               coming
               to
               South
               ,
               which
               is
               called
               the
               Moons
               hour
               .
            
             
               
                 Then
                 ,
                 by
                 the
                 Line
                 of
                 24
                 Hours
                 ,
                 say
              
               ;
            
             
               As
               the
               Suns
               Right
               Ascention
               ,
               is
               to
               the
               Moons
               Right
               Ascention
               ;
            
             
               So
               is
               the
               Moons
               hour
               last
               found
               ,
               to
               the
               true
               hour
               .
            
             
               Example
               .
            
             
               Suppose
               that
               on
               the
               8th
               of
               Ianuary
               ,
               about
               40
               min.
               after
               3
               ,
               there
               is
               a
               New
               Moon
               ;
               then
               note
               ,
               That
               the
               Suns
               true
               place
               ,
               is
               the
               Moons
               true
               place
               ;
               and
               consequently
               ,
               their
               Right
               Ascentions
               ;
               and
               the
               Moons
               Hour
               and
               Altitude
               is
               the
               same
               with
               the
               Suns
               .
            
             
             
               Therefore
               ,
            
             
               As
               8
               hours
               04
               min.
               the
               Suns
               Right
               Ascention
               ,
               is
               to
               8-04
               ,
               the
               Moons
               Right
               Ascention
               ;
            
             
               So
               is
               the
               Moons
               hour
               at
               any
               Altitude
               ,
               to
               the
               Suns
               true
               hour
               .
            
             
               Again
               ,
            
             
               Suppose
               that
               on
               the
               1st
               Quarter-day
               ,
               the
               Moon
               being
               gone
               90
               degrees
               from
               the
               Sun
               ,
               to
               find
               her
               place
               ;
            
             
               
                 Then
                 do
                 thus
              
               ;
            
             
               Set
               one
               Point
               in
               the
               Moons
               place
               the
               Change-day
               ,
               and
               open
               the
               other
               to
               the
               beginning
               or
               the
               end
               of
               the
               Line
               of
               24
               hours
               ,
            
             
               Then
               ,
            
             
               The
               same
               Extent
               applied
               the
               contrary
               way
               from
               6
               hours
               ,
               or
               7
               dayes
               and
               a
               half
               ,
               the
               Moons
               Age
               ,
               shall
               give
               28
               deg
               .
               58
               min.
               ♈
               ;
               to
               which
               you
               must
               add
               7
               degrees
               and
               30
               minuts
               (
               the
               Suns
               place
               )
               between
               ,
               and
               the
               sum
               shall
               be
               the
               Moons
               true
               place
               required
               ,
               viz.
               6-28
               degrees
               in
               ♉
               .
            
             
               Example
               .
            
             
               If
               the
               Moon
               Change
               on
               the
               8th
               day
               ,
               the
               First
               Quarter
               being
               7
               dayes
               and
               a
               half
               after
               ,
               will
               be
               on
               the
               15th
               day
               later
               at
               night
               ;
               then
               the
               difference
               between
               the
               Sun
               and
               Moons
               Right
               Ascention
               ,
               will
               be
               found
               to
               be
               near
               6
               hours
               ;
               for
               the
               Suns
               Right
               Ascention
               ,
               
               Ianuary
               15
               ,
               is
               8-32
               ;
               and
               the
               Moons
               Right
               Ascention
               ,
               the
               same
               day
               ,
               being
               about
               8
               degrees
               and
               a
               half
               in
               ♉
               ,
               is
               2
               hours
               and
               28
               minuts
               ;
               if
               you
               take
               the
               distance
               between
               them
               ,
               on
               the
               24
               hours
               ,
               it
               is
               near
               6
               hours
               ;
               which
               is
               the
               difference
               of
               time
               between
               the
               Moon
               and
               the
               Suns
               hour
               .
            
             
               Again
               ,
            
             
               For
               the
               Full
               Moon
               ;
               on
               the
               22
               day
               ,
               near
               4
               hours
               after
               noon
               ,
               the
               Moons
               Age
               being
               14
               dayes
               ¾
               ;
               if
               you
               add
               12
               hours
               ,
               or
               6
               signs
               ,
               to
               the
               Moons
               place
               a●
               the
               Change
               ,
               you
               shall
               find
               ♋
               29-0
               ;
               to
               which
               if
               you
               add
               14-45
               ,
               the
               dayes
               between
               the
               New
               and
               Full
               ,
               you
               shall
               find
               ♌
               13
               deg
               .
               45
               min.
               for
               the
               Moons
               place
               ;
               the
               Suns
               Right
               Ascention
               the
               22
               day
               is
               9
               hours
               ,
               and
               the
               Moons
               the
               same
               day
               at
               1
               afternoon
               ,
               is
               9
               hours
               also
               (
               or
               rather
               12
               difference
               )
               so
               that
               the
               Suns
               hour
               and
               the
               Moons
               is
               equal
               ;
               only
               one
               is
               North
               ,
               and
               the
               other
               South
               .
            
             
               Again
               ,
            
             
               For
               the
               Last
               Quarter
               22
               ¼
               dayes
               ,
               or
               18
               hours
               added
               ,
               and
               22
               degrees
               also
               together
               ,
               makes
               ♏
               22
               deg
               .
               11
               min.
               for
               the
               Moons
               place
               ,
               by
               help
               of
               which
               ,
               to
               find
               the
               Moons
               hour
               by
               her
               Altitude
               above
               the
               Horizon
               found
               by
               observation
               .
            
             
             
               Or
               ,
            
             
               Without
               regarding
               the
               Sun
               or
               Moons
               Right
               Ascention
               ,
               having
               her
               true
               Age
               ,
               and
               Hour
               ,
            
             
               
                 Say
                 thus
              
               ;
            
             
               As
               12
               on
               the
               Line
               of
               24
               hours
               ,
               is
               to
               the
               Moons
               Age
               in
               the
               Line
               of
               her
               Age
               ;
            
             
               So
               is
               the
               Moons
               hour
               ,
               to
               the
               true
               hour
               .
            
             
               For
               ,
            
             
               The
               Extent
               from
               12
               in
               the
               middle
               ,
               to
               the
               Moons
               Age
               under
               or
               over
               the
               middle
               ,
               shall
               reach
               the
               same
               way
               ,
               on
               the
               same
               Line
               ,
               from
               the
               Moons
               hour
               ,
               to
               the
               true
               hour
               .
            
             
               The
               like
               work
               serves
               to
               find
               the
               hour
               of
               the
               night
               by
               any
               Planets
               ,
               as
               
                 Saturn
                 ,
                 Mars
              
               ,
               or
               Iupiter
               ,
               which
               are
               seen
               to
               shine
               very
               brave
               and
               bright
               in
               Winter
               evenings
               ;
               and
               having
               learned
               their
               Place
               by
               their
               distance
               from
               the
               fixed
               Stars
               ,
               or
               by
               the
               Ephemerides
               ,
               then
               their
               Altitude
               and
               Place
               will
               find
               their
               hour
               from
               the
               Meridian
               ,
               and
               the
               comparing
               their
               Right
               Ascentions
               with
               the
               Suns
               ,
               gives
               the
               true
               hour
               ,
               as
               before
               ,
               in
               the
               Fixed-Stars
               .
            
          
           
             
             
               Use
               XLV
               .
               To
               find
               the
               Moons
               Place
               and
               Declination
               ,
               without
               the
               Ephemerides
               ,
               somewhat
               near
               .
            
             
               First
               ,
               observe
               when
               the
               Moon
               is
               in
               the
               Meridian
               ,
               and
               then
               find
               her
               Altitude
               ,
               and
               take
               the
               same
               from
               the
               particular
               Scale
               between
               your
               Compasses
               ;
               then
               set
               one
               Point
               in
               the
               hour
               12
               ,
               and
               lay
               the
               Thred
               to
               ND
               ,
               and
               on
               the
               degrees
               it
               shall
               shew
               the
               Moons
               declination
               ;
               and
               in
               the
               Line
               of
               the
               Suns
               Place
               ,
               the
               Moons
               present
               Place
               ,
               counting
               her
               Progress
               orderly
               from
               the
               last
               Change-day
               ,
               or
               New
               Moon
               ,
               when
               she
               was
               with
               the
               Sun.
               
            
             
               
                 Otherwise
                 thus
              
               ;
            
             
               Observe
               what
               Hour
               the
               Moon
               sheweth
               on
               any
               Sun-dial
               ,
               at
               the
               same
               instance
               by
               the
               Fixed
               Stars
               ,
               or
               other
               wayes
               ,
               find
               the
               true
               Hour
               ;
            
             
               Then
               ,
            
             
               The
               Extent
               from
               the
               Moons
               Hour
               ,
               to
               the
               the
               true
               Hour
               ,
               shall
               reach
               the
               same
               way
               from
               12
               ,
               to
               the
               Moons
               Age
               ,
               right
               against
               which
               is
               her
               coming
               to
               South
               ,
               at
               which
               time
               you
               may
               find
               her
               true
               Altitude
               ,
               and
               so
               come
               by
               her
               Declination
               .
            
             
             
               
                 Yet
                 again
                 ,
                 for
                 her
                 Age
                 and
                 Place
                 ,
                 according
                 to
                 Mr.
                 Street
                 ,
                 and
                 Mr.
                 Blundevil
                 .
              
            
             
               Add
               the
               Epact
               ,
               the
               Month
               ,
               and
               Day
               of
               Month
               in
               one
               sum
               ,
               counting
               the
               Months
               from
               March
               ,
               by
               calling
               March
               the
               first
               Month
               ,
               April
               the
               second
               ,
               &c.
               then
               that
               sum
               ,
               if
               under
               30
               ,
               is
               the
               Moons
               Age
               ;
               but
               if
               the
               sum
               be
               above
               30
               ,
               then
               substract
               30
               ,
               and
               the
               remainder
               is
               the
               Moons
               Age
               ,
               when
               the
               Month
               hath
               31
               dayes
               ;
               but
               if
               the
               Month
               hath
               but
               30
               ,
               or
               less
               than
               30
               dayes
               ,
               then
               substract
               but
               29
               ,
               and
               the
               remainder
               is
               the
               Moons
               Age.
               
            
             
               
                 Or
                 thus
              
               ;
            
             
               Add
               to
               the
               Epact
               for
               the
               present
               year
               ,
               and
               in
               Ianuary
               0
               ,
               in
               February
               2
               ,
               in
               March
               1
               ,
               in
               April
               2
               ,
               in
               May
               3
               ,
               in
               Iune
               4
               ,
               in
               Iuly
               5
               ,
               in
               August
               6
               ,
               in
               September
               8
               ,
               in
               October
               8
               ,
               in
               November
               10
               ,
               in
               December
               10
               ;
               and
               the
               sum
               ,
               if
               under
               30
               ,
               or
               the
               excess
               above
               30
               ,
               added
               to
               the
               day
               of
               the
               Month
               ,
               abating
               30
               ,
               if
               need
               be
               ,
               gives
               the
               Moons
               Age
               that
               day
               ;
               but
               substracted
               from
               30
               ,
               leaves
               the
               day
               of
               her
               Change
               in
               that
               Month
               ,
               or
               from
               the
               〈…〉
               ●onth
               .
            
             
             
               
                 Example
                 .
                 
                   July
                   10.
                   1668.
                
              
               
            
             
               The
               Epact
               that
               year
               is
               26
               ,
               and
               the
               Number
               for
               Iuly
               is
               5
               ,
               the
               Excess
               above
               30
               ,
               is
               1
               ;
               which
               added
               to
               any
               day
               of
               the
               Month
               as
               to
               10
               ,
               gives
               11
               ,
               for
               the
               Moons
               Age
               ,
               Iuly
               10.
               1668.
               
            
             
               
                 Then
                 for
                 the
                 Moons
                 Place
              
               .
            
             
               Multiply
               the
               Moons
               Age
               by
               4
               ,
               and
               the
               Product
               divided
               by
               10
               ,
               the
               Quotient
               giveth
               the
               signs
               ;
               and
               the
               remainder
               multiplied
               by
               3
               ,
               gives
               the
               degrees
               ,
               which
               you
               must
               add
               to
               the
               Suns
               place
               that
               day
               ,
               to
               find
               out
               the
               Moons
               place
               for
               that
               day
               of
               her
               Age.
               
            
             
               Example
               .
            
             
               On
               Iuly
               10.
               1668
               ,
               the
               Moons
               Age
               is
               11
               ,
               which
               multiplied
               by
               4
               ,
               makes
               44
               ;
               and
               44
               divided
               by
               10
               ,
               gives
               4
               signs
               in
               the
               Quotient
               ;
               and
               4
               ,
               the
               remainder
               ,
               multiplied
               by
               3
               ,
               makes
               12
               degrees
               more
               ;
               which
               added
               to
               Cancer
               ,
               29
               degrees
               ,
               the
               Suns
               place
               on
               the
               10th
               day
               of
               Iuly
               ,
               makes
               11
               degrees
               in
               Sagittarius
               ,
               the
               Moons
               place
               the
               same
               day
               ,
               
                 propè
                 verum
              
               .
            
             
             
               
                 Or
                 rather
                 by
                 the
                 Rule
                 thus
                 ,
                 on
                 the
                 Line
                 of
                 24
                 hours
                 by
                 particular
                 Scale
                 ,
                 having
                 the
                 Moons
                 place
                 ,
                 to
                 find
                 her
                 Age
                 by
                 the
                 Line
                 of
                 24
                 hours
                 .
              
            
             
               The
               Extent
               from
               the
               Suns
               true
               place
               ,
               to
               the
               Moons
               true
               place
               ,
               shall
               reach
               the
               same
               way
               ,
               from
               0
               day
               ,
               to
               the
               day
               of
               her
               Age.
               
            
             
               
                 Or
                 contrarily
                 ,
                 having
                 the
                 Moons
                 true
                 Age
                 ,
                 to
                 find
                 her
                 true
                 Place
                 .
              
            
             
               The
               Extent
               from
               0
               day
               old
               ,
               to
               the
               Moons
               true
               Age
               ,
               shall
               reach
               the
               same
               way
               from
               the
               Suns
               true
               Place
               to
               the
               Moons
               .
            
             
               
                 Or
                 ,
                 having
                 the
                 Moons
                 true
                 Place
                 at
                 the
                 New
                 Moon
                 ,
                 to
                 find
                 her
                 Place
                 any
                 day
                 of
                 her
                 Age
                 after
                 .
              
            
             
               The
               Extent
               from
               ♈
               ,
               to
               the
               Moons
               true
               Place
               at
               the
               Change
               ,
               shall
               reach
               the
               same
               way
               ,
               from
               the
               day
               of
               her
               true
               Age
               ,
               to
               her
               true
               Place
               ,
               adding
               as
               many
               degrees
               to
               the
               Number
               found
               ,
               as
               the
               Moon
               is
               dayes
               old
               .
            
             
               Then
               ,
            
             
               Having
               her
               Place
               ,
               and
               Age
               ,
               it
               is
               easie
               to
               find
               the
               Moons
               Hour
               ,
               and
               then
               her
               true
               Hour
               ;
               but
               I
               fear
               I
               spend
               herein
               too
               much
               time
               on
               an
               uncertain
               subject
               .
            
          
           
             
             
               Use
               XLVI
               .
               The
               Right
               Ascention
               and
               Declination
               of
               any
               Star
               ,
               with
               the
               Suns
               Right
               Ascention
               ,
               and
               the
               Hour
               of
               the
               Night
               given
               ,
               to
               find
               the
               Altitude
               and
               Azimuth
               of
               that
               Star
               ,
               and
               thereby
               to
               know
               the
               Star
               ,
               if
               you
               knew
               it
               not
               before
               .
            
             
               Set
               one
               Point
               of
               the
               Compasses
               in
               the
               Stars
               Right
               Ascention
               ,
               found
               in
               the
               Line
               of
               twice
               12
               hours
               ;
               and
               open
               the
               other
               to
               the
               Suns
               Right
               Ascention
               ,
               found
               in
               the
               same
               Line
               ;
               then
               this
               Extent
               shall
               reach
               ,
               in
               the
               same
               Line
               ,
               from
               the
               true
               hour
               of
               the
               Night
               ,
               to
               the
               Stars
               hour
               from
               the
               Meridian
               ;
               then
               laying
               the
               Thred
               to
               the
               Stars
               Declination
               ,
               the
               ND
               from
               the
               Stars
               hour
               ,
               in
               the
               Line
               of
               hours
               ,
               to
               the
               Thred
               ,
               measured
               on
               the
               particular
               Scale
               of
               Altitudes
               ,
               gives
               the
               Stars
               Altitude
               ;
               then
               by
               his
               Declination
               and
               Altitude
               ,
               you
               may
               soon
               find
               his
               Azimuth
               ,
               by
               Use
               27.
               
            
             
               And
               if
               the
               Instrument
               be
               neatly
               fixed
               to
               a
               Foot
               ,
               to
               set
               North
               and
               South
               ,
               and
               turn
               to
               any
               Azimuth
               and
               Altitude
               ,
               you
               may
               find
               any
               Star
               ,
               at
               any
               time
               convenient
               and
               visible
               .
            
          
           
             
             
               Use
               XLVII
               .
               The
               Altitude
               and
               Azimuth
               of
               any
               Star
               being
               given
               ,
               to
               find
               his
               Declination
               .
            
             
               Lay
               the
               Thred
               to
               the
               Altitude
               on
               the
               degrees
               ,
               counted
               from
               600
               toward
               the
               end
               ,
               then
               setting
               one
               Point
               on
               the
               Stars
               Azimuth
               ,
               counted
               in
               the
               Azimuth
               Line
               ,
               and
               take
               the
               ND
               from
               thence
               to
               the
               Thred
               ;
               which
               distance
               measured
               from
               the
               beginning
               of
               the
               particular
               Scale
               of
               Altitudes
               ,
               shall
               give
               the
               Declination
               .
            
             
               If
               the
               Compasses
               stand
               on
               the
               right-side
               of
               the
               Thred
               ,
               then
               the
               Declination
               is
               North
               ;
               if
               on
               the
               left
               ,
               it
               is
               South
               ;
               according
               as
               you
               work
               for
               the
               Suns
               Azimuth
               in
               a
               particular
               Latitude
               .
            
          
           
             
               Use
               XLVIII
               .
               The
               Altitude
               and
               Declination
               of
               any
               Star
               ,
               with
               the
               Right
               Ascention
               of
               the
               Sun
               ,
               and
               the
               true
               Hour
               of
               the
               Night
               given
               ,
               to
               find
               the
               Right
               Ascention
               of
               that
               Star.
               
            
             
               First
               ,
               by
               the
               43d
               Use
               ,
               find
               the
               Stars
               Hour
               ,
               viz.
               How
               many
               hours
               and
               minuts
               it
               wants
               of
               coming
               to
               ,
               or
               is
               past
               the
               Meridian
               ;
               then
               the
               Extent
               of
               the
               Compasses
               (
               on
               
               the
               Line
               of
               24
               hours
               on
               the
               Head-leg
               )
               from
               the
               Stars
               hour
               to
               the
               true
               hour
               ,
               shall
               reach
               the
               same
               way
               from
               the
               Suns
               Right
               Ascention
               ,
               to
               the
               Stars
               Right
               Ascention
               ,
               on
               the
               Line
               of
               twice
               12
               ,
               or
               24
               hours
               .
            
          
           
             
               Use
               XLIX
               .
               To
               find
               when
               any
               Fixed-Star
               cometh
               to
               South
               ,
               by
               the
               Line
               of
               twice
               12
               ,
               or
               24
               hours
               .
            
             
               In
               Use
               42
               ,
               Section
               4
               ,
               you
               have
               the
               way
               by
               Substraction
               ,
               with
               its
               Cautions
               :
               But
               by
               the
               Line
               of
               twice
               12
               ,
               or
               24
               hours
               ,
               
                 work
                 thus
              
               ;
            
             
               Count
               the
               Suns
               Right
               Ascention
               on
               that
               Line
               ,
               and
               take
               the
               distance
               from
               thence
               to
               the
               next
               12
               backward
               ,
               viz.
               that
               at
               ♈
               ,
               at
               the
               beginning
               of
               the
               Line
               ,
               when
               the
               Suns
               Right
               Ascention
               is
               under
               12
               hours
               ;
               or
               ,
               to
               the
               next
               12
               in
               the
               middle
               of
               the
               Rule
               at
               ♎
               ,
               when
               the
               Suns
               Right
               Ascention
               is
               above
               12
               hours
               ,
               (
               which
               is
               nothing
               but
               a
               rejecting
               12
               for
               more
               conveniency
               )
               .
            
             
               Then
               ,
            
             
               The
               same
               Extent
               laid
               the
               same
               way
               from
               the
               Stars
               Right
               Ascention
               ,
               shall
               reach
               to
               the
               Stars
               coming
               to
               South
               .
            
             
             
               Or
               ,
            
             
               The
               Extent
               from
               the
               Sun
               ,
               to
               the
               Stars
               Right
               Ascention
               ,
               shall
               reach
               the
               same
               way
               from
               12
               ,
               to
               the
               Stars
               coming
               to
               South
               .
            
             
               
                 Example
                 ,
                 for
                 the
                 Lyons-Heart
                 ,
                 August
                 20.
                 
              
            
             
               The
               Suns
               Right
               Ascention
               the
               20th
               of
               August
               ,
               is
               10
               hours
               36
               minuts
               ;
               the
               Right
               Ascention
               of
               the
               Lions-Heart
               ,
               is
               9
               hours
               and
               50
               min.
               
            
             
               Therefore
               ,
            
             
               The
               Extent
               from
               10
               hours
               35
               min.
               to
               the
               beginning
               ,
               shall
               reach
               the
               same
               way
               from
               9
               hours
               50
               min.
               (
               by
               borrowing
               12
               hours
               )
               because
               the
               Suns
               Right
               Ascention
               is
               more
               than
               the
               Stars
               )
               to
               11
               hours
               13
               min.
               of
               the
               next
               day
               ,
               viz.
               at
               a
               quarter
               past
               11
               ;
               or
               ,
               at
               11
               hours
               and
               13
               min.
               the
               same
               day
               ;
               where
               you
               may
               observe
               ,
               that
               the
               remainder
               being
               above
               12
               ,
               if
               you
               add
               24
               hours
               ,
               the
               time
               of
               Southing
               is
               between
               mid-night
               ,
               and
               mid-day
               next
               following
               .
            
          
           
             
             
               Use
               L.
               To
               find
               what
               two
               dayes
               in
               the
               year
               are
               of
               equal
               length
               ,
               and
               the
               Suns
               Rising
               and
               Setting
               .
            
             
               Lay
               the
               Thred
               on
               any
               one
               day
               in
               the
               upper
               Line
               of
               Months
               and
               Dayes
               ,
               and
               at
               the
               same
               time
               the
               Thred
               cuts
               in
               the
               lower-Line
               of
               Months
               the
               day
               that
               is
               answerable
               to
               it
               in
               length
               ,
               rising
               ,
               setting
               ,
               and
               declination
               ,
               and
               other
               requisites
               .
            
             
               Example
               .
            
             
               The
               1st
               of
               April
               ,
               and
               the
               21
               of
               August
               ,
               are
               dayes
               of
               equal
               length
               ;
               and
               the
               Suns
               Rising
               and
               Setting
               is
               the
               same
               on
               both
               those
               dayes
               ;
               only
               in
               the
               upper-Line
               ,
               the
               dayes
               are
               increasing
               in
               length
               ,
               and
               in
               the
               lower-Line
               they
               are
               decreasing
               .
            
          
           
             
               Use
               LI.
               To
               find
               how
               many
               degrees
               the
               Sun
               is
               under
               the
               Horizon
               at
               any
               Hour
               ,
               the
               Declination
               and
               Hour
               being
               given
               .
            
             
               Count
               the
               Suns
               Declination
               on
               the
               degrees
               ,
               the
               contrary
               way
               ,
               viz.
               for
               North
               Declination
               ,
               count
               from
               600
               toward
               the
               end
               ;
               and
               count
               for
               Southern
               Declination
               toward
               the
               Head
               ,
               and
               thereunto
               lay
               the
               Thred
               ;
               
               then
               take
               the
               nearest
               distance
               from
               the
               hour
               given
               to
               the
               Thred
               ;
               this
               distance
               measured
               in
               the
               particular
               Scale
               of
               Altitudes
               ,
               shall
               shew
               the
               Suns
               Depression
               under
               the
               Horizon
               at
               that
               hour
               .
            
             
               
                 Example
                 .
                 January
                 the
                 10th
                 at
                 8
                 at
                 Night
                 ,
                 how
                 many
                 degrees
                 is
                 the
                 Sun
                 under
                 the
                 Horizon
                 .
              
            
             
               On
               that
               Day
               and
               Hour
               ,
               the
               Suns
               Declination
               is
               about
               20
               degrees
               South
               ;
               then
               if
               I
               lay
               the
               Thred
               to
               20
               degrees
               of
               Declination
               North
               ,
               and
               take
               the
               nearest
               distance
               from
               8
               to
               the
               Thred
               ,
               that
               distance
               ,
               I
               say
               ,
               measured
               in
               the
               particular
               Scale
               ,
               gives
               34
               degrees
               and
               9
               min.
               for
               the
               Suns
               Depression
               under
               the
               Horizon
               of
               8
               afternoon
               .
            
             
               To
               do
               this
               in
               other
               Latitudes
               ,
               you
               are
               to
               find
               the
               Suns
               Altitude
               at
               8
               in
               Northern
               Declination
               ,
               by
               Use
               37.
               
            
          
        
         
           
           
             CHAP.
             XVII
             .
             The
             use
             of
             the
             Trianguler
             Quadrant
             ,
             in
             finding
             of
             Heights
             and
             Distances
             ,
             accessable
             or
             inaccessable
             .
          
           
             
               Use
               I.
               To
               find
               an
               Altitude
               at
               one
               Station
               .
            
             
               FIrst
               ,
               The
               
                 Trianguler
                 Quadrant
              
               being
               rectified
               ,
               and
               fixed
               to
               a
               Ball
               and
               Socket
               and
               three-legged-staff
               ,
               being
               necessary
               in
               these
               Operations
               to
               perform
               them
               exactly
               ,
               especially
               for
               Distances
               ;
               look
               up
               to
               the
               object
               as
               you
               would
               to
               a
               Star
               ;
               and
               observe
               what
               degree
               and
               minut
               the
               Thred
               cuts
               ,
               and
               set
               it
               down
               :
               Also
               ,
               observe
               the
               place
               where
               you
               stand
               at
               the
               time
               of
               Observation
               ,
               and
               the
               distance
               from
               your
               Eye
               to
               the
               ground
               ,
               and
               the
               place
               on
               the
               object
               that
               is
               level
               with
               your
               eye
               also
               ;
               as
               the
               playing
               of
               the
               Thred
               and
               Plummet
               will
               plainly
               shew
               .
            
             
             
               Also
               ,
               you
               must
               have
               the
               measure
               from
               the
               place
               where
               you
               stood
               observing
               ,
               to
               the
               Point
               exactly
               right
               under
               the
               object
               ,
               whose
               height
               you
               would
               have
               in
               Feet
               ,
               Yards
               ,
               Perch
               ,
               or
               what
               you
               please
               ,
               to
               Integers
               ,
               and
               Fractions
               in
               Decimals
               ,
               if
               it
               may
               be
               .
            
             
               
                 Also
                 Note
              
               ,
               That
               in
               all
               Right-Angle-Triangles
               ,
               one
               Acute
               Angle
               is
               alwayes
               the
               complement
               of
               the
               other
               ;
               so
               that
               observing
               or
               finding
               one
               by
               Observation
               ,
               by
               consequence
               you
               have
               the
               other
               ,
               by
               taking
               that
               from
               90.
               
            
             
               These
               things
               being
               premised
               ,
               the
               Operation
               followes
               ,
               by
               the
               Artificial
               Numbers
               ,
               Sines
               and
               Tangents
               ,
               and
               also
               by
               the
               Natural
               .
            
             
               
                 Note
                 also
                 by
                 the
                 way
              
               ,
               That
               in
               regard
               the
               complement
               of
               the
               Angle
               observed
               is
               frequently
               used
               ,
               if
               you
               count
               the
               degrees
               the
               contrary
               way
               ,
               that
               is
               to
               say
               from
               the
               Head
               ,
               you
               shall
               have
               the
               complement
               required
               ;
               as
               hath
               been
               oftentimes
               hinted
               before
               .
            
             
               Then
               ,
            
             
               As
               the
               sine
               of
               the
               Angle
               ,
               opposite
               to
               the
               measured
               side
               ,
               is
               to
               the
               measured
               side
               ,
               counted
               on
               the
               Numbers
               ;
            
             
               So
               is
               the
               sine
               of
               the
               Angle
               found
               ,
               to
               the
               Altitude
               or
               Height
               required
               on
               Numbers
               .
            
             
             
               
                 Example
                 at
                 one
                 station
              
               .
            
             
               Standing
               at
               C
               ,
               I
               look
               up
               to
               B
               the
               object
               ,
               whose
               Height
               is
               required
               ,
               and
               I
               find
               the
               Thred
               to
               fall
               on
               41
               degrees
               and
               45
               minuts
               ;
               but
               if
               you
               count
               from
               the
               Head
               ,
               it
               is
               48-15
               ,
               the
               complement
               thereof
               ,
               as
               in
               the
               Figure
               you
               see
               .
            
             
               Also
               ,
               the
               measure
               from
               C
               to
               A
               ,
               is
               found
               to
               be
               218
               foot
               .
            
             
               Then
               ,
            
             
               As
               the
               sine
               of
               48-15
               ,
               the
               Angle
               at
               B
               ,
               being
               the
               complement
               of
               the
               Angle
               at
               C
               ,
               is
               to
               218
               on
               the
               Line
               of
               Numbers
               ;
            
             
               So
               is
               the
               sine
               of
               the
               Angle
               at
               C
               ,
               41-45
               ,
               to
               195
               the
               Altitude
               of
               AB
               the
               height
               required
               ,
               found
               on
               the
               Line
               of
               Numbers
               .
            
             
             
             
               
                 A
                 second
                 Example
                 standing
                 at
                 D.
              
               
            
             
               
               But
               if
               I
               were
               standing
               at
               D
               ,
               129
               foot
               and
               a
               half
               from
               A
               ,
               and
               would
               find
               the
               height
               AB
               ,
               the
               complement
               of
               the
               Angle
               at
               D
               ,
               that
               is
               to
               say
               ,
               the
               Angle
               at
               B
               is
               33-30
               .
            
             
               
                 This
                 being
                 prepared
                 ,
                 then
                 say
              
               ;
            
             
               As
               the
               sine
               of
               33-30
               ,
               the
               Angle
               at
               B
               ,
               to
               the
               measured-side
               DA
               ,
               129
               ½
               counted
               on
               the
               Numbers
               ;
            
             
               So
               is
               56-30
               ,
               the
               sine
               of
               the
               Angle
               at
               D
               ,
               to
               195
               ,
               the
               Altitude
               required
               ,
               AB
               ,
               and
               5
               foot
               more
               ,
               the
               usual
               height
               of
               the
               eye
               from
               the
               Level
               to
               the
               ground
               ,
               makes
               200
               ,
               the
               whole
               height
               required
               .
            
             
               
                 To
                 work
                 this
                 by
                 the
                 Trianguler
                 Quadrant
                 ,
                 say
                 thus
              
               ;
            
             
               As
               —
               129
               ½
               ,
               taken
               from
               any
               Scale
               ,
               is
               to
               the
               =
               sine
               of
               33
               deg
               .
               30
               min.
               laying
               the
               Thred
               to
               the
               nearest
               distance
               ;
            
             
               So
               is
               the
               =
               sine
               of
               56-30
               ,
               the
               Angle
               at
               D
               ,
               to
               the
               —
               measure
               of
               195
               on
               the
               Scale
               you
               took
               129
               ½
               from
               .
            
             
             
               
                 The
                 like
                 manner
                 of
                 work
                 is
                 by
                 the
                 Sector
                 ,
                 as
                 thus
                 ,
                 in
                 the
                 foregoing
                 Example
                 .
              
            
             
               As
               218
               ,
               taken
               from
               the
               Line
               of
               Lines
               ,
               to
               the
               =
               sine
               of
               48
               deg
               .
               15
               min.
               
            
             
               So
               is
               the
               =
               sine
               of
               41-45
               ,
               to
               195
               on
               the
               Line
               of
               Lines
               latterally
               .
            
             
               
                 And
                 yet
                 further
              
               ,
            
             
               So
               is
               the
               =
               sine
               of
               90
               ,
               to
               291
               ,
               the
               Line
               CB.
               
            
          
           
             
               Use
               II.
               To
               find
               an
               Altitude
               at
               two
               stations
               .
            
             
               But
               if
               you
               cannot
               come
               to
               measure
               to
               the
               foot
               of
               the
               object
               ,
               
               then
               you
               must
               observe
               at
               two
               places
               .
            
             
               
                 As
                 thus
                 for
                 Example
              
               .
            
             
               First
               ,
               as
               before
               ,
               find
               the
               Angle
               at
               D
               ,
               or
               rather
               the
               complement
               thereof
               ,
               viz.
               33-30
               ;
               then
               go
               further
               backward
               in
               a
               right
               Line
               with
               the
               object
               and
               first
               station
               ,
               any
               competent
               Number
               of
               feet
               ,
               as
               suppose
               88
               ½
               to
               C
               ;
               there
               also
               observe
               the
               Altitude
               or
               Complement
               ,
               viz.
               the
               Angle
               ABC
               ,
               48-15
               .
            
             
               Then
               ,
            
             
               Find
               the
               difference
               between
               48-15
               ,
               and
               33-30
               ,
               and
               it
               is
               14-45
               .
            
             
             
               Then
               ,
            
             
               As
               the
               sine
               of
               the
               difference
               last
               found
               ,
               viz.
               the
               Angle
               CBD
               ,
               14-45
               ,
               to
               88
               ½
               ,
               on
               the
               Line
               of
               Numbers
               ;
            
             
               So
               is
               the
               sine
               of
               the
               Angle
               at
               C
               ,
               41-45
               ,
               to
               the
               measure
               of
               the
               side
               DB
               ,
               233
               ,
               on
               the
               Line
               of
               Numbers
               .
            
             
               
                 Again
                 ,
                 for
                 the
                 second
                 Operation
                 .
              
            
             
               As
               the
               sine
               of
               90
               ,
               the
               Angle
               at
               A
               ,
               to
               the
               Hypothenusa
               DB
               ,
               233
               ;
            
             
               So
               is
               the
               sine
               of
               56-30
               ,
               the
               Angle
               at
               D
               ,
               to
               195
               ,
               the
               Altitude
               required
               .
            
             
               
                 The
                 same
                 by
                 the
                 Trianguler
                 Quadrant
                 ,
                 or
                 Sector
                 .
              
            
             
               As
               —
               88
               ½
               ,
               the
               measured
               distance
               CD
               ,
               to
               the
               =
               sine
               of
               14-45
               ,
               CBD
               ;
            
             
               So
               is
               =
               sine
               of
               41-45
               ,
               to
               the
               —
               measure
               of
               233
               ,
               the
               opposite-side
               DB.
               
            
             
               Again
               ,
            
             
               As
               —
               233
               ,
               taken
               from
               the
               Line
               of
               Lines
               ,
               to
               =
               sine
               of
               90
               ;
            
             
               So
               is
               the
               =
               sine
               of
               56-30
               ,
               the
               Angle
               at
               D
               ,
               to
               —
               195
               ,
               on
               the
               Line
               of
               Lines
               ,
               the
               height
               required
               .
            
          
           
             
             
               Use
               III.
               Another
               way
               to
               save
               one
               Operation
               from
               IC
               .
            
             
               First
               ,
               
               observe
               the
               complement
               of
               the
               Angle
               at
               D
               ,
               and
               also
               the
               complement
               of
               the
               Angle
               at
               C
               ;
               then
               count
               these
               two
               complements
               on
               the
               Line
               of
               Natural
               Tangents
               ,
               on
               the
               loose-piece
               ,
               or
               moving-leg
               ,
               and
               take
               the
               distance
               between
               them
               ,
               and
               measure
               it
               on
               the
               same
               Tangent-line
               from
               the
               beginning
               thereof
               ,
               and
               note
               what
               Tangent
               the
               Compass-point
               stayeth
               at
               ,
               and
               count
               that
               for
               the
               first
               term
               ,
               in
               degrees
               and
               minuts
               .
            
             
               Then
               ,
            
             
               As
               the
               Tangent
               of
               this
               first
               term
               ,
               to
               the
               measured
               distance
               CD
               ,
               88
               ½
               ,
               on
               the
               Line
               of
               Numbers
               ;
            
             
               So
               is
               the
               Tangent
               of
               45
               ,
               to
               the
               Altitude
               required
               .
            
             
               
                 Thus
                 in
                 our
                 Example
              
               ;
            
             
               The
               distance
               measured
               is
               88
               ½
               ,
               the
               two
               complements
               33-30
               ,
               and
               48-15
               ;
               the
               distance
               between
               them
               makes
               the
               Tangent
               of
               24-34
               ,
               to
               be
               used
               as
               a
               first
               term
               .
            
             
               Then
               ,
            
             
               As
               the
               Tangent
               of
               ●4-34
               ,
               the
               first
               term
               last
               found
               ,
               to
               88
               ½
               on
               the
               Numbers
               ;
            
             
               So
               is
               the
               Tangent
               of
               45
               ,
               to
               195
               ,
               ferè
               ,
               on
               the
               Numbers
               ,
               the
               height
               required
               .
            
             
             
               
                 But
                 if
                 the
                 distance
                 from
                 D
                 or
                 C
                 ,
                 to
                 A
                 ,
                 the
                 foot
                 of
                 the
                 Object
                 ,
                 were
                 required
                 ,
                 then
                 the
                 manner
                 of
                 Calculation
                 runs
                 thus
              
               ;
            
             
               As
               the
               Tangent
               of
               the
               difference
               of
               the
               Co-tangents
               first
               found
               ,
               24-34
               ,
               is
               to
               the
               distance
               between
               D
               and
               C
               88
               ½
               ;
            
             
               So
               is
               the
               Co-tangent
               of
               the
               greater
               Ark
               48-15
               ,
               to
               the
               greater
               distance
               CA
               218.
               
            
             
               Or
               ,
            
             
               So
               is
               the
               Co-tangent
               of
               the
               lesser
               Ark
               33-30
               ,
               to
               the
               lesser
               distance
               DA
               ,
               129
               ½
               .
            
             
               
                 But
                 if
                 the
                 Hypothenusaes
                 be
                 required
                 ,
                 then
                 reason
                 thus
              
               ;
            
             
               As
               the
               Tangent
               of
               the
               difference
               first
               found
               is
               24-34
               ,
               to
               the
               distance
               between
               the
               stations
               D
               and
               C
               ,
               88
               ½
               ;
            
             
               So
               is
               the
               Secant
               of
               the
               Angle
               at
               B
               the
               greater
               ,
               viz.
               48-15
               ,
               counted
               beyond
               90
               ,
               to
               CB
               291.
               
            
             
               Or
               ,
            
             
               So
               is
               the
               Secant
               of
               33-30
               ,
               the
               lesser
               Angle
               at
               B
               ,
               to
               233
               the
               lesser
               distance
               DB
               ,
               the
               Hypothenusa
               required
               .
            
             
             
               
                 To
                 work
                 these
                 two
                 last
                 by
                 the
                 Trianguler
                 Quadrant
                 .
              
            
             
               First
               ,
               prick
               off
               the
               Tangents
               and
               Secants
               to
               be
               used
               parallelly
               ,
               from
               the
               loose-piece
               ,
               on
               the
               greater
               general
               Scale
               ;
               and
               note
               those
               Points
               for
               your
               present
               use
               .
            
             
               
                 As
                 thus
              
               ;
            
             
               The
               Tangent
               of
               24-34
               ,
               taken
               from
               the
               loose-piece
               from
               60
               ,
               counted
               as
               00
               will
               reach
               to
               the
               sine
               of
               10-40
               ,
               on
               the
               general
               Scale
               .
            
             
               Secondly
               ,
            
             
               The
               Secant
               of
               33-30
               ,
               being
               the
               measure
               from
               the
               Tangent
               of
               33-30
               ,
               on
               the
               loose-piece
               (
               counting
               from
               60
               )
               to
               the
               Center
               ,
               will
               reach
               on
               the
               general
               Scale
               from
               the
               Center
               ,
               to
               28-50
               .
            
             
               Thirdly
               ,
            
             
               The
               measure
               from
               the
               Tangent
               of
               48-15
               ,
               on
               the
               loose-piece
               ,
               to
               the
               Center
               ,
               being
               the
               Secant
               of
               48-15
               ,
               will
               reach
               from
               the
               Center
               to
               32-5
               ,
               on
               the
               general
               Scale
               .
            
             
               
                 This
                 being
                 prepared
                 ,
                 the
                 work
                 is
                 thus
              
               ;
            
             
               As
               —
               distance
               between
               the
               two
               stations
               ,
               to
               =
               Tangent
               ,
               of
               the
               first
               term
               ,
               at
               10-40
               ;
            
             
               So
               is
               =
               Tangent
               of
               45
               ,
               to
               the
               Altitude
               required
               .
            
             
             
               
                 Again
                 ,
                 for
                 the
                 Distance
              
               .
            
             
               As
               —
               distance
               between
               the
               two
               stations
               ,
               to
               the
               =
               Tangent
               of
               the
               first
               term
               ;
            
             
               So
               is
               the
               =
               Tangent
               of
               the
               greater
               Angles
               complement
               ,
               at
               26-36
               ,
               to
               the
               greatest
               distance
               CA
               218.
               
               Or
               ,
            
             
               So
               is
               the
               =
               Tangent
               of
               the
               lesser
               Angles
               complement
               ,
               at
               15-25
               ,
               to
               the
               lesser
               distance
               DA
               ,
               129
               ½
               ;
               Or
               ,
            
             
               So
               is
               the
               =
               Secant
               of
               the
               greater
               Angles
               complement
               ,
               at
               32-5●
               to
               the
               greater
               Hypothenusa
               CB
               ,
               291.
               
               Or
               ,
            
             
               So
               is
               the
               =
               Secant
               of
               the
               lesser
               Angles
               complement
               ,
               at
               28-50
               ,
               to
               the
               lesser
               Hypothenusa
               DB
               ,
               233.
               
            
          
           
             
               Use
               IV.
               Another
               way
               for
               Altitudes
               ,
               by
               the
               Line
               of
               Shadows
               ,
               either
               accessable
               or
               unaccessable
               ,
               by
               one
               or
               two
               stations
               .
            
             
               If
               this
               way
               be
               desired
               ,
               it
               may
               be
               put
               on
               this
               ,
               as
               well
               as
               any
               other
               Quadrants
               .
            
             
               
                 Then
                 the
                 use
                 is
                 thus
                 ;
                 
                   Figure
                   II.
                
                 
              
            
             
               Suppose
               that
               AB
               be
               the
               height
               of
               a
               Tree
               ,
               or
               other
               Object
               to
               be
               found
               ;
               go
               so
               far
               back
               
               from
               it
               ,
               as
               suppose
               to
               C
               ,
               till
               looking
               up
               by
               the
               two
               Pins
               put
               for
               sights
               ,
               the
               Thred
               falls
               on
               45
               degrees
               on
               the
               Quadrant
               ,
               or
               on
               1
               on
               the
               Line
               of
               Shadows
               ;
               then
               ,
               I
               say
               ,
               that
               the
               height
               AB
               ,
               is
               equal
               to
               the
               distance
               CA
               ,
               more
               by
               the
               height
               of
               your
               eye
               from
               the
               ground
               .
            
             
               But
               if
               you
               go
               further
               back
               still
               to
               D
               ,
               till
               the
               Thred
               falls
               on
               2
               on
               the
               Line
               of
               Shadows
               ;
               that
               is
               to
               say
               ,
               at
               26
               deg
               .
               34
               min.
               the
               Altitude
               will
               be
               but
               half
               the
               distance
               from
               A
               ;
               but
               if
               you
               remove
               to
               E
               ,
               the
               Thred
               falling
               on
               3
               on
               the
               Shadows
               ,
               the
               Altitude
               will
               be
               but
               one
               third
               part
               of
               the
               distance
               EA
               .
            
             
               From
               hence
               you
               may
               observe
               ,
               that
               observing
               at
               C
               ,
               and
               at
               D
               ,
               where
               the
               Thred
               falls
               on
               1
               ,
               and
               on
               2
               ,
               the
               distance
               between
               C
               and
               D
               ,
               is
               equal
               to
               the
               Altitude
               ;
               so
               likewise
               at
               D
               and
               at
               E
               ,
               and
               so
               by
               consequence
               at
               1
               ½
               and
               2
               ½
               and
               3
               ½
               ,
               or
               any
               other
               equal
               parts
               .
               This
               is
               an
               excellent
               easie
               way
               .
            
             
               The
               like
               will
               be
               if
               you
               observe
               at
               D
               and
               C
               ,
               looking
               up
               to
               F
               ,
               where
               the
               Altitude
               AF
               is
               twice
               the
               distance
               AC
               .
            
          
           
             
               Use
               V.
               Another
               way
               ,
               by
               the
               Line
               of
               Shadows
               ,
               at
               one
               station
               .
            
             
               Measure
               any
               distance
               ,
               as
               feet
               ,
               yards
               ,
               
               or
               
               the
               like
               from
               any
               object
               ;
               as
               suppose
               from
               A
               to
               D
               were
               200
               foot
               ,
               and
               looking
               up
               to
               B
               ,
               the
               Thred
               cuts
               the
               stroke
               by
               2
               on
               the
               Line
               of
               Shadows
               .
            
             
               
                 Then
                 by
                 the
                 Line
                 of
                 Numbers
                 ,
                 say
              
               ;
            
             
               As
               2
               ,
               the
               parts
               cut
               ,
               is
               to
               1
               ;
            
             
               So
               is
               200
               ,
               the
               distance
               measured
               ,
               to
               100
               the
               height
               .
            
             
               Or
               ,
            
             
               Suppose
               I
               measured
               any
               other
               uneven
               Number
               ,
               and
               the
               Thred
               fall
               between
               00
               on
               the
               Loose-piece
               ,
               and
               1
               on
               the
               Shadows
               ,
               commonly
               called
               contrary
               Shadow
               .
            
             
               
                 The
                 Rule
                 is
                 alwayes
                 thus
              
               ;
            
             
               As
               the
               parts
               cut
               by
               the
               Thred
               ,
               are
               to
               1
               ;
            
             
               So
               is
               the
               measured
               distance
               ,
               to
               the
               height
               required
               ,
               being
               less
               than
               the
               measured
               distance
               .
            
             
               
                 But
                 when
                 the
                 Thred
                 falls
                 between
                 1
                 and
                 90
                 at
                 the
                 Head
                 ,
                 called
                 right
                 Shadow
                 ;
                 then
                 the
                 Rule
                 goes
                 thus
              
               ;
            
             
               As
               1
               ,
               to
               the
               parts
               cut
               by
               the
               Thred
               ;
            
             
               So
               is
               the
               measured
               distance
               ,
               to
               the
               height
               ,
               being
               alwayes
               more
               than
               the
               measured
               distance
               from
               the
               foot
               of
               the
               object
               ,
               to
               the
               station
               .
            
          
           
             
             
               Use
               VI.
               Another
               way
               by
               the
               Line
               of
               Shadows
               ,
               and
               the
               Sun
               shining
               .
            
             
               When
               the
               Sun
               shineth
               ,
               find
               his
               Altitude
               ,
               and
               also
               as
               the
               Thred
               lies
               ,
               see
               what
               division
               on
               the
               Line
               of
               Shadows
               is
               cut
               by
               the
               Thred
               ,
               and
               then
               straightway
               measure
               the
               shadows
               length
               on
               the
               ground
               ;
               and
               if
               the
               Sun
               be
               under
               45
               degrees
               high
               ,
               the
               shadow
               is
               longer
               than
               the
               length
               of
               that
               object
               which
               causeth
               the
               shadow
               ;
               but
               if
               the
               Sun
               be
               above
               45
               degrees
               high
               ,
               then
               the
               object
               is
               longer
               than
               the
               shadow
               ;
               and
               the
               Operation
               is
               thus
               by
               the
               Line
               of
               Numbers
               ,
               only
               with
               a
               pair
               of
               Compasses
               .
            
             
               
                 The
                 Height
                 of
                 the
                 Sun
                 being
                 under
                 45
                 ,
                 say
              
               ;
            
             
               As
               the
               parts
               cut
               by
               the
               Thred
               on
               the
               Shadows
               ,
               is
               to
               1
               ;
            
             
               So
               is
               the
               Shadow
               measured
               ,
               to
               the
               height
               required
               .
            
             
               
                 The
                 Height
                 of
                 the
                 Sun
                 being
                 above
                 45
                 ,
                 say
              
               ;
            
             
               As
               1
               ,
               to
               the
               parts
               cut
               by
               the
               Thred
               on
               the
               Line
               of
               Shadows
               ;
            
             
               So
               is
               the
               measure
               of
               the
               shadow
               ,
               to
               the
               height
               in
               the
               same
               parts
               .
            
          
           
             
             
               Use
               VII
               .
               To
               find
               an
               inaccessable
               Altitude
               ,
               by
               the
               Quadrat
               and
               Shadows
               ,
               otherwise
               .
            
             
               Observe
               the
               Altitude
               at
               both
               stations
               ,
               and
               count
               the
               observed
               Altitudes
               at
               both
               stations
               ,
               on
               the
               Quadrat
               or
               Shadows
               ,
               according
               as
               it
               happens
               to
               be
               either
               above
               or
               under
               45
               degrees
               ;
               and
               take
               the
               lesser
               out
               of
               the
               greater
               ,
               noting
               the
               remainder
               for
               the
               first
               term
               ;
               and
               the
               Divisor
               to
               divide
               the
               distance
               between
               the
               stations
               ,
               increased
               with
               Cyphers
               ,
               if
               need
               be
               ;
               and
               the
               Quotient
               is
               the
               Answer
               required
               .
            
             
               
                 But
                 by
                 the
                 Line
                 of
                 Numbers
                 ,
                 work
                 thus
              
               ;
            
             
               The
               Extent
               from
               the
               difference
               to
               1
               ,
               shall
               reach
               the
               same
               way
               from
               the
               measured
               distance
               ,
               to
               the
               height
               required
               .
            
             
               
                 Example
                 .
                 
                   Figure
                   II.
                
              
               
            
             
               Let
               ABCDE
               represent
               the
               Object
               and
               three
               Stations
               ;
               let
               the
               Line
               AC
               represent
               the
               Altitude
               ;
               the
               Point
               B
               one
               station
               ,
               50
               foot
               from
               A
               ;
               D
               another
               station
               ,
               100
               foot
               from
               A
               ,
               or
               50
               from
               B
               ;
               and
               E
               another
               station
               ,
               73
               foot
               from
               D
               ,
               or
               173
               foot
               from
               A
               ;
               all
               which
               measures
               you
               need
               not
               know
               before
               ,
               but
               only
               BD
               ,
               and
               DE
               ;
               Also
               ,
               the
               
               Angle
               at
               B
               ,
               63-27
               ,
               and
               his
               complement
               ,
               counting
               the
               other
               way
               ,
               being
               the
               Angle
               at
               C
               26
               degrees
               33
               minuts
               ;
               the
               Angle
               at
               D
               45
               ,
               and
               his
               complement
               so
               also
               ;
               the
               Angle
               at
               E
               30
               ,
               and
               his
               complement
               60.
               
               Now
               mind
               the
               Operation
               by
               either
               of
               these
               ,
               First
               lay
               the
               Thred
               on
               26-33
               ,
               and
               in
               the
               Quadrat
               it
               cuts
               50
               ;
               lay
               the
               Thred
               on
               45
               ,
               and
               in
               the
               Shadows
               ,
               or
               Quadrat
               ,
               it
               cuts
               100
               ,
               or
               1
               ;
               or
               ,
               if
               you
               lay
               the
               Thred
               to
               60
               ,
               then
               in
               the
               shadows
               it
               cuts
               173.
               
            
             
               The
               difference
               between
               173
               ,
               and
               100
               ,
               is
               73.
               
            
             
               Then
               ,
            
             
               As
               73
               ,
               the
               difference
               in
               Tangents
               between
               the
               two
               observations
               ,
               is
               to
               the
               distance
               in
               feet
               73
               ;
            
             
               So
               is
               Radius
               100
               ,
               or
               the
               side
               of
               the
               Quadrat
               ,
               to
               100
               ,
               the
               hight
               required
               .
            
             
               
                 Again
                 ,
                 for
                 the
                 two
                 nearest
                 Observations
                 ,
                 whose
                 difference
                 of
                 Tangents
                 ,
                 is
                 50.
                 
              
            
             
               As
               50
               ,
               the
               difference
               in
               Tangents
               ,
               to
               50
               foot
               the
               measured
               distance
               ▪
            
             
               So
               is
               100
               ,
               the
               side
               of
               the
               Quadrat
               ,
               to
               100
               the
               height
               .
            
             
             
               
                 Again
                 ,
                 lastly
                 by
                 the
                 observations
                 at
                 B
                 &
                 E
                 ,
                 the
                 difference
                 of
                 Tangents
                 being
                 123.
                 
              
            
             
               As
               123
               ,
               the
               difference
               in
               Tangents
               ,
               to
               123
               ,
               the
               measured
               distance
               ;
            
             
               So
               is
               100
               ,
               the
               Radius
               or
               side
               of
               the
               Quadrat
               ,
               to
               100
               ,
               the
               height
               required
               .
            
             
               Or
               ,
            
             
               In
               the
               
                 first
                 Figure
              
               ,
               the
               Angles
               at
               the
               top
               being
               33-30
               ,
               and
               48-15
               ;
               and
               the
               measured
               distance
               88
               foot
               and
               a
               half
               ,
               the
               difference
               in
               Tangents
               will
               be
               45-8
               .
            
             
               Then
               ,
            
             
               As
               45-8
               ,
               to
               100
               ,
               the
               side
               of
               the
               Quadrat
               ;
            
             
               So
               is
               88
               ½
               ,
               the
               measured
               distance
               to
               194
               ,
               the
               Altitude
               required
               .
            
             
               This
               way
               is
               general
               for
               any
               Station
               ,
               though
               both
               of
               right
               shadow
               ,
               or
               both
               of
               contrary
               ,
               or
               mixt
               of
               right
               and
               contrary
               ,
               and
               done
               by
               the
               Line
               of
               Numbers
               ,
               or
               by
               Multiplication
               and
               Division
               .
            
             
               
                 Also
                 Note
              
               ,
               That
               you
               may
               find
               this
               difference
               in
               Tangents
               or
               Secants
               ,
               by
               the
               Natural
               Tangents
               ,
               or
               Natural
               Secants
               on
               the
               Sector
               ,
               and
               the
               Scale
               of
               equal
               parts
               belonging
               to
               them
               .
            
             
             
               Thus
               ;
            
             
               Take
               the
               distance
               between
               the
               compleplement
               of
               the
               two
               observations
               ,
               on
               the
               greater
               or
               lesser
               Line
               of
               Tangents
               ,
               (
               as
               is
               most
               convenient
               )
               and
               measure
               this
               distance
               in
               the
               Line
               of
               Lines
               ,
               or
               equal
               parts
               equal
               to
               that
               Radius
               ;
               and
               that
               shall
               be
               the
               difference
               in
               Tangents
               required
               .
               The
               like
               for
               the
               Secants
               .
            
             
               Also
               ,
               By
               the
               Artificial
               Numbers
               ,
               Sines
               ,
               and
               Tangents
               ,
               you
               may
               come
               by
               this
               differences
               in
               Tangents
               ,
               or
               Secants
               ,
               very
               well
               thus
               ;
            
             
               Just
               right
               against
               the
               Tangent
               of
               the
               Co-altitude
               ,
               counted
               on
               the
               Line
               of
               Tangents
               ,
               in
               the
               Line
               of
               Numbers
               ,
               is
               one
               Number
               ;
               and
               against
               the
               Tangent
               of
               the
               complement
               of
               the
               other
               Angle
               ,
               is
               the
               other
               Number
               ;
               only
               with
               this
               Caution
               ,
               That
               if
               the
               Tangent
               be
               above
               45
               ,
               then
               take
               the
               distance
               from
               45
               to
               the
               Tangent
               ,
               as
               it
               is
               counted
               backward
               ,
               with
               Compasses
               ,
               and
               set
               the
               same
               the
               increasing
               way
               from
               1
               ,
               on
               the
               Numbers
               ,
               to
               the
               other
               Number
               required
               ;
               then
               the
               lesser
               taken
               from
               the
               greater
               ,
               leaves
               the
               difference
               in
               Tangents
               that
               was
               required
               .
               In
               the
               same
               manner
               ,
               the
               Sines
               counted
               from
               90
               ,
               and
               laid
               the
               contrary
               way
               from
               1
               increasing
               ,
               will
               give
               the
               difference
               in
               Secants
               ,
               to
               measure
               the
               Ba●●
               ,
               and
               Hypothenusa
               by
               Numbers
               only
               .
            
          
           
             
             
               Use
               VIII
               .
               Another
               pretty
               way
               by
               Scale
               and
               Compass
               ,
               without
               Arithmetick
               ,
               from
               
                 T.
                 S.
              
               
            
             
             
               Then
               draw
               the
               Line
               ,
               that
               the
               Thred
               maketh
               ,
               on
               the
               Board
               ;
            
             
               Then
               measure
               from
               your
               standing
               ,
               to
               the
               foot
               of
               the
               Object
               ,
               and
               take
               the
               number
               of
               feet
               ,
               or
               yards
               from
               any
               Scale
               ,
               and
               lay
               it
               from
               the
               right
               Angle
               on
               the
               other
               Line
               ,
               and
               raise
               a
               Perpendiculer
               from
               thence
               to
               the
               Plumb-line
               made
               by
               the
               Thred
               ,
               and
               that
               shall
               be
               the
               Altitude
               required
               ,
               being
               measured
               on
               the
               same
               Scale
               .
            
             
               Example
               .
            
             
               Let
               ABGD
               represent
               the
               Boards
               end
               ,
               or
               Trencher
               ,
               and
               on
               that
               ,
               let
               AB
               be
               one
               streight
               Line
               ,
               and
               AG
               another
               Perpendiculer
               
               to
               it
               ;
               in
               the
               Point
               A
               ,
               knock
               in
               one
               Pin
               ;
               and
               in
               B
               ,
               or
               any
               where
               toward
               the
               end
               ,
               another
               ;
               On
               the
               Pin
               at
               A
               ,
               hang
               a
               Thred
               and
               Plummet
               ;
               and
               standing
               at
               I
               ,
               any
               convenient
               station
               ,
               look
               up
               by
               the
               two
               Pins
               at
               B
               and
               A
               ,
               till
               they
               bourn
               in
               a
               right
               Line
               with
               the
               Point
               H
               ,
               the
               object
               whose
               height
               is
               to
               be
               measured
               ;
               then
               the
               Plummet
               playing
               well
               and
               even
               ,
               make
               a
               Point
               just
               therein
               ,
               and
               draw
               the
               Line
               AD
               ,
               as
               the
               Thred
               shewed
               .
            
             
               Then
               ,
               having
               measured
               the
               distance
               from
               G
               the
               foot
               of
               the
               Object
               ,
               to
               I
               the
               station
               ,
               take
               it
               from
               any
               first
               Scale
               ,
               and
               lay
               it
               from
               A
               to
               G
               ;
               then
               on
               the
               Point
               G
               ,
               raise
               a
               Perpendiculer
               to
               AG
               ,
               till
               it
               intersect
               the
               Plumb-line
               AD
               ;
               then
               ,
               I
               say
               ,
               the
               distance
               CD
               ,
               measured
               on
               the
               same
               Scale
               you
               took
               AC
               from
               ,
               shall
               be
               equal
               to
               the
               Altitude
               GH
               ,
               which
               was
               required
               .
            
          
           
             
               Use
               IX
               .
               The
               same
               work
               at
               two
               stations
               .
            
             
               But
               if
               you
               cannot
               come
               to
               measure
               from
               I
               ,
               the
               first
               station
               to
               G
               ;
               
               then
               measure
               from
               I
               to
               K
               ;
               and
               having
               observed
               at
               I
               ,
               and
               drawn
               the
               Plumb-line
               AD
               ,
               take
               the
               measure
               between
               I
               and
               K
               ,
               the
               two
               stations
               ,
               from
               any
               fit
               Scale
               of
               equal
               parts
               ,
               and
               lay
               it
               on
               the
               
               Line
               AC
               ,
               from
               A
               to
               C
               ,
               viz.
               79
               parts
               and
               in
               the
               Point
               C
               ,
               knock
               another
               Pin
               ,
               and
               hang
               the
               Thred
               and
               Plummet
               thereon
               ,
               and
               observe
               carefully
               where
               this
               last
               Plumb-line
               doth
               cross
               the
               other
               ,
               as
               suppose
               at
               E
               ;
               then
               from
               E
               ,
               let
               fall
               a
               Perpendiculer
               to
               the
               Line
               AC
               ,
               which
               Line
               AC
               shall
               be
               the
               height
               GH
               required
               ;
               (
               or
               thus
               ,
               the
               nearest
               distance
               from
               E
               to
               AC
               is
               the
               height
               required
               )
               viz.
               120
               of
               the
               same
               parts
               that
               IK
               is
               79
               ;
               Note
               the
               Figure
               ,
               and
               behold
               that
               ACFE
               ,
               the
               small
               Figure
               on
               the
               Board
               is
               like
               and
               proportional
               to
               AA
               ,
               GH
               ,
               the
               greater
               Figure
               .
            
             
               
               Other
               wayes
               there
               be
               ,
               as
               by
               a
               Bowl
               of
               Water
               ,
               or
               a
               Glass
               ,
               or
               a
               Plash
               of
               Water
               ,
               or
               a
               Square
               ;
               but
               these
               set
               down
               ,
               are
               as
               convenient
               and
               ready
               as
               any
               whatsoever
               ;
               As
               in
               the
               next
               Figure
               you
               may
               see
               the
               way
               by
               the
               Glass
               ,
               and
               Square
               .
            
             
               
                 As
                 thus
              
               ;
            
             
               Let
               C
               represent
               a
               Glass
               ,
               a
               Bowl
               ,
               or
               Plash
               of
               Water
               ,
               wherein
               the
               Eye
               ,
               at
               A
               ,
               sees
               the
               picture
               or
               reflection
               of
               the
               Object
               E.
               
            
             
               
                 Then
                 ,
                 by
                 the
                 Line
                 of
                 Numbers
              
               ;
            
             
               As
               CB
               ,
               the
               measure
               from
               your
               foot
               to
               the
               Glass
               ,
               is
               to
               AB
               ,
               the
               height
               from
               your
               eye
               ,
               to
               the
               ground
               at
               your
               foot
               ;
            
             
             
               So
               is
               the
               measure
               from
               C
               to
               D
               ,
               to
               the
               height
               DE.
               
                 See
                 Figure
              
               VI.
               
            
             
               
                 Again
                 ,
                 to
                 find
                 a
                 distance
                 by
                 the
                 Square
                 ,
                 that
                 is
                 not
                 over-long
                 .
              
            
             
               Let
               C
               represent
               the
               upper-corner
               of
               a
               Square
               ,
               hung
               on
               a
               staff
               at
               F
               ;
               then
               the
               one
               part
               of
               the
               Square
               directed
               to
               E
               ,
               and
               the
               other
               to
               A.
               
            
             
               
                 The
                 Proportion
                 will
                 hold
                 ,
                 by
                 the
                 Line
                 of
                 Numbers
                 .
              
            
             
               As
               FA
               11-37
               ,
               to
               FC
               50
               ;
            
             
               So
               is
               FC
               50
               ,
               to
               FE
               220.
               
            
             
               
                 That
                 is
              
               ,
            
             
               So
               many
               times
               as
               you
               find
               AF
               in
               FC
               ;
            
             
               So
               many
               times
               is
               FC
               in
               FE
               ,
               and
               the
               like
               .
            
             
               Note
               ,
               That
               you
               must
               conceive
               AFE
               to
               be
               the
               Ground
               ,
               or
               Base-line
               in
               this
               Operation
               by
               the
               Square
               ;
               C
               being
               the
               top
               of
               an
               upright
               Staff
               ,
               5
               foot
               long
               ,
               called
               50
               for
               Fraction
               sake
               .
            
          
           
             
               Use
               X.
               To
               find
               a
               Distance
               not
               approachable
               by
               the
               Trianguler
               Quadrant
               .
            
             
             
             
               First
               ,
               I
               plant
               my
               
                 Trianguler
                 Quadrant
              
               ,
               set
               upon
               a
               three
               legged
               Staff
               and
               Ball
               socket
               ,
               right
               over
               the
               place
               A
               ;
               and
               then
               bring
               the
               Index
               with
               two
               sights
               in
               it
               ,
               laid
               or
               fastened
               to
               the
               Center
               of
               the
               
                 Trianguler
                 Quadrant
              
               ,
               right
               over
               the
               Lines
               of
               Sines
               ,
               and
               Lines
               cutting
               90
               at
               the
               Head
               ;
               the
               Index
               and
               sights
               so
               placed
               ,
               hold
               it
               there
               ,
               and
               bring
               it
               and
               the
               Instrument
               together
               ,
               till
               you
               see
               the
               mark
               at
               C
               ,
               through
               the
               two
               sights
               ,
               by
               help
               of
               the
               Ball-socket
               ,
               and
               then
               there
               keep
               it
               ;
               then
               remove
               the
               Index
               only
               to
               0-60
               on
               the
               loose-piece
               ,
               which
               makes
               a
               right
               Angle
               ;
               and
               set
               up
               a
               mark
               in
               that
               Line
               ,
               at
               any
               convenient
               distance
               ;
               as
               suppose
               at
               B
               ,
               102
               foot
               from
               A
               ;
               then
               remove
               the
               Instrument
               to
               B
               ,
               and
               laying
               the
               Index
               on
               the
               Center
               ,
               and
               0-60
               on
               the
               loose-piece
               ,
               direct
               the
               sights
               to
               A
               ,
               the
               first
               station
               ,
               by
               help
               of
               a
               mark
               left
               there
               on
               purpose
               ;
               Then
               remove
               the
               sights
               till
               you
               see
               the
               mark
               at
               C
               ,
               and
               note
               exactly
               on
               what
               degree
               the
               Index
               falleth
               ,
               as
               here
               on
               60
               ,
               counting
               from
               060
               on
               the
               loose-piece
               ;
               or
               on
               30
               ,
               counting
               from
               the
               Head
               ,
               which
               is
               the
               Angles
               at
               B
               ,
               and
               at
               C.
               
            
             
             
               
                 Then
                 by
                 the
                 Artificial
                 Numbers
                 ,
                 Sines
                 and
                 Tangents
                 on
                 the
                 edge
                 ,
                 say
              
               ;
            
             
               As
               the
               sine
               of
               30
               ,
               the
               Angle
               at
               C
               ,
               to
               102
               ,
               the
               measured
               distance
               counted
               on
               the
               Numbers
               ;
            
             
               So
               is
               the
               sine
               of
               60
               ,
               the
               Angle
               at
               B
               ,
               to
               117
               ,
               on
               the
               Numbers
               ,
               the
               distance
               required
               .
            
             
               So
               also
               is
               90
               ,
               the
               Angle
               at
               A
               ,
               to
               206
               ,
               the
               distance
               from
               B
               to
               C.
               
            
             
               
                 Or
                 ,
                 by
                 the
                 Lines
                 and
                 Sines
                 on
                 the
                 Quadrant-side
                 ,
                 as
                 it
                 lies
                 ,
                 thus
              
               ;
            
             
               As
               the
               —
               measure
               of
               102
               ,
               taken
               from
               any
               Scale
               ,
               as
               the
               Line
               of
               Lines
               doubling
               ,
               to
               the
               =
               sine
               of
               30
               ,
               laying
               the
               Index
               ,
               or
               a
               Thred
               ,
               to
               the
               nearest
               distance
               ;
            
             
               So
               is
               the
               =
               sine
               of
               60
               ,
               to
               117
               ,
               measured
               latterally
               on
               the
               same
               Line
               of
               Lines
               .
            
             
               And
               ,
            
             
               So
               is
               the
               =
               sine
               of
               90
               ,
               to
               206
               ,
               the
               distance
               from
               B
               to
               C.
               
            
             
               
                 So
                 also
              
               ,
               If
               you
               observe
               at
               B
               ,
               and
               at
               D
               only
               ,
               you
               must
               be
               sure
               to
               set
               your
               Instrument
               at
               one
               station
               ,
               at
               the
               same
               scituation
               ▪
               
               as
               at
               the
               other
               ,
               as
               a
               looking
               back
               from
               station
               to
               station
               will
               do
               it
               ,
               and
               the
               same
               way
               of
               work
               will
               serve
               .
            
             
               For
               ,
            
             
               As
               the
               Sine
               of
               20
               ,
               to
               110
               ;
            
             
               So
               is
               the
               Sine
               of
               40
               ,
               to
               206.
               
            
             
               And
               ,
            
             
               So
               is
               the
               Sine
               of
               120
               ,
               to
               the
               Line
               DC
               278
               ,
               &c.
               
            
          
           
             
               Use
               XI
               .
               To
               find
               a
               Breadth
               and
               a
               Distance
               at
               any
               two
               Stations
               .
            
             
               
               Let
               AB
               be
               two
               marks
               ,
               as
               two
               corners
               of
               a
               House
               or
               Wall
               ,
               and
               let
               the
               breadth
               between
               them
               be
               demanded
               ,
               and
               their
               distance
               from
               C
               and
               D
               ,
               the
               two
               stations
               ;
               First
               ,
               set
               up
               two
               marks
               at
               the
               two
               stations
               ,
               then
               setting
               up
               the
               Instrument
               at
               C
               ,
               set
               the
               fiducial
               Line
               on
               the
               Rule
               to
               D
               ,
               the
               other
               mark
               ;
               then
               direct
               the
               sights
               exactly
               to
               B
               ,
               and
               to
               A
               ;
               observe
               the
               Angles
               DCB
               45
               ,
               and
               DCA
               113-0
               ,
               as
               in
               the
               Figure
               .
            
             
               Secondly
               ,
               Remove
               the
               Instrument
               to
               D
               ,
               the
               other
               station
               ,
               and
               set
               the
               fiducial-Line
               of
               the
               Quadrant
               (
               viz.
               the
               Line
               of
               Lines
               and
               Sines
               )
               directly
               to
               C
               ;
               then
               fix
               it
               there
               ,
               and
               remove
               the
               Index
               and
               sights
               to
               A
               ,
               and
               to
               B
               ,
               to
               get
               the
               Angles
               CDA
               42-30
               ,
               and
               
               CDB
               109-0
               ;
               Then
               observe
               ,
               that
               the
               3
               Angles
               ,
               of
               every
               Triangle
               ,
               being
               equal
               to
               180
               degrees
               ;
               having
               got
               the
               Angles
               at
               C
               113
               ,
               and
               the
               Angle
               at
               D
               42-30
               ,
               by
               consequence
               ,
               as
               you
               take
               155
               ,
               the
               sum
               of
               the
               Angles
               at
               C
               and
               D
               ,
               out
               of
               180
               ,
               then
               there
               remains
               24-30
               ,
               the
               Angle
               at
               A.
               
            
             
               
                 So
                 also
              
               ,
               Taking
               109
               and
               45
               from
               180
               ,
               rests
               26
               ,
               the
               Angle
               at
               B
               ;
               then
               also
               ,
               taking
               45
               ,
               the
               Angle
               BCD
               ,
               out
               of
               113
               ,
               the
               Angle
               DCA
               ,
               rests
               68
               degrees
               ;
               the
               Angle
               BCA
               ,
               in
               like
               manner
               ,
               taking
               42-30
               from
               109
               ,
               the
               Angles
               at
               D
               ,
               rests
               66-30
               ,
               the
               Angle
               ADB
               ;
               and
               let
               the
               distance
               measured
               ,
               between
               the
               two
               stations
               ,
               be
               100
               ,
               viz.
               CD
               .
               These
               things
               thus
               prepared
               by
               the
               Artificial
               Numbers
               ,
               Sines
               and
               Tangents
               on
               the
               edge
               ,
            
             
               Say
               ,
            
             
               As
               the
               Sine
               of
               24-30
               ,
               the
               Angle
               at
               A
               ,
               to
               100
               ,
               on
               the
               Numbers
               ,
               the
               measured
               side
               CD
               ;
            
             
               So
               is
               the
               Sine
               of
               the
               Angle
               at
               D
               42-30
               ,
               to
               164
               ,
               on
               the
               Numbers
               ,
               the
               side
               CA.
               
            
             
               So
               is
               the
               Sine
               of
               113
               ,
               the
               Angle
               ACD
               ,
               to
               222
               ,
               on
               the
               Numbers
               ,
               the
               distance
               from
               C
               to
               B.
               
            
             
             
               
                 Also
                 ,
                 for
                 the
                 other
                 Triangle
                 ,
                 at
                 the
                 other
                 Station
                 D.
                 
              
            
             
               As
               the
               Sine
               of
               26
               ,
               the
               Angle
               CBD
               ,
               to
               100
               ,
               on
               the
               Numbers
               ,
               the
               measured
               distance
               CD
               ;
            
             
               So
               is
               the
               Sine
               of
               45
               ,
               to
               161
               ,
               on
               the
               Line
               of
               Numbers
               ,
               the
               distance
               from
               D
               to
               B
               ;
            
             
               So
               is
               the
               Sine
               of
               109
               ,
               the
               Angle
               CDB
               ,
               to
               216
               ,
               on
               the
               Numbers
               ,
               the
               distance
               from
               D
               to
               A.
               
            
             
               
                 Then
                 ,
                 having
                 the
                 Sides
                 
                   DB
                   161
                
                 ,
                 and
                 
                   AD
                   222
                
                 ,
                 and
                 ADB
                 the
                 Angle
                 included
                 66-30
                 ,
                 to
                 find
                 the
                 Angles
                 DAB
                 ,
                 or
                 ABD
                 ,
                 use
                 this
                 Proportion
                 .
              
            
             
               As
               the
               sum
               of
               the
               two
               sides
               given
               ,
               is
               to
               the
               difference
               between
               the
               two
               sides
               ;
            
             
               So
               is
               the
               Tangent
               of
               half
               the
               sum
               of
               the
               two
               Angles
               sought
               ,
               to
               the
               Tangent
               of
               half
               their
               difference
               .
            
             
               Example
               .
            
             
               222
               ,
               and
               161
               ,
               make
               383
               for
               a
               sum
               ;
               and
               161
               ,
               taken
               from
               222
               ,
               rest
               61
               for
               a
               difference
               .
            
             
             
               Again
               ,
            
             
               66-30
               ,
               taken
               from
               180
               ,
               rest
               113-30
               ,
               for
               a
               sum
               of
               the
               Angles
               sought
               ,
               whose
               half
               56-45
               ,
               is
               the
               third
               Number
               in
               the
               proportion
               .
            
             
               As
               383
               ,
               the
               sum
               of
               the
               two
               known
               sides
               ,
               is
               to
               061
               ,
               the
               difference
               on
               the
               Numbers
               ;
            
             
               So
               is
               the
               Tangent
               of
               56-45
               ,
               the
               half
               -
               sum
               of
               the
               two
               Angles
               sought
               ,
               to
               the
               Tangent
               of
               half
               the
               difference
               13-40
               ;
               which
               half
               -
               difference
               ,
               13-40
               ,
               added
               to
               56-45
               ,
               makes
               70-25
               ,
               the
               greater
               Angle
               required
               at
               B
               ,
               viz.
               ABD
               .
            
             
               
                 Then
                 also
              
               ,
            
             
               If
               you
               take
               13-40
               ,
               from
               56-45
               ,
               the
               half
               -
               sum
               of
               the
               Angles
               inquired
               ,
               rest
               43-05
               ,
               the
               Angle
               BAB
               ;
               the
               like
               may
               you
               do
               with
               the
               other
               Triangle
               ABC
               ,
               being
               needless
               in
               our
               Proposition
               .
            
             
               Thus
               having
               found
               the
               Angles
               ,
               and
               one
               side
               ,
               the
               Sines
               of
               the
               Angles
               ,
               as
               proportional
               to
               their
               opposite
               sides
               .
            
             
               As
               the
               Sine
               of
               44-33
               ,
               the
               Angle
               ABC
               ,
               is
               to
               the
               side
               AC
               146
               ,
               on
               the
               Numbers
               ;
            
             
             
               So
               is
               the
               Sine
               of
               68
               ,
               the
               Angle
               a●
               C
               ,
               to
               217
               ,
               the
               distance
               between
               the
               marks
               required
               .
            
             
               Or
               ,
            
             
               As
               Sine
               43-05
               ,
               the
               Angle
               at
               A
               ,
               to
               161
               ;
            
             
               So
               is
               Sine
               of
               66-30
               ,
               the
               Angle
               at
               D
               ,
               to
               217
               ,
               the
               distance
               between
               the
               marks
               required
               .
            
             
               
                 Also
                 note
              
               ,
            
             
               That
               if
               this
               manner
               of
               Calculation
               be
               tedious
               or
               difficult
               ,
               then
               on
               a
               Slate
               ,
               or
               sheet
               of
               Paper
               ,
               you
               may
               do
               it
               by
               protraction
               ,
               by
               the
               Line
               of
               Lines
               and
               Chords
               ,
               or
               half
               Sines
               ,
               very
               near
               the
               matter
               with
               care
               ;
            
             
               Thus
               :
               Draw
               CD
               the
               Station-Line
               ,
               or
               measured-distance
               ;
               and
               make
               AD
               100
               ,
               from
               any
               fit
               Scale
               .
               Then
               ,
               on
               C
               and
               D
               draw
               a
               Circle
               ,
               and
               on
               that
               Circle
               lay
               off
               from
               C
               and
               D
               the
               Angles
               ,
               found
               by
               observation
               ,
               and
               draw
               those
               Lines
               ,
               and
               where
               they
               cross
               one
               another
               ,
               as
               at
               A
               and
               B
               ,
               draw
               the
               Line
               AB
               :
               those
               Lines
               and
               Angles
               measured
               on
               the
               same
               Scales
               and
               Chords
               ,
               shall
               be
               the
               Sides
               ,
               breadth
               ,
               and
               distances
               required
               ;
               as
               you
               see
               in
               the
               Figure
               .
            
          
           
             
               Use
               XII
               .
               Another
               way
               for
               a
               long
               Distance
               .
            
             
               
               Let
               C
               be
               your
               standing
               place
               to
               set
               your
               
               Instrument
               ,
               and
               let
               E
               be
               the
               mark
               afar
               off
               ,
               whose
               distance
               from
               you
               C
               would
               know
               :
               first
               ,
               move
               in
               a
               right
               Line
               between
               C
               and
               E
               to
               A
               ,
               any
               number
               of
               Yards
               or
               Perches
               ,
               as
               suppose
               50
               Perch
               ,
               and
               set
               a
               mark
               at
               A
               ;
               Then
               move
               in
               a
               Perpendiculer-Line
               to
               CE
               ,
               from
               A
               to
               B
               any
               distance
               ,
               and
               there
               set
               up
               a
               mark
               at
               B
               ,
               as
               suppose
               66
               Perches
               from
               A.
               
            
             
               Then
               come
               back
               again
               to
               C
               ,
               and
               remove
               in
               a
               Perpendiculer-Line
               to
               CE
               ,
               till
               you
               see
               the
               mark
               set
               up
               at
               B
               ,
               and
               the
               enquired
               point
               at
               the
               distance
               E
               in
               a
               Right-Line
               ;
               and
               note
               that
               place
               at
               D
               ,
               getting
               the
               exact
               distance
               thereof
               from
               C
               ,
               suppose
               76.
               
            
             
               Then
               substract
               the
               measured
               distance
               AB
               from
               the
               measured
               distance
               CD
               ,
               and
               note
               the
               difference
               10.
               
               Then
               ,
               by
               the
               Line
               of
               Numbers
               ,
               or
               by
               the
               Rule
               of
               Three
               ,
               say
               ,
            
             
               As
               the
               Difference
               between
               AB
               and
               BC
               ,
               10
               ,
               is
               to
               the
               distance
               between
               A
               &
               C
               50
               :
            
             
               So
               is
               the
               measured
               distance
               CD
               76
               ,
               to
               the
               distance
               CE
               380.
               
            
             
               Or
               ,
            
             
               So
               is
               the
               measured
               distance
               AB
               66
               ,
               to
               the
               distance
               AE
               330
               ,
               the
               distance
               required
               .
            
             
               Note
               ,
               That
               you
               must
               be
               careful
               and
               exact
               in
               measuring
               the
               Distances
               AC
               ,
               AB
               ,
               &
               CD
               ,
               and
               the
               Answer
               will
               be
               the
               more
               exact
               accordingly
               .
            
          
           
             
             
               Use
               XIII
               .
               To
               find
               an
               Altitude
               of
               a
               House
               or
               Tower
               ,
               by
               knowing
               part
               of
               it
               .
            
             
               If
               you
               divide
               the
               inside-edge
               of
               the
               Loose-peice
               into
               inches
               ,
               
               or
               any
               equal
               parts
               ,
               such
               as
               the
               nearest
               distance
               from
               the
               Rectifying-Point
               to
               that
               inside-edge
               may
               be
               1000
               ,
               and
               for
               this
               use
               two
               small
               sliding
               sights
               may
               be
               convenient
               :
               Then
               the
               use
               is
               thus
               for
               any
               Angle
               under
               30
               degrees
               ;
            
             
               Fix
               the
               Instrument
               to
               the
               Ball-socket
               and
               Staff
               ,
               and
               turn
               it
               toward
               the
               Object
               ,
               causing
               the
               Plummet
               to
               play
               on
               30
               degrees
               ;
               for
               then
               the
               Loose-piece
               is
               perpendiculer
               .
               Then
               one
               pin
               or
               sight
               set
               in
               the
               Rectifying-Point
               ,
               slip
               on
               a
               sight
               along
               the
               inner-edge
               of
               the
               Loose-piece
               ,
               till
               you
               see
               the
               Object
               at
               the
               upper
               part
               of
               the
               Altitude
               ,
               and
               another
               sight
               at
               the
               lower
               part
               of
               the
               Altitude
               known
               ;
               and
               observe
               the
               precise
               distance
               in
               parts
               between
               the
               two
               sights
               ,
               on
               the
               Loose-piece
               ;
               Or
               ,
               the
               several
               parts
               cut
               by
               the
               Index
               at
               each
               Observation
               :
            
             
               Then
               ,
            
             
               As
               the
               distance
               between
               the
               two
               sights
               ,
               is
               to
               the
               distance
               between
               the
               remotest
               sight
               from
               the
               middle
               of
               the
               Loose-piece
               ;
            
             
               So
               is
               the
               height
               of
               the
               known
               part
               ,
               to
               the
               
               whole
               height
               required
               above
               the
               level
               of
               the
               eye
               .
            
             
               Example
               .
            
             
               Let
               CI
               represent
               the
               Altitude
               of
               a
               Pyramid
               on
               the
               Tower
               of
               a
               Steeple
               30
               foot
               high
               ,
               and
               ,
               standing
               at
               B
               ,
               I
               would
               know
               the
               height
               of
               IA
               from
               the
               level
               of
               the
               eye
               upward
               .
            
             
               Fix
               the
               
                 Trianguler
                 Quadrant
              
               on
               the
               Staff
               and
               Ball-socket
               ,
               with
               the
               Head-Center
               at
               B
               ,
               with
               the
               Plummet
               playing
               on
               30
               degrees
               ,
               and
               the
               Loose-piece
               perpendiculer
               :
               Then
               slip
               two
               sights
               on
               the
               Loose-piece
               ,
               one
               in
               a
               Right-Line
               to
               C
               ,
               the
               other
               to
               I
               ;
               and
               note
               the
               parts
               between
               ,
               and
               the
               parts
               the
               furthest
               sight
               cuts
               ,
               from
               the
               middle
               stroak
               on
               the
               Loose-piece
               ,
               from
               whence
               the
               parts
               are
               numbred
               ,
               which
               in
               our
               Example
               let
               be
               500
               ,
               the
               sight
               of
               H
               ,
               and
               the
               sight
               at
               G
               to
               cut
               359
               ;
               then
               the
               distance
               between
               the
               sights
               will
               be
               143
               ,
               and
               the
               remotest
               from
               the
               middle
               of
               the
               Loose-piece
               to
               be
               500
               ;
               and
               the
               known
               Altitude
               ,
               being
               part
               of
               the
               whole
               ,
               to
               be
               30
               foot
               .
            
             
               
                 Then
                 ,
                 by
                 the
                 Line
                 of
                 Numbers
                 ,
                 say
                 ,
              
            
             
               As
               143
               ,
               the
               distance
               betwixt
               the
               sights
               at
               G
               &
               H
               ,
               to
               500
               the
               remotest
               sight
               from
               the
               level
               or
               middle
               ,
               viz.
               FH
               :
            
             
               So
               is
               30
               foot
               ,
               part
               of
               the
               Altitude
               known
               ,
               CI
               ,
               to
               105
               ,
               the
               whole
               Altitude
               unknown
               ,
               AC
               .
            
             
             
               Or
               ,
            
             
             
               So
               is
               75
               ,
               the
               height
               of
               the
               lower
               part
               ,
               to
               105
               the
               whole
               height
               AC
               .
            
             
               Or
               ,
            
             
               As
               143
               ,
               the
               distance
               between
               the
               sights
               ,
               to
               IC
               the
               part
               of
               the
               height
               known
               30
               :
            
             
               So
               is
               357
               ,
               the
               parts
               cut
               between
               F
               and
               G
               ,
               to
               75
               the
               height
               AI
               unknown
               ,
               &c.
               
            
          
           
             
               Use
               XIV
               .
               Having
               the
               Height
               ,
               to
               find
               a
               Distance
               .
            
             
               Let
               CA
               be
               the
               Altitude
               given
               ,
               and
               AB
               the
               distance
               required
               .
               Then
               I
               standing
               at
               C
               ,
               observe
               the
               Angle
               CAB
               ,
               by
               setting
               the
               end
               of
               the
               Head-leg
               to
               my
               eye
               ,
               and
               the
               Head-end
               downwards
               ,
               and
               set
               down
               ,
               as
               the
               Thread
               cuts
               ,
               numbring
               both
               wayes
               ,
               for
               the
               Angle
               at
               C
               and
               at
               B
               his
               complement
               .
            
             
               
                 Then
                 say
              
               ,
            
             
               As
               the
               Angle
               at
               B
               ,
               30
               deg
               .
               40
               minutes
               ,
               counted
               on
               the
               Sines
               ,
               to
               105
               the
               height
               of
               the
               Tower
               :
            
             
               So
               is
               59
               deg
               .
               20
               min.
               the
               Angle
               at
               C
               on
               the
               Sines
               ,
               to
               176
               the
               distance
               required
               on
               the
               Numbers
               .
            
             
               
                 Also
                 note
                 by
                 the
                 way
              
               ,
            
             
               That
               if
               you
               take
               an
               Altitude
               at
               two
               stations
               ,
               
               as
               suppose
               at
               E
               and
               at
               B
               ;
               if
               the
               Angle
               observed
               at
               B
               ,
               be
               found
               to
               be
               the
               half
               of
               the
               Angle
               at
               E
               ;
               as
               here
               in
               Figure
               VIII
               ,
               the
               Angle
               at
               E
               ,
               being
               61-20
               ,
               and
               the
               Angle
               at
               B
               30-40
               ,
               the
               just
               half
               thereof
               ;
               then
               ,
               I
               say
               ,
               that
               the
               distance
               between
               the
               two
               stations
               ,
               is
               equal
               to
               the
               Hypothenusa
               EC
               ,
               at
               the
               first
               station
               ,
               viz.
               EB
               is
               equal
               to
               EC
               ;
               which
               being
               observed
               ,
               say
               ;
            
             
               As
               the
               sine
               of
               90
               ,
               to
               120
               ,
               on
               the
               Numbers
               .
            
             
               So
               is
               61-20
               on
               the
               Sines
               ,
               to
               105
               ,
               the
               height
               required
               on
               the
               Numbers
               .
            
             
               
                 A
                 further
                 proof
                 hereof
                 ,
                 take
                 in
                 this
                 following
                 Figure
                 IX
                 .
              
            
             
               Let
               AB
               be
               a
               breadth
               of
               a
               Wall
               ,
               or
               Fort
               ,
               not
               to
               be
               approached
               unto
               ;
               then
               by
               the
               degrees
               on
               the
               in-side
               of
               the
               loose-piece
               ,
               to
               find
               that
               breadth
               one
               way
               ,
               is
               thus
               ;
               Put
               two
               Pins
               into
               the
               two
               holes
               in
               the
               Head
               and
               Moving-leg
               ,
               (
               or
               set
               the
               sights
               there
               in
               large
               Instruments
               )
               ;
               then
               move
               nearer
               or
               further
               from
               the
               objects
               ,
               till
               your
               eye
               ,
               fixed
               at
               the
               rectifying
               Point
               ,
               can
               but
               just
               see
               the
               marks
               A
               and
               B
               by
               the
               two
               Pins
               in
               each
               Leg
               ,
               which
               will
               only
               be
               at
               the
               mark
               C
               ,
               at
               an
               Angle
               of
               60
               degrees
               ;
               for
               so
               the
               Rule
               is
               made
               to
               that
               Angle
               :
               then
               the
               Instrument
               being
               still
               fixed
               at
               C
               ,
               look
               backward
               in
               a
               right
               Line
               from
               the
               middle
               of
               the
               loose-piece
               ,
               and
               rectifying
               
               Point
               toward
               D
               ,
               putting
               up
               a
               mark
               either
               in
               ,
               or
               over
               ,
               or
               beyond
               the
               Point
               D
               ;
               and
               also
               be
               sure
               to
               leave
               a
               mark
               at
               C
               ,
               the
               first
               place
               of
               observation
               ▪
               Then
               remove
               the
               sights
               to
               15
               degrees
               ,
               the
               half
               of
               30
               ,
               counting
               from
               the
               middle
               ,
               and
               go
               back
               in
               a
               right
               Line
               from
               C
               ,
               toward
               D
               ,
               till
               you
               can
               just
               see
               the
               marks
               by
               the
               two
               sights
               set
               at
               15
               degrees
               each
               way
               ;
               for
               then
               ,
               I
               say
               ,
               that
               the
               measure
               between
               the
               two
               stations
               ,
               C
               and
               D
               ,
               shall
               be
               exactly
               equal
               both
               to
               AB
               ,
               the
               breadth
               required
               ,
               and
               also
               to
               CB
               ,
               or
               CA
               ,
               the
               Hypothenusaes
               ;
               then
               ,
               having
               the
               sides
               CB
               ,
               and
               CD
               ,
               and
               the
               Angles
               BCE
               ,
               and
               CBE
               ,
               and
               BDC
               ,
               it
               is
               easie
               to
               find
               all
               the
               other
               Sides
               and
               Angles
               ,
               by
               the
               Rules
               before
               rehearsed
               ,
               by
               the
               Lines
               of
               Artificial
               Nmmbers
               and
               Sines
               .
            
             
               For
               ,
            
             
               As
               the
               Sine
               of
               15
               degrees
               ,
               the
               Angle
               at
               D
               ,
               viz.
               BDC
               ,
               to
               108
               on
               the
               Numbers
               ;
            
             
               So
               also
               is
               the
               Angle
               at
               B
               ,
               viz.
               DBC
               15
               ,
               to
               108
               on
               the
               Sines
               and
               Numbers
               .
            
             
               So
               also
               is
               the
               Sine
               of
               150
               ,
               the
               Angle
               at
               C
               ,
               viz.
               DCB
               ,
               to
               BD
               208
               ½
               on
               the
               Numbers
               .
            
             
               
                 Note
                 also
              
               ,
               That
               if
               the
               Angles
               of
               60
               and
               30
               be
               inconvenient
               ,
               then
               you
               may
               make
               use
               of
               52
               and
               26
               ,
               or
               48
               and
               24
               ,
               or
               40
               and
               20
               ,
               or
               any
               other
               ,
               and
               the
               half
               thereof
               ;
               and
               
               then
               the
               measured
               distance
               ,
               and
               the
               Hypothenusa
               BC
               ,
               at
               the
               nearest
               station
               ,
               will
               alwayes
               be
               equal
               ;
               but
               not
               equal
               to
               the
               breadth
               at
               any
               other
               Angle
               ,
               except
               30
               and
               60
               ,
               as
               in
               the
               Figure
               .
               But
               having
               the
               Angles
               ,
               and
               those
               Sides
               ,
               you
               may
               soon
               find
               all
               the
               others
               by
               the
               Artificial
               Numbers
               ,
               Sines
               and
               Tangents
               ,
               by
               the
               former
               directions
               .
            
          
        
         
           The
           End
           of
           the
           First
           Part.
           
        
      
    
     
       
         
           The
           Table
           or
           Index
           of
           the
           things
           contained
           in
           this
           Book
           .
        
         
           
             
               TRianguler
               Quadrant
            
             ,
             why
             so
             called
             ,
             
               Page
               2
            
          
           
             The
             Lines
             on
             the
             ou●ter-Edg
             ,
             
               N.
               T.
               S.
               VS
               ,
               Page
               2
            
          
           
             The
             Line
             on
             the
             inner-edge
             ,
             
               I.
               F.
               112
               ,
               Page
               3
            
          
           
             The
             Lines
             on
             the
             Sector-side
             ,
             
               L.S.T.
               Sec.
               Page
               3
            
          
           
             Lesser
             Sines
             ,
             Tangents
             ,
             and
             Secants
             ,
             
               Page
               5
            
          
           
             The
             Lines
             on
             the
             Quadrant-side
             ,
             
               Page
               6
            
          
           
             The
             180
             degrees
             of
             a
             semi-Circle
             variously
             accounted
             ,
             as
             use
             and
             occasion
             requires
             ,
             
               Page
               7
            
          
           
             60
             Degrees
             on
             the
             Loose-piece
             ,
             as
             a
             fore-Staff
             for
             Sea-Observation
             ,
             
               Page
               7
            
          
           
             The
             Line
             of
             right
             Ascentions
             ,
             
               Page
               8
            
          
           
             The
             Line
             of
             the
             Suns
             true
             Place
             ,
             ibid.
             
          
           
             The
             Months
             and
             Dayes
             ,
             ibid.
             
          
           
             
             The
             Hour
             and
             Azimuth-line
             for
             a
             Particuler
             Latitude
             ,
             
               Page
               9
            
          
           
             Natural
             versed
             Sines
             ,
             ibid.
             
          
           
             Lines
             and
             Sines
             ,
             or
             the
             general
             Scale
             of
             Altitudes
             for
             all
             Latitudes
             ,
             
               Page
               10
            
          
           
             The
             particular
             Scale
             of
             Altitudes
             ,
             or
             Sines
             ,
             for
             one
             Latitude
             only
             ,
             
               Page
               11
            
          
           
             The
             Degreees
             of
             a
             whole
             Circle
             ,
             12
             Signs
             ,
             12
             Inches
             ,
             or
             24
             Hours
             ,
             and
             Moons
             Age
             ,
             ibid.
             
          
           
             The
             Appurtenances
             to
             this
             Instrument
             ,
             ibid.
             
          
           
             Numeration
             on
             Decimal-lines
             ,
             
               Page
               12
            
          
           
             Three
             Examples
             thereof
             ,
             
               Page
               13
            
          
           
             Numeration
             on
             Sexagenary
             Circular-lines
             ,
             with
             Examples
             thereof
             .
             
               Page
               17
            
          
           
             How
             Right
             Sines
             ,
             Versed
             Sines
             and
             Chords
             ,
             are
             counted
             on
             the
             Rule
             ,
             
               Page
               20
            
          
           
             Of
             a
             Circle
             ,
             Diameter
             ,
             Chord
             ,
             Right
             Sine
             ,
             Sine
             Complement
             ,
             or
             Co-sine
             ,
             Versed
             Sine
             ,
             Tangent
             ,
             Secant
             ,
             what
             it
             is
             ,
             
               Page
               23
               ,
               24
            
          
           
             Two
             good
             Notes
             or
             Observations
             ,
             
               Page
               25
            
          
           
             Of
             the
             division
             of
             a
             Circle
             ,
             ibid.
             
          
           
             What
             a
             Radius
             is
             ,
             
               Page
               26
            
          
           
             What
             an
             Angle
             ,
             a
             Triangle
             ,
             Acute
             ,
             Right
             ,
             or
             Obtuse
             ;
             Plain
             ,
             or
             spherical
             Angle
             is
             ,
             
               Page
               26
               ,
               27
            
          
           
             Parallel-lines
             ,
             and
             Perpendiculer-lines
             ,
             what
             they
             are
             ,
             ibid.
             
          
           
             The
             usual
             Names
             of
             Triangles
             ,
             ibid.
             
          
           
             Of
             four
             sided
             Figures
             ,
             and
             many
             sided
             ,
             
               Page
               28
            
          
           
             
             Terms
             in
             Arithmetick
             ,
             as
             Multiplicator
             ,
             Product
             ,
             Quotient
             ,
             &c.
             what
             they
             mean
             ,
             
               Page
               29
            
          
           
             Geometrical
             Propositions
             ,
             
               Page
               31
            
          
           
             To
             draw
             a
             right
             Line
             ,
             ibid.
             
          
           
             To
             raise
             a
             Perpendiculer
             on
             any
             line
             ,
             ibid.
             
          
           
             To
             let
             fall
             a
             Perpendiculer
             any
             where
             ,
             
               Page
               33
            
          
           
             To
             draw
             Parallel-lines
             ,
             
               Page
               34
            
          
           
             To
             make
             one
             Angle
             equal
             to
             another
             ,
             
               Page
               35
            
          
           
             To
             divide
             a
             Line
             into
             any
             number
             of
             parts
             ,
             ib.
             
          
           
             To
             bring
             any
             3
             Points
             into
             a
             Circle
             ,
             
               Page
               36
            
          
           
             To
             cut
             any
             two
             Points
             in
             a
             Circle
             ,
             and
             the
             Circle
             into
             two
             equal
             parts
             ,
             
               Page
               37
            
          
           
             A
             Segment
             of
             a
             Circle
             given
             ,
             to
             find
             the
             Center
             and
             Diameter
             ,
             
               Page
               38
            
          
           
             A
             Segment
             of
             a
             Circle
             given
             ,
             to
             find
             the
             length
             of
             the
             Arch
             ,
             
               Page
               39
            
          
           
             To
             draw
             a
             Helical-line
             ,
             and
             to
             find
             the
             Centers
             ,
             of
             the
             Splayes
             ,
             of
             Eliptical
             arches
             ,
             and
             Key-stones
             ,
             
               Page
               41
            
          
           
             To
             draw
             an
             Oval
             ,
             ibid.
             
          
        
         
           
             Explanation
             of
             Terms
             particularly
             belonging
             to
             this
             Instrument
             .
          
           
             Radius
             ,
             how
             taken
             ,
             
               Page
               41
            
          
           
             Right
             Sines
             ,
             how
             taken
             and
             counted
             ,
             
               Page
               42
            
          
           
             Tangents
             ,
             Secants
             and
             Chords
             ,
             how
             taken
             ,
             ib.
             
          
           
             Sine
             complement
             ,
             or
             co-sine
             Tangent
             ,
             complement
             or
             co-tangent
             ,
             how
             taken
             and
             counted
             on
             this
             Instrument
             ,
             ibid.
             
          
           
             
             Latteral
             Sine
             and
             Tangent
             ,
             
               Page
               43
            
          
           
             Parallel
             Sine
             and
             Tangent
             ,
             ibid.
             
          
           
             Nearest
             Distance
             ,
             what
             it
             means
             ,
             ibid.
             
          
           
             Addition
             on
             Lines
             ,
             ibid.
             
          
           
             Substraction
             on
             Lines
             .
             
               Page
               44
            
          
        
         
           
             Of
             Terms
             used
             in
             Dialling
             .
          
           
             Plain
             ,
             and
             Pole
             of
             the
             Plain
             ,
             
               Page
               45
            
          
           
             Declination
             ,
             Reclination
             ,
             and
             Inclination
             of
             a
             Plain
             ,
             what
             it
             is
             ,
             
               Page
               46
            
          
           
             What
             the
             Perpendiculer-line
             ,
             and
             Horizontal-line
             of
             a
             Plain
             are
             ,
             ibid.
             
          
           
             Meridian-line
             ,
             Substile-line
             ,
             and
             Stile-line
             ,
             Angle
             of
             12
             and
             6
             ,
             and
             the
             Inclination
             of
             Meridians
             ,
             what
             they
             are
             ,
             
               Page
               47
            
          
           
             Parallels
             and
             Contingent-lines
             ,
             what
             ,
             
               Page
               48
            
          
           
             Vertical-line
             and
             Point
             ,
             what
             ,
             ibid.
             
          
           
             Nodus
             Apex
             and
             foot
             of
             the
             Stile
             ,
             what
             ,
             ibid.
             
          
           
             Axis
             of
             the
             Horizon
             ,
             what
             ,
             
               Page
               49
            
          
           
             Erect
             ,
             Direct
             ,
             what
             ,
             ibid.
             
          
           
             Declining
             ,
             Reclining
             ,
             or
             Inclining
             ,
             what
             ,
             ibid.
             
          
           
             Circles
             of
             Position
             ,
             what
             ,
             ibid.
             
          
        
         
           
             Of
             Terms
             in
             Astronomy
             .
          
           
             What
             a
             Sphear
             is
             ,
             
               Page
               50
            
          
           
             Of
             ten
             Points
             ,
             and
             ten
             Circles
             of
             the
             Sphear
             ,
             
               Page
               51
            
          
           
             The
             2
             Poles
             of
             the
             World
             or
             Equinoctial
             ,
             ibid.
             
          
           
             The
             2
             Poles
             of
             the
             Zodiack
             ,
             
               Page
               52
            
          
           
             
             The
             2
             Equinoctial-points
             ,
             ibid.
             
          
           
             The
             2
             Solstitial-points
             ,
             
               Page
               53
            
          
           
             The
             Zenith
             and
             Nadir
             ,
             
               Page
               54
            
          
           
             The
             Horizon
             ,
             the
             Meridian
             ,
             the
             Equinoctial
             ,
             the
             Zodiack
             ,
             the
             2
             Colures
             ,
             the
             2
             Tropicks
             ,
             and
             2
             Polar
             Circles
             ,
             
               Page
               55
               ,
               56
               ,
               58
            
          
           
             Hours
             ,
             Azimuths
             ,
             Almicanters
             ,
             Declination
             ,
             Latitude
             ,
             Longitude
             ,
             Right
             Ascention
             ,
             
               Page
               59
               ,
               60
            
          
           
             Oblique
             Ascention
             ,
             Difference
             of
             Ascentions
             ,
             Amplitude
             ,
             Circles
             ,
             and
             Angles
             of
             Position
             ,
             what
             they
             are
             ,
             
               Page
               61
               ,
               62
            
          
           
             To
             rectifie
             the
             Trianguler
             Quadrant
             ,
             
               Page
               63
            
          
           
             To
             observe
             or
             find
             the
             Suns
             Altitude
             ,
             
               Page
               64
            
          
           
             To
             try
             if
             any
             thing
             be
             level
             ,
             or
             upright
             ,
             
               Page
               66
            
          
           
             To
             find
             what
             Angle
             the
             Sector
             stands
             at
             ,
             at
             any
             opening
             ;
             or
             to
             set
             the
             Sector
             to
             any
             Angle
             required
             ,
             
               Page
               67
               ,
               68
            
          
           
             The
             day
             of
             the
             Month
             given
             ,
             to
             find
             the
             Suns
             Declination
             ,
             true
             Place
             ,
             Right
             Ascention
             ,
             or
             Rising
             and
             Setting
             ,
             by
             inspection
             only
             ,
             
               Page
               71
            
          
           
             To
             find
             the
             Suns
             Amplitude
             ,
             and
             difference
             of
             Ascentions
             ,
             and
             Oblique
             Ascention
             ,
             
               Page
               73
            
          
           
             To
             find
             the
             Hour
             of
             the
             Day
             ,
             
               Page
               74
            
          
           
             To
             find
             the
             Suns
             Azimuth
             ,
             
               Page
               75
            
          
        
         
           
           
             The
             use
             of
             the
             Line
             of
             Numbers
             ,
             and
             the
             use
             of
             the
             Line
             of
             Lines
             ,
             both
             on
             the
             Trianguler
             Quadrant
             and
             Sector
             ,
             one
             after
             another
             ,
             in
             most
             Examples
             .
          
           
             To
             multiply
             one
             Number
             by
             another
             ,
             
               Page
               78
            
          
           
             A
             help
             to
             Multiply
             truly
             ,
             
               Page
               85
            
          
           
             A
             crabbed
             Question
             of
             Multiplication
             ,
             
               Page
               90
            
          
           
             Precepts
             of
             Reduction
             ,
             
               Page
               94
            
          
           
             To
             divide
             one
             Number
             by
             another
             ,
             
               Page
               95
            
          
           
             A
             Caution
             in
             Division
             .
             
               Page
               97
            
          
           
             To
             2
             Lines
             or
             Numbers
             given
             ,
             to
             find
             a
             3d
             in
             Geometrical
             proportion
             ,
             
               Page
               98
            
          
           
             Any
             one
             side
             of
             a
             Figure
             being
             given
             ,
             to
             find
             all
             the
             rest
             ;
             or
             to
             find
             a
             proportion
             between
             two
             or
             more
             Lines
             or
             Numbers
             ,
             
               Page
               99
            
          
           
             To
             lay
             down
             any
             number
             of
             parts
             on
             a
             Line
             to
             any
             Radius
             ,
             
               Page
               100
            
          
           
             To
             divide
             a
             line
             into
             any
             number
             of
             parts
             ,
             
               Page
               102
            
          
           
             To
             find
             a
             Geometrical
             mean
             proportion
             between
             two
             Lines
             or
             Numbers
             ,
             three
             wayes
             ,
             
               Page
               104
            
          
           
             To
             make
             a
             Square
             equal
             to
             an
             Oblong
             ,
             
               Page
               107
            
          
           
             Or
             to
             a
             Triangle
             ,
             ibid.
             
          
           
             To
             find
             a
             Proportion
             between
             unlike
             Superficies
             ,
             
               Page
               108
            
          
           
             To
             make
             one
             Superficies
             like
             another
             Superficies
             ,
             and
             equal
             to
             a
             third
             ,
             
               Page
               109
            
          
           
             The
             Diameter
             and
             Content
             of
             a
             Circle
             being
             given
             ,
             to
             find
             the
             Content
             of
             another
             Circle
             by
             having
             his
             Diameter
             ,
             
               Page
               111
            
          
           
             
             To
             find
             the
             Square-root
             of
             a
             Number
             ,
             ibid.
             
          
           
             To
             find
             the
             Cube-root
             of
             a
             Number
             ,
             
               Page
               113
            
          
           
             To
             find
             two
             mean
             Proportionals
             between
             two
             Lines
             or
             Numbers
             given
             ,
             
               Page
               116
            
          
           
             The
             Diameter
             and
             Content
             of
             a
             Globe
             being
             given
             ,
             to
             find
             the
             Content
             of
             another
             Globe
             ,
             whose
             Diameter
             also
             is
             given
             ,
             
               Page
               118
            
          
           
             The
             proportion
             between
             the
             Weights
             and
             Magnitudes
             of
             Metals
             ,
             
               Page
               119
            
          
           
             The
             Weight
             and
             Magnitude
             of
             a
             body
             of
             one
             kind
             of
             Metal
             being
             given
             ,
             to
             find
             the
             Magnitude
             of
             a
             body
             of
             another
             Metal
             of
             equal
             weight
             ,
             
               Page
               121
            
          
           
             The
             magnitudes
             of
             two
             bodies
             of
             several
             Metals
             ,
             having
             the
             weight
             of
             one
             given
             ,
             to
             find
             the
             weight
             of
             the
             other
             ,
             
               Page
               122
            
          
           
             The
             weight
             and
             magnitude
             of
             one
             body
             of
             any
             Metal
             being
             given
             ,
             and
             another
             body
             like
             unto
             the
             former
             ,
             is
             to
             be
             made
             of
             any
             other
             Metal
             ,
             to
             find
             the
             diameters
             or
             magnitudes
             of
             it
             ,
             
               Page
               123
            
          
           
             To
             divide
             a
             Line
             ,
             or
             Number
             ,
             by
             extream
             and
             mean
             proportion
             ,
             
               Page
               124
            
          
           
             Three
             Lines
             or
             Numbers
             given
             ,
             to
             find
             a
             fourth
             in
             Geometrical
             proportion
             ,
             
               Page
               128
            
          
           
             The
             nature
             &
             reason
             of
             the
             Golden
             Rule
             ,
             
               Page
               129
            
          
           
             The
             Rule
             of
             Three
             inversed
             ,
             with
             several
             Cautions
             and
             Examples
             ,
             
               Page
               132
            
          
           
             The
             double
             and
             compound
             Rule
             of
             Three
             Direct
             
             and
             Reverse
             ,
             with
             Examples
             ,
             
               Page
               139
            
          
           
             The
             Rule
             of
             Fellowship
             with
             Examples
             ,
             
               Page
               148
            
          
           
             The
             use
             of
             the
             Line
             of
             Numbers
             in
             Superficial
             measure
             ,
             and
             the
             parts
             on
             the
             Rule
             ,
             
               Page
               154
            
          
           
             The
             breadth
             given
             in
             Foot-measure
             ,
             to
             find
             the
             length
             of
             one
             Foot
             ,
             
               Page
               156
            
          
           
             The
             bredth
             given
             in
             Inches
             ,
             to
             find
             how
             much
             in
             length
             makes
             one
             Foot
             ,
             ibid.
             
          
           
             The
             bredth
             given
             ,
             to
             find
             how
             much
             is
             in
             a
             Foot-long
             ,
             
               Page
               157
            
          
           
             Having
             the
             length
             and
             bredth
             given
             in
             Foot-measure
             ,
             to
             find
             the
             Content
             in
             Feet
             ,
             ibid.
             
          
           
             Having
             the
             bredth
             given
             in
             Inches
             ,
             and
             length
             in
             Feet
             ,
             to
             find
             the
             Content
             in
             Feet
             ,
             
               Page
               158
            
          
           
             Having
             the
             length
             &
             bredth
             given
             in
             Inches
             ,
             to
             find
             the
             content
             in
             superficial
             Inches
             ,
             
               Page
               160
            
          
           
             Having
             the
             length
             &
             bredth
             given
             in
             Inches
             ,
             to
             find
             the
             Content
             in
             Feet
             superficial
             ,
             
               Page
               161
            
          
           
             The
             length
             and
             bredth
             of
             an
             Oblong
             given
             ,
             to
             find
             the
             side
             of
             a
             Square
             equal
             to
             it
             ,
             
               Page
               163
            
          
           
             The
             Diameter
             of
             a
             Circle
             given
             ,
             to
             find
             the
             Circumference
             ,
             Square
             ,
             equal
             Square
             ,
             inscribed
             and
             Content
             ,
             
               Page
               164
            
          
           
             The
             Content
             of
             a
             Circle
             given
             ,
             to
             find
             the
             Diameter
             or
             Circumference
             ,
             
               Page
               166
               ,
               167
            
          
           
             Certain
             Rules
             to
             measure
             several
             figures
             ,
             
               Page
               108
            
          
           
             A
             Segment
             of
             a
             Circle
             given
             ,
             to
             find
             the
             true
             Diameter
             and
             Area
             thereof
             ,
             
               Page
               169
            
          
           
             A
             Table
             to
             divide
             the
             Line
             of
             Segments
             ,
             
               Page
               170
            
          
           
             
             The
             use
             of
             it
             in
             part
             ,
             
               Page
               171
            
          
           
             The
             measuring
             of
             Triangles
             ,
             Tapeziaes
             ,
             Romboides
             ,
             Poligons
             ,
             and
             Ovals
             ,
             
               Page
               172
               ,
               173
            
          
           
             A
             Table
             of
             the
             Proportion
             between
             the
             Sides
             and
             Area's
             of
             regular
             Poligons
             ,
             and
             the
             use
             thereof
             for
             any
             other
             ,
             
               Page
               174
               ,
               175
            
          
           
             To
             make
             an
             Oval
             equal
             to
             a
             Circle
             ,
             and
             the
             contrary
             ,
             two
             wayes
             ,
             
               Page
               175
               ,
               176
            
          
           
             The
             length
             and
             bredth
             of
             any
             Oblong
             Superficies
             given
             in
             Feet
             ,
             to
             find
             the
             Content
             in
             Yards
             ,
             
               Page
               177
            
          
           
             The
             length
             and
             bredth
             given
             in
             feet
             and
             parts
             ,
             to
             find
             the
             Content
             in
             Rods
             ▪
             
               Page
               179
            
          
           
             The
             nearest
             way
             to
             measure
             a
             party
             Wall
             ,
             
               Page
               180
            
          
           
             To
             multiply
             and
             reduce
             any
             length
             ,
             bredth
             ,
             or
             thickness
             of
             a
             Wall
             to
             one
             Brick
             and
             a
             half
             at
             one
             Operation
             ,
             
               Page
               183
            
          
           
             Examples
             at
             six
             several
             thicknesses
             ,
             
               Page
               184
            
          
           
             To
             find
             the
             Gage-points
             for
             this
             reducing
             ,
             
               Page
               185
            
          
           
             At
             one
             opening
             of
             the
             Compasses
             ,
             to
             find
             how
             many
             Rods
             ,
             Quarters
             ,
             and
             Feet
             in
             any
             sum
             under
             10
             Rods
             ,
             
               Page
               186
            
          
           
             The
             usual
             and
             readiest
             equal
             wayes
             to
             measure
             Tileing
             and
             Chimnyes
             ,
             
               Page
               187
            
          
           
             Of
             Plaisterers-work
             ,
             or
             Painters-work
             ,
             
               Page
               188
            
          
           
             Of
             particulars
             of
             work
             ,
             usually
             mentioned
             in
             a
             Carpenters-Bill
             ,
             with
             Cautions
             ,
             
               Page
               189
               ,
               190
            
          
           
             At
             any
             bredth
             of
             a
             House
             ,
             to
             find
             the
             Rafters
             ,
             and
             Hip-rafters
             ,
             length
             and
             Angles
             ,
             by
             the
             
             Line
             of
             Numbers
             readily
             ,
             
               Page
               191
            
          
           
             The
             price
             of
             one
             Foot
             being
             given
             ,
             to
             find
             the
             price
             of
             a
             Rod
             ,
             or
             a
             Square
             of
             Brick-work
             ,
             or
             Flooring
             ,
             by
             inspection
             ,
             
               Page
               193
            
          
           
             At
             any
             length
             of
             a
             Land
             given
             ,
             to
             find
             how
             much
             in
             bredth
             makes
             one
             Acre
             ,
             
               Page
               194
            
          
           
             A
             useful
             Table
             in
             measuring
             Land
             ,
             and
             the
             use
             thereof
             in
             several
             Examples
             ,
             
               Page
               196
               ,
               197
            
          
           
             The
             length
             and
             bredth
             given
             in
             Perches
             ,
             to
             find
             the
             Content
             in
             Squares
             ,
             Perches
             ,
             Poles
             ,
             or
             Rods
             ,
             
               Page
               200
            
          
           
             The
             length
             and
             bredth
             in
             Perches
             ,
             to
             find
             the
             Content
             in
             Acres
             ,
             ibid.
             
          
           
             The
             length
             and
             bredth
             given
             in
             Chains
             ,
             to
             find
             the
             content
             in
             square
             Acres
             ,
             Quarters
             ,
             and
             Links
             ,
             
               Page
               201
            
          
           
             To
             measure
             a
             Triangle
             at
             once
             ,
             without
             halfing
             the
             Base
             or
             Area
             ,
             ibid.
             
          
           
             To
             reduce
             Statute-measure
             ,
             or
             Acres
             ,
             to
             Customary
             ,
             and
             the
             contrary
             ,
             ibid.
             
          
           
             A
             Table
             to
             make
             Scales
             to
             do
             it
             by
             measuring
             or
             inspection
             ,
             with
             Examples
             ,
             
               Page
               204
            
          
           
             Knowing
             the
             content
             of
             a
             piece
             of
             Land
             plotted
             out
             ,
             to
             find
             by
             what
             Scale
             it
             was
             done
             ,
             
               Page
               206
            
          
           
             The
             same
             Rule
             applied
             to
             the
             measuring
             of
             Glaziers
             Quarries
             ,
             
               Page
               208
            
          
           
             A
             Table
             of
             all
             the
             usual
             sizes
             of
             quarries
             ,
             
               Page
               210
            
          
           
             The
             bredth
             &
             depth
             of
             any
             solid
             body
             being
             given
             ,
             to
             find
             the
             side
             of
             the
             square
             equal
             ,
             
               Page
               211
            
          
           
             
             The
             bredth
             and
             depth
             ,
             or
             square
             equal
             given
             ,
             to
             find
             how
             much
             in
             length
             makes
             one
             foot
             solid
             ,
             four
             manner
             of
             wayes
             ,
             according
             to
             the
             wordin●
             of
             the
             question
             ,
             
               Page
               212
            
          
           
             The
             bredth
             an●
             depth
             ,
             or
             the
             side
             of
             the
             square
             of
             any
             Solid
             given
             ,
             to
             find
             how
             much
             is
             in
             a
             Foot
             long
             solid
             measure
             ,
             three
             wayes
             ,
             according
             to
             the
             wording
             the
             question
             ,
             
               Page
               219
               ,
               220
            
          
           
             The
             bredth
             depth
             and
             length
             of
             any
             solid
             body
             given
             ,
             to
             find
             the
             solid
             Content
             ,
             four
             wayes
             ,
             according
             to
             the
             wording
             the
             question
             ,
             
               Page
               221
               ,
               222
               ,
               223.
               
            
          
           
             The
             3
             last
             Probl.
             wrought
             by
             the
             Sector
             ,
             
               Page
               226
            
          
           
             The
             Diameter
             of
             a
             Cillender
             given
             ,
             to
             find
             how
             much
             in
             length
             makes
             1
             foot
             ,
             4
             wayes
             ,
             
               Page
               230
            
          
           
             The
             diameter
             of
             a
             Cillender
             given
             ,
             to
             find
             how
             much
             is
             in
             a
             foot
             long
             ,
             3
             wayes
             ,
             
               Page
               232
            
          
           
             The
             diameter
             of
             a
             Cillender
             with
             the
             length
             given
             ,
             to
             find
             the
             Content
             3
             wayes
             ,
             
               Page
               233
            
          
           
             The
             Circumference
             given
             ,
             to
             find
             a
             foot
             ,
             3
             wayes
             ,
             
               Page
               234
            
          
           
             The
             Circumference
             ,
             to
             find
             how
             much
             in
             a
             foot
             ,
             3
             wayes
             ,
             
               Page
               236
            
          
           
             The
             Circumference
             and
             Length
             ,
             to
             find
             the
             Content
             3
             wayes
             ,
             
               Page
               238
            
          
           
             The
             customs
             &
             allowances
             in
             measuring
             round
             Timber
             ,
             as
             Oak
             or
             Elm
             ,
             &
             the
             like
             ,
             
               Page
               240
            
          
           
             The
             use
             of
             2
             Points
             for
             that
             allowance
             ,
             
               Page
               242
            
          
           
             
             To
             measure
             a
             round
             Pyramid
             or
             Steeple
             ,
             ibid.
             
          
           
             A
             nicity
             in
             measuring
             round
             Timber
             ,
             stated
             ,
             
               Page
               246
            
          
           
             To
             measure
             Globes
             ,
             and
             Segments
             of
             Globes
             ,
             both
             superficially
             round
             about
             ,
             and
             with
             th●
             solidity
             several
             wayes
             ,
             by
             Arithmetick
             ▪
             and
             the
             Line
             of
             Numbers
             ,
             and
             solid
             Segments
             ;
             with
             a
             small
             Table
             of
             solid
             Segments
             ,
             
               Page
               252
               ,
               253
            
          
           
             The
             Experimented
             Proportions
             ,
             between
             a
             Cube
             ,
             a
             Cillender
             ,
             a
             Sphear
             ,
             a
             Cone
             ,
             a
             Prism
             ,
             a
             Square
             and
             Trianguler
             Pyramid
             ,
             
               Page
               257
            
          
           
             The
             use
             of
             the
             sliding
             (
             cover
             or
             )
             Rule
             ,
             
               Page
               259
            
          
           
             The
             description
             ,
             
               Page
               260
            
          
           
             The
             Gage-points
             ,
             and
             places
             of
             them
             ,
             
               Page
               261
            
          
           
             The
             Uses
             ,
             to
             square
             a
             Piece
             ,
             to
             find
             how
             much
             in
             length
             will
             make
             1
             foot
             of
             square
             Timber
             ,
             
               Page
               263
            
          
           
             To
             find
             how
             much
             is
             in
             a
             foot
             long
             ,
             
               Page
               264
            
          
           
             The
             square
             and
             length
             given
             ,
             to
             find
             the
             Content
             ,
             
               Page
               265
            
          
           
             The
             Diameter
             of
             round
             Timber
             given
             ,
             to
             find
             how
             much
             is
             in
             a
             foot
             long
             ,
             
               Page
               267
            
          
           
             To
             find
             how
             much
             in
             length
             makes
             1
             foot
             ,
             
               Page
               268
            
          
           
             Diameter
             and
             length
             given
             ,
             to
             find
             the
             Content
             ,
             
               Page
               269
            
          
           
             The
             Circumference
             given
             ,
             to
             find
             how
             much
             is
             in
             a
             foot
             long
             ,
             
               Page
               271
            
          
           
             The
             Circumference
             given
             ,
             to
             find
             how
             much
             
             makes
             a
             foot
             ,
             ibid.
             
          
           
             The
             Circumference
             and
             length
             given
             ,
             to
             find
             the
             Content
             ,
             
               Page
               272
            
          
           
             To
             Gage
             round
             Cask
             by
             the
             Rule
             or
             Square
             ,
             counting
             6
             Foot
             for
             a
             Barrel
             of
             Beer
             ,
             or
             one
             Foot
             for
             6
             Gallons
             ,
             or
             one
             Foot
             for
             7
             Gallons
             and
             a
             half
             of
             Wine
             measure
             ,
             
               Page
               273
            
          
           
             The
             diameter
             and
             length
             of
             a
             Cask
             given
             ,
             to
             find
             the
             Content
             in
             Wine-gallons
             ,
             or
             Ale-gallons
             ,
             ibid.
             
          
           
             To
             Gage
             Brewers
             great
             round
             Tuns
             ,
             and
             to
             have
             the
             Content
             in
             Barrels
             at
             one
             work
             ,
             
               Page
               274
            
          
           
             The
             use
             of
             the
             other-side
             in
             superficial
             measure
             ,
             Golden
             Rule
             ,
             and
             Division
             ,
             
               Page
               275
            
          
        
         
           
             To
             make
             and
             measure
             the
             5
             regular
             bodies
             ,
             with
             the
             Declination
             and
             Reclination
             of
             every
             side
             ,
             at
             any
             scituation
             of
             them
             .
          
           
             The
             Cube
             ,
             
               Page
               277
            
          
           
             The
             Tetrahedron
             ,
             
               Page
               279
            
          
           
             The
             Octahedron
             ,
             
               Page
               281
            
          
           
             The
             Dodecahedron
             ,
             
               Page
               283
            
          
           
             The
             Icosahedron
             ,
             
               Page
               286
            
          
           
             A
             Figure
             and
             a
             Table
             of
             all
             the
             Sides
             and
             Angles
             ,
             
               Page
               294
            
          
           
             Gaging
             by
             the
             Line
             of
             Numbers
             ,
             
               Page
               295
            
          
           
             To
             Gage
             great
             square
             Vessels
             ,
             and
             round
             Vessels
             ,
             
               Page
               297
            
          
           
             
             —
             Artificially
             and
             Naturally
             ,
             with
             Examples
             ,
             
               Page
               300
            
          
           
             To
             find
             the
             mean
             Diameter
             ,
             and
             Gage-point
             ,
             
               Page
               303
            
          
           
             To
             find
             the
             Contents
             of
             Cask
             otherwise
             ,
             ibid.
             
          
           
             The
             Content
             and
             mean
             Diameter
             given
             ,
             to
             find
             the
             length
             of
             the
             Cask
             ,
             &
             contrary
             ,
             
               Page
               308
            
          
           
             To
             find
             the
             wants
             &
             nullage
             ,
             two
             wayes
             ,
             
               Page
               311
            
          
           
             A
             Table
             of
             the
             Wants
             in
             a
             Beer
             Barrel
             ,
             in
             Beer
             and
             Wine
             Gallons
             ,
             at
             any
             Inches
             ,
             wet
             or
             dry
             ,
             
               Page
               317
            
          
           
             The
             use
             of
             the
             Line
             of
             Numbers
             in
             Interest
             ,
             and
             several
             Examples
             thereof
             ,
             many
             wayes
             useful
             ,
             
               Page
               324
            
          
           
             The
             use
             of
             the
             Line
             in
             Military
             questions
             ,
             
               Page
               332
            
          
           
             The
             use
             of
             the
             Line
             in
             solid
             Proportions
             ,
             as
             the
             weights
             and
             measures
             of
             Rope
             ,
             and
             Burthen
             of
             Ships
             ,
             
               Page
               336
            
          
           
             The
             way
             to
             use
             the
             Logarithmal
             
               Tables
               ,
               Page
               340
            
          
           
             The
             use
             of
             the
             Rule
             in
             Geometry
             &
             Astronomy
             ,
             in
             50
             Propositions
             ,
             or
             Uses
             ,
             by
             the
             perticular
             Scale
             or
             Quadrant
             ,
             the
             general
             Scale
             or
             Quadrant
             ,
             the
             Sector
             and
             Artificial
             Numbers
             ,
             Sines
             ,
             and
             Tangents
             ,
             
               Page
               345
               ,
               to
               448
            
          
           
             The
             use
             of
             the
             
               Trianguler
               Quadrant
            
             ,
             in
             finding
             of
             Heights
             and
             Distances
             ,
             accessable
             or
             inaccessable
             ,
             in
             14
             Uses
             ,
             
               Page
               449
               ,
               to
               483.
            
             
          
        
         
           FINIS
           .
        
         
      
       
         Notes, typically marginal, from the original text
         
           Notes for div A29761-e3430
           
             Radius
             .
          
           
             Right-Sine
             .
          
           
             Tangent
             .
          
           
             
               Secant
               ,
               Chord
            
             .
          
           
             Cosine
             .
          
           
             Lateral-Sine
             .
          
           
             Parallel
             .
          
           
             Nearest-Distance
             .
          
           
             
               Addition
               on
               Lines
            
             .
          
           
             
               Substraction
               on
               Lines
            
             .
          
           
             
               Rectifying
               Point
            
             .
          
           
             Plain
             .
          
           
             
               Pole
               of
               the
               Plain
            
             .
          
           
             Declination
             .
          
           
             
               Perpendiculer-line
               on
               the
               Plain
            
             .
          
           
             Horizontal-line
             .
          
           
             
               Reclination
               and
               Inclination
            
             .
          
           
             Meridian-line
             .
          
           
             Substile
             .
          
           
             Stile
             .
          
           
             
               Angle
               between
            
             12
             and
             6.
             
          
           
             
               Inclination
               of
               Meridians
            
          
           
             
               Parallels
               ▪
            
          
           
             Contingent
             .
          
           
             Vertical-line
             .
          
           
             
               Nodus
               or
               Apex
            
             .
          
           
             
               Perpendiculer
               height
               of
               the
               Stile
            
             .
          
           
             
               Foot
               of
               the
               Stile
            
             .
          
           
             Virtical-point
             .
          
           
             
               Axis
               of
               the
               Horizon
            
             .
          
           
             Erect
             .
          
           
             Direct
             .
          
           
             
               Declining
               Reclining
               or
               Inclining-plains
            
          
           
             Oblique
             .
          
           
             
               Circles
               of
               Position
            
             .
          
           
             Poles
             .
          
           
             
               Poles
               of
               the
               Zodiack
            
             .
          
           
             Equinoctial-Points
             .
          
           
             Solsticial-Points
             .
          
           
             Zenith
             .
          
           
             Nadir
             .
          
           
             
               Horizon
               .
               1.
            
             
          
           
             Zodiack
          
           
             4.
             
          
           
             Colures
             .
          
           
             5.
             
             &
             6.
             
          
           
             Tropicks
             .
          
           
             7.
             
             &
             8.
             
          
           
             Polar-Circles
             .
          
           
             9.
             
             &
             10.
             
          
           
             Hours
             .
          
           
             Azimuths
          
           
             Almicanters
             .
          
           
             Declination
             .
          
           
             Latitude
             .
          
           
             Longitude
             .
          
           
             Right-Ascention
             .
          
           
             Oblique-Ascention
             .
          
           
             
               Ascentional
               Difference
            
             .
          
           
             Amplitude
             .
          
           
             
               Circles
               and
               Angles
               of
               Position
            
             .
          
           
             *
             
               In
               both
               these
               the
               inversed
               Proportion
               is
               in
               th
               lower
               line
            
          
           
             Circle
             .
          
           
             Half-Circle
             .
          
           
             
               Quadrant
               ,
               or
               the
               quarter
            
             .
          
           
             Lesser-parts
             .
          
           
             Segments
             .
          
           
             Triangles
             .
          
           
             Rhombus
             .
          
           
             Trapeziaes
             .
          
           
             Regular-Polligons
             .
          
           
             By
             th'
             Trianguler-Quadrant
             .
          
           
             Sines
             .
          
           
             Tangent
             .
          
           
             Secant
             .
          
           
             Chord
             .
          
           
             Sines
             .
          
           
             Tangents
             .
          
           
             Secants
             .
          
           
             Chords
             .
          
           
             
               By
               the
               Sector-side
            
             .
          
           
             Sine
             .
          
           
             Tangent
             .
          
           
             
               Tan.
               to
            
             76.
             
          
           
             Secant
             .
          
           
             
               By
               the
               Lines
               on
               the
               Edge
               .
            
          
           
             Sines
             .
          
           
             Tangent
             .
          
           
             
               Secants
               &
               Tangents
               beyond
            
             45
             degrees
             .
          
           
             Artificial-logarithms
          
           
             
               The
               proof
               of
               the
               truth
               of
               the
               Instrument
               .
            
          
           
             Quadrant
             .
          
           
             Sector
             .
          
           
             Quadr.
             
          
           
             Sector
             .
          
           
             Quadrant
             .
          
           
             
               Sector
               ▪
            
          
           
             Quadr.
             
          
           
             Sector
             .
          
           
             Quadr.
             
          
           
             Sector
             .
          
           
             Quadr.
             
          
           
             Sector
             .
          
           
             Sector
             .
          
           
             Sector
             .
          
           
             
               Quadr
               ▪
            
          
           
             Sector
             .
          
           
             Quadr.
             
          
           
             Sector
             .
          
           
             Quadr.
             
          
           
             Quadr.
             
          
           
             
               By
               the
               Quadr.
               particularly
            
             .
          
           
             
               Quad.
               Generally
            
             .
          
           
             
               Quadr·
               Particularly
            
             .
          
           
             
               Artificial-S
               .
               &
            
             T.
             
          
           
             
               Quad.
               Generally
            
             .
          
           
             Sector
             .
          
           
             
               Particular
               Quadr.
            
             
          
           
             Artificial-S
             .
             &
             T.
             
          
           
             
               Quad.
               Generally
            
             .
          
           
             
               Sector
               ▪
            
          
           
             
               Art.
               Sine
            
             .
          
           
             Quadr.
             
          
           
             Sector·
          
           
             
               Partic.
               Quadr.
            
             
          
           
             
               Art.
               Sines
            
             .
          
           
             
               Quadr.
               generally
            
             .
          
           
             Sector
             .
          
           
             
               Art.
               Sine
            
             .
          
           
             Quadr.
             
          
           
             Sector
          
           
             Partic.
             Q.
             
          
           
             Artific
             .
             S.
             
          
           
             
               Gen.
               Quad.
            
             
          
           
             Sector
             .
          
           
             Part.
             Q.
             
          
           
             Artificial
             S.
             &
             T.
             
          
           
             
               Gen.
               Quad.
            
             
          
           
             Sector
             .
          
           
             Partic.
             Q.
             
          
           
             Oblique-Ascention
             .
          
           
             
               Artificial
               S.
            
             &
             Tan.
             
          
           
             
               G.
               Quad.
            
             
          
           
             Sector
             .
          
           
             Artificial-S
             .
             &
             T.
             
          
           
             Partic.
             Q.
             
          
           
             
               Gen.
               Quad
            
          
           
             Sector
             .
          
           
             Partic.
             Q.
             
          
           
             
               Particular
               Quadr.
            
             
          
           
             
               Particular
               Quadrant
            
             .
          
           
             
               Particular
               Quadrant
            
             .
          
           
             
               Artificial-S
               .
               &
            
             T.
             
          
           
             
               Gen.
               Quad.
            
             
          
           
             Sector
             .
          
           
             
               Particular
               Quadrant
               ▪
            
          
           
             
               Gen.
               Quad.
            
             
          
           
             Sector
             .
          
           
             
               Gen.
               Quad
            
          
           
             
               By
               Artificial
               Sines
               &
               Tang.
            
             
          
           
             
               Particular
               Quadrant
            
             .
          
           
             General-Quadr
             .
          
           
             
               By
               the
               Artificial-Sines
               and
               Tangents
            
             .
          
           
             
               By
               the
               Sector
            
             .
          
           
             
               By
               Artifl
            
             .
             S.
             &
             T.
             
          
           
             
               General-Quadr
               ▪
            
          
           
             
               Gen.
               Quad
            
             ,
          
           
             
               By
               the
               Sector
            
             .
          
           
             
               Gen.
               Q●ad
            
          
           
             Sector
             .
          
           
             
               Artificial
               S.
            
             &
             Tat.
             
          
           
             
               Artificial
               S.
            
             &
             T.
             
          
           
             
               By
               the
               General-Quadrant
               &
               Sector
            
             .
          
           
             
               Gen.
               Quad.
            
             
          
           
             
               By
               Artificial
               Sines
               &
               Tang.
            
             
          
           
             
               By
               Artifi
               .
               S.
               &
            
             T.
             
          
           
             
               By
               the
               Sector
            
             .
          
           
             
               General-Quadr
               .
               Or
               ,
               Sector
            
             .
          
           
             
               Particular
               Quadrant
            
             .
          
           
             
               By
               the
               Artificial-Sines
               and
               Tangents
            
             .
          
           
             General-Quadr
             .
          
           
             
               Particular
               Quadrant
            
             .
          
           
             Artificial-S
             .
             &
             T.
             
          
           
             Artificial-S
             .
             &
             T.
             
          
           
             Partic.
             Q.
             
          
           
             
               Gen.
               Quad
            
          
           
             
               Particular
               Quadrant
            
             .
          
           
             Fig.
             I.
             
          
           
             Fig.
             I.
             
          
           
             Fig.
             I.
             
          
           
             Fig.
             II.
             
          
           
             Fig.
             V
             ▪
             
          
           
             Fig.
             VI.
             
          
           
             Fig.
             IV.
             
          
           
             Fig.
             VII
             .
          
           
             Fig.
             VIII
             .