Dary's miscellanies examined and some of his fundamental errors detected by authority of ancient and modern mathematicians ... : to which is added a task for Mr. Dary of his own setting / by Robert Anderson.
         Anderson, Robert, fl. 1668-1696.
      
       
         
           1670
        
      
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             Dary's miscellanies examined and some of his fundamental errors detected by authority of ancient and modern mathematicians ... : to which is added a task for Mr. Dary of his own setting / by Robert Anderson.
             Anderson, Robert, fl. 1668-1696.
          
           [3], 13 p.
           
             Printed for Philip Brooksby ...,
             London :
             1670.
          
           
             Reproduction of original in Bodleian Library.
          
        
      
    
     
       
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         eng
      
       
         
           Dary, Michael. -- Dary's miscellanies.
           Mathematics -- Early works to 1800.
        
      
    
     
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           DARY'S
           Miscellanies
           Examined
           ;
           And
           some
           of
           his
           Fundamental
           Errors
           DETECTED
           .
        
         
           BY
           Authority
           of
           Ancient
           and
           Modern
           MATHEMATICIANS
           .
        
         
           The
           Ancient
           :
           
             Euclid
             ,
             Diophantus
             ,
             Apollonius
          
           and
           Archimedes
           .
        
         
           The
           Modern
           :
           
             Xylander
             ,
             Bachetus
             ,
             Stevin
             ,
             Albert
             Girard
             ,
             Torricellius
          
           and
           Regeomontanus
           .
        
         
           To
           which
           is
           added
           ,
           
             A
             Task
             for
             Mr.
          
           Dary
           
             of
             his
             own
             setting
          
           .
        
         
           By
           
             Robert
             Anderson
          
           .
        
         
           London
           ,
           Printed
           for
           
             Philip
             Brooksby
          
           ,
           next
           to
           the
           
             Golden
             Ball
          
           ,
           near
           the
           Hospital
           gate
           ,
           in
           West-Smithfield
           .
           1670.
           
        
      
    
     
       
         
         
         
           DARY'S
           Miscellanies
           EXAMINED
           .
           And
           some
           of
           his
           Fundamental
           Errours
           DETECTED
           .
        
         
           WHilst
           those
           miscellanies
           were
           printing
           ,
           I
           met
           with
           two
           of
           Master
           Darys
           friends
           together
           ,
           at
           their
           office
           in
           Holborn
           and
           they
           related
           to
           me
           that
           Master
           Dary
           had
           a
           Book
           in
           the
           press
           ,
           in
           the
           Preface
           whereof
           he
           had
           a
           quarrel
           with
           me
           ,
           and
           that
           he
           was
           resolved
           to
           vex
           me
           ;
           my
           answer
           was
           ,
           that
           if
           he
           gave
           me
           bad
           language
           I
           would
           lay
           it
           under
           my
           feet
           ;
           but
           if
           he
           gave
           bad
           mathematicks
           I
           would
           return
           it
           to
           him
           again
           ;
           therefore
           ,
           all
           those
           calumnies
           ,
           that
           bad
           and
           scurrilous
           language
           ,
           (
           for
           such
           are
           their
           only
           demonstrations
           )
           either
           by
           him
           or
           by
           any
           of
           his
           crew
           in
           that
           preface
           given
           ,
           or
           may
           be
           hereafter
           given
           in
           any
           of
           their
           writings
           ,
           I
           shall
           take
           no
           further
           notice
           of
           ,
           but
           shall
           ever
           lay
           them
           as
           dirt
           under
           foot
           ;
           but
           I
           shall
           prosecute
           a
           close
           conviction
           of
           their
           erronious
           principles
           in
           Geometry
           .
           That
           preface
           those
           
           authors
           divides
           into
           two
           parts
           ,
           the
           first
           against
           
             Stereometrical
             Propositions
          
           ,
           the
           second
           in
           defence
           of
           the
           ART
           of
           
             practical
             gaging
          
           ;
           and
           as
           they
           have
           little
           to
           say
           against
           the
           first
           ,
           they
           have
           as
           little
           to
           say
           in
           defence
           of
           the
           latter
           ,
           but
           in
           both
           I
           shall
           easily
           subvert
           their
           crippled
           arguments
           .
           To
           the
           first
           ,
        
         
           In
           the
           first
           page
           of
           the
           preface
           ,
           Master
           Dary
           hath
           it
           thus
           ;
           
             in
             the
             tail
             of
             which
             Book
             there
             is
             a
             whole
             broad
             side
             .
          
           Here
           he
           is
           outragious
           because
           he
           was
           so
           perfectly
           confuted
           in
           the
           tail
           of
           
             the
             guide
             to
             the
             young
             gager
             :
          
           truly
           ,
           as
           it
           was
           the
           confutation
           
             of
             the
             ART
             of
             practical
             gaging
          
           ;
           it
           deserved
           no
           better
           preferment
           ,
           than
           to
           be
           put
           in
           the
           tail
           of
           the
           young
           gagers
           guide
           :
           however
           ,
           if
           I
           find
           Master
           Dary's
           understanding
           improved
           by
           my
           instructions
           there
           given
           ,
           in
           these
           his
           Miscellanies
           ;
           I
           shall
           to
           encourage
           him
           ,
           commend
           him
           in
           the
           tail
           of
           this
           .
        
         
           In
           the
           second
           page
           of
           the
           Preface
           ,
           he
           hath
           these
           words
           ,
           
             the
             word
             frustum
             pyramide
             I
             cannot
             understand
             ,
             But
             if
             he
             had
             said
             frustum
             of
             a
             pyramid
             ,
          
           &c.
           this
           complaint
           may
           consist
           of
           three
           parts
           ,
           first
           ,
           frustum
           pyramide
           ;
           second
           ,
           I
           cannot
           understand
           ;
           third
           ,
           frustum
           of
           a
           pyramide
           ;
           if
           we
           compare
           the
           first
           with
           the
           last
           ,
           we
           shall
           find
           them
           both
           of
           one
           signification
           ,
           for
           frustum
           signifies
           a
           broken
           piece
           ,
           therefore
           it
           is
           as
           well
           sense
           to
           say
           a
           broken
           pyramid
           ,
           as
           to
           say
           a
           piece
           of
           a
           pyramid
           ;
           one
           familiar
           example
           for
           many
           ,
           a
           broken
           knife
           ,
           as
           to
           say
           a
           piece
           of
           a
           knife
           .
           Such
           expressions
           are
           brief
           and
           well
           understood
           ,
           both
           signifying
           the
           same
           thing
           ,
           and
           he
           himself
           using
           the
           same
           expressions
           in
           the
           31
           and
           32.
           89
           pag
           ,
           of
           
             the
             Art
             of
             practical
             gaging
          
           :
           thus
           ,
           
             is
             the
             content
             of
             the
             frustum
             pyramid
             ,
          
           and
           in
           29
           page
           of
           that
           
             Book
             of
             Art
          
           ,
           you
           have
           it
           thus
           ,
           
             Master
             Michael
             Dary
             ,
             an
             ingenious
             
             Artist
             and
             practised
             gager
             :
             when
             it
             is
             the
             frustum
             of
             a
             Cylindriod
             :
          
           here
           we
           find
           him
           a
           giving
           himself
           a
           good
           character
           ;
           and
           then
           telling
           us
           of
           a
           frustum
           of
           a
           Cylindriod
           ,
           and
           in
           page
           32
           of
           that
           Book
           of
           Art
           ,
           thus
           ,
           
             to
             cut
             this
             Cylindriod
          
           ,
           here
           we
           may
           observe
           this
           ingenious
           artist
           ,
           how
           in
           one
           page
           he
           calls
           it
           a
           frustum
           of
           a
           Cylindriod
           ,
           and
           immediately
           he
           calls
           the
           same
           solid
           a
           Cylindriod
           ,
           so
           he
           makes
           no
           difference
           betwixt
           the
           part
           and
           the
           whole
           .
           Further
           ,
           as
           for
           those
           mighty
           words
           of
           art
           ,
           to
           wit
           ,
           Cylindroid
           ,
           prismoid
           and
           peripetasma
           ;
           I
           shall
           say
           only
           this
           ,
           
             
               Oft
               have
               I
               known
               some
               men
               of
               no
               great
               parts
               ,
            
             
               Stuff
               up
               their
               mouths
               with
               mighty
               words
               of
               arts
               .
            
          
           For
           his
           2.
           complaint
           ,
           that
           is
           ,
           
             I
             cannot
             understand
          
           ;
           it
           troubles
           me
           to
           hear
           it
           ,
           yet
           I
           see
           his
           understanding
           mend
           a
           little
           ,
           as
           we
           shall
           observe
           hereafter
           .
           In
           the
           same
           page
           ,
           he
           is
           angry
           because
           that
           irregular
           frustum
           is
           cut
           into
           so
           many
           parts
           ,
           if
           that
           do
           not
           please
           his
           worship
           he
           may
           take
           one
           of
           the
           other
           ways
           which
           cuts
           that
           solid
           into
           fewer
           parts
           ;
           for
           there
           are
           four
           ways
           every
           one
           lesse
           work
           than
           other
           .
           But
           the
           gunner
           and
           his
           crew
           must
           be
           a
           shooting
           though
           but
           with
           pot-guns
           .
        
         
           In
           the
           third
           page
           ,
           our
           gunner
           gives
           the
           
             seventh
             prop.
          
           of
           the
           5
           of
           Diophantus
           a
           Broad-side
           ;
           thus
           ,
           
             the
             stress
             of
             his
             argument
             is
             weak
             and
             infirm
             .
             Though
             we
             should
             grant
             Z
             equal
             to
             ⅗
             ,
             it
             is
             yet
             to
             demonstrate
             that
             Z
             is
             3
             and
             A
             5.
             
             A
          
           by
           supposition
           was
           an
           unit
           ,
           then
           reduce
           them
           to
           one
           denomination
           ,
           and
           that
           denomination
           being
           rejected
           ,
           Z
           will
           be
           equal
           to
           3
           ,
           and
           A
           equal
           to
           5
           ▪
           this
           he
           looks
           upon
           as
           an
           hard
           demonstration
           ,
           
             which
             I
             am
             not
             bound
             to
             tell
             him
             how
             to
             do
             ,
          
           saith
           he
           .
           Further
           ,
           he
           dwindles
           to
           his
           Reader
           ,
           hoping
           for
           glory
           and
           would
           know
           ,
           
             Whether
             this
             proposition
             
             hath
             any
             relation
             at
             all
             to
             gaging
             .
          
           I
           answer
           yes
           ,
           and
           argue
           thus
           ,
           numbers
           have
           relation
           to
           gaging
           ,
           this
           Proposition
           is
           of
           Numbers
           ;
           therefore
           this
           Proposition
           hath
           relation
           to
           Gaging
           .
           Again
           ,
           triangles
           have
           relation
           to
           gaging
           ,
           this
           proposition
           is
           of
           triangles
           ;
           therefore
           ,
           this
           proposition
           hath
           relation
           to
           gaging
           .
           This
           prop.
           which
           he
           quarrels
           with
           ,
           is
           the
           7.
           of
           5.
           of
           Diophantus
           ,
           as
           is
           cited
           in
           the
           106
           page
           of
           
             Stereo
             .
             Prop.
          
           and
           seeing
           he
           hath
           so
           much
           immodesty
           as
           to
           say
           his
           arguments
           are
           weak
           and
           infirm
           ,
           I
           shall
           set
           down
           the
           text
           as
           Bachetus
           hath
           it
           .
           
             Esto
             primus
          
           1
           N.
           
             secundus
             unitatum
             quotlibet
             ,
             puta
             1.
             &
             est
             productus
             eorum
             multiplicatione
          
           1
           N.
           
             summa
             vero
             quadratorum
             est
          
           1
           Q
           +
           1.
           adde
           1
           N.
           fit
           1
           Q
           +
           1
           N
           +
           1
           
             aequalis
             quadrato
             .
             Esto
             latus
             ejus
          
           1
           N
           −
           2.
           
             fit
             quadratus
          
           1
           Q
           +
           4
           −
           4
           N.
           aequalis
           1
           Q
           +
           1
           N
           +
           1.
           
             &
             fit
          
           1
           N.
           ⅗
           
             ad
             positiones
             ,
             Erit
             primus
             ⅗
             ,
             secundus
             5
             /
             5
             :
             &
             abjecto
             denominatore
             ,
             erit
             primus
             3.
             secundus
             5.
             &
             postulatis
             respondent
             .
          
           So
           then
           I
           have
           these
           witnesses
           on
           my
           side
           ,
           1.
           
           Diophantus
           the
           author
           of
           the
           proposition
           ,
           2.
           
           
             Xylander
             .
             3.
             
             Bachetus
             .
             4.
             
             Stevin
             .
             5.
             
             Albert
             Girard
             .
          
           Those
           four
           commentators
           upon
           Diophantus
           every
           one
           of
           them
           setting
           it
           as
           above
           .
           6.
           
           
             Truth
             it self
             ,
             and
             it
             will
             prevail
             .
          
           So
           then
           ,
           if
           Mr.
           Dary
           cannot
           bring
           better
           authority
           then
           these
           on
           his
           side
           for
           the
           stress
           of
           his
           argument
           ;
           I
           shall
           conclude
           ,
           that
           pride
           and
           ignorance
           is
           baffled
           ;
           and
           where
           he
           saith
           I
           fling
           dirt
           in
           the
           face
           of
           
             Van
             Schooten
          
           ,
           I
           may
           very
           well
           say
           he
           flingeth
           dirt
           in
           the
           face
           of
           these
           5
           authors
           ,
           yea
           and
           in
           the
           face
           of
           
             truth
             it self
          
           .
           Further
           ,
           had
           this
           7.
           and
           8.
           prop.
           of
           the
           5
           of
           Diop
           been
           observed
           by
           the
           proposer
           and
           resolver
           of
           that
           question
           ,
           it
           is
           very
           likely
           it
           would
           not
           have
           been
           proposed
           by
           the
           one
           ,
           nor
           resolved
           
           by
           the
           other
           ;
           however
           ,
           what
           I
           said
           concerning
           Van-Schooten
           and
           
             des
             Cartes
          
           is
           true
           and
           just
           ,
           therefore
           no
           dirt
           .
           In
           the
           fourth
           page
           ,
           our
           gunner
           hath
           more
           fire-works
           ,
           to
           wit
           ,
           
             his
             note
             for
             progressions
             is
             invalid
             and
             of
             no
             force
             .
          
           For
           saith
           he
           ,
           
             there
             is
             no
             need
             of
             unity
             for
             the
             first
             terme
             of
             this
             progression
             .
          
           My
           answer
           is
           ,
           that
           note
           for
           progression
           is
           of
           force
           and
           truth
           ,
           and
           unity
           of
           use
           ;
           thus
           ,
           the
           question
           it self
           requires
           whole
           numbers
           ;
           the
           seventh
           of
           the
           fifth
           of
           Dioph.
           finds
           whole
           numbers
           ;
           therefore
           greathan
           an
           unit
           ;
           therefore
           well
           limited
           .
        
         
           The
           second
           part
           ,
           in
           defence
           of
           the
           
             Art
             of
             Practicall
             gaging
          
           ,
           and
           it
           begins
           in
           the
           fifth
           page
           ,
           and
           there
           he
           telleth
           his
           Reader
           how
           he
           hath
           
             been
             commended
             by
             divers
             artists
             in
             this
             City
             .
          
           Here
           he
           appeals
           to
           men
           as
           ignorant
           ,
           as
           himself
           is
           vain
           glorious
           .
           In
           the
           6
           pag.
           he
           flingeth
           dirt
           in
           the
           face
           of
           the
           printer
           ,
           thus
           ,
           
             in
             which
             I
             see
             there
             are
             many
             press-faults
          
           ;
           that
           is
           false
           ,
           they
           are
           the
           segment
           makers
           faults
           ,
           for
           the
           segments
           are
           the
           complement
           of
           one
           to
           the
           other
           to
           100000
           &c.
           therefore
           no
           printers
           faults
           .
           In
           the
           sixth
           and
           seventh
           pages
           ,
           he
           sheweth
           how
           to
           calculate
           a
           table
           of
           segments
           ,
           and
           here
           his
           understanding
           mends
           a
           little
           ,
           for
           he
           works
           pretty
           well
           since
           the
           last
           time
           I
           taught
           him
           ;
           so
           then
           ,
           as
           one
           mends
           in
           his
           Rules
           ,
           so
           I
           hope
           the
           other
           will
           mend
           in
           his
           calculation
           ,
           (
           with
           that
           instruction
           I
           formerly
           gave
           him
           )
           so
           we
           may
           expect
           a
           better
           table
           of
           segments
           some
           time
           or
           other
           .
           In
           the
           8.
           page
           he
           again
           dwindles
           and
           would
           fain
           insinuate
           into
           the
           affection
           of
           his
           Reader
           ;
           and
           make
           him
           believe
           that
           I
           did
           not
           know
           that
           there
           was
           a
           third
           &c.
           differences
           in
           the
           table
           of
           segments
           ,
           to
           speak
           the
           truth
           ,
           that
           table
           of
           segments
           was
           calculated
           so
           falsly
           that
           the
           first
           differences
           
           did
           manifestly
           shew
           it
           ;
           further
           ,
           If
           Mr.
           Dary
           had
           known
           that
           way
           ,
           or
           any
           other
           way
           better
           to
           examine
           Tables
           by
           ,
           before
           he
           published
           those
           segments
           ;
           more
           shame
           to
           him
           to
           publish
           such
           false
           tables
           ,
           without
           examination
           .
           In
           the
           9
           and
           10.
           pa.
           the
           Gunner
           has
           fire
           and
           gun-powder
           ,
           
             viz.
             know
             ye
             not
             that
             the
             Table
             for
             wine
             ,
             ale
             and
             beer
             ,
             are
             capable
             but
             only
             of
             the
             first
             and
             second
             differences
             .
          
           If
           so
           ,
           more
           shame
           to
           the
           Calculator
           that
           they
           have
           more
           diff
           .
           and
           they
           so
           much
           confusedly
           put
           .
           As
           for
           that
           Book
           entituled
           
             A
             guide
             to
             the
             young
             gager
             ,
          
           I
           knew
           not
           the
           man
           nor
           heard
           of
           the
           Book
           untill
           a
           great
           part
           of
           it
           was
           printed
           ,
           neither
           did
           I
           see
           one
           line
           of
           that
           part
           of
           it
           ,
           till
           it
           was
           publickly
           exposed
           to
           sale
           .
           Thus
           have
           I
           passed
           through
           this
           fiery
           conflict
           ,
           and
           have
           not
           heard
           the
           bounce
           of
           one
           gun
           ,
           nor
           received
           any
           harme
           ,
           which
           makes
           me
           conclude
           our
           gunner
           and
           his
           crew
           are
           as
           bad
           marks
           men
           ,
           as
           they
           are
           segment
           makers
           ,
           for
           he
           promi●ed
           at
           the
           beginning
           of
           his
           preface
           to
           charge
           his
           guns
           and
           pepper
           me
           .
           Thus
           have
           I
           considered
           him
           as
           a
           Gunner
           with
           his
           Crew
           ;
           now
           will
           I
           consider
           him
           as
           a
           Geometer
           ,
           with
           his
           famous
           Companions
           .
        
         
           
             These
             famous
             men
             ,
             whose
             true
             descent
             doth
             run
          
           
             From
             aged
             Neptune
             ,
             and
             the
             glorious
             Sun.
             
          
        
      
       
         
         
           AN
           EXAMINATION
           OF
           Dary's
           Miscellanies
           .
        
         
           IN
           the
           first
           page
           of
           the
           Preface
           ,
           he
           saith
           ,
           
             Most
             whereof
             have
             lain
             by
             me
             many
             years
             :
          
           If
           so
           ,
           I
           hope
           very
           true
           .
        
         
           1
           In
           the
           second
           page
           of
           the
           Preface
           ,
           saith
           he
           ,
           
             For
             although
             the
             sides
             thereof
             be
             continued
             ,
             they
             would
             never
             be
             included
             or
             terminated
             in
             one
             point
             ,
             as
             the
             Pyramide
             is
          
           ;
           that
           is
           ,
           the
           sides
           of
           a
           Pyramide
           are
           included
           in
           one
           point
           ,
           which
           I
           deny
           ,
           thus
           ;
           a
           point
           hath
           no
           part
           ,
           by
           1
           def
           .
           1
           Euclid
           .
           A
           Superfices
           (
           for
           such
           are
           the
           sides
           of
           a
           Pyramide
           )
           have
           length
           and
           breadth
           5
           def
           .
           1
           Euclid
           .
           That
           which
           hath
           no
           part
           ,
           to
           include
           that
           which
           hath
           length
           and
           breadth
           ,
           is
           absurd
           ;
           that
           's
           a
           lumping
           point
           for
           an
           able
           Anylist
           .
        
         
           2
           In
           the
           fourteenth
           page
           ,
           saith
           he
           ,
           
             The
             3
             Angles
             of
             any
             Spherical
             Triangle
             being
             given
             ,
             there
             are
             likewise
             three
             sides
             of
             another
             Spherical
             Triangle
             given
             ,
             whose
             Angles
             are
             equal
             to
             the
             sides
             of
             the
             former
             Triangle
             .
          
           Here
           the
           Gentlemen
           forgot
           to
           complement
           ,
           and
           I
           presume
           in
           the
           next
           they
           will
           forget
           all
           good
           manners
           .
           Further
           ,
           the
           sum
           of
           the
           sides
           of
           any
           spherical
           
           triangle
           ,
           are
           less
           then
           two
           semi-circles
           ,
           Reg.
           39
           ▪
           of
           3.
           
           The
           sum
           of
           the
           three
           angles
           of
           any
           spherical
           triangle
           ,
           are
           greater
           then
           two
           right
           angles
           ,
           but
           less
           then
           six
           ,
           Reg.
           49
           of
           3.
           therefore
           the
           Rule
           is
           false
           ,
           except
           the
           sum
           of
           the
           three
           sides
           be
           greater
           then
           two
           right
           angles
           ;
           but
           the
           Rule
           is
           set
           down
           general
           ,
           therefore
           a
           general
           error
           .
        
         
           3
           In
           page
           21.
           we
           have
           it
           thus
           ;
           
             If
             a
             sphere
             be
             by
             a
             plain
             touch'd
             ,
             and
             the
             eye
             be
             placed
             at
             the
             center
             of
             the
             sphere
             ,
             then
             a
             right
             line
             infinitely
             extended
             from
             the
             eye
             to
             any
             assigned
             point
             in
             the
             spherical
             surface
             ,
             shall
             project
             the
             assigned
             point
             upon
             the
             plain
             .
          
           Here
           the
           Radius
           of
           the
           Sphere
           is
           taken
           to
           be
           infinite
           ,
           for
           ,
           saith
           he
           ,
           then
           a
           right
           line
           infinitely
           extended
           from
           the
           eye
           to
           any
           assigned
           point
           in
           the
           spherical
           surface
           ;
           but
           the
           plain
           is
           without
           the
           sphere
           ,
           therefore
           beyond
           infiniteness
           it self
           ,
           which
           is
           absurd
           :
           however
           this
           proves
           them
           to
           be
           infinite
           Projectors
           .
        
         
           4
           In
           page
           29.
           at
           the
           18th
           it
           is
           thus
           ;
           
             If
             a
             sphere
             be
             inclosed
             in
             a
             cylinder
             ,
             and
             that
             cylinder
             be
             cut
             with
             plains
             parallel
             to
             its
             base
             ,
             then
             the
             intercepted
             rings
             of
             the
             cylinder
             are
             equal
             to
             the
             intercepted
             surfaces
             of
             the
             respective
             segments
             of
             the
             sphere
          
           ;
           that
           is
           false
           :
           For
           ▪
           
             Hemisphaerii
             superficies
             aequalis
             est
             superficiei
             curvae
             cylindri
             eadem
             ipsi
             basim
             ,
             &
             eadem
             altitudinem
             habentis
             ,
          
           saith
           Torricellius
           at
           the
           18.
           
           Prop.
           
             de
             sphaera
             ,
             &
             solidis
             sphaeralibus
             lib.
             prim
             .
          
           and
           as
           the
           whole
           ,
           so
           the
           parts
           ,
           by
           the
           19.
           
           Prop.
           of
           the
           same
           .
           Here
           we
           have
           a
           combat
           betwixt
           Torricellius
           and
           our
           Geometers
           :
           First
           ,
           they
           say
           the
           intercepted
           rings
           of
           the
           cylinder
           are
           equal
           to
           the
           intercepted
           surfaces
           of
           the
           respective
           segments
           of
           the
           sphere
           .
           Torricellius
           proves
           ,
           that
           the
           intercepted
           superfices
           of
           the
           cylinder
           ,
           are
           equal
           to
           the
           intercepted
           superfices
           
           of
           the
           respective
           segments
           of
           the
           sphere
           .
           2.
           
           These
           Geometers
           say
           ,
           if
           a
           sphere
           be
           inclosed
           in
           a
           cylinder
           ,
           here
           we
           may
           make
           the
           Diameters
           of
           the
           base
           of
           the
           cylinder
           of
           any
           magnitude
           ,
           greater
           then
           the
           diameter
           of
           the
           sphere
           ,
           and
           yet
           the
           sphere
           be
           inclosed
           ▪
           Torricellius
           proves
           ,
           that
           the
           cylinder
           and
           hemisphere
           must
           have
           the
           same
           base
           .
           3.
           
           These
           Geometers
           regard
           it
           not
           ,
           whether
           the
           sphere
           and
           cylinder
           are
           upright
           or
           inclining
           .
           Torricellius
           by
           construction
           makes
           them
           upright
           .
           Thus
           do
           these
           Geometers
           make
           solid
           superfices
           ,
           for
           a
           ring
           is
           solid
           .
        
         
           5
           In
           page
           33.
           they
           set
           down
           a
           rule
           for
           the
           sphere
           ,
           and
           conclude
           it
           will
           hold
           in
           the
           spheroid
           ;
           
             this
             rule
             will
             also
             hold
             if
             it
             were
             the
             Frustum
             of
             a
             Spheroid
             ,
             putting
             d●d
             equal
             to
             the
             fact
             of
             the
             right
             angled
             conjugates
             in
             the
             base
             .
          
           That
           is
           false
           ,
           by
           21
           of
           1
           of
           Apollonius
           ,
           and
           31
           and
           33
           of
           Archimedes
           of
           conoid
           and
           spheroid
           ;
           for
           the
           diameter
           of
           the
           base
           one
           way
           ,
           or
           the
           right
           angled
           conjugates
           of
           the
           base
           the
           other
           ,
           with
           the
           height
           of
           either
           ,
           will
           not
           limit
           a
           spheroid
           ,
           as
           the
           diameter
           of
           the
           base
           and
           the
           height
           doth
           a
           sphere
           .
           This
           very
           rule
           Crowns
           all
           their
           endeavours
           ;
           for
           before
           they
           had
           made
           a
           point
           bigger
           then
           any
           superfice
           ,
           a
           line
           longer
           then
           infiniteness
           ,
           a
           solid
           superfice
           ,
           but
           now
           they
           are
           come
           to
           an
           unlimited
           solid
           .
        
         
           6
           In
           page
           39.
           they
           write
           thus
           ;
           
             But
             if
             such
             a
             solid
             have
             not
             its
             Zons
             made
             by
             circles
             or
             ellipses
             ,
             but
             by
             four
             flat
             sides
             at
             right
             angles
             to
             the
             foresaid
             conjugates
             ,
             then
             it
             is
             a
             prismoid
             ;
             nevertheless
             ,
             the
             rules
             before
             prescribed
             ,
             hold
             to
             all
             intents
             and
             purposes
             :
          
           that
           is
           false
           to
           all
           intents
           and
           purposes
           at
           the
           first
           appearance
           ;
           for
           if
           two
           right
           lines
           be
           at
           right
           angles
           ,
           
           and
           they
           be
           at
           right
           angles
           with
           four
           plains
           ,
           those
           plains
           wil
           be
           the
           4
           sides
           of
           a
           Parallelepipedon
           ,
           by
           the
           2
           ,
           3
           ,
           and
           30
           def
           .
           of
           11
           Euclid
           .
           a
           Parallelepipedon
           being
           calculated
           gradually
           ,
           can
           have
           but
           a
           first
           difference
           ,
           and
           not
           a
           second
           and
           third
           :
           But
           this
           is
           like
           the
           rest
           of
           these
           Famous
           Geometers
           works
           .
           Our
           Master
           Geometer
           telleth
           his
           Reader
           thus
           ,
           
             Most
             whereof
             have
             lain
             by
             me
             many
             years
             .
          
           And
           in
           the
           Title
           Page
           he
           saith
           ,
           they
           are
           brief
           Collections
           from
           divers
           Authors
           :
           If
           so
           ,
           why
           so
           many
           Fundamental
           Errors
           ?
           Further
           ,
           seeing
           M.
           Dary
           ,
           and
           his
           Companions
           will
           assert
           any
           thing
           ,
           and
           demonstrate
           nothing
           ,
           except
           they
           are
           required
           in
           print
           ;
           therefore
           I
           desire
           them
           to
           demonstrate
           these
           six
           following
           assertions
           of
           their
           own
           ,
           and
           and
           shall
           call
           it
        
         
           
             A
             Task
             for
             Mr.
             Dary
             of
             his
             own
             setting
             .
          
           
             To
             wit
             :
          
           
             1
             The
             sides
             of
             a
             Pyramide
             being
             32000.
             
             I
             desire
             Mr.
             Dary
             to
             give
             one
             point
             to
             include
             that
             superfice
             ,
             as
             he
             asserts
             in
             page
             2.
             
          
           
             2
             The
             three
             sides
             of
             a
             Spherical
             Triangle
             ,
             being
             6
             ,
             8
             ,
             and
             10
             degrees
             ,
             their
             sum
             24
             degrees
             ,
             I
             desire
             Mr.
             Dary
             to
             give
             a
             Spherical
             Triangle
             ,
             whose
             sum
             of
             the
             three
             angles
             are
             24
             degrees
             ,
             as
             he
             asserts
             in
             page
             14.
             
          
           
             3
             In
             the
             Gnomonick
             Projection
             ,
             the
             Radius
             of
             the
             Sphere
             being
             infinite
             ,
             and
             the
             arch
             from
             the
             touch
             point
             to
             the
             assigned
             point
             be
             30
             degrees
             ,
             I
             desire
             Mr.
             Dary
             to
             extend
             a
             line
             from
             the
             center
             of
             that
             sphere
             ,
             by
             the
             assigned
             point
             ,
             to
             
             the
             touching
             plain
             ,
             that
             is
             further
             then
             infiniteness
             ,
             as
             he
             asserts
             in
             page
             21.
             
          
           
             4
             If
             a
             Cylinder
             and
             Hemisphere
             be
             of
             one
             height
             ,
             but
             the
             diameter
             of
             the
             base
             of
             the
             Cylinder
             be
             greater
             then
             ,
             or
             equal
             to
             the
             diameter
             of
             that
             Sphere
             ,
             and
             they
             concentrick
             ,
             this
             Hemisphere
             is
             inclosed
             in
             that
             Cylinder
             ;
             let
             that
             Hemisphere
             and
             Cylinder
             be
             cut
             with
             Plains
             parallel
             to
             their
             bases
             I
             desire
             Mr.
             Dary
             to
             prove
             ,
             that
             those
             intercepted
             rings
             of
             the
             Cylinder
             (
             that
             is
             solid
             rings
             )
             are
             equal
             to
             the
             intercepted
             surfaces
             of
             the
             respective
             segments
             of
             the
             Sphere
             ,
             as
             he
             asserts
             in
             page
             29
          
           
             5
             In
             a
             Spheroid
             ,
             let
             6
             be
             the
             perpendicular
             height
             of
             the
             Frustum
             ,
             8
             the
             diameter
             of
             the
             base
             ,
             when
             cut
             by
             a
             Plain
             at
             right
             angles
             with
             the
             Axis
             ;
             let
             10
             and
             12
             be
             the
             right
             angled
             conjugates
             in
             the
             base
             (
             as
             he
             calls
             them
             )
             when
             the
             cutting
             Plain
             is
             parallel
             to
             the
             axis
             ,
             the
             altitude
             of
             the
             Frustum
             4.
             
             I
             desire
             Mr.
             Dary
             to
             give
             one
             example
             in
             each
             ,
             if
             but
             one
             ;
             if
             more
             then
             one
             ,
             to
             give
             them
             all
             ;
             that
             is
             ,
             to
             prove
             it
             a
             limited
             Proposition
             ,
             as
             he
             asserts
             in
             page
             33.
             
          
           
             6
             If
             two
             right
             lines
             ,
             to
             wit
             ,
             one
             6
             ,
             the
             other
             8
             ,
             be
             at
             right
             angles
             ,
             and
             these
             two
             lines
             be
             at
             right
             angles
             with
             four
             Plains
             ,
             the
             height
             of
             these
             plains
             may
             be
             12.
             those
             will
             be
             the
             limits
             of
             a
             solid
             ,
             which
             Euclid
             names
             a
             Parallelepipedon
             ,
             at
             the
             30
             def
             .
             of
             11.
             
             I
             desire
             Mr.
             Dary
             to
             prove
             such
             a
             solid
             to
             be
             a
             Prismoid
             ,
             and
             to
             have
             second
             and
             third
             differences
             ,
             as
             he
             asserts
             in
             page
             39.
             
          
        
         
           
           
             Now
             to
             commend
             him
             .
          
           
             THose
             six
             Assertions
             of
             Mr.
             
             Dary's
             ,
             may
             well
             be
             termed
             
               A
               Task
            
             for
             him
             for
             six
             daies
             ;
             which
             Assertions
             being
             performed
             according
             to
             the
             Rules
             of
             Geometry
             ,
             I
             shall
             ever
             conclude
             Mr.
             Dary
             to
             be
             great
             ,
             yea
             greater
             ;
             nay
             the
             greatest
             Geometer
             of
             all
             mortal
             men
             .
          
           
             But
             if
             Mr.
             Dary
             ,
             with
             the
             help
             of
             his
             Companions
             ,
             cannot
             or
             will
             not
             fairly
             demonstrate
             these
             their
             Assertions
             ,
             but
             still
             cavil
             and
             quarrel
             it
             out
             .
             I
             may
             well
             conclude
             ,
             his
             or
             their
             Geometry
             is
             not
             ,
             nor
             will
             not
             be
             worth
             taking
             notice
             of
             for
             the
             future
             ;
             for
             that
             Micellanea
             Riff-raff
             
               having
               lain
               by
               him
               many
               years
            
             ;
             and
             we
             may
             be
             sure
             ,
             often
             thumb'd
             over
             with
             much
             care
             and
             prudence
             ,
             like
             an
             ingenious
             Artist
             ,
             and
             a
             practised
             Gager
             ;
             being
             his
             whole
             stock
             of
             Mathematical
             knowledge
             ,
             is
             now
             made
             publick
             ,
             to
             prove
             himself
             what
             he
             is
             ,
             to
             wit
             ,
             a
             Geometer
             full
             of
             errors
             ,
             and
             a
             Mathematician
             altogether
             without
             demonstration
             ;
             therefore
             I
             shall
             imploy
             my
             idle
             time
             better
             then
             in
             confuting
             such
             unwise
             ridiculous
             Assertions
             ;
             for
             this
             we
             may
             be
             sure
             of
             ▪
             that
             whatever
             Mr.
             Dary
             writes
             ,
             will
             be
             full
             of
             Fundamental
             Errors
             .
          
           
             Although
             I
             am
             well
             assured
             ,
             that
             whatever
             Mr.
             Dary
             writes
             will
             be
             so
             full
             of
             Fundamental
             Errors
             ,
             that
             it
             will
             not
             be
             worth
             taking
             notice
             of
             :
             yet
             seeing
             one
             deeply
             swears
             by
             his
             Maker
             he
             would
             have
             us
             never
             agree
             ,
             because
             it
             will
             be
             good
             sport
             for
             them
             ;
             and
             another
             of
             Mr.
             
             Dary's
             friends
             is
             
             desirous
             to
             see
             Paper
             Battels
             ,
             therefore
             I
             shall
             the
             rather
             desist
             :
             However
             ,
             if
             I
             take
             pen
             in
             hand
             again
             ,
             I
             shall
             be
             as
             ready
             to
             bring
             them
             into
             the
             Lift
             ,
             by
             examining
             their
             works
             ,
             as
             they
             are
             desirous
             that
             we
             should
             make
             them
             sport
             .
             Further
             ,
             Mr.
             Dary
             hath
             related
             to
             several
             of
             my
             acquaintance
             ,
             that
             those
             his
             Miscellanies
             were
             published
             as
             a
             snare
             for
             me
             ;
             and
             one
             of
             his
             Crew
             hath
             told
             me
             to
             my
             face
             ,
             that
             he
             could
             be
             revenged
             on
             me
             ,
             and
             never
             appear
             in
             it
             himself
             :
             I
             asked
             him
             how
             ;
             He
             answered
             ,
             he
             could
             hires
             a
             stab
             to
             be
             given
             for
             a
             very
             small
             matter
             :
             My
             answer
             to
             these
             two
             ,
             and
             the
             rest
             of
             them
             is
             ,
             I
             value
             the
             snares
             of
             one
             ,
             the
             stab
             of
             the
             other
             ,
             and
             the
             envy
             of
             the
             rest
             ,
             no
             more
             then
             the
             dirt
             of
             my
             shoes
             ;
             my
             seconds
             shall
             be
             
               Euclid
               ,
               Diophantus
               ,
               Apollonius
            
             and
             Archimedes
             ,
             and
             my
             Weapons
             Truth
             and
             Demonstration
             .
          
           
             FINIS
             .