1837
UNIVERSITY OF MICHIGAN
wahrt
QUEROS CUNINSULA AMOHAM
SCIENTIA
ARTES
VERITAS
LIBRARY
OF THE
TIEBOR
GO
UU
HINDI TIMUR
-
ܙ.ܫ
Crap
.
VeryFor
wrw
با من
望
​,
:
SYSTEM
A
MATHEMATICS
OF THE
CONTAINING
The EUCLIDEAN GEOMETRY, Plane and Spherical
TRIGONOMETRY; the Projection of the SPHERE, both
Orthographic and Stereographic, ASTRONOMY, the
USE of the GLOBES and NAVIGATION:
The Manner of Computing the APPULSES of the MOON to
the Fixed Stars, and their Occultations by the Interpoſition of
Her Body, very uſeful for determining the Difference of Longitude
between Places. With an Account of the ſeveral METHODS
Propoſed and made Uſe of, by the moſt celebrated Aſtronomers
for aſcertaining the ſame.
New Solar TABLES, with their Conſtruction and Uſe, Tables
of the Sun's Place, Right Aſcenſion, Declination, Equation of Natural
Days for every Four Years; with Tables of Variation to make them
ſerve tor a Hundred Years to come; and a Catalogue of the Right Afcenfions,
Declinations, Egece of the moſt Eminent Fixed Stars : Deduced from the
FLAMS TEDIAN OBSERVATION S.
The Conſtruction of the Meridional Parts, Logarithms, Sines, Tangents
and Secants, both Natural and Artificial, by the NEWTONIAN SERIES.
With an Account of the Cycles, Periods, Epoch's, Epacts, Kalendars, Egor
A L SO
A TABLE of Meridional Parts for every Degree and Minute
of Latitude to the Ten Thouſandth Place in Decimals, calculated de Noro.
Together with a Large and very Uſeful TABLE of the Latitudes
and Longitudes of Places; the whole being deſigned for the Uſe of the
MATHEMATICAL SCHOOL, founded by King CHARLES IT.
VOL. I.
By 7 A MES HODGSON,
Maſter of the Royal Mathematical School in Chriſt's Hoſpital,
and Fellow of the ROYAL SOCIETY,
London: Printed for THOMAS PAGE, WILLIAM and
Fisher Mount, at the Poſtern on Tower-Hill. 1723.
中​,
亲
​”。
.
;
To his moſt Sacred MAJESTY
G E OR GE,
King of Great-Britain, France,
and Ireland, &c.
.
THIS
Syſtem of the Mathematicks
Defigned for che
Use of the Royal Mathematical School,
in Chriſt's Hofpital LONDON;
Founded by King CHARLES II.
Is with the greateſt Submiſſion Dedicated
By His Majeſty's moſt Humble,
Moſt Dutiful and moſt Obedient
SUBJECT and SERVANT,
James Hodgſon
了​。
“:
*;
t
“
,
生
​i
SERIES
Isa
The PRE FACE
T is now ſome Years ſince I publiſhed a Trea-
tiſe, intituled, The Theory of NAVIGATION
I Demonſtrated ; and having ſince thatTime been
choſen Maſter of the Royal Mathematical
School, where the Inſtructing of Youth in Aſtro-
nomy and Navigation fell more immediately un-
der my Care; and as the Methods at that Time in Uſé, did
not appear to me fo Rational and Inſtructive as I could have
wiſhed for, I let my ſelf in good Earneſt to make ſuch Al-
terations and Additions in the Form and Manner as I judg'd
moſt convenient; and as the Method made Uſe of in the
above-mentioned Book, met with a kind Reception from
thoſe Perſons whoſe Inclination or Buſineſs led them to Con-
template theſe parts of the Mathematics, I refolved to make
the Method there obſerved, my Rule to walk by, and to
Explain and Amend the Scheme in General : And hence
aroſe what is contained in the following Sheets, which is
an Account of the Method that I have made Ule of with
ſome Succeſs ſince that Time, in the common Courſe of
my Practice; and that the Reader may the better be led in-
to the Nature of the Deſign, it will be neceſſary to give
him ſome ſhort Account of the Work it felf.
The ist Part contains an Abridgment of the firſt Six
Books of the Euclidean Geometry, after a Manner different
from what has hitherto appeared abroad, or that I have
met with; wherein the Chief and Primary Propoſitions are
demonſtrated intirely after his Manner ; and thoſe which
b
more
ii
The Preface.
morcimmediately depend upon them are deduced as Corol-
Tarys, (except in the Fifth Book, wliere I ſlave taken the
Liberty to introduce the Algebraic Way of Reaſoning, and De-
monfrated the Laws of Proportion from the Nature of Ratio's
conſidered; having found it much more ealily attained by
young Beginners this way, thian from the Conlideration of
Equiva'sipkes, &c.) and though it is impoſſible that ſo ſtrict
a Connection and Dependence of the leverai Propoſitions can
be here cbſerved as in Euclid it felf; yet it is abundant-
tly ſuficient for my Purpoſe, and the more inquiſitive
Reader has nething more to do, when he meets with any
Dificulty »: than to apply to Euclid himſelf; and in rea-
ding this Abridgement, he has this Advantage above read-
irg
of others, that the Canal by which his Knowledge is
conveyed, is the fame, and his Way and Manner of Rea-
ſoning and thinking perfectly the ſame; for as in my pri-
vate Opinion the Manner of Demonſtrating the Elements of
Geometry as delivered by Euclid himſelf is not to be mend.
ed; fo I have endeavoured to keep as cloſe to his Method
as the Nature of my Deſign would admit of.
In tlie ad Part there is an Account of various Methods
for Computing the Natural Siñes, Tangents and Secants
, with
an Inveſtigation of the ſeveral Series invented by Sir Iſaaa
Newton, for finding the Length of the Circumterence of
the Circle, in equal Parts of the Radius, of the Sine, Tangent
and Secant, of any Arch in the ſame Parts, with the Appli-
cation of the fame Series to the conſtructing of ihe Trian.
gular Canon, the Quadrature of the Circle, and the Soluti-
on of the Archimedian Problems relating to the Sphere, the
Cylinder, the Cone, &c. Alloa Demonítration of the ſeve.
ral Propoſitions commonly called Axiumi, for the Solution of
the feveral Cafeș in Piane Tringles; and the actual Solution
of the fame Cales in all their Varieties and Ambiguities.
AV. ***
Thet
The Preface.
ili
The 3d Part to the End of the 18th Section, contains
what is properly called Navigation; and begins with an
Introduction preliminary, to the various kinds of Sailing ;
the Solution of the ſeveral Caſes relating to a ſingle Courſe;
and the Solution of ſome uſeful Traverſes, with a large
Traverſe Table, and its Uſes, very neceſſary for the ready
working of Traverſes.
In the Oblique Sailing, you have the Solution of the moſt
uſeful Problems, fuch as thoſe for meaſuring inacceſſible
Diſtances, Surveying of Harbours, finding the Courſe
and Diſtance to hit a Place upon a Wind, determi-
ning the Leeway, and allowing for Currents, with various
Ways of expreſſing the ſame Problem, and variety of unre-
folved Problems to exerciſe the Learner.
The Way and Manner of finding the Bearing and Diſ-
tance of Places by the firing of Guns, may be uſeful
fometimes in diſcovering how far Ships are diſtant from
each other, after they have been ſeparated, or in Times of
Engagement; and for meaſuring the Diſtances of Places,
when more certain Methods cannot take Place.
After this follows the Solution of ſuch Problems as arife
from Sailing under a Parallel, with a Table fhewing the
Breadth of a Degree of Longitude, in any Parallel of La-
titude ; with the Application of both to the finding of the
Difference of Longitude, by the help of the Middle Latitude,
commonly called Middle Latitude Sutling.
In the Mercator's Sailing, you will meet with a Two-fold
Solution of each of the Problems, die Conſtruction and Uſe of
the Chart ſhewn, and ſome Endeavours to explain ſuch Diffi-
culties as Perſons not well skill'd in Geometry and young
Beginners are apt to meet with, and that I might render
this truly uſeful Part of Navigation more Compleat, I have
given a Table of Meridional Parts, calculated a-new, and
independent of each other, to much greater Degrees of Ex-
acneſs than has ever yet appeared; and to induce the Nia-
viga-
b 2
---
..
iv
The Preface.
ever:
vigator to make Uſe of this kind of Sailing, I liave ſhewn
him how a Traverſe may be work’d, by the common Ta-
bles of Difference of Latitude and Depari ure, and the true Dif.
ference of Longitude thence found, in as little Time almoſt
as is required to perform it the common Way.
In Great Circle Sailing I have given Examples in all the
Varieties of Poſitions of Places, touched upon the Difficulty
of applying it to Practice, and ſhewn the near Agreement
of it with the Mercator's Sailing ; and in Order to lead the
Reader into a true and perfect Knowledge of The Theory of
the Art of Navigation, I have given a Solution of the two
principal Problems in Navigation, from the infinite Series ;
by the help of Multiplication and Diviſion, without the
Aliſtance of the Tables of Logarithms, Sines, Tangents Se-
canis, Meridional Parts, or any Artificial Numbers whatſo-
In the next Place there are varieties of Ways for finding
the Latitudes of Places from proper Data; with a very Cor-
rect Table of Refractions, and the Law of Refraction it ſelf
laid down and demonſtrated.
The Methods propoſed for diſcovering the Difference of
Longitudes of Places, are, each of them confirmed by Ex-
amples, demonſtrably true; and fuchas have been approved of
by the ableſt Aſtronomers and Mathmaticians of all Times, by
which the Deſcription of this our Earth has been obtained,
and may be often put in Practice if thoſe whoſe Buſineſs
and Intereit it is, would apply themſelves to it,
The Rules for finding the Variation of the Compaſs ſuc-
ceed, together with proper Methods for meaſuring the
Ships Way, for correcting the Courſe, by making pro-
per Allowances for Leeway, Variation, and other Accidents.
The Manner of working of Days Work, and Journalizing
the ſame, with an Example of a ſix Days kun, which
compleats that Part which falls properly under the Head of
Navigation.
The
The Preface.
V
The remaining Sections which follow in this Part,
and compleat the Volume, were deſigned to come im-
mediately after the End of the 5th Part ; but as they could
not well be placed there without cauſing the 2d Volume to
fwell to an unuſual Size, it was thought proper to place
them where they are ; and ſince they follow in Order, it
may not be amiſs to ſpeak of them in this place.
And firſt you have an Account of Time, and of the
ſeveral ſorts of Days made Uſe of by Aſtronomers ,
wherein conſiſts the Inequality of the Solar Day and its
juſt Quantity enquired into and ſtated;
into and ſtated; and Tables
Shewing how much it differs from the Mean Day, to every
Day in the Years 1724, 1725, 1726, 1727, with a Table
forwing how much the Equation varies in One Hundred Years;
all calculated a-new, from the beſt Solar Tables ; very
neceſſary to be well underſtood by every one,who intends to
apply himſelf to the Calculation of Solar and Lunar Eclipſes;
the Appulſes of the Moon to the Fixed Stars, and the apply-
ing of Aſtronomical Obſervations to Uſe.
And becauſe the Knowledge of the Sun's Place in the
Ecliptic is the Principal Poſtulatum in all Problems of the
Sphere, I have given new Solar Numbers, deduced from the
Flamſteedian Obſervations made at Greenwich, with their Con-
Atruétion and Ufe: And inaſmuch as a Knowledge of the
Declination of the Sun is the next Thing wanting, and an
Ingredient in almoſt every Astronomical Problem, I have
added for the Eaſe of Calculation, Tables Shewing the Decli-
nation of the Sun 10 Seconds, to every two Minutes of the Qua-
drant ; as alſo Tables ſewing the Right Aſcenſion of the Sun
to the ſame Degree of Exa&tneſs, whereby the Sun's Place
being known or given, you have his correſponding Right
Aſcenſion and Declination by Inſpection.
The Laws of Orthographic Projełtion, are laid down and
demonſtrated in a more General Way than is to be found
in
vi
The Preface.
:
in the Books now in Uſe, with their Application to the
Projection of the Sphere upon any Plane; and the Uſe of
thoſe Projections in the Solution of thecommon Aſironomical
Problems,
The Solution of all the Principal Problems in Aſtronomy,
ariſing from the Revolution of the Earth about her Axis,
upon the Copernican Hypotheſis, are delivered in as plain á
Manner as I believe the Subject will admit of
In treating upon the Globes, I have explained the Na.
ture of the ſeveral Circles and Appurtenances,and Exem.
plified their Uſe in the Solution of the chief Problems in
Navigation and Aſtronomy.
And inaſmuch as a Knowledge of the Epacts, Cycles, &c.
are of very great Uſe in many Caſes, and the Ways of find-
ing them delivered generally by Rules only, I have made
ſome Enquiry into the Nature of them, ſhewn from whence
they ariſe, and the Manner how they may be found, de-
duced the Rules for making the Kalendar, and determyi.
ning the Times when the New and Full Moons and move.
able Feafts, Etc. will happen.
After this you have a Diſcourſe of the Nature and Con-
ſtruction of Logarithms, with various Exumples of making the
ſame; and the Manner of computing the Artificial or Lo-
garithmic Sines, Tangents, and Secants, from the Length of
the Arch of the Circle firft given, in cqual Parts of the Ra-
dius , independent of the Tables of Logarithms, Natural
Sines, Tangents and Secants with Examples; in the Per-
formance of which, conſidering the uſefulneſs of the Mat-
ter, I have endeavoured to deliver my ſelf with as much
Clearneſs as I was capable of.
The Fourth Part begins the Second Volume, and contains the
Projection of the Sphere,in which the Laws of Stereographic Pro-
jection are explained and demonſtrated, and applied to the
drawing of Great and SmallCircles upon any Plane Stereogra-
phically,
.
The Preface.
vii
*
phically, to the laying off any Number of Degrees, and of
ineaſuring any part or Portion of a Great or Small Circle
when projected, different ways, and in all the Varieties.
Alſo Spherical Trigon metry, in which the Nature and
Properties of Spherical Triangles are explained at large,
all the Axioms deinonitrated, and the actual Solutions of all
the various Caſes, as well Stereographically as Logarichmical-
ljo various Ways, and to as great Degrees of Exactneſs
as may be.
The 5th and laſt Part, contains the Application of
the Projection of the Sphere, and Spherical Trianeles, to
the Solution of the moſt uleni Altronomical Problems ;
wherein are shown the Proj'ition of the Sphere Stereo
giurbically, upon the Pianes of the ſeveral Great Circles
of the Sphere, with their Uſes ; and the Solutions of all the
chief Problems in Aſtronomy, relating to the Sunor Stars ariſing
from the Diurnal Motion of the Sun, according to the Ptole-
maic Sylein, in all their Varieties and Ambiguities; with the
ſeveral Methods made Uſe of by Aſtronomers for aſcertaining
the Places of the Fixed Stars and Planets, : illuſtrated
with Examples of actual Obſervations: The Doctrine of Pa-
rallaxes, with the Manner of computing the Times of the
Appulies of the Moon to the fixed Stars and their Occulta-
tions, by the Interpoſition of her Body, with an Example of
each; likewiſe new Tables of the Sun's Place, Declination, Ec*c.
for the next Hundred Years to come, computed from the
belt Solar Numbers, to the greateſt Exactneſs poſſible.
A very Correct and Exact Catalogue of the moſt Eminent
Fixed Stars, and the largeſt and (I believe, correctelt Ta-
ble of the Latitudes and Longitudes of Places yet pub.
bliſhed.
As the only End of Writing is Information, ſo I have
endeavoured thro'out the Whole Courſe of this work to de
liver every thing with as much Plainneſs as pollible, and if
f
in
viii
The Preface.
in my eager Purſuit after this, I may have ſometimes over
done it, and by ſtriving to make it very plain, have ren-
dered it not quite ſo clear (a Thing not impoſſible) I am
ſatisfied the Candid and Ingenuous Reader will readily for-
give me, eſpecially ſince 'tis a Fault ſeldom to be met with
in Mathematical Books now-a Days
As to the Work in General, I can't recollect that there
is one Thing left undemonttrated, that is capable of it; and
as the Calculations are all done to the greateſt Degree of
Exactneſs, and have all gone through my own Hands, if
any ſmall miſtake may have happened, which in a Work of
this Bulk is not altogether unlikely ; I am ſure the
gene-
rous and honeſt Part of Mankind will readily excuſe it; and
the rather, in that the Papers have not been ſubjected to
the View and Judgment of any one Perſon ; and as for
thoſe Perſons whoſe Excellency lies in finding Fault with
every Thing, and very often when there is no real Occa-
fion, and of diſcovering Errors where there are none; all
thať ſeems needful to be ſaid to them is, that they would
produce ſomething better.
THE
E
CATA332 33 34
POLIZIDIZEIFETIZZNESE
THE
CONTENTS
P A R T I.
C
Ontains the principal and moſt uſeful Propoſitions of the firſt
fix Books of Euclid's Elements, demonitrated after the Euclidean
Maoner.
From Page i to Page 54.
Geometrical Definitions
PI
Poſtulara's and Axioms
p. 5
Propoſition the ift of Euclid, &c.
p. 7
The Rules for railing and letting fall Perpendiculars, biſecting Right-
lines and Angles, Gr. demonſtrated,
P. 10
The Method of drawing Parallel Lines demonſtrated
p. 14
Several Properties of Triangles demoſtrated
p. 16
A Demonſtration of the Method of meaſuring any Right-lined Fi-
gure
p. 17
The Method for Trinsformation of Planes demonſtrated
p. 18
The Method of adding and lubſtracting Squares, to and from each
other, demonſtrated
P 19
The Abſurdity of the com non Way of multiplying Money by Mo-
ney, ſhewn and demonſtrated
p. 21
The Method of Extracting the square Root out of any Number,
P 22
fhewn and demonſtrated
p. 23
The Method of drawing Tangents to the Circle, and finding the
Centers of Circles, ſhewn and demonſtrated.
p. 26
Uſeful Properties of the Circle demonſtrated
P 27
The Method of raiſing and letting fall Perpendiculars, at, and over
the End of a given Right-line, Thewn and demonſtrated
b
Se-
p. 28
1
ii
CONTENT S.
3
Several other uſeful Properties of the Circle demonſtrated p. 30
Methods for inſcribing in and circumſcribing Figures about a
Circle, ſhewn and demonſtrated
p. 34
The chief and primary Properties of Proportion, fhcwn and demen-
ſtrated
p. 39
The Method for finding the Sum of any infinite Series, in a. Geome-
tric Progreſſion, ſhewn and demonſtrated
P. 40
How to find the Sum of any infinitely decreaſing Series, in a Geo-
metric Progreſſion.
P. 41
How to divide Lines and Triangles in any given Ratio
p. 40
How to find a mean Proportional between any two given Lines, and
to divide a Triangle in any given Ratio, by a Line drawn paral-
lel to one of its Sides
Univerſal Laws for Transformation of Planes, laid down and de-
monſtrated
P. So
An eaſier Way of demonſtrating ſome Properties of the Circle p. si
How to enlarge or diminiſha Right-lined Figures in any given Ratio,
ſhewn and demonſtrated
p. 52
How to find the Ratio that like Figures have to each other, mewn
and demonſtrated.
p. 53
How to find the Sum or Difference of any two ſimilar Figures p. 54.
P 48
P AR T. II.
Contains the Science of Plane Trigonometry From p. 54 to p. 134
SECTION 1.
Contains the Definition and Conſtruction of the Lines uſually appli-
ed to the Circle, ſuch as Chords, Sines, &c. From p. 54 to p. 59.
Trigonometrical Definitions
p. 54
The Gecmetrical Conſtruction of Sines, Tangents, &c.
P. 56
SECTION II.
Contains ſeveral Methods for conſtructing the Tables of Sines, Tan-
genrs and Secants, Squaring the Circle, and the Menſuration of
luch Superficies and Solids as depend upon the Circle. (From : 59
to p: 93)
The
.
CONTENTS.
iii
merit
P. 60
P. 61
P. 61
P. 62
The Sine of an Arch being given, to find the Sine of its Comple-
pi 59
The Sine of an Arch being given, to find the Sine of half the Arch.
The Sine of an Arch being given, to find the Sine of the double of
that Arch
р. бо
The Sines of two Arches being given, to find the Sine of the Sum
or Difference of thoſe Arches
The Sines to 30 Degrees being given, to find from thence the Sines
to 60 Degrees
The Sines to 60 Degrees being giveni, the Sines from thence to 90
Degrees may be found, by Addition of thoſe already found, the
Manner how, ſhewn and demonſtrated
The ancient Method of making the Table of Sines, ſhewn and de-
monſtrated
p. 63
The Difficulty attending this Method
P 63
The ancient Way of finding the Proportion of the Diameter of any
Circle to its Circumference
p. 64
The Modern Way of doing the ſame
P64
An Inveſtigation of a Series for finding the Length of any Arch
from its Right Sine.
p. 65
The Series it felf for finding the Length of any Arch from its Right
Sines, ſhewn and demonſtrated
The Application of the ſame Series for finding the Length of the
Arch of 30 Degrees, to 10 places in Decimals, from the Sine of
the ſame Arch
The Length of the Circumference in ſuch Parts as the Diameter of
the Circle is Unity, inveſtigated to 10 places in Decimals p. 67
Another Inveſtigation of the ſame
p. 68
That by this Method the Proportion of the Circumference of a
Circle to its Diameter, may be inveſtigated to 's places in Deci.
mals, which is ſuicfficient for almoſt all Uſes, in a very few Mi-
nutes, ſhewn by an Example
The Advantage of this Method above the ancient Method P. 69
How to Extra& the Root of an infinite Equation,
P. 69
The Application of this Method in the Inveſtigation of the Series,
for finding the Right-fine of any Arch from the Arch it ſelf.
P. 69
b 2
That
P 66
p. 66
p. 68
iv
CONTENT S.
That the Arch and Sine of 1 Minute do not differ from each other
in Length, to 10 places in Decimals, and are both expreffed by
the ſame Figures
p. 69
The Application of the former Series, to the actual finding the Na-
tural Sine of 10 Degrees to ro Places
P70
The Application of the fame Series, to the a&ual finding the Na-
tural Sine of 20 Degrees to 10 Places
P. 70
The ſeeming Difficulty of applying theſe Series to Pra&ice removed
p. 70
Examples of compendious Methods in Multiplication, for the more
eaſy raiſing of Powers, and applying the former Series to Prac-
tice
p. 71
The Series for finding the Co-fine of any Arch frorn the Arch it felf,
ſhewn and inveſtigaced
p. 72
The Application of this Series to the finding the Co-lines of i min.
and 20 deg. or the Sincs of 89 deg. 59 min. and 70 deg. p. 73
The Sines of 40 Degrees and 50 Degrees found, by fubftracting the
Sines of 20 Degrees and 10 Degrees from the Sines of 80 De-
grecs and 70 Degrees
P 74
Thai A Table of Natural Sines to 7 places in Decimals, which is as
ex. Et us common Uſe requires, may be done with great Eaſe by
this Method
P. 74
The Scries for finding the Verſed Sine of any Arch from the Arch
it felt, inveſtigated and ſhewn
P74
The Series for finding the Verſed Sine of the Supplement of any
Arch inveſtigated
Þ: 74
The Sinical Proportion demonſtrated
p. 75
The Uſe and Excellency of this Merliod, and how it may be applied
to Practice
p. 77
An Inveſtigation of the Series for finding the Natural Tangent of any
Arch from the Arch it felf
P. 78
An Inveſtigation of the Series for finding the Natural Co-tangent of
p. 79
The Series for finding the Secont of any Arch
p. 79
The Series for finding the Co-ſecant of any Arch
P. 79
How by the Help of theſe ſeveral Series, ſo many different Series
may be inveſtigated, for finding the Length of the Circumference of
a Circle in equal Parts of the Radius
p. 79
A Series for finding the Length of any Arch, and conſequently the
Length of the whole Circumference in equal Parts of the Radius,
from
any Arch
CONTENT S.
P.81
P. 82
p. 83
from the Tangent of the ſame Arch firſt known or given, ſhewn
and inveſtigated
P 80
The Application of the ſame Series to the inveſtigating of the Cir-
cumference of the Circle, - 10 10 places in Decimals
The Proportion of the Circumference of a Circle to its Diameter,
to 100 places in Decimals
The Proportion of the Circumference of the Circle to its Diameter,
applied (and the Rules demonſtrated.)
P.
ift, to the finding of the Circumference of any Circle, its Diame-
ter being given, and the contrary
2d, To the finding the Area of the Circle, different Ways from pro-
per Data, and the Reaſon of the Rules fhewn
3d, To the finding of the Area of an Ellipſis, or any Segment of it
different Ways
4th, To the finding of Surfaces of Cones arid Cylinders
5th, To find the Superficies of a Sphere different Ways
6th, To find the Surface of any Sigment of a Globe
7th, To the finding of the Solidity of a Cylinder
8th, To the finding of the Solidity of a Cone leveral Ways p. 89
9th, To the finding of the Solidity of the Sphere different Waysp: 90
10th, To the finding of the Solidity of the Spheroid, ſeveral Ways
P. 92
P. 84
p. 85
P. 86
P. 87
P. 88
p. 88
SECTION III.
!
Contains a Demonſtration of ſeveral Theorems commonly called Ax-
ioms, for the Solution of the ſeveral Caſes of Right and Oblique-
angled Plane Iriangles
From p. 93 to p. 102
A Demonſtration of the ift Theorem or Axiom
P: 93
A Demonſtration of the 2d I heorem or Axiom
p. 54
A Demonſtration of the 3d Theorem or Axiom
P. 95
A Demonſtration of the 4th Theorem or Axiom
P 96
A Demonſtration of the 5th Theorem or Axiom
p. 97
A Demonítration of the 6th Theorem or Axiom
P98
Several Methods deduced from the 6th Axiom, for finding the
Angles from the 3 Sides given
P. 100
SECT.
:
vi
CONTENTS
SECTION. IV.
Contains the Solution of the Seven Caſes of Right-angled Plane Tri-
angles, ſeveral Ways
From p. 102 to p. 119
That the whole Science of Trigonometry is centained in one ſingle
Problem
P. 102
Some uſeful Properties of Triangles
P. JO2
What the Tables of Sines, Tangents and Secants are, and their Ap-
plication
P. 103
The Solution of the itt Cale by Natural Sines &c.
104
The Solution of the ſame Caſe by the Artifical Sines, Tangents,
and Secants
The General Rule for working by Logarithms
p. 107
The Uſe of both Kinds of Solutions
p. 107
The Solutions of the ad, 3d, and 4th Cafes, ſeveral Ways P. 106
P. 106
SECTION V.
Contains the Solution of the Six Caſes of Oblique-angled Plane Iri-
angles various ways, and in all their Ambiguities (From p. 119
to p. 133)
Various Ways of Reſolving the oth Care
Rules for reſolving the fame Cafe by the Gunter's Scale
P. 133
P. 128
PART III.
SECTION I.
Contains an Introduction to Navigation, the Definitions of all the
Terms made uſe of, and ſeveral Corollaries drawn from thence
for the Solution of feveral uſeful and introductory Problems
in Sailing
p. 134, 10 145
Some Errors in the Plane Chart detected
P. 145
SECTION II.
Contains the Application of the Do&rine of Right Angled Plane
Triangles to the Solutions of the ſeveral Caſes in Plain Sailing,
re.
CONTENT S.
vii
P. 161
p. 162
p. 163
p. 165
relating to a Single Courſe, with their Uſes, from p. 145 to p. 182
Queſtions Preparatory to the Underſtanding of Compound Tra-
verſes
p. 156
A farther Explication and Illuſtration of the ſeveral preceeding Caſes
SECTION III,
Contains the Application of Plane Triangles to the Solution of a
Traverſe or Co npound Courſe
from p. 162 to p. 172
What a Traverſe or Compound Courſe is:
The iſt Example with its Solution
Dire&tions for making the Traverſe Table
P.163
The ſecond Example, with its Solution
p. 165
The Geometrical Solution
The third. Example with irs Anſwer, by making Allowance for the
Variation
p. 169
The 5th Example with its Anſwer, by making Allowance for Lee-
Way
p. 170
The 6th Example of a Traverſe, wherein the Lee-Way and Varia-
tion are both conſidered
p. 170
Two more Examples of Compound Traverſes, wherein the Leeway
and Variation are both given, with their Anſwers P. 171
A Table of Difference of Latitude and Departure in Minutes and
tenth Parts to every Degree and Quarter Point of the Compaſs
for the exact working of Traverſes
p. 173
The Uſe of the preceeding Table in the Solution of Traverſes and
Queſtions in Plain Sailing
p. 194
SECTION IV.
Contains the Doctrine of Oblique Angled, Planc Triangles applied
to Problems of Sailing
P. 197, to 210
The ſame Queſtions expreſſed in different Terms, and dreſled up
af.
ter different Manners to inlarge the Learners Way of thinking,
and give him a better Inſight into the Nature of Queſtions, p. 199.
A Problein of great Uſein drawing the Draught of any Haven, Ri-
ver or Bay, and in laying down of Sands, Rocks or Shoals in
Sea-Charts
P 205
Methods for diſcovering the Diſtances of Places by the firing of
Guns, how far Ships are afunder, and for meaſuring of Inaccesible
Diſtance's
SECT
.
Pi 280
viii
CONTENT S.
SECTION V.
Contains Queſtions relating to Turning to Windward, with the
Manner of their Solution
P. 210 to 218,
How to find the Lee-way a Ship makes
P. 213
SECTION VI.
Contains an Explication of the Nature of Currents, and the Me-
thods of allowing for the ſame
pe 218, to 232
What Currents are, and how to allow for them
A Traverſe Current, with its Solution
p. 221
A Queſtion of Turning to Windward in a Current
P 225
Several uſeful Queſtion, with their Solutions for finding the Settings
and Drifts of Currents
p. 225
p. 218
SECTION VII.
Contains a Deſcription of the Plain Chart with its Uſes, from
p. 232 to 235
SECTION VIII.
Contains Queſtions, with their Solutions relating to Sailing under a
Parallel of Latitude, uſually called Parallel Sailing, froin p 235 to 242
The Errors of the Plain Chart ſhewn, and the Quantity of a Degree
of Longitude in any Parallel of Latitude determined
P 236
Queſtions in Parallel Sailing, with their Solutions,
p. 237
A Table fhewing how many miles anſwer to a Degree of Longitude
at every Degree of Latitude
p. 240
he Uſe of the former Table
P 241
SECTION IX:
Contains the Application of the former Se&ion to the Solution of
the moſt uſeful Queſtions in failing by the Middle Latitude,
P. 242, 253
What the Middle Latitude is, and the Manner of the Operation,
p. 242
re
CONTENTS.
ix
The Solution of a Traverſe by this Method,
The near Agreement of this Way with the true Sailing
How far this Method may be relied upon
SECTION X.
P.252
P.253
p. 153
tor
Conrains a farther Demonſtration of the Errors of the Plane Chart,
and of the truth of the Principles upon which the Mercator's Chart
is founded, together with ſeveral Methods for computing the
Length of a Degree of Latitude at all Diſtances from the Equa-
from p. 25€ to p. 260
The Errors of the Plane Chart farther ſhewn
p. 553
Fo re&ify this Error was the great Study of the Ancients, and not
accompliſhed till undertaken by Mr. Wright
P. 254
The Principles upon which the true Chart is founded, demonſtra-
ted
p. 254
The Methods made uſe of by Mr. Wright and Mr. Oughtred for com-
puting the Merid. Parts
p. 255.
How the ſame may be found independently of the Table of Secants,
from the Length of the Arch of Latitude firſt given by the Help
of an Infinite Series ſhewn and demonſtrated
p. 255
The Meridional Parts anſwering to 5 and 10 degrees of Latitude, to
6 places in Decimals, inveſtigated by the help of the former Series
p. 256
How the fame Meridional Parts may be found by the help of the
Logarithmic Tangents
p. 257
The Application of the ſame to the finding of the Meridional
Parts of 's and 10 deg. of Latitude
p. 258
A large and very Correa Table of Meridional Parts to erery degree
and Minute of Latitude to 4 places in Decimals computed Indepen-
dently of the Table of Secants, after the former Manner p. 261
NB. The Tables of Meridional Parts being computed after the
Sheet beginning at Page 285 was printed off, and there being not
fufficient room left for them, is the reaſon why the laſt Folio of
the Tables does not agree with the Folio of the Sheet Pp.
SECT.
C
:
:
х
CONTENT S.
SECT 10 N XI.
The Application of the former Seation to the a&ual Solution of the
ſeveral Cafes in ſailing, relating to a ſingle Courle, uſually called
Mercultor's Sailing,
from p. 285, to p 295
The Solution of a Traverſe or Compound Courſe, after the former
Manner
p 295
How the difference of Longitude in Mercator's Sailing may be found
by the common Tables of Difference of Latitude and Departure
po 299
Examples
p. 300
That the common Method (made uſe of and taught in Books) of
finding the difference of Longitude by the whole Departure made
in one or more days is falſe
p. 300
Proved by an caſy Inſtance
P.301
Farther proved by ſolving the former Traverſe after this Way
p. 301
A Traverſe Current reſolved after the Mercatorian Way
p. 302
SECTION XII.
i
Contains a Solution of all the Mercatorian Problems, without the
Affiſtance of Meridional Parts, by the help of the Logarithmic
Tangents only
from p. 303, to 310
What is meant or to be underſtood by Difference of Longitude,
Meridian Diſtance and Departure, and wherein they effentially
differ from each other
p. 311
The Errors in Plain Sailing farther exhibited, and the Neceſſity and
Occaſion of the true ſailing ſhewn
p. 312
Thar Departure has no part or ſhare in Mercator's Sailing, nor does
it properly belong to it, tho' the Concluſions drawn from it are
P. 313
SECTION XIII.
true
Contains the Conſtruction and Uſe of the Mercator's Chart, from
P 313, to p. 325
That upon this Chart all places may be laid down true' according to
their Latitudes and Longitudes, and what gives it the Preference to
+
all
CON TEN T S.
xi
P: 316
all other Contrivances whatſoever for Nautical Uſes is, that the
Rumb Line is repreſented by a perfect ſtreight Line P. 315
That this Reailinearity of the Rumb Line is the chief and prima-
ry Property of the Mercator's Projection
That by the help of this Property all Nautical Problems are reſolved
to the greateſt Degree of Exa&neſs, and eafier than by the Globe
itſelf
That it is impoſſible by any Contrivance to repreſent any part or
portion of a Spherical Surface upon a plain, but that it will be
diftorted
p. 316
That if the distorted Parts when projected retain between themſelves
the lame Ratio that the Diinenſions of the Parts they repreſent
upon the Globe do, all Solutions performed by them will be
equally true with thoſe that are performed by the Globe it ſelf.
P316
P. 316
That therefore the Mercator's Chart is the moſt apt and proper In-
ſtrument, and the beſt adapted to the Mariner's Uſe that can be,
whilſt Mankind makes uſe of the Compaſs
p. 317
SECTION XIV.
Contains the Solution of ſuch Problems as ariſe from Sailing, by or
upon the Arch of a Great Circle
from p. 325, top: 353
The Difficulty or rather Impoſſibility of failing exa&ly by or upon
the Arch of a Great Citcle, according to the Dire&tion of the
Compaſs ſhewn
p. 330
That the Arch of 5 Degrees differs but i:oo from the Correſpon-
dent: Tangent
P 330
Whence the Rumb Line and Arch of s degrees differ but inſenGibly
from each other in Length
p. 330
Exemplified in finding the Arch of neareſt Diſtance between the
Lizard and Newfoundland
P.331
Thar che Length of the Arch of neareſt Diſtance between 2 places
5 degrees alunder, differ but one fiftieth part of a Mile from the
true length of the Rumb Line in the firſt Example fhewn p. 332
Examples and Solutions in all the various Problems of Great Circle
Sailing
That
P. 333, &c.
:
xii
CONTENTS
That the Length of the Arch of meareft Diſtance between the Li-
zard and Barbadoes, differs but 371 Miles from the Length of
the Rumb Line, paſſing thro' the two Places
p. 345
That the Arch of neareſt Diſtance between Barbadoes and St. Helena,
differs but it of a Mile from the Length of the Rumb Line,
which ſhews the Excellency of Mercator's Sailing
p. 349
A Solution of the two chief and primary Problems in Mercator's Sail-
ing, independent of the Tables of Logarithms, Sines, Tangents, Sex
cants, or Meridional Parts, or any Artificial Number, by the help
of the infinite Series
p. 350
That all Nautical Problems may be reſolved after this Manner.
p. 353;
The Uſe of theſe Kinds of Solutions
P3531
SECTION XV.
Containing the Method of finding the Latitudes of Places or Heights
of the Pole, by the Meridional Height of the Sun or Stars, and
their Declinations
from p. 353 to pa 373,
A Demonſtration of the Fundamental Law of Refraktion, by the
Maxima do Minima.
P.
More conciſe and general Ways of Reſolving the preceding Pros
blemas
PA
SECTION XVI.
Contains an Account of the ſeveral Methods propoſed and made
Uſe of, by the moſt skilful Aſtronomers and Geographers, for find
ing thợ Difference of Longitude between Places (from p. 373 to .
p. 386).
The Difficulty of finding the Difference of Longitude true, by the
Courſe Steer'd and Diſtance Run, faewn.
P: 373:
What is meant by finding out the Longitude
p. 373
What thư Longitude is, and how it may be found.
P: 374
11,- By Ecliples of the Moor, and an Inſtance given of the Diference
of the Longitude between London and Paris determined, by Ob-
fervations made at each Place
p. 375
An Inſtance of the Difference of Longitude between Greenwich and
Lisbon,, determined after the ſame Manner
p. 376,
Another Inſtance of the Difference of Longitude, being determined
atser..
CONTENT S.
XA1
ner how
after the ſame manner, between London and Botox in New England
p. 376
Longitude of Places may be determined by Solar Eclipſes
p. 376
Longitudes of Places by the Eclipſes of Jupiter's Satellites, the man-
p. 377
The Longitude between Rome and Greenwich determined by this
Method
P. 378
That the frequency of thele Eclipſes, which happen ſometimes twice
or :hrice in a Night, the great Eaſe with which they are made,
and the great Exactneſs with which they may be obſerved, render
them one of the beſt Expedients hitherto known for accomplifhing
Ile ſame
p. 378
Longitude of Places inay be found by the Appules of the Moon, c.
P. 378.
Example of the Difference of Longitude between London, Dantzick,
and Ballafore in the Eaſt Indies, determined by this Method
f.: 379
That the Longitude of Places may be found by the Immerſions and
Emerſions of the Planets, c.
p. 380
Inſtances of the Difference of Longitude between Paris, London,
Nöremberg, and Canton in China, determined by this Method
p. 380
That the great Number of Fixed Stars that lye within the Zodiac,
render the Appulſes and Occultations of the Stars, 6c, very fre-
quent
P.-3811
Longitudes of Places may be derermined by Pendulum Watchés p 381
An Inſtance of the Uſe of theſe in a Voyage from the Coaft of Gui-
nea to the Ile of Fuego
p. 382
The Objeđion againſt the Uſe of the Pendu’um-Watch exhibited
p. 382
The principal Thing wanting to render thele Methods more pradi-
cable
p. 382.
Not conſidered by the modern Pretenders to the Diſcovery of the
Longirude
P. 383
How this may in ſome meaſure be obtained
p. 383
That there are frequent Opportunities at Sea of putting ſome of theſe
Methods in pra&ice
p. 384
That theſe or no other Methods, how true foever, can be put in a
Prađice, till we have a new and mort corre& Deſcription of the
Sea-Coafis and Iſlands, &c...
p. 384
A
*.
xiv
CONTENTS,
A Method propoſed how this may be effected in a few Years by
the Government, with little or no Expence
1. 385
That if the Commanders of Merchant-men would joyn in it, and lend
their helping Hand, it mnight be very ſoon accompliſhed p. 385
SECTION XVII.
Contains an Account of the Variation, and Methods of obtaining it
p. 386 to P. 399
What is meant by the Variation
p. 386
Various Methods how it may be obtained
f. 387
A more conciſe and general Way of finding it
f. 397
That the Variation has conſtantly changed ſince it was firſt obſerved,
and that to obtain a good Theory of it, there wants a greater Stock
of Obſervations, and how uſeful theſe may be
p. 398
SECTION XVIII,
Contains an Account of the Methods m. d: Uſe of for meaſuring
the Ships Way, for correcting the Courſe Steer'd, by making pro-
per Allowances, 6c.
from p. 399 to p. 437
The Diſtance between Knot and Knot aſcertained
p. 490
How to corre& Errors that ariſe from not having the Diſtance be-
tween Knot and Knot true
p. 401
How to corre& Errors ariſing from the half Minute Glaſs, being ei-
ther too long or too ſhort
p. 402
How to try whether the Half Minute Glaſs be of a juſt Length
0 403
How to corre& Errors both in the Log line and Half-minute Glaſs
p. 403
What is meant by Leeway, and from what Cauſes it ariſes
P. 405
The uſual Allowances made for it
p. 406
How to correct the Courſe ſteerd, the Quantity of the Leeway
being known
P 407
How to corre& the Courſe ſteer'd, by making proper Allowance for
the Variation
p. 408
How to corre& a Reckoning, by making proper Allowances for an
Error in the Courſe, for an. Error in the Diſtance, and for Errors
in both Courſe and Diitance, c. with Examples P. 409
A
+
CONTENT S.
XV
3
Dy, and of what Uſe the Tables of
A Log-Book, with the Form and Manner of working of Day's Work
at Sea as an Example to the foregoing Rules
po 417
Rules and Directions for keeping a Journal
P. 434
An Example for the foregoing fix Days Work
P. 436
SECTION. XIX.
Contains an Account of the Equation of Time, or the Inequality of
the Solar Days, with the manner of computing it, from p. 437 to
iP 458
What is meant by the Equatorial Sydereal and Solar Day, p. 438
The quantity of the Tropical, the Solar and Sydereal Day deter-
mined
P. 439
That one Cauſe of the Equation of days ariſes from the Inclination
of the Earth's Axis to the Plane of the Ecliptic
p. 439,
That another Cauſe of the Equation of days ariſes from the Ex-
centricity of the Great Orb
p. 441
That the Inequality of the days or the Equation of time ariſes from,
theſe two Cauſes, and is a Compound of them both
p. 444.
How the quantity of it may be found
p. 444
Tables for the more ready finding the abſolute Equation of time
P: 446
The Uſe of theſe Tables
P. 447
Tables ſhewing the Equation of time for every day of the 4 Years
p. 448
The Uſe of theſe Tables in finding the Apparent or Solar time by
a well regulated Clock
p. 452
A Table Chewing how much the former Tables willvary in 100 Years
for every day ip the Year
P: 454
The Uſe of the former Table.
P. 455
A Table ſhewing the Equation of time, anſwering to every Sign
and Degree of the Ecliptic for the Year 1726,
P: 456
Why Aſtronomers have pitch'd upon a mean Solar day to adjuſt the
Equation of cime are in Aſtronomical Computations p. 457
SECTION XX.
1724, c.
Contains the Theory of the Sun, and the Conſtruction and Uſe of the
Solar Tabies
from p. 458 to p. 492
Thar:
Xvi
CON TEN T'S.
That 'the Knowledge of the Length of the Solar Year is the firſt
thing neceffary to be known, and how this may be obtained from
Obſervations
P.459
The Length of the Solar Year determined from the Flameedian ob-
fervations
P. 402
That there is a Præcefſion of the Equinoxes, and the quantity of it
determined by Obſervations
P. 402
The quantity of the Periodical or Sydereal Year determined p. 462
How the mean Motion of the Sun for one day, and conſequently
for any Number of days is found
p. 463
How the mean Motion for Hours and Minutes may be determined
p. 463
How the mean Motion from common Julian Years is determined
p. 463
How the common Solar Tables are made to ſerve in Leap-Years
p. 463
How the Radix's of mean Motion of the Sun are obtained
P. 465
That the Solar Tables are adjuſted to mean or equal time
p. 406
That Aſtronomers by comparing the places of the Sun deduced from
Obſervations have found that his apparent Motion is unequal
p. 467
That he is not placed in the Center of the great Orb
P 467
The Proportion of his greateſt and leaſt Diſtance from the Earth de-
termined, and thence his Excentricity, or how far he is removed
from the Center of the @rb
P. 467
The ancient Method made Uſe of, for allowing for the Inequality
of the Sun's Motion ariſing from this Cauſe
How theſe Inequalities may bc accounted for, from the Diſcovery
p. 468
of Kepler, that the Earth mov'd in an Eliptic Orb; and in moving
deſcribes Areas proportionable to the Time, by a Ray drawn
from the Center of the Earth to the Sun
p. 469
The Keplerian Way of computing the Profthapbærefis P. 475
Examples to the foregoing Method
p. 476
The Newtonian Way of finding the ſame direaly
P. 481
Examples to the foregoing Method
p. 485
The Place of the Aphelion determined from the Flamſteedian Obler-
vations
p. 486
The mean Motion of the Apoge
P. 486
A
*
CONTENTS.
A dire&t Method for finding the Place of the Sun from theſe Da-
ta, without the help of the Tables
P.487
An Example
p. 487
The Manner' of computing the Diſtance of the Sun from the Earth,
ſhewn and demonſtrated
p. 489
New Solar Tables deduced from the Flanſieedian Obſervations, and
co:ſtructed after the former manner
p. 493
The Uſe of theſe Tables for the ready computing of the Sun's
Place, with an Example
p. sor
Another Example of the Uſe of the fame Tables
P. 503
The Uſe of the following Tables ſhewn
p. 503
A Table fhewing the Declination of the Sun to every two Minutes
of the Eclipric, the greateſt Declination being fixed at 23.29.
P: 506
A Table ſhewing the Right Aſcenſion of the Sun to every two Mi-
nutes of the Eciiptic, the greateſt Declination being fixed at 23.29.
p.525
SECTION XXI,
Contains general Laws for the deſcribing of any great or finall Cir-
cle upon any Plane Orthographically, for laying off any
Num-
ber of Degrees upon, or for meaſuring any Part or Portion of a
great or ſmall Circle when projected
P.537
The Application of the preceding Rules to the Projection of the
Sphere Orthographically upon the Plane of the Meridian p. 548
The Application of the ſame Laws to the Proje&ion of the Sphere
upon the Plane of the Horizon Orthographically
p. 553
The Reaſon why in many Caſes the Stereographic Repreſentations
of the Sphere are preferr'd to the Orthographic.
That the Projection of the Sphere upon each particular Plane has
its peculiar Uſe
P: 558
That the Orthographic Proje&ion of the Sphere upon the Plane of
the Meridian is the moſt apt and ready for giving Solutions to
Aſtronomical Problems.
The Uſe of the Analemma or Orthographic Proje&ion of the
Sphere in the Solution of the common Aſtronomical Problems
P558
p. 558
P 562
That the common Problems of the Sphere are reſolved with more
Eaſe by the Orthographic than by the Stereographic Projection,
with an Example
P. 567
d
SECT
xviii
CONTENTS.
SECTION XXII.
Contains a Solution of the chief and primary Problems in Aftronomy
according to the ancient Pythagorean or Copernican Syſtem P.568
That the great Orb in which the Earth moves is but as a Phyſical
Point with regard to the immenſe Ditance of the neareſt fixed
Star
P. 577
That the Hypotheſis of the Earth's Motion is the moſt ancient Hy-
potheſis, &c.
p.591
SECTION XXIII.
P. 598
Contains the Uſe of both Globes,
from 2:593 to p. 627
What Globes are, the Manner of their Conſtruction, with a De
ſcription of their ſeveral Appurterances
p. 593
To find the Latitudes of Places
To find the Diference of Latitude between two Places
P. 599
To find the Longitude of any Place
That the fixing of a firſt Meridian being Arbitrary occaſions great
Diſagreement between the Longitudes of Places determined by dif-
ferent Globes or Maps
The Scituation of the firſt Meridian, according to Prolomy, &c.
po 600
p.600
p. 600
p. 601
P. 601
. p. 602
To find the Difference of Longitude between two Places
To find the Angle of Poſition between 2 places
A Solution of a common Geographical Paradox
How to lay down the Running of a Ship upon the Globe P. 603
To find the Diſtance between two places upon the Globe P. 604
To find the Hour of the Day at any place, the Hour at another
place being given
P.600
To find the Sun's place in the Ecliptic
p.605
To find the Sun's Declination
p. 605
To frud the Sun's Right Aſcenſion
To find the Sun's Oblique Aſcenſion and Deſcenſion
p.607
To fitd the Sun's Amplitude
p. 607
To find the Merid. Altitude or Merid. Zenith Diſtance
To find the Aſcentional Difference
p. 619
To find the time of the Sun's Riſing, &c.
P. OTO
To find the Length of the Day, 6c, at any place within the Artic
Circle
P: 613
Το
P: 606
P. 608
CONTENT S.
xix
p. 616
To find the Beginning and End of Twilight, c.
t. 614
To find the Sun's Height and Azimuth, c.
1.615
To find the Hour of the Day, 6.
To account for the General Phenomena by one Poſition of the
Globe
1.617
To find the Longitude of a Star
p. 619
To find the Right Aſcenſion, &c of a Star
P. 620
To find the time of a Star's Culmination
1.620
To find the time of a Star's Riſing,
p.622
To find the Altitude and Azimuth of a Star, c.
P 623
To find the Hour of the Night, &c.
p.624
To find whae Stars are Riſing, doc.
p. 624
To determine all thoſe places upon the Earthi where an Eclipſe of
the Moon or of Jupiter's Satellites is Viſible
P: 625
p. 628
P 628
P. 628
SECTION XXIV:
Contains an account of the Diviſion of Time and its Parts, of the
Cycles, Epoch's, Era's, Periods, Moveable Feafts, &c. from
P.627 to p.655
The Diviſion of Time
What a Scruple is, and its various kinds
What an Hour is, and its various Kinds
What a Week is, and the Names of the ſeveral Days contained in it
P 630
What a Month is, and its ſeveral Kinds explained
p. 631
What a Year is, and its ſeveral Kinds
p. 632
The Roman Way of dividing their Months
P 635
What the Kalendar is, and how it is formed
p. 636
What is meant by the Solar Cycle, and whence it ariſeth
P. 637
The Uſe of the Solar Cycle, and how it may be found
p. 638
What the Dominical Letter is, and how it may be found
p. 638
What the Metonic Cycle is, and whence it ariſes
p. 640
The Uſe of the Metonic Cycle or Golden Number in finding the times
of the Full-moons, New and Eaſter-Day, &c.
*P: 642
What the Epact is, and whence it ariſes
p. 645
The Uſe of the Epact in finding the Age of the Moon, &c p. 646
The Uſe of the Epa&.in finding when Eaſter Day and the reſt of the
moveable Feaſts will happeni
p. 619
What the Julian, Viktorian, and Dionyhan Periods are, and whence
they ariſe
P651
An Account of the moſt remarkable Epocha's or Era's
P 652
SOI
XX
CONTENTS
SECTION XXV.
Of the Nature and Conſtruction of Logarithms, from p. 655 to
p. 687
The Nature of Logarithms explained
p. 656
An Inveſtigation of Sir Saac Newton's Binomial Series
P. 669
The ſame Series applied to the extracting the infinite Root p. 670
The Application of the Method of extracting tire infinite Root to the
making of Logarithms
P. 671
A General Rule thence deduced, for the making of Logarithms p. 672
The Application of the ſame Rule, to the making of Logarithms
P 673
How the Logarithmic Sine, Tangent, &c. of any Arch may be found
independently of the Tables of Logarithms, &c. from the Length of
the Arch firſt given
An Inveſtigation of the Series for finding the Number from the Lo-
garithm
p. 690
An Inſtance of the Uſe of Logarithms in finding the Value of Annui-
P. 694
p. 686
ties, &c.
As it is almoſt impoſſible to avoid Typographical Errors, ſo not-
withſtanding the greateſt Care has been taken, no doubt, ſeveral
of that Kind have crept into this work; but they are ſuch I dare
ſay, as the Reader may eaſily, amend as he goes on, which no doubt
for his own Advantage, he will not fail to do.
GEOMETRY
1
Ooo 0000000000000000000000000000000000
SAAAAAAAAAAAAAAAAA
GOG COCO0000 DO 2000 0000 0000 0000 000000
GEOMETRY
:
PARTI
DCZE
Definitions.
HE Subje& of Geometry, is Magnitude or Quan
tity continued ; ics Buſineſs to diſcover the Nature
and Properties of thoſe Quantities about which it
T
is employed, in order to find out the Proportion
that they have to each other, that ſo from the
Knowledge of one, we may be led to the Know-
ledge of the other, according to Quantity.
Magnitudes are of three Kinds, viz. Lines, Superficies and Solids.
A Solid is that which hath Length, Breadth and Depth, or
Thickneſs ; its Limits or Boundaries, are called Superficies ; fo that,
3. A Superficies hath Length and Breadth, without Thickneſs; the
Limits of which are Lines.
4. And if it lie equally betwixt it Lines, 'tis called a Plain Sue
perficies.
5. A Line then hath only Length, and is bounded by Points ; fo
that,
6. A Point Mathematical, is incapable of being divided, and
therefore hath no Parts.
7. A Right or Straight Line, is that which
lies equally betwixt, or is the neareſt Tract
А.
-B
between its Points; as AB.
8. A Plain Angle, is the Inclination of two Lines, the one to the
other ; the one touching the other in the ſame Plain, yet not lying
ill the ſame ſtrait Line.
B
9. And
2
Euclides Elements.
9. And if the Lines which contain the Angle be Right-lines, it
is called a Right-lined Angle.
10. When a Right Line CG, ſtanding upon a
C
Right Line AB, makes the Angles on either ſide
thereof CGA, CGB, equal one to the other,
then both thoſe Angles are Right Angles ; and
the Right Line CG, which ſtandeth on the o-
A-
B ther, is termed a Perpendicular to that (AB)
G
whereon it ſtandeth.
Note ; When ſeveral Angles meet at the ſame Point as at (G) each
particular Angle is deſcribed by three Letters, whereof the middle Letter
Sheweth the Angular Point, and the other two Letters ihe Lines that make
that Angle; as the Angle which the Right Lines CG, AG, make at G,
is called CGA, or AGC.
11. An Obtuſe Angle, is that which is great-
А.
er than a Right Angle; as ACD.
12. An Acute Angle, is that which is leſs
than a Right Angle; as ACB.
B-
-D
13. A Limit, or Term, is the End of any
C с
thing.
14. A Figure, is that which is contained under one or more Terms.
15. A Circle, is a plain Figure, contained under one Line, which
is called a Circumference ; unto which all Lines drawn from one Point
within the Figure, and falling upon the Circumference thercot, are
cqual the one to the other.
16. And that Point is called the Center of the
Circle.
'17. A Diameter of a Circle, is a Right Line
drawn through the Center thereof, and ending at
A
E E
the Circumference on either ſide, dividing the
Circle into two equal Parts.
18. A Semi-circle, is a Figure which is contain-
D
ed under the Diameter, and
under that part of the
Circumference which is cut off by the Diameter.
N
B
In the Circle EABCD, E is the Center, AC the Diameter, ABC
the Semi-circle.
N, B.
Euclides Elements.
3
N. B. The equal Diviſions of the Circumference of every Circle,
into 360 Parts, are called Degrees; the equal Diviſion of each of theſe
again into 60 Parts more, are called Minutes ; and the equal Diviſion
of each of theſe into 6o Parts more, are called Seconds, &c. So that
the Arch of the Semi-circle contains 180 Degrees, and the Quadrant is
equal to 90 Degrees.
19. Right Lined Figures are ſuch as are contained under Right Lines.
20. Three Sided or Trilateral Figures, are ſuch as are contained un-
der three Right Lines.
21. Four Sided or Quadrilateral Figures, are ſuch as are contained
under four Right Lines.
22. Many Sided Figures, or Poligons, are ſuch as are contained un-
der more Right Lines than four.
23. Of Trilateral Figures, that is, an Equilateral
Triangle, which hath three equal Sides; as the Tri-
angle A.
A
24. Iſoſceles, is a Triangle which hath only two
Sides equal; as the Trianglc B.
B
25. Scalenum, is a Triangle whoſe three Sides
are all unequal; as C.
C
26. Of theſe Trilateral Figures, A Right-angled
Triangle is that which has one Right Angle; as the
Triangle r.
A
27. An Amblygoniun, or obtuſe-angled Tri-
angle, is that which has one Angle Ob fe;
as B.
B
A 2
28 An
4
Euclides Elements.
28. An Oxygonium, or Acute-angled Triangle, is
that which has three Acute Angles; as C.
C
$
An Equiangular, or Equal-Angled Figure, is that whereof all the
Angles are equal: Two Figures are Equiangular, if the ſeveral An-
gles of the one Figure be equal to the ſeveral Angles of the other.
# The fame is to be underſtood of Equilateral Figures.
A
29. Of Quadrilateral, or Four-Sided Figures, a Square
is that whoſe Sides are equal and Angles right, as A.
B
30. A Figure on one part longer, or a Long
Square, or Oblong, is that which hath Right Angles,
but not equal Sides, as B.
31. A Rhombus, or Diamond Figure, is that
which has four equal Sides, but is not Right
Angled ; as C.
D
32. A Rhomboides, or Diamond-like Figure, is
that whoſe oppofite Sides and oppoſite Angles
are equal; but has neither Equal nor Right An-
gles; as D.
33. All other Quadrilateral Figures beſides theſe
are called Trapezia, or Tables, as E.
E
А
34: Parallel, or Equi-diſtant Right Lines,
are ſuch, which being in the fame Superficies,
if infinitely produced, would never meet; as A
and B.
B-
35. A
Euclide's Elenients.
35: A. Parallelogram, is a Quadrilateral Figure, whoſe oppoſite
Sides are parallel or equi-diſtant; as the Figures A. B. C. D.
A Problem, is when ſomething is propoſed to be done or effected.
A Theorem, is when ſomething is propoſed to be demonſtrated:
A Corollary, is a Conſe&ary, or ſome conſequent Truth gain'd
from a preceding Demonſtration.
A Lemma, is the Demonſtration of some Premiſe, whereby the.
Proof of the thing in hand becomes the ſhorter.
Poſtulates, or Pecitions.
1: From any point to any Point, to draw a Right Line.
2. To produce a Right Line finite, ftrait forth continually.
3. Upon any Center, and at any Diſtance, to deſcribe a Circle.
AXIOMS
1. Things equal to the ſame thing, are alſo equal one to the other.
As A=B=C. Therefore A=C. Qr therefore all, A, B, C, are
equal the one to the other.
Note, When ſeveral Quantities are joined the one to the other, con-
tinually with this Mark=, the firſt Quantity is by vertue of this Axiom,
equal to the laſt, and every one to every one :
In which Cafe we often
abſtain from citing the Axion, for brevity fake ; although the force of the
Conſequence depends thereon.
2. If to equal things you add the ſame, or equal things, the wholes
fhall be equal. And fince Multiplication is nothing but an Addition of
the Thing, a certain number of Times to its ſelf, it is equally true and
equally inteligible; that, if equal things are multiplied by the ſame, or
equal things, the Produ£ts or Reſults will be equal.
3. If from equal things you take away the ſame, or equal things,
the things remaining will be equal. And fince Diviſion is nothing elſe
but the taking away the Diviſor, a certain number of times from the Di-
vidend, it is equally true, and equally inteligible, that if equal things are
divided by the ſame or equal things, the Quotients or Reſults will be equal.
4. If to unequal things you add equal things, the Wholes will be
unequal.
5. If from unequal things, you take away equal things, the Re-
6. Things
mainders will be unequal.
8
Euclides Elements.
1
6. Things which are double to the same thing, or to equal things,
are equal one to the other. Underſtand the ſame of Triple, Qua-
druple, &c.
2. Things which are half of one and the ſame thing, or of things
equal, are equal the one to the other. Conceive the ſame of Subtriple,
Subquadruple, &c.
8. Things which agree together, are equal one to the other.
„The Converse of this Axiom is true in Right Lines and Angles, but not
in Figures unleſs they be alike.
Moreover, Magnitudes are ſaid to agree, when the parts of the one being
applied to the parts of the other, they fill up an equal, or the ſame place,
9. Every whole is greater than its part.
10. Two Right Lines cannot have one and the fame Segment, (or
part) corninon to them both.
11. Two Lines meeting in the ſame Point, if they be both pro-
duced, they ſhall neceſſarily cut one the other in that point.
12. All Right-Angles are equal the one to the other.
13. If a Line BA, falling on two Right
iB
Lines AD, CB, make the internal Argles on
the ſame Side BAD, ABC, leſs than two Right
C Angles ; thoſe two Right Lines produced thall
meet on that Side, where the Angles are leſs
A
D
than two Right Angles.
14. Two Right Lines do not contain a ſpace:
15. If to equal things, you add things unequal, the Exceſs of the
Wholes ſhall be equal to the Exceſs of the Additions.
16. If to unequal things equal be added, the Excels of the Wholes
ſhall be equal to the Exceſs of thoſe which were at firſt.
17. It from equal things, unequal things be taken away, the Exceſs
of the Remcinders ſhall be equal to the Exceſs of the Wholes. :
18. If from things unequal, things equal be taken away, the Ex-
ceſs of the Remainders ſhall be equal to the Exceſs of the Wholes.
19. Every Whole is equal to all its parts taken together.
20. If one Whole be double to another, and that which is taken a-
way from the Firſt, to that which is taken away from the Second,
the Remainder of the Firft ſhall be double to the Remainder of the
Second.
Prop.
E
5
Euclides Elements
7
Propoſition I. id Problems,
of Euclid, the ift of the iſt.
Upon a finite Right Line AB, to deſcribe an Equilateral Triangle
ACB.
Conſtr. From the Centers A and B, at the di-
C
ſtance of AB, or B4, deſcribe two Circles curting
each other in the Point C; from whence draw
two Right Lines, CA, and CB, and the thing is A
B
done.
Demonſtration. For AC=AB (by the 15th det. (and AB=BC) by
the ſame.) Wherefore AC=CB (by the 1. Ax.) and the Triangle ABC
Equilateral (by the 23d defin.) W. W. D.
Scholium.
If AC, and CB, the Radius's of the equal Circles be taken grear-
er or leſs than AB, the Triangle will be lſoſceles; by the 24th def.
Again ; If AC, and CB, the Radius's of the two Circles, deſcribed
about the Points A, and B, of the Right Line AB, be taken equal to
two given unequal Right Lines, then the Triangle will be Scalene
by the 25th def (which is the 22d of the 187")
Propoſition II.
A Theorem,
of Euclid, the 4th of the iſt.
If two Triangles BAC, EDF, have two sides of the one BA, AC
equal to two sides of the other, ED, DF, each to its correſpondent
Side (that is BA=ED, and AC=DF) and have
A
the Angle A equal to the Angle D contained
under the
equal Right Lines, they ſhall have
the Baſe BC, equal to the Baſe EF; and the
Triangle BÁC equal to the Triangle EDF; and B
the remaining Angles B, C, ſhall be equal to
D
the remaining Angles E, F, each to each ;
under which the equal Sides are fubtended.
E4
F
Demon
8
Euclide's Elements
ſhall fall upon $placed upon the
.
som
6. Alſo, from the Propofition it follows, that it two Ar
est to
Demonſtration. If the Point D be applied to the Point A, and
DF fhall fall upon AC, becauſe the Angle A=D (by. Hyp.) More-
over the Point F fhall fall upon the Point C, becauſe AC=DF (by
Hyp:) Herefore the Right Lines EF, BC, fhall agree (by the 54th Ax.)
becauſe they have the fame Terms, and conſequently are equal ;
wherefore the Triangle BAC, DEF, do agree, and are equal, (by the
E:' 4xiom.) W.W. D. Whence,
Cor.
1. If twọ Triangles ABC, DEF, have two Sides AB, AC, equal
to two Sides, DE, DF, each to each, and the Baſe BC equal to the
Baſe EF, then the Angles contained under the equal Right Lines
Niall be equal, (which is the 8th of the ift.)
2. Triangles mutually Equilateral, are alſo mutually Equian-
gular.
3. Triangles mutually Equilateral, are equal one to the other.
А
Triangle are equal one to the other.
For let D be the middle Point of the Line.BC,
and draw the Line AD (by the 1ſt Poft.) Then
are the Triangles ABD, and ACD, mutually
Equilateral, and conſequently mutually Equi-
C
angular; whence the Angle B will be equal to
D
the Angle C.
5. Hence every Equilateral Triangle, is alſo Equiangular.
and DEF, have two sides of the one Triangle AB, AC, equal to
two Sides of the other Triangle, DE, DF, cach to each ; and have
the Angle A greater than the Angle D, contained under the equal
Right Lines, they ſhall alſo have the Baſe "BC, greater than the Baſe
EF, (which is the 24th of the ift) and conſequently,
If two Triangles ABC, DEF have two Sides AB, AC, equal
to two Sides DE, DF, each to each, and have the Baſe BC greater
than the Bafe EF; they ſhall alſo have the Angle A contained un-
der the equal Right Line, greater than the Angle D; (whicb is the
25th of the in)
B
7.
8. Again,
Euclide's Elements.
୨
8. Again, From the Propoſition it follows, that if two Triangles
BAC, and EDF, have two Angles of the one B and C, equal to
two Angles of the other E and F, each to its correſpondent Angle,
and have alſo one side of the one equal to one Side of the other;
either that Side which lieth betwixt the cqual Angles, or that which
is ſubtended under one of the equat Angles. The other Sides alſo of
the one ſhall be equal to the other sides of the other, each to his cor-
reſpondent Side; and the other Angle of the one ihall be equal to
the other Angle of the other; (Which is the 26th of the ſt.)
But it does not follow from hence, that if 2 Sides of one Triangle
be equal to two sides of another Triangle, and the Angle oppoſite
to one of the given Sides iti one, be equal to the Angle oppoſite to
one of the given Sides of the other, that the remaining parts of the
one Triangle, be equal to the remaining parts of the other Tri-
angle; becauſe the Angle oppoſite to the other given Side may be
either Acute or Obtuſe, and conſequently the remaining Side of
the one, may be greater or leſs than the remaining Side of the other.
9. And becauſe the Sides BD, DA, and the contained Angle D,
of the Triangle BDA (See the Figure belonging to the 4th Cor.) are
equal to the Sides DC, DA, and the contained Angle D, of the
Triangle CDA; therefore the remaining Side B A of the one, fall
be equal to the remaining Side CA of the other. Which is the 6th
Prop. of the sft. viz. That if two Angles ABC, ACB, of a Triangle
ABC, be equal the one to the other, then the Sides AC, AB, fab-
tended under the equal Angles, fhall be equal one to the other.
And
10. Hence every Equiangular Triangle is alſo Equilateral.
11. From the iſt Cor. we are taught how to biſect or divide
into two equal parts a given Right Lined Angle BAC. (Which is the
9th of the ift.)
Conſtr. On A as a Center, with any
A
diſtance deſcribe a Circle, cuting AB
and AC, in D and E, (by the 3d Pos7.)
and joyn the points D and E, (by the if
D
E
Poft.) upon which make an Equilateral
or Iſoſceles Triangle DFE, (by the 1ſt
Prop.) and draw the Line AF, which
B
C
will biſect the given Angle DAE.
For
:
10
Euclide's Elements.
D
For the Triangles ADF, and AEF, are mutually Equilateral
, by
Conſtruction, and conſequently the Angle BAF=CAF W.W.D.
12. From the ſame Cor. we are alſo taught how to biſect a given
Right Line. (Which is the 10th of the ift.)
Conftr. Upon AB make an Equilateral or Iſof-
с
celes Triangle ACB, and biſe& 'the Angle C
(by the former Cor.) with the Line CD, and
the thing is done. For AC=CB (by Conſtr.)
and the Side CD common, and the Angle
А
B ACD=BCD (by Conſtr.) therefore AD DB
(by the ad Prop.) W.W. D.
13. We are again taught from the ſame Cor. how from a
Point C, in a Right Line given AB, to ere& a Perpendicular or
Right Line CF, at Right Angles. (Which is the rith of the ift.)
F
Conſtr. Take CD equal to ce,
upon DE make an Equilateral or
Iſoſceles Triangle, (by the ift.) and
draw the Line FC (by the ift Posl.)
and it will be the perpendicular
A
B В required:
D с
Dem. For the Triangles DCF, and ECF, are mutually Equilateral
(by Conſtr.) therefore, <DCF=<ECF (by the 1ſt Cor.) therefore,
CF'is a Perpendicular (by the 10th Def.) W. W. D.
14. We are alſo taught agair, from the fame Cor. how, upon an
infinite Right Line given AB, from a Point given C, that is not in
it; to let fall a Perpendicular, (Which is the 11th of the itt:)
C
Conftr. From the Center C, deſcribe
a Circle, cuting the Right Line gi-
ven AB, in the points E and F, (by
the 3d Poſt.) then biſect EF in G, (by
the 12th Cor.) and draw the Line CĞ
LG
( by the iftPoft.) which will be the
А
B Perpendicular required.
E
F
D
12
E
M
Dem.
Euclide's Elements.
11
Dem. Let the Lines CE and CF, be drawn (by the 1ſt Poft.) then
are the Triangles EGC, and FGC, mutually Equilateral (by Constr.)
therefore, the Angles EGC and FGC are Equal, and conſequently
Right (by the 10th Def.) wherefore GC is a Perpendicular (by the ſame
Def.) W: W. D.
15. From the fame Cor. We are likewiſe taught, how at a Point
A in a Right Line given AB, to make a Right Lined Angle A, e-
qual to a Right Lined Angle given, D. (Which is the 23d of the ift.)
Conſtr. Draw the Right Line
CF, cutting the Sides of the
Angle given any ways, take
D
AG=CD, upon AG make a
Triangle Equilateral to the for-
mer CDF (by the Scholium of
the 1st.) So that AH be equal
C4
F G
to DF, and GH equal to CF;
H
then will the Angle A be equal
E B
to D. (by the il Cor. of the 2d.)
W. W.D.
Propoſition III.
A Theorem,
of Euclide, the 13th of the ift.
E
When a Right Line AB, ſtanding upon a
A
Right Line CĎ, maketh Angles ABC, ABD,
it maketh either two Right Angles, or two
Angles equal to two Right.
C.
B
Dem. If the Angles ABC, ABD, be equal, then they make two
Right Angles (by the roth Def.) If unequal, then from the Point E
let there be ereded a Perpendicular BE, (by the 13th Cor.)
Becauſe the Angle ABC=to a Right <+<ABE (by the 19th Ax.)
and the ABD=toa Right <-ABE (by the 3d Axı) therefore,
<ABC+< ABD=to two Right <, TABE--<ABE (by the
2d Ax.) therefore <ABC+<ABD=two right < W.W. D.
Corollaries.
1. Hence if one Angle ABD, be Right, the other Angle ABC is
alſo Right; if one Angle be Acute the other is Obtuſe, and ſo on
B 2
*
V
the contrary.
2. If
1 2
Euclide's Elements.
С
ES
D
2. If more Right Lines than one ſtand upon the ſame Right Line
at the ſame Point, the Angles ſhall be equal to two Right.
3. Two Right Lines cutting each other, make Angles equal to
four Right.
4. All the Angles made about one point, make four Right, as
appears by the ad Cor.
5. Hence we ſee the Reaſon of the 15th
Prop. that if two Right Lines interſee each
A
B
other, tho two Angles CEB and AED,
which are oppoſite, will be equal to each
other
For the <* AEC+ CEB = two Right <by the 3d. and the
AEC+AED=two Right by the ſame. Therefore, <'AEC+
CEB=<'AEC+ AEĎ, by the ift. Ax. Wherefore, AEC and DEB
are equal (by the 3d Ax.) W. W. D.
6. Wherefore, If the oppoſite Angles CEB and AED, allo AEC
and DEB be equal, then ſhall AB, and CD be ſtrait Lines.
7. Hence follows the Reaſon of the 16th Prop that if one Sida
A
F
BC, of a Triangle ABC, be produced, the
outward Angle ACD, will be greater than
either of the inward and oppoſite Angles CAB,
B
-D
Let the Right Line BE, bifect the Side
AC, and be produced till EF be equal to BE
and joyn the Points F and C. (by the 1s7 Poft.)
G
Becauſe EC=EA (by Conſtr.) and BE=EF, by the ſame, and the
<CEF=<AEB (by the 5th Cor.) Therefore, <ECF=EAB (by the
2d Prop.) but ACD is greater than ECF, (by the 9th Ax.) Therefore,
greater than BAC. By the like way of reaſoning the <ÁCD may be
proved greater than FCD=HCI='ABC.
8. Hence and from the (2d. Prop.) it follows, that any two Angles
of any Triangle, which way ſoever they are taken, are leſs than two .
Right Angles, which is the 17th of the 11t.)
9. Hence it follows, that in every Triangle, wherein one Angle
is either Right or Obruſe, the other two are Acute Angles.
10. If a Right Line make unequal Angles with another Right Line,
one Acute the other Obruſe, a Perpendicular let fall from any Point
to the Baſe Line, fhall fall on that Side the Acute Angle is of.
or CBA.
I
I
II, A11
..
Euclide's Elements.
13
11. All the Angles of an Equilateral Triangle, and the iwo An-
gles of an Iſoſceles Triangle that are upon the Baſe, are Acute.
Propofition IV. A Theorem,
of Euclid, the 18th of the iſt.
The greateſt Side AC, of every Triangle ABC, ſubtends the greateft.
Angle ABC.
Let AD be equal to AB, and joyn BD (by A
the ift post.) The <ADB=ABD (by the
4th Cor. of the 2d.) but ADB-ACB (by the
7th Cor. of the 3d.) Therefore, ABD - ACB,
D
and conſequently the whole <ABC-<4CB
B
(by the 9tb. Ax.) W. W. D.
С
1. Hence in every Triangle ABC, under the greateſt Angle B,
is ſubtended the greateſt Side CA, (which is the 19th of the ift.)
2. Hence likewiſe, of every Triangle BDC, two Sides BD, DC,
any way taken, are greater than the Side BC that remains, (which is
the 20th of the ift.)
For, produce CD till DA=DB, and draw BA, then will <DAB
KABD (by the 4th Cor. of the 2d.) but <ABC ABD (by the 9th Ax.)
Therefore, <ABC <BAC; therefore, BD+DC (=AC) --BC:
W. W. D.
Propoſition V. A Theorem,
of Euclid, the 29th of the ift.
1
If a Right Line EF, fall upon two parallel Lines AB, CD, it
will both make the alternate Angles DHG, AGH, equal to each o-
ther, and the outward Angle BĞE, equal to the inward and oppo-
Lite Angle, on the ſame Side DHE, as álfо the inward Angles on the.
ſame Side AGH, CHG, equal to two Right Angles.
14
Euclide's Elements,
E
D
H
F
the ift.)
It is evident that AGH+CHG=twoRight
Angles; otherwiſe AB, CD, would not be
G
parallel (by the 13th Ax.) which is contrary
A-
B
to the Hypotheſis. But moreover, the
<DHG+CHG= two Right Angles (by the
C-
13th of the ift.) Therefore, is < DHG=<
AGH (by the 1ſt and 2d Ax.) =<BGE (by
the 5th Cor. of the 3d.) W. W. D.
iſt. Hence if a Right Line EF, falling upon two Right Lines
AB, CD, make the alternate Angles DHG, AGH, equal the one
to the other, then are the Right Lines parallel, which is the 27th of
2. If a Right Line EF, falling upon two Right Lines AB, CD,
makes the outward Angle AGE of the one Line equal to CHG, the
inward and oppoſite Angle of the other, on the ſame"Side, or make
the inward Angles on the ſame Side AGH, CHG, equal to two
Right Angles, then are the Right Lines AB, CD, parallel (which
is the 28th of the ift.)
3. Hence it follows, that every Parallelogram having one Angle
Right, the reſt are alſo Right.
4. Hence Right Lines parallel to one and the ſame Right Line, are
alſo parallel the one to the other, (which is the zoth of the 11t.)
5. Hence we are taught, how from a Point given A, to draw a
Right Line parallel to a Right Line given BC, (which is the 31ſt of
the ift.)
Conſtr. From the Point A, draw a Right Line
AD, to any Point of the given Right Line
BC, (by the ift Poft.) with which at the Point
c thereof A, make an Angle DAE=<ADC
(by the 15th Cor. of the 2d.) then will AE and BC be parallel (by the
iſt Cor. of the 5th). W.W. D.
A
6. Hence and from the 2d Prop. it fol-
lows, that if two equal and parallel Lines
AB, CD, be joyned together, with two
other right Lines, AC, BD, then are thoſe
C
Right Lines alſo equal and parallel (which
is the 33d of the ift.)
E
B
D
B
D
7. Allo
*Euclide's Elements.
15
7. Alſo, Hence and from the 8th Cor. of the ad. it follows, that
in Parallelograms as ABCD, the oppoſite Sides AB, CD, and AC,
BD, are equal to each other; and the oppoſite Angles A, D, and
AB D; ACD, are allo equal ; and the Diameter BC biſects the
fame (which is the 34th of the st.)
8. Hence we are taught another way how from a Point given C,
to draw a Right Line CD, parallel to a Right Line given AB.
С
D
E
F
Conſtr. Take in the Line AB any Point E,
from the Centers E and c, at any diſtance
draw two equal Circles EF, CD, from the A-t
+B
Center F, by the ſpace of EC, draw a Circle
FD, which ſhall cut the former Circle CD, in the Point D, ther
fhall the Line drawn CD, be parallel to AB, For CEFD is a Pa-
rallelogram; therefore, &c.
Propoſition. VI. A Theorem,
of Euclide, the 32d of the ift.
N
Of any Triangle ABC, one Side BC
E
A
being produced, the outward Angle
ACD ſhall be equalto the two inward
and oppoſite Angles A, B, and the
three inward Angles of the Triangle B
A, B, ACB, ſhall be equal to 2 Right
C
D
Angles.
Conſtr. From C draw CE, parallel to AB, (by the 5th Cor. of the
sth.) Then is the <A=ACE (by the 5th) and the <B=ECD (by the
fame.) Therefore, <A+<B = ACETECD (by the 2d Ax.)
ACD (by the 13th Ax.) W. W. D.
Again, ACD+ ACB =two Right Angles (by the3d.) Therefore,
<A+<B+<ACB=two Right Angles (by the ift Ax.)W.W.D.
Coro
1. The three Angles of any Triangle taken together, are equal
to the three Angles
of any other Triangle taken together; whence
it follows,
2. That
16
Euclide's Elements.
---
2. That if in one Triangle,two Angles (taken ſeverally or together)
be equal to two Angles of another Triangle, (taken ſeverally or toge-
ther) then is the remaining Angle of the one, equal to the remaining
Angle of the other. In like manner, if two Triangles have one An-
gle of the one, equal to one Angle of the other, then is the Sum of
the remaining Angles of the one Triangle, equal to the Sum of the
remaining Angles of the other Triangle.
3. If one Angle in a Triangle be Right, the other two are equal
to a Right Angle ; likewiſe, that Angle in a Triangle which is e-
qual to the other two, is it ſelf a Right Angle.
4. When in an Iſoſceles Triangle, the Angle made by the equal
Sides is Right ; the other two upon the Baſe, are each of them half
. Right Angle.
5. An Angle of an Equitateral Triangle, makes two thirds of a
Right Angle, for one third of two Right Angles, is equal to two
thirds of one.
6. The Angles of a Right Lined Figure, do together, make twice
as many Right Angles bating four, as there are sides of the Figure;
as will be manifeſt, if from any Point within, Lines be drawn to
every Angle.
7. All the outward Angles of any Right Lined Figure taken toge-
ther, make up four Right Angles.
8. All Right Lined Figures, of whatfoever Species, have the Sum
of their outward Angles equal.
The 6th Cor. is of great Uſe in the Conſtruction of regular. Polygons,
and in taking the Plot of any irregular Polygon, for by it the good-
neſs of the Inſtrument, and care of the Artiſt is diſcovered.
Propoſition. VII. A Theorem,
of Euclid, the 35th of the iſt, ***,
Ά
D
E
h
G
Parallelograms BCDA, BCFE
which ſtand upon the fame Baſe BC,
and between the ſame Parallels AF,
BC, are equal one to the other.
B
C
HT
For
Euclide's Elements.
17
For AD=BC=EF (by the 7th Cor. of the sth.) ad ! DE common
to both, then is AE-DF (by the ad Ax.) alſo, AB DC (by the
9th Cor. of the sth.) and the A=D (by the sth ) therefore is
the Triangle BAE=CDF: take away from each the Triangle
GDE common to both Triangles, and there will remain the Ti ape-
zium BADG=CGEF(by the 3d Ax.) Add to each-Trapezium the
Triangle BGC common to both, then is the Parallelogram BADC=
BEFC (by the ad Ax.) W. W. D.
i. Hence, and from the iſt Ax, ir follows, that Parallelograms
ABDC and EFIH, ſtanding upon equal Baſes BC and IH, and be-
twixt the ſame Parallels AF, BH, are equal to each other (which is
the 36th of the ift.).
2. Hence we are taught how to find the Area of any Parallelogram,
for fince the Area of a Right-Angled Parallelogram is found by the
drawing or multiplication of any two contiguous Sides, one into a-
nother; the Area of any other Parallelogram will be found by the
multiplication of the Baſe into the Perpendicular Height.
3. Hence, and from the 7th
E A D F
Ax. it follows, that Triangles
BCA, BCD, ſtanding upon the
ſame Baſe BC, and between the
ſame Parallels BC, EF, are e-
P
qual one to the other (which is
B
C
the 37th of the ift.) For they
are the Halves of the equal Parallelograms BEAC, and BDFC.
4. Hence, and from the iſt Ax. it follows, that Triangles EBA,
DCF, ſtanding upon equal Baſes EA, DF, and between the ſame
Parallels EF, BC, are equal one to the other (which is the 38th of
the ift.)
s. From the 3d Cor. and 6th Ax. it follows, that if a Parallelogram
EBCA (Soe the Figure to the 3d Cor.) have the ſame Baſe BC, with
the Triangle BDC, and be between the ſame Parallels BC, EF, then
is the Parallelogram EBCA, double to the Triangle BDC. (Which
is the 41ſt of the 1st.)
For the Triangle BDC=BCA = half the Parallelogram EBCA.
therefore, O.
6. Hence we are taught how to find the Area of any Triangle, for
fince every Triangle is half the Parallelogram, having the fame Bale
D
and
8
Euclide's Elements
F
D
and Altitude, the Area thereof will be found by the multipli-
cation of the Baſe into half the Perpendicular Height, or of half
the Baſe into the whole Height; or of half the Product of the Baſe
into the Height.
7. Hence we are taught how to find the Area of any Right-lined
Figure ; for it be reduced into Triangles, by drawing Diago-
nals from Angle to Angle, the Sum of the Area's of all the Tri.
angles, ſhall be the Area of the Figure (by the 19th Ax.)
А.
G
8. Hence we are taught how to
make a Parallelogram ECGF, equal
P
to a Triangle given ABC, having an
Angle equal to a Right-lined Angle
B E с
given D (which is the 42d of the 1ſt.)
Thro' A, draw AG parallel to BC (by the 5th Cor. of the 5th.)
make the <BCG=D (by the 15th Cor. of the 2d) biſect the Baſe BC
in E (by the 12th Cor. of the 2d.) and draw EF parallel to CG, and
the thing is done. For AE being drawn, the ECG=D (by Conſtr.)
and the Triangle BAC=(by the 2d Ax.) twice AEC=Parallelogram.
ECGF. W. W. D.
9. Hence we are taught how to make a Triangle BAC, equal to
a given Parallelogram ECGF, having an Angle ACB = to a Right-
lined Angle given P.
Produce ĞF (by the ift Poft.) make the <ACB=<P (by the 15th
Cor, of the 2d.) ſet off CE, from E to B, and draw AB (by the 1ſt
Poſt.) and the thing is dorfe. For the Triangle BAC=Parallelogram
ECGF (by the 5th Cor.of the 7th.)
10. (From the 3d Cor. of the 7th) we are taught how to make a
Triangle BDC = Triangle BẠC, having an Angle BCD = to a
Right-lined Angle given P. (See the Figure of the 3d Cor. of the 7th.)
Thro' A, draw AD parallel to BC (by the sth Cor. of the 5th)
make the <BCD=<P (by the į sth Cor. of the 2d.) and draw the
Line DB (by the ift Poft.) and the thing is done. For the Triangles
ABC, DBC, are equal (by the 3d Cor. of the 7th)
How upon a given Right Line to make a Parallelogram; equal to a given
Triangle, having a given Angle (which is the 44th of the 11t) and thence
to make a Parallelogram equal to any Right-lined Figure given (which is
the 45th of the ift) may be ſeen in the 6th Cor. of the 19tb.
Prop.
.
41
Euclide's Elements.
19",
H
F
Propoſition. VIII. A Theorem,
of Euclide, the 47th of the 1ſt.
· In Right-Angled Triangles BAC,
G
the Square BE which is made of the
Side BC, that ſubtends the Right An-
gle BAC, is equal to both the Squares
BG, CH, which are made of the Sides
AB, AC, containing the Right An-
B
gle.
Conſtr. Joyn AE and AD (by the 1ſt
Poft.) and draw AM parallel to Ce
(by the 5th Cor. of the sth.)
D
A
I
с
L
ME
Dem. Becauſe the DBC=FBA (by the 12 Ax.) add the <ABC
common to both, then is the <ABD=FBC (by the zd Ax.) more-
over, AB=BF (by the 29th Def.) and BD=BC (by the ſame) there-
fore, is the Triangle ABD=Triangle FBC (by the ad) but the Pa-
rallelogram BM = twice the Triangle ABD (by the 8th Cor. of the 7th)
and the Square BG = twice the Triangle FBC (by the ſame.) For,
GAC is one Right Line, (by Hyp.) therefore is the Parallelogram BM
=Square BG (by the 6th Ax.) by the ſame way of Argument is the
Parallelogram CM=Square CH; therefore is the Square BE =Pa-
rallelogram BM+Parallelogram CM (by the 19th Ax.) =Square BG
+CH (by the ift Ax.) W. W.D.
1. Hence we are taught how to make one Square equal to any
number of Squares given.
B
Let the thiree Squares be given, whereof
the Sides are AB, BC, CE, make the Right-
angle FBZ (by the 13th of the ad.) and transfer
X
BĂ from B to A, and BC from B to C, and
joyn AC (by the 1ſt Post.) then is ACq=ABą
+BCq (by the 8th Prop) then transfer AC À BO
from B to X, and CE from B to E, and joyn
EX bythe ift. Poft.) then is EXq=EBq (ECE)
+BXqCby the8th)=ACq=CE7+ABq+BCI FA
B
(by the ad Ax.) W. W. D.
D 2
2. Hence
C С.
A.
:
E
#
20
Euclide's Elements.
E
2. Hence we are taught, how from two unequal Lines being given,
AB, BC, to make a Square equal to the Difference of the two
Squares, made upon the given Lines AB, BC.
From the Center B, at the diſtance of BA,
deſcribe a Circle (by the 3d Poft.) from the Point
Cere& a PerpendicularCE (by the 13th of the 2d.)
cutting the Circumference in E, and draw CE
A BC (by the 1ſt Poft.) which will be the side of the
Square required. For BÉq=BAq=BCo +CEq (by the 8th) where-
fore, BA9-BCq=CEq (by the 3d Ax.) W.W:D.
3. From the 8th and by the help of the Square Root, we are
taught how_from any two Sides being given, the third may eaſily
be found : For fuppoſe the two Sides containing the Right Angle
were given, to find the Side fubtending the Right Angle; in this
Cale, to the Square of one of the Sides, add the Square of the o-
ther, the Square Root of their Sum will be the Side required.
Again, ſuppoſe the longeſt Side, or Side fubtending the Right
Angle were given, and one of the containing Sides to find the other ;
- in this caſe, from the Square of the longeſt Side, fubtract the Square
of the other side, the Square Root out of the Remainder, will be
the remaining Side; as is abundantly manifeſt from what went be-
fore, which is the Theorem of the 47th of the ift.
How the Square Root may be performed, ſhall be thewn in its due
Place.
Propoſition IX. A Theorem,
of Euclide, the 1ſt of the ad.
T H I G If two Right Lines AF, AB, be given, and
one of them AB, divided into as many Parts or
Segments as you pleaſe, the Rectangle compre-
hended under the two whole Right Lines AB, AF,
A D E B
ſhall be equal to all the Rectangles contained under
the whole Line AF, and the leveral Segments AD, DE, EB.
Set
Euclide's Elements.
21
Set AF perpendicular to AB by the 13th of the 2d) thro' F draw an
infinite Line FG, perpendicular to AF (by the ſame) from the Points
D, E, B, erect Perpendiculars, DH, EI, BG, (by the fame) then is
AG a Rectargle, comprenended under AF, AB, and is equal to
the Rectangles AH, DI, EG, (by the 19th Ax.) that is, becauſe
DH, EI, AF, are equal (by the 7th of the 5th) to the Rectangles
under AF, AD, under AF, DE, under AF, EB, W. W. D.
1. Hence if two Right Lines given, be both divided into how
many parts ſoever, the product of the two whole Lines multiplied
into themſelves, ſhall be the ſame with that of the parts multiplied
into themſelves.
What has been ſaid of Lines, holds good of Quantitys in
general; thus, if the Quantities btc bc to be multiplied by
dte, the product will be adt.d64dcae teb tec.
And here I cannot omit taking notice of a very notorious Blunder, or rather
Error in Judgment, that prevails amongſt almoſt all the Teachers of Arith-
metick, viz. That of multiplying Money by Money, the Reſult of which they
call Money, which is in its own Nature abſurd; for as Multiplication is
no other but the Addition or Repeating the ſame thing a certain number of
times, and conſequently produces the Poſition or Affirmation of the thing,
ſo many times more than it was before : So therefore to multiply 4p(where
p in this caſe is put for a Pound Sterling) by the Number 4, is to add 4p
or repeat the Poſition or Affirmation of 4p, 4 times; and to repeat
the Pofition or Affirmation of 4p 4 times, is to make it 16p; and con-
Sequently the Préduct is in this caſe 16 Pounds. Whereas, to multiply
4p or 4 Pounds, by 4p or 4 Pounds, produces (by the preceding Corol-
lary) io pp, which cannot be called 16 Pounds, but in the Phraſe of the
Mathematicians is called a Quantitas Sui Generis; and is of a quite diffe-
rent Nature from that, ariſing from the Multiplication of 4 p or 4 Pounds,
by the Number 4, which Produt alone is properly called 16 Pounds. For
of the Produkt ariſing from the Multiplication of 4 Pounds by 4 Pounds,
be 16 Pounds, then it follows, that 4p is equal to 4 PP, a Part to the
Whole, which is impoſſible (by the 9th Ax.)
Upon, the 9th Propofition, the reaſon of Algebraic Multiplication is
founded, and therefore whoever goes about to prove this, and conſe-
quently the reſt of the Propofitions of the ad Bvok Algebraically (ms fome
of
:
22
Euclide's Elements.
·Z::
of late have done) take the thing for granted, which it is intended
to prove, and leave the Reader in reality in as much Ignorance as they
found him.
2. Hence, if a Right Line Z, be cut any-wiſe into two parts, the
Square made of the whole Line 2, is equal both to the Squares made
of the Segments A, E, and to twice a Rectangle made of the parts
A, E.
I ſay Zq=A9+2 AE+Eq.
For becauſe 2=A+E (by the 19th Ax.)
therefore, Za=ZA+ZE (by the 2d Ax.)
4..
E
but, ZA= Aq+AE (by the ſame) and ZE
-AE+Eq (by the ſame) therefore,Zq=ZA
+ZE=Aq+24E+ Eq (by the 2d Ax.) and conſequently, Zg=A9
+2AE+Ēq. W. W. D.
3. Hence, we are taught how to Extra& the Square Root out of
any given Number.
Let the given Number whoſe Root is to be found be 576, which put
equal to Zq=A9+2AE+-Eq, then will A+E=Z=V
576. Aſume
A=20, then will Aq=400, and conſequently 576–400=176=2AE
+ EE=2A+ExE. Now in order to find out E, the other part of the
Root, enquire how oft 2A=40 may be had in 176, with this condition,
that the Anſwer being added to 2A=40, and that Sum multiplied by the
Number found, may not exceed 176=2A + ExE, and the Anſwer in
this caſe will be 4 ; for 40+ 4x4=2A+ExE=160+16=2AE+ Eq
=176, wherefore Z=A+E=20+4=24 is the Root required.
If any one compare the ſteps here taken, with the Rules uſually delivered
for Extracting the Square Root will find them exactly to agree.
But to ſhew
the univerfality of the Method, and that the Practitioner can-
not be out in his Allumption of the value of A, we will aſſume A=16, then
will Aq=256, and conſequently 2A + ExE=576—256=320 ; enquire
therefore bow oft 2 A or 32 may be had in 320, with the proper limita-
tion, and the Anſwer will be 8, for 32+ 8x8=2A + ExE=256-464
=2AE+Eq=320. Wherefore, 16+8=A+E=24=2, the Root re-
quired.
-
Again,
Euclide's Elements.
23
Again, Let us take A=30, which is manifeftly more than the true Roots
then will Aq be equal to 900, and conſequently 2A-+-ExE=576--900
5-324, whence E will be found equal to -6, and 2A + ExE=50-6
x-65-360+36=–324=2AE+Eq. Wherefore, Z=A+ E=3€
-6=24, the true Root as before found.
4. Hence it follows, that the Ređangle inade of the Sum and
Difference of two Lines, is equal to the Difference of their Squares,
(which is the sth of the 2d.) and of Uſe in Trigonometry for finding one
of the sides of a Right-Angled Triangle, when the Hypotenuſe and
other Side is given.
Let A and E repreſent two Lines, put Z=A E=Sum, and X-A
-E= Difference, then by the ad Ax.) will ZX=Aq+AE-AE+
Eq=Aq+Eq (by the 3d Ax.) W. W. D.
5. Hence it follows that if a Right Line A, be divided into equal
parts, and another Right Line E, be added to the fame dire&ly in
öne Line, then I ſay, QATE=1 Aq+AE+Eq (by the ad Cor. of
the 9th.) which is the 6th of the 2d.
6: Hence if three Right Lines be in Arithmetical Proportion, the
Re&angle contained under the Extreams, together with the Square
of the Difference, is equal to the Square of the middle Term.
7. Hence if a Right Line Z, be divided any-wiſe into two parts
A and E, I ſay 29+Eq=2ZE+ Aq (which is the 7th of the 2d.)
For, 2q=Aq+Eq+2AE (by the ad of the 9th.) and 2 ZE=2Eq
+2AE (by the ad Ax.) therefore, 29-2ZE=49- Eq (by the 2d
Ax.) Therefore, Zq+ Eq=2ZE+AZq. W.W.D.
8. Hence it follows, that the Square of the Difference of two
Lines, is equal to the Squares of both the Lines, leſs by a double
Ređangle comprehended under the Lines.
9. Hence we are taught how to cut a Right Line given AB, in
the point G, ſo that the Rectangle comprehended under the whole
Line AB, and one of the Segments BG, ſhall be equal to the Square
that is made of the other Segment AG, (which is the 11th of the 2d.)
and is the fame with the 30th of the 6th, viz. To cut a finite Right
Line, according to Extream or Mean Ratio.
Conftr.
.
24
Euclide's Elements.
С
B
G
)
Conſtr.
Upon AB form the Square AC, bife & the
Н
Side AD in E, and draw EB; make EF
equal to EB, on AF make the Square AH,
then is AH=ABXBG.
D
А.
F
Dem. For HG being produced to 1(by the ad Poft.) the Rect. DHto
EAq=EFq (by the 5th of the yth.)=EBq (by Conſtr.) =B Aq+EAq(by
the 8th.) therefore, is Rect. DH=RAI (by' the 3d Ax.) = to the
Square AC, take away the Rect. Al common to both, and there re-
mains the Square AH equal to the Rea. GC; that is, AGI=ABX
BG. W. W. D.
10. From the 8th and 2d of the 9th it follows, that in an Obtuſe
Angled Triangle ABC, the Square that is made of the Side ſubrend-
ing the Obtuſe Angle, is greater than the Squares of the Sides BC,
AB, that contain the Obtuſe Angle ABC, by a double Rectangle
contained under one of the Sides BC, which are about the Obruſe
Angle ABC, on which Side produced the Perpendicular AD falls,
and under the Line RD taken without the Triangle, from the point
on which the Perpendicular Að falls, to the Obtufe Angle ABC.
I ſay, Acq=BCq+ABq-t-2CBxBD.
For ACq=CDq+ADI (by the 8th)=CBQ---
2CBD |-BDq+ADq (by the 2d of the 9th) :
CBq.1-2CBD TABq (by the 8th.)" W. W.D.
A
.
D.
„С
B
A
11. In Acute-Angled Triangles ABC, the
Square made of the Side AB, ſubtending the
Acute Angle ACB, is leſs than the Squares
made of the Sides AC, CB, comprehending the
Acute Angle ACB, by a double Redangle
C
contained under one of the Sides BC, which
are about the Acute Angle ACB on which the Perpendicular AD falls,
and ander the Line DC, taken within the Triangle from the Perpen-
dicular AD, to the Acute Angle ACB.
I fay that Acq+ECq=AB9 + 2BCD.
For
co
D
Euclide's Elements.
25
For ACq+ BCq=ADq+DCq+ BCq (by the 8th) =ADq+BDq
+ 2BCD (by the 7th of the gih) = ABq+ 2BCD (by the 8th.)
W.W. D.
12. Hence the three sides of any Right Lined Triangle being
known, the external Baſe BD in the 10th, and the Segments DC, BD,
in the 11th, as alſo the Perpendicular (by the 3d of the 8th) may be
calily found; and from thence the Angles of each Triangle, as alſo
their Areas, as fhall be ſhewn hereafter.
:
Propoſition X. A Theorem,
of Euclide, the 16th of the 3d.
ty
I
G
B
H
A Line CD, drawn from the extream Point C F А.
D
of the Diameter HA, of a Circle BALH, Per-
pendicular to the ſaid Diameter, ſhall fall with-
L
out the Circle and between the ſame Right
Line, and the Circumference cannot be drawn
another Line AL, and the Angle of the Semi-
circle BAI is greater than any Right Lined A-
cute Angle BAL. And the remaining Angle without the Circumfe-
rence DAI, is lets than any Right Lined Angle.
From the Center B to any Point F, in the Right Line AC, draw
the Right Line BF, the Side BF ſubtending the Right Angle BAF,
is greater than the Side BA, which is oppoſite to the Acute Angle
BFA (by the 1st Cor. of the 41h.) therefore, whereas BA=BG) reach-
es to the Circumference, BF ſhall reach farther; and to the Point F
and for the ſame Reaſon is any other Point of the Line AC, placed
without the Circle. W.W. D.
2. Draw BE Perpendicular to AL (by the 14th Cor. of the 2d.) the
Side BA oppoſite to the Right Angle BEA, is greater than the Side
BE, which ſubrends the Acute Angle BAE (by the iš7 Cor. of the 4th)
therefore the Point E, and ſo the whole Line EA falls within the
Circle. W.W.D.
3. Hence it follows, that any Acute Angle, viz. EAD is greater
than the Angle of Contaa DAI, and that any Acute Angle BAL, is
leſs than the Angle of a Semicircle B AI. W.W.D. Hence,
E E
1st
26
Euclide's Elements.
{
E
I
A
с
F
D
G
ift. A Right Line drawn from the Extremity of the Diameter, and
at Right Angles to it, is a Tangent to the ſaid Circle.
2. Wherefore, If any Right Line FD, touch a Circle, and from
the Center to the point of Contact A, a Right Line BA be drawn,
that Line BA ſhall be Perpendicular to the Tangent FD. (Which is
the 18th of the 3d.)
B
3. Hence, and from the 4th and 8th Cor. of
the ad it follows, that if from the Point A, a
K
Right Line CA be drawn parallel to the Tan-
gent GH, it ſhall be biſcated by the Diameter
BD, paſſing through the Point of Contact.
(Which is the ſubſtance of the 3d of the 3d.)
H
4. Hence all Lines drawn within the Circle parallel to the Tan-
gent, are bifected by the Diameter paffing through the Point of
Contact:
5. Wherefore if a Line be drawn any-wiſe within the Circle, and
through the middle of it, and at Right Angles to it, another Right
Line be drawn, that Line ſhall paſs through the Center.
6. Hence we are taught how to find the Center of any given Cir-
cle. (U'hich is the ift of the 3d.) viz.
1. Draw any Line as AC'within the Circle, biſca it by the Line
DB (by the 12th Cor. of the 2d.) and the Center of the Circle will be
found in the middle of this Line (by the 5th of the roth) wherefore,
if this Line be again bifected (by the 12th of the 2d.) with the Line
IK, you will have the Center E required.
7. From the ift Cor. we are taught, how from a Point given A, to
draw a Right Line AC, which ſhall touch a Circle given DBC.
(Which is the 17th of the 3d.)
From the Center D, draw DA cutting the Circumference in B,
E
(by the If Pos7.) from B draw a Perpendicular BE
(by the 13th of the 2d) on D, with the Diſtance
DA deſcribe a Circle (by the 3d Poft.) cutting
BE in E; from whence to the Center draw DE,
cuting the Circumference in C; laſtly, draw AC
and the thing is done.
A А
B)
For,
Euclide's Elements.
27
For; DC=DB, and DA=DE (by the 15th def.) and < D com-
mon, therefore is <DCA=DBE=Right (by the zd.) and AC
a Tangent (by the ift Cor.)
Propofition. XI. A Theorem,
of Euclide, the 20th of the 3d.
+
A
In a Circle DABC, the Angle BDC at the Cen-
ter, is double the Angle BAC at the Circumference ;
when the ſame Arch of the Circle BC, is the Baſe
of the Angles.
B В
If one of the Lines paſs through the Center, as
in the ift Caſe, then the <BDC=DAC+DCA
(by the 6th) but <DAC=DCA (by the 4th of the
2d therefore, <BDC=2DAC WW: D.
A
D
P
E
E
A
If neither paſs through the Center, as in the 2d.
and 3d Caſes; draw the Diameter ADÉ, then the
outward Angle BDE=DAB+DBA (by the 6th)
=2DAB (by the 4th of the 2d.) alſo, <EDC
2DAC by the ſame, therefore in the 2d Caſe, the
D
whole <BDC=2DAC (by the ad. Ax.) and in the
3d. Caſe, the remaining <BDC=2BAC (by the
B
20th Ax.) W. W. D.
1. Hence and from the 7th Ax. it follows, that
in a Circle EDAC, the Angles DAC and DBC, A
which are in the ſame Segment are equal.
(Which is the 2 ift of the 3d) or which ſtand upon
equal parts of the Circumference, in equal Cir-
cles, (Which is the 26th and 27th of the 3d) are
D
equal.)
E
B
E 2
2. From
:
:
1
28
Euclide's Elements
E
D
с
2. From Prop. 6th and iſt it follows, that the
B
Angles ADC, ABC, of a Quadrilateral Fi-
gure ABCD, inſcribed in a Circle, which are
A
oppoſite one to the other, are equal to two
Right Angles.
с
D
For, ABG +(BC A=) BDA+(BAC=)
BDC= two Right Angles (by the 6th) =ABC
+ ADC (by the ad Ax.) Which is the 2 21 of the 3d
3. In a Circle, the < ABC which is in
the Semicircle, is a Right Angle ; but the
Angle which is in the greater Segment BAC,
B
is leſs than a Right Angle; and the Angle
which is in the lefler Segment BFC, is great-
А
er than a Right Angle. Moreover, the An-
gle of the greater Segment is greater than a
Right Angle; and the Angle of the lefler
Segment is leſs than a Right Angle.
Draw from the Center D, DB; becauſe AD, DB, DC, are e-
qual (by the 15th Def.) therefore is the A=<DBA (by the 4th of
the 2d.) and <DCB=< DBC by the lame, therefore, <ABC
KA+<ACB (by the ad Ax.)= a Right Angle(by the 3d Cor. of the
6th) =EBC (by the 6th.)
Again, becauſe <ABC is Right, the <BAC is Acute (by the 9th
Cor. of the 3d.) and becauſe (by the 2d Cor of the 11th)<BAC+BFC
=two Right, therefore <BFC is an Obtuſe Angle.
Laſtly, the Angle contained under the Right Line CB, and the
Arch BAC, is greater than the Right <AEC ; but the <made
by the Right Line CB, and the Periphery of the leſſer Segment BFC,
is leſs than the Right < ABC. (by the 9th Ax.) W.W. D.
4. Whence, In a Right Angled Triangle, if the Hypotenuſe AC
be biſected in D, a Circle drawn from the Center through the Point
4, ſhall alſo paſs thro' the Point B, which is of Uſe in the Con-
ftru&ion of Quadratic Equations.
B
5. Hence we are taught, how from a given
Point P, at the end of a Right Line DP, to e-
reet a Perpendicular PB, or Line at Right An-
gles.
Conar,
D A
F
Euclide's Elements.
29
38)
Conſt. Placing one Foot of the Compaſſes in P the Point given,
with any diſtance CP, placing the other Foot in C, deſcribe the Circle
BPA (by the 3d Poft.) and through A and C draw the Right Line AC
(by the iſt Poft.) and produce it (by the 2d Pos7.) til it cuť the Cir-
cumference in B, and draw BP (by the 1ſt Poft.) and the thing is
done.
For becauſe ACB is a Diameter, the Angle .APB is a Right Angle
(by the 3d of the 11th) and BP a Perpendicular (by the roth Def.)
6. Hence we are taught, how from a Point given B, over the end
of a Right Line DP, to let fall a Perpendicular BP.
For from the given Point B, to any Point A in the Right Line
BD, draw the Line BA (by the 1ſt Poſt.) and biſex AB in the Point
C (by the 12th of the 2d.) on C as a Center, with the diſtance CA or
CB, deſcribe the Circle CBPA (by the 3d Poft.) cutting the Line DP
in P, and draw BP (by the ift Poft.) and the thing is done.
For, becauſe C is the Center of the Circle CABP, and AB a Di-
ameter, therefore is the Angle APB a Right Angle, by the (3d of
the lith) and BP a Perpendicular (by the roth Def.)
7. It a Right Line AB touch a Circle, and
from the Point of Contaa be drawn a Right
Line CE cuting the Circle, the Angles ECB,
EÇA, which it makes with the Tangent Line,
are equal to thoſe Angles EDC, EĚC, which
are made in the Alternate Segment of the Baſe.
(Which is the 32d of the 3d.)
A
B
Let CD Be Perpendicular to AB, then is the <CED Right (by
the 3d of the sith.) and therefore <D+DCE= Right (by the 3d
Cor. of the 6th) =<ECB+DCE (by Conftr.) therefore <D=ECB
by the 3d. Ax. Again, becauſe ECB+ECĀ= two Right (by the
=D+F (by the ad of the sitb) therefore is <F=ECA (by the
3d Ax.) W.W. D
8. Hence we are taught, how upon a Right Line AB, to deſcribe
the Segment of a Circle, which ſhall contain an Angle equal to a
Right Lined Angle given C, viz. (Which is the 33d of the 3 d.)
D
F
с
Ву
30
Euclide's Elements.
I
By making the <BAD=C (by the 15th of
the 2d) and drawing AE Perpendicular to AD
H
E by the IIth of the 2d) then, making the < ABF
= BAF (by the 15th of the 2d) and drawing
EF to cut AE in F, which will be the Center
А
of a Circle, about which if a Circle be drawn,
with the Radius FA or FB, the Segment AIB
Ic will be the Segment required. For becauſe AE
is Perpendicular to HD, the <AEB (=AIB)
by the 1ſt of the 11th) =BAD(by the 5th of the 11th) =<C by Con-
ſtruction. W. W. D.
B
A
B
9. Hence we are taught likewiſe, how
from a Circle given ABC, to cut off a Seg-
ment ABC ; containing an Angle B, equal
to a Right Lined Angle given D (which is the
/
34th of the 3d.)
For draw the Tangent EF, touching the Circle in A (by the 7th
of the icth) let AC be drawn, making an Angle FAC=D (by the
15th of the 2d.) this Line ſhall cut off ABC, containing an<B=CAF
(by the 5th of the 11th)=<D (by Conſtruction.). W. W. D.
E
A
F
D
Propoſition. XII. A Theorem,
of Euclide, the 35th of the 3d.
с
E
If in a Circle FBC A, two Right Lines AB, DC,
cut each other, the Rectangle comprehended un-
der the Segments AE, ER, of the one, ſhall be
equal to the Re&angle comprehended under the
Segments CE, ED, of the other..
B
1. Cafe, If the Right Lines cut one the other in the Center, the
thing is evident.
2. Caſe,
:
Euclide's Elements.
31
A
2. Cafe, If one Line AB, paſs thro' the
Center F, and biſe& the orher DC, then draw
FD (by the iſt Poft.) now, AEⓇEB+ FEq=FBq
(by the 4th of the 9th)=FD9 - EDq+ FEI (by the
8th) therefore, AE EB=EDa (by the 3d Ax.)
=DEXEC. W. W. D.
F
C
E
B
3. Caſe, If one of the Lines paſs thro' the
Center, and cut the other unequally, biſect CD
by FG (by the 12th of the 2d) a Perpendicular from
the Center, then is, AEXEB 1. FEq=(by the
4th of the 9th) FBq=FDq=FGq GDq (by the
8th) = FGq+GEq --CEXED (by the 4th of the
9th)=FEq+CEXED (by the 8th) therefore, AE
XEB=CEXED (by the 3d Ax.) W. W. D.
F
C
GE
D
B
G
A
с
4. Caſe, If neither of the Right Lines AB,
CD, paſs thro' the Center, then thro”
the Point of Interſection E, draw a Diameter
GH, and it will be (by the laſt Cor.) AEXEB
GEXEH=CEXED. W. W. D.
EVE
D
H
B
D
Propoſition XIII. A Theorem,
of Euclide, the 36th of the 3d.
If any Point D, be taken without a Circle
EBC, and from that Point two Right Lines
DA, DB, fall upon the Circle, whereof one
DA, cuts the Circle, the other DB touches it,
the Rectangle comprehended under the whole
Line DA that cuts the Circle, and under DC,
that part which is taken from the Point given
D, to the part of the Periphery, ſhall be equal
to the Square made of the Tangent Line.
с
#
B
А.
I ſay,
32
Euclide's Elements.
E
F
I ſay, ADxDC=DBq.
D
1. Cafe, If AD the Secant Line, paſs thro'
the Center, joyn EB, this will make a Right
Angle with the Line DB, wherefore, DBq+
EBq (=ECI)=EDq (by the 8th) = ADxDC,
+ĒCq (by the 5th of the 9th) therefore, DB9
B
=ADxDC (by the 3d Ax.) W. W. D:
2. Cafe, If AD paſs riot thro' the Center
then draw. EC, EB, ED, (by the ift Poft.) and
EF a Perpendicular to AD (by the 13th of the
2d) which will biſect AC (by the 3d of the roth.)
Becauſe, BDq+EBq=EDq (by the 8th)=EFq+FDg by the ſame
=EFq+ADxDC+FCq (by the 5th of the oth)=ADxDC+ (ECq=)
EBq (by the 8th) therefore, is BDq=ADxDC (by the 3d Axiom.)
W. W. D.
1. Hence, If from any Point A, taken with-
out a Circle, there be ſeveral Lines AB, AC
А
drawn, which cut the Circle, the Rectangles com-
prehended under the whole Lines AB, AC, and
the external Parts AE, AF, are equal between
D
themſelves.
A
F F
E
P
А.
For if the Tangent Line AD be drawn, then is
CAXAFE ADq=BAXAE (by the 13th) where-
fore, CAXAF-BAXAE (by the 1ſt Ax.)
2. It appears alſo from hence, that if two
Lines AB, AC, drawn from the ſame Point
do touch the Circle, thoſe two Lines are
equal. For ABq = AEXAF = ACI (by the
13th) wherefore, AB=AC.
3. It is alſo evident, that from a Point ta-
ken without a Circle, there can be drawn but
two Lines AB, AC, that ſhall touch the Cir-
cle.
F
B
C
의
​4. And
!
1
Euclide's Elements.
33
4. And on the contrwy, If two equal Right Lines AB, AC, fall
from any Point A, upon the Convex part of the Periphery of a Cir-
cle, that if one of thoſe equal Lines AB, touch the Circle, the other
AC, touches the Circle aſſo.
5. Alſo, hence it is manifeſt, that if AEX.4F=ABq then AB will
touch the Circle that AE cuts. (Which is the 37th of the 3d.)
Propoſition XIV. A Problem
of Euclide, the 2d of the 4th.
A
1.
D
In a Circle given ABC, to
infcribe a Triangle ABC, E-
quiangular to a Triangle given
DEF.
c
Let the Right Line GH, E
B
touch the Circle given in A,
(by the 7th Cor. of the 10th)
make the <HAC=E (by the 15th of the 2d) and the GAB=F
by the ſame, and joyn BC (by the if Poft.) and the thing is done.
For <B=HAC (by the 5th of the lith)=E (by Conftru&tion) and
the <C=GAB=F (by the ſame) whence, the BAC=D (by
the ad of the 6th) therefore, the Triangle BAC inſcribed in the Cir-
cle, is Equiangular to the Triangle DEF. W. W. D.
This.Propoſition is of great Uſe in the infcribing of any regular Poligon
in a Circle, for by the help of Iſoceles Triangles, whoſe Angles at the Baſe
are the multiples of the Angles at the Vertex, may Figures of odd Num-
ber of Sides be inſcribed ; and Figures of even Number of Sides are in-
fcribed in Circles, by the help of Iſoceles Triangles, whoſe Angles at the
Baſe are multiples Siſquialter of thoſe at the Top.
F
Thus
Euclide's Elements.
34
H.
A
B
1. Thus by inſcribing an Iſoceles Triangle,
whoſe Angle at the Baſe ACB, is once and an
half the Angle at the Vertex A; we ſhall have
one Side CB of a Square ACBD, in cribed in a
Circle. (Which is the 6th of the 4th )
E D
F
H
A
G
B
E
2. Again, by inſcribing an Iſoceles Triangle,
whoſe Angle at the Bale ACD, is doublc of
the Angle at the Top CAD; we ſhall have
one Side CD of a Pentagon, to be inſcribed
in a Circle. (Which is the irth of the 4th.)
I
L
D
F
D
A
3. By inſcribing an Equilateral Triangle ACE
P
F
in a Circle, and biſecting each Arch in the Points
B, F, D, and drawing the Lines AB, BC, CD,
DE, EF, and FA; we ſhall have an Hexagon,
AFEDCB, inſcribed in a Circle. (Which is the
15th of the 4th.)
4.Again, by inſcribing an
Equilateral Triangle ABC in
a Circle, and from the ſame
point A, an Equilateral Pen-
tagon AEFGH, the dif-
H ference BF,between the Arch
AF and AB, will be the
Side of a Quindecagon ;
for } - {==i}
Or, if an Iſoceles Triangle
be inſcribed, whoſe Angle at
the Baſe is 7 times the An-
gle at the Vertex; the Baſe
thereof will be the side of
the Quindecagon. (which is
the 16th of the 4th.) and ſo for any other.
E
D
B
C с
F
5. Hence
Euclide's Elements.
35
I
s. Hence we are taught, into what parts a Circle may be divided
Geometrically, and what rot. For,
A Circle is 4. 8. 16. &c. by the iſt of the 14th, and 11th of the ad.
Geometrically) 3. 6. 12. &c. by the 3d of the 14th, and 11th of the 2d.
divided into 5. IC. 20. &c. by the ad of the 14th, and IIth of the 2d. 3
Payts. 15. 30. 60. &c. by the 4th of the 14th, and 11th of thead.
Any other ways of dividing the Circumference into any parts given, is
as yet unknown to Geometers.
6. By drawing the Tangent Lines HG, GF, FE, EH, (Sce the
Figure belonging to the ist Cor. of the 14th Prop.) to the ſeveral
Points C, B, D, A, we ſhall have a Square circumſcribed about
the Circle. (Which is the 7th of the 4th.)
7. Ili like manner, by drawing Tangents HG, GL, LF, &c.
(See the Figure belonging to the 2d Cor.) to the ſeveral Angular
Points A, E, D, of the Pentagon inſcribed in the Circle, we ſhall
have a Pentagon circumſcribed about a Circle, (Which is the 12th
of the Iſt.) and ſo for any other.
8. By biſeding the Sides AB, AC.
BC, of the Triangle ABC, inſcribed
in a Circle, and drawing HI, IK, KH,
A
to the ſeveral Points of Inrerſection L,
M, N, we ſhall have a Triangle cir-
M
cumſcribed about the Circle, ſimilar to
a given Triangle. (by the 14th and 5th)
(Which is the 3d of the 4th.)
H
N
K
9. Hence we are táugbt, how about any Triangle to circumſcribe
a Circle ; (which is the 5th of the 4th) or which is the ſame Problem
to draw a Circle that ſhall paſs through any three Points, not ſcitu-
ated in a ſtraight Line, viz. by biſecting any two Sides BA, AC, of
the Triangle BAC, by the Lines DF, EF, and the Point of Inter-
fe&tion E, will give the Center fought.
10. In like manner, if any two contiguous Sides of a Square be
biſe&ed by two other Lines, their interfe&ion will give the Center
of the Circle to be circumſcribed: (Which is the 9th of the 4th.)
And likewiſe, if any contiguous Sides of a Pentagon be bifected,
you will by that means have the Center of the circumſcribing Circle.
F 2
Which
!
T
D
E
F
B
G
;
36
Euclide's Elements.
(Which is the 14th of the 4th.) And proceding in the ſame manner,
you may find the Center of a Circle, that ſhall circumſcribe any re-
gular Poligon.
11. The Center of the Circle circumſcribing any Equilateral Fi-
gure, is the Center of the Circle to be inſcribed in the fame Figure:
thus, by biſeaing the two contiguous Sides of the Square, you have
the Center of the Circle to be inſcribed (which is the 8th of the 4th,)
and by biſe&ting any two sides of the Pentagon, you will have the
Center of the inſcribing Circle, (which is the 13th of the 4th) and ſo
for any other.
A
E
G
12. By biſeeting any two Angles B, C,
of a Triangle ABC, with the Lines BD, DC,
we ſhall have the Center D, of a Circle EFG,
to be inſcribed in the Triangle BAC. (Which
is the 4th of the 4th.)
D
B
F
C
13. The following Problem is of Uſe in making a Pentagon upon
a given Right Line, and inſcribing the ſame in a Circle, viz. To
make an Iſoſceles Triangle ABD, having cach Angle B, D, at the
Baſe double to the remaining Angle A at the Vertex. Which is
the roth of the 4th.)
1. Take any Right Line AB, and di-
vide it in C (by the 9th of the 9th) ſo that
ABXBC=ACq; from the Center A, thro'
B, deſcribe the Circle ABD (by the 3d
A
Poft.) and make BD=BA, and joyn AD,
and the thing is done; and ABD the Tri-
angle required.
С
B
Conſtr. Draw DC by the iſt Poſt. and
about the Points. C, A, D, deſcribe a
Circle (by the 9th of the 14th.)
Now becauſe ABXBC=ACq=BDg, BD touches the Circle that
AB cuts (by the 5th of the 13th) therefore, <BDC=<A (by the 5th
of the IIth) therefore, <BDC+<CDA=CDA+<A (by the 2d
Ax.).
but
!
--
Euclide's Elements.
37
but <BDG+CDA><A+<CDA=BCD (by the 6th) but <
BDC + CDA=<CBD therefore, <BCD=<CBD (by the 1ſt Ax.)
therefore, DC= DB= AC therefore, <CDA=A=<BDC there-
fore, <ADB=CA=ABD. W.W. D.
numeremony
Propoſition XV. Of Proportion.
The ſubſtance of the 5th Book of Euclide.
Two Numbers, Lines or Quantities, A and B, being propoſed;
their Relation one to another may be conſidered under one of theſe
two Heads.
1. How much A exceeds B, or B exceeds A, and this is found
by taking A from B, or B from A, and is called Arithmetic Reaſon
or Ratio. Or,
2. How many times or parts of a time A cortains B, or B con-
tains A, and this is called Geometric Reaſon, or Ratio (or as Euclide
defines it, (in the 3d Def. of the 5th; it is the mutual Habitude or Re-
Spect of two Magnitudes of the ſame kind according to Quantity; that is,
as to how oft the one contains or is contained in the other) and is found by
dividing A by B, or B by A; and here Note, that that Quantity which
is referr'd to another Quantity, is called the Antecedent of the Ratio ;
and that to which the other is referr'd, is called the Conſequent of the
Ratio; as in the Ratio of A to B, A is the Antecedent, and B the Cone
Sequent.
3. Therefore, any Quantity as Antecedent, divided by any Quan-
tity as a Conſequent, gives the Ratio of that Antecedent, to that Con-
fequent.
A
B
Thus, the Ratio of A to B is but the Ratio of B to A is
B
A
and in Numbers the Ratio of 12 to 4 is
--=3 or Triple, but the
www
12
4
Ratio of 4 to 12 is 4-5 or Subtriple, and in the following is noted
by the Letter :
I 3
And
38
Euclide's Elements.
4
And here Note, that the Quantitys thus compared, muſt be of
the ſame kind, that is, ſuch winch by Multiplication may be made
to exceed one the other, or as (it is exprefled in the 4th Def. of the
Sth) thoſa Quantitys are j.lid 10 have a Ratio between them, which being
multiplied, may be made to exceed one another. Thus, a Line how
ſhort focver may be multiplied, that is, produced ſo long as to ex-
ceed in Length any given Right Line, and conſequently theſe may
be compared together, and the Ratio exprefled, but as a Line can
never by any Multiplication whatſoever, be made to have Breadth,
that is, be made equal to a Superficies, how ſmall ſoever; theſe can
therefore never be compared together, and conſequently have no Ra-
to or Reſpea one to another according to Quantity, that is, as
to how oft the one contains, or is contained in the other.
5. Magnitudes or Quantitys, A, B, C, D, are' ſaid to be Propor-
tional, the ift A, to the ad B, and the 3d C, to the 4th D, when
the Rario of the ift A, to the 2d B, 15 the ſame with the Ratio of the
3d C, to the 4th D; or which is the ſame thing, when the firſt A,
as oft contains or is contained in the ſecond B, as the third C, con-
tains or is contained in the fourth D. Thus, the Numbers 4,
8, 12,
24, are proportional Numbers, becauſe the firſt Number 4 is as oft
contained in the ſecond Number 8, as the third Number 12, is
contained in the fourth Number 24, that is twice, and which is
therefore called the common Ratio
Whence it follows that Magnitudes which to one and the fame
Magnitude have the ſame Ratio, are equal the one to the other, and
if a Magnitude have the ſame Ratio to other Magnitudes, thoſe Mag-
nitudes are equal one to the other. (Which is the oth of the sth.)
6. Now in comparing the Quantitys A, B, C, D, together, if
the Ratio of the ſecond B, to the third C, be found to be the ſame
with the Ratio of the first A, to the ſecond B, and of the third C,
to the fourth D, then thoſe four Quantitys A, B, C, D, are ſaid to
be in Geometric Proportion continued, or ſome times in Ceometric
Progreſſion, and are therefore called Progreſſional Numbers; and ſo
for any greater Number, viz. 5, 6, 7, c. thus, the Numbers #, 1,
1, 2, 4, 8, 16, 32, c. or, 81, 27, 9, 3, 1, 1, 5, c. are in Geome-
tric Proportion continued, becauſe there is the ſame conſtant Ratio
between them, and are therefore called Progreſſional Numbers, or
Geo-
Euclide's Elements.
399
8, 24, 48, becauſe the. Ratio of the ſecond 8; to the third 24, is.
Geometrical Series; the firſt of which is called an increaſing Seriesyn
becauſe it is generated by a continual Multiplication, the ſecond is
called a decreaſing Series, in as inuch as it is generated by continual
Diviſion.
7. When three, four, or more Quantitys A, B, C, D, &c. are come
tinually proportional, the firſt A, ſhall have a duplicate Ratio ta
the tiird, of that it has to the fecond B, as being compounded of
tiie Ratio's of the firſt to the ſecond, and of the ſecond to the third.
In like manner, the firſt A, ſhall have a triplicate Ratio to the fourth
D, of what it had to the ſecond B, as being compounded of the
Ratio's of the firſt to the fecond, of the ſecond to the third, and of
third to the fourth; and ſo always in order one more, as the Pro-
portion ſhall be exprefled.
8. In comparing the Quantitys A, B, C, D, together, if the
Ratio of the ſecond Quantity B, to the third C, be greater or lefler
than the Ratio of the firſt A, to the ſecond B, or of the third C, to
the fourth D; then thoſe Quantitys A, B, C, D, are ſaid to be in
Geometric Proportion diſcontinued ; thus, in the Numbers
479
triple; greater than the Ratio of the firſt 4, to the ſecond 8, or of
the third 24, to the fourth 48, which is but double ; therefore thoſe:
four Numbers are in Geometric Proportion diſcontinued; and the
Sign for diſcontinued Proportion is :: and where this Sign is placed
it hews that there the Proportion is interrupted, and begins a new,
and according where four Quantitys are thus expreſſed A: B:::C:D
they are thus to be read, as A is to B, that is in Proportion to B,
fo is C to D.
the Means.
From theſe Premiſes duly conſidered, all the various Proç:rties of
proportional Numbers or Quantitys, will be eaſily inveſtigated.
ift. If four Quantitys be in Proportion, either continued or diſ--
continued, the Product of the Extreams, is equal to the Product of
Thus, if in diſcontinued Proportion it be ra:a:: rb.: 6 the Pro-
dud of the Extreams is exaxb=rab evidently equal to that of the
Again, if in continued Proportion it be a:ar:: arr: arrr the
Product of the Extreams is axarrr=aarrr, evidently equal to the.
Product of the Means arxarr=aarrr. Whence on the contrary,
2. JE
Means axrxb=arb.
40
Euclide's Elements.
2. If of four Quantitys, the Product of the Means is equal to the
Product of the Extreams, then thoſe four Quantitys are in Geome-
tric Proportion.
3. If three Quantitys are in continued Proportion Geometrical,
the Product of the Extreams is equal to the Square of the Mean or
middle Term. Thus, if the three Terms be a, ar, arr, the Product
of the Extreams, is axarr=aarr evidently equal to the Square of the
Mean, or middle Term arxar=aarr.
4 It there be never ſo many continual Proporcionals, thc Product
of the Extreams, is equal to the Product of any two of the Means,
equally diſtant from the Extreams; as alſo to the Square of the mid-
dle Term, If the Number of Terms be odd ; thus in the Series,
a, ar, arr, ar”, art, ar', ar', &c. the Product of the Extrcams,
and of any two Terms equally remote from them, and the Square of
the middle Term is =aar as is manifeſt, by multiplying them toge-
ther.
5. If ſeveral Quantitys are proportional continued, or diſcontinued,
as one Antecedent is to its Confequent, fo is the Sum of all the An-
tecedents, to the Sum of all the Conſequents.
1. If they are diſcontinued, as : ar::b: br::C:c then, as
a : ar::a+b+c:ar+brter for, a-76 texar=artbr tcrxa = aar
tabr + acr evidently equal.
2. If they are continued, as a, ar, ar?, ar?, art, ar, &c. then
I ſay, as a : ar::a+ar+ara tari tart &c. : ar tarFar+ art
tars, &c. For, axar tara tari tart tars=arxatartara tar?
tart=aar+aara taari+aart +aars evidently equal.
6. Hence we may eaſily find the Sum of any Geometric Progreſ-
fion. For ſuppoſe G the greateſt Term, L the leaſt, and the Pro-
portion of the greateſt to the next leffer, as R to 1. Suppoſe the
Sum of the whole Progreſſion S, now ſince the Antecedent is to the
Conſequent, as R tos,
and the Sum of all the Antecedent, is S-L,
and the Sum of all the Conſequents S-G, it will be R:1::S-
:S-G, and therefore by the 1ſt RS--RG=S-L and taking away
Son both Sides, and adding RG, we ſhall have RS-S=RG-L, and
dividing both sides of the Equation by R-1, we ſhall have s=
RGL
RI
Examp. ,
..
Euclide's Elements.
41
I
Example. Suppoſe 64, 32, 16, 8, 4, 2, 1, 1, 4, and it be re-
quired to find its Sun.
Here R=2, G=64 L=., and conſequently RxG-L=1274, and.
ſince R-I=1, therefore the Sum is 127*.
9. By this Theorem, may the Sum of any infinite Progreſſion
decreaſing be found. For here [ the laſt Term, being leſs than
any affignable Quantity, may be rejected as nothing; for in this .
RxG
Caſe it realy vaniſhes, and then the 'Theorem will be se
R-1
Example. Suppoſe this decreaſing Series, \, ;, 17, 32, 64, 128,
&c. infinitely continued, the whole Sum will be 1, or Unity, for RxG
=2x==1, which divided by R-I=1, makes i = Sum, Again,
Suppoſe this Series }, }, zi, ai, &c. infinitely continued, the
Sum will be i ; for R is 3, and G , and therefore RXG=3x}=1,
which divided by R-1=2, makes , for the Sum ; and univerſally,
let any Quantity be propoſéd, if I take a part thereof denominated
by a, viz. and of this a part denominated by a, and ſo on infinitely,
whatever be the Number a, the whole Progreſſion will be a part
denominated by a I, that is,
44 &c. infinitely con-
tinued ==
8. If there be two Ranks of Quantitys in Geometrical Proportion,
their Products and Quotients ſhall likewiſe be proportional, thus,
Suppoſe a : ar::b: br
And c:cs::d:ds
Their Produas ac: acts:: bd: bdrs; and quotes
a ar b.br
dº ds
dently proportional; for in the firſt, the Product of the Extreams acbdrs
is equal to the Produd of the Means acrsbd, and in the ſecond, the
Products of the Extreams
abr
arb
cds is equal to the Produ& of the Means
asd
9. If there be two Ranks of Quantitys, a, ar as, where the firſt
b br
to the ſecond in the upper Rank, is as the firſt to the ſecond in the
G
under
I
I
I
ааа
a-I
are evi
C
CS
42
Euclide's Elements
...
under Rank, and the ſecond to the third in the upper Rank, is as the
fecond to the third in gre under Rank, &c. then thall the firſt, to the
lait in the upper Rank, be as the firſt to the laſt in the under Rank;
that is, a:as : 6:bs evidently proportional, for axbs=asxb=abs
which is called Proportion of Equality, orderly placed.
10. If there be two Ranks of Quantitys oli en, a; where the
eob ob eb
firſt to the ſecond in the upper Rank, is as the ſecond to the third
in the under Rank, and the ſecond to the third in the upper Rank,
is as the firſt to the lecond in the under Rank ; then ſhall the firſt
to the laſt in the upper Rank, be as the firſt to the laſt in the under
Rank; that is, it ſhall be as on:a::eob: eb which are evidently propor-
tional; and this is called Proportion of Equality, diſorderly placed.
11. Proportionals which are proportional to the ſame, are pro-
portional one to another. That is,
If it be as a : ar::b: br
And as a:ar::c:cr
It ſhall be as b: br: :c: cr which are evidently proportional
.
2:b::c:d
12. If there be two Ranks of proportionals
where
e:b::c:n
the ſecond and third Terms in each Rank, are the ſame, the firſt
in the ſecond Rank, ſhall be to the firſt in the firſt Rank, as the
tourth in the firſt Rank, to the fourth in the ſecond Rank.
For, axd=bxc=exn (by the 1ſt of the 15th.) Wherefore, axd=exn
(by the 1st Ax.) and conſequently, e:a::d:n (by the ad of the 15th.)
· If four Quantitys, a : ar::b: br are proportional, they ſhall be
proportional.
13. When Alternated, (which is comparing of Antecedent to Ante-
cedent, and Conſequent to Conſequent) (which is the 16th A che sth.)
tbat is, a :b: :ar: br.
For becauſe, axbr=bxar the Quantities are Proportional, (by the
2d of the 15th.)
14. When Inverted, (which is a taking the Conſequent, as the Ante-
cedent, and ſo comparing it with the Antecedent, as the Conſequent.)
(Which is the 4th of the sth.)
That is, if it be a: ar::b: br
It ſhall be ar: a::br: 6
For
chan
Euclide's Elements.
43
For becauſe, arxb=axbr the Quantitys are proportional, (by the
lü of the 15th.)
is. When compounded (which is, when the Antecedent and Confe-
quent taken both as one, are compared to the Conſequent it felt.)
Which is the 18th of the 5th.)
That is, if it be a: ar::b: br
It ſhall be atar: ar::b + br : br
For becauſe, brxa tar=arxb br=arb tarbr, the Quantitys are
proportional (by the ad of the 15th.)
16. When divided, (which is when the Excèfs, wherein the Anime
tecedent exceeds che conſequent, is compared to the Conſequent.) W’hich
is the 17th of the sth.)
That is, if it be a :ar::b: br
It ſhall be aar: ar:: b-br:br
For, becauſe a-arxbr=arxb-br=arb-arbr the Quantitys are
proportional (by the ad of the 5th.)
17. When Converted, (which is when the Antecedent is compared
to the Exceſs, wherein the the Antecedent exceeds the Conſequent.
(Which is the Cor. of the 19th of the 5th.)
That is, if it be a: ar::b: br
It ſhall be a:a-ar::b1b-br
For, becauſe axb-br=a-arxb=ab-abr, the Quantitys are
proportional (by the ad of the 15th.)
18. When Mixed, (which is when the Sum of the Antecedent and
Conſequent, is compared to the Excefs, wherein the Antecedent ex-
eeds the Conſequent.)
That is, if it be 4: ar:: b: br
I ſay, it ſhall be at ar: amar:: b + br:b—br
For, becauſe at arxb-br=a-arxh 1 by, the Qantitys are pro-
portional (by the ad of the 15th.)
19. Any Number of Magnitudes being taken, the Proportion of
the firſt to the laft, is Compound of the Proportion of the firſt to the
fecond, of the ſecond to the third, of the third to the fourth, de.
For, Suppoſe a, ar, ar?, ar?, art, & it is evident, that the
Proportion of a to ar*,' is Compound of a to ar, of ar to ar?, &c.
and this alſo holds good, whither the Numbers be proportional or not.
¿
G2
20, of
""
1
44
Eudclide's Elements.
20. Of the Rule of Proportion.
It has been ſhewn in the firſt Cor. concerning Proportion, that if
there be four Quantitys in Geometric Proportion, that the Product
of the Extreams, is equal to that of the Means. So that if I was to find
a fourth proportional to theſe three, a, b, c, let us imagine this tourth
Term to be x, then it will be a:b:::*; therefore ax=bc, and dividing
be
both sides of the Equation by a, we ſhall have s="(by the 3d
Ax.)ihat is, tothe Product of the ſecond and third Terms divided by
the firſt, which gives the Reaſon of the Rule of Three in Arithmetic.
Propoſition XVI. A Theorem,
of Euclide, the 1st of the 6th.
Triangles ABC, ACD, and Parallelograms BCAE, CDFA,
which have the ſame Heighth, are in Proportion one to the other,
as their Baſes BC, CD, are,
AF
E
Take as many as you pleaſe, BG, GH,
=BC, and alſo DI=CD, and joyn AG,
AH, AI, (by the aft Poft.)
HI
GB CD I
Becauſe the Triangles ACB, ABG, AGH, are equal (by the 3d
of the 7th.) and the Triangle ACD=ADI by the ſame, therefore the
Triangle ACH, as of contains the Triangle ACB, as the Baſe HC
contains the Baſe BC; and therefore, (by the 5th of the 15th) the
Triangle ACH: Tri. ACB::HC:BC.
Again, the Triangle CAI, as oft contains the Triangle CAD, as
the Baſe CI contains the Baſe CD, and therefore (by the 5th of the
15th) the Tri. ACI: Tri. ACD::CI: CD, now..comparing theſe
two Ranks of proportionals together, it will follow, from the ixth
and 13th Prop. of Pro. in the 15th) that the Tri. ABC: Tri. ACD ::
BC:CP. W.W.D.
Again, becauſe (by the 5th of the 7th) the Parallelograms CE, CF,
are double of the Triangles ABC, ACD. Therefore, Pgr. CE:
Pgr. CF::BC:CD, W.WD.
1. Hence, Triangles and Parallelograms whoſe Baſes are equal,
are in the ſame Proportion as their Altitudes.
Prop.
Euclide's Elements.
45
Propoſition. XVII. A Theorem,
of Euclide, the ad of the 6th.
A
C
If to one Side BC, of a Triangle ABC, be drawn
a parallel Right Line DE, the ſame ſhall cut the
Sides of the Triangle proportionally. (AD:BD: :
AE:EC) and if the Sides of the Triangle be pro-
portionally cut, (AD: BD::AE: EC ) then a Right D
Line DE, conne&ting the Se&iors D, E, ſhall be
parallel to the remaining Side of the Triangle BC. B
1. Hyp. Becauſe the Triangle DEB=DEC (by the 3d of the 7th.)
therefore ſhall the Triangle A DE: DBE::ADE: ECD (by the sth of
the 15th.) but, the Triangle ADE:DBE: :AD: DB (by the 16th) and
the Triangle ADE: DEC::AE:EC (by the Same) therefore, (by the
11th of the 15th.) AD: DB::AE: EC. W.W. D.
2. Hyp. Becauſe_AD: DB::AE: EC therefore,_(by the 16th)
Triangle A DE: DBE::ADE: ECD; therefore is Triangle DBE=
ECD, by the 5th and 13th of the 15th.) and therefore DE, BC, are
parallel (by the 3d of the 7th.) W.W.D
Scholium.
If there be drawn many Parallels to one Side of any Triangle,
then all the Segments of the Sides ſhall be proportional; as is eafily
deduceable from the precedent.
1. Hence, and from the (4th of the ad ard
61h.) it follows, that if an Angle BAC, of a
Triangle BAC, be biſected, and the Right
Line AD, that bifects the Angle cut the Baſe
alſo, then thall the Segments of the Baſe have
the (ame Ratio, that the other Sides of the Tri-
angle have; that is, BD: DC: :AB:AC; and
the contrary, Which is the 3d of the 6th.)
A
B
D
2. Hencs
.
1
1
ang
40
Euclide's Elements.
2. Hence we are taught, how from a
Right Line AB, to cut off any part required,
as a third AG, (which is the grh of the 6th.)
E
D
B
D
F
Conſtr. From the Point A, draw an in-
finite Line AC, at pleaſure, in which ſet
A G
off any three equal parts, as AD, DE, EF,
joyn FB, and thro' D draw DG, parallel to FB (by the sth of the 5th)
and AG will be the part required.
For, (by the 15th of the 15th, and 17th.) AD: AF: : AG: AB.
W. W. D.
3. Hence we are taught, how to divide a Right Line given AB,
as another Line given AC is, (which is the roth of the 6th.)
Conſtr. Joyn the Points C, B, and from the
Points E, D, draw the Lines EG, DF, paral-
lel to CB,(by the 5th of the 5th) which will cut the
Line AB, in the Points Fand G, required; as
A
G B is abundantly manifeſt from the precedent.
4. Hence we are taught, how to cur a Right Line given AB, in-
to as many equal Parts as you pleaſe, ſuppoſe three.
Conſtr. Draw an infinite Line BC, and ano
ther AD, parallel to it (by the 5th of the 5th)
and ſet off in each of theſe equal Parts, BE,
EF, AG, HG, ineach Line leſs parts by one,
B than are required in AB, and let the Lines
FG, EH, be drawn, which ſhall cut the Line
AB in I, and K, into three equal Parts; as was
required, &c. which is abundantly manifeſt,
from the (17th and 6th of the stb.)
5. Hence and from the 16th we are
taught, how to divide a Triangle ABC, into
any Number of equal Parts, (as ſuppoſe
three) by dividing the Baſe AB into three
B equal Parts, and drawing the Lines CH,
CG, which will divide the Triangle ac-
cordingly. Or, as two Right Lines AE,
EF, by dividing the Baſe AB in G, as theſe
two Right Lines, and drawing CG, and
the thing is done.
6. Hence
E
K
H
D
A
H
G
D
E
F
Euclide's Elements.
47
E
D
6. Hence we are taught, how from two
Right Lines AB, AD, being given, to find
out a third DE, in proportion to them. (Which
is the sith of the 6th.)
B
Joyn DB, and ſet off AD from E to C, and draw CE parallal to
BD, and DE is the third Proportion. For by the 17th AB: BC
(=AD): : AD: DE. W. W. D.
7. Hence we are taught, how from three Right Lines being gi-
ven, DE, EF, DG, co find out a fourth Proportional GH. (Which
is ths 12th of the 6th.)
Conſtr. Joyn EG, by the 1ſt Poft. and thro'
H
F, draw FH parallel to EG (by the 5th of the
5th) meeting DG produced in H, then is
GH the fourth Proportional.
D
For it is evident, that DE: EF: :DG:GH
(by the 17th) wherefore, &c. W.W. D.
E
F
Propoſition. XVIII. A Theorem,
of Euclide, the 4th of the 6th.
E
D
A
с
E
Of Equiangular or Similar Triangles ABC,
DCE, the Sides are Proportional which are a-
bout the equalAngles B, DCE,(AB:BC::DC
:CE, &c.) and the Sides AB, DC; &c. which
are ſubtended under the equal Angles ACB,
E, &c. are Homologus, or of like Ratio.
BO
Set the Side BC in a dired Line to the Side CE, and produce
BA and ED till they meet.
Becauſe the <B=ECD (by Hyp.) therefore BF, CD, are pa-
rallel (by the ad of the 5th) alſo, becauſe <BCA=CED, therefore
are CÀ, EF, parallel (by the ſame) therefore the Figure CAFD is a
Parallelogram ; therefore AF=CD, AC=FD, (by the 7th of the 5th.)
whence it is evident (by the 17th) AB:(AF:-) CD::BC: CE there-.
fore
48
Euclide's Elements.
fore (by the 13th of the sth). AB: BC::CD:CE; alſo, BC:CE::
(FD =) AC:DE, therefore (by the 13th of the 5th) BC: AC::CE: ED
wherefore alſo by Equality, AB: AC::CD:DE. Therefore, C.
W. W.D.
1. Hence, AB: DC::BC:CE:: AC:DE.
2. Hence, if in a Triangle FBE, there be drawn AC parallel to
one Side FE, the Triangle ABC, ſhall be like to the whole FBE.
3. Hence, if two Triangles ABC, DCE, have one Angle B, e-
qual to one Angle DCE, and the Sides about the equal Angles B,
DCE, proportional AB:BC:: DC:CE) then thoſe Triangles ABC,
DCE, are Equiangular, and have thoſe Angles equal ; under which
are ſubtended the Homologus Sides. (Which is the 6th of the 6th.)
4. Hence we are taught, how from two
Right Lines being given AE, EB, to find
out a Mean Proportional EF
ht
E
Upon the whole Line AB as a Diame-
ter, deſcribe a Semicircle AFB, and from
B E, ere& a Perpendicular EF, meeting
with the Periphery in F, then is EF the
Mean Proportional. For let AF and BF be drawn.
Becauſe the Angles AEF, FEB, are Right Angles (by Coiiftr.)
and therefore equal, and the A=<EFB (by the 1st of the 11th.)
alſo <B=AFE (by the ſame) therefore, the Triangles AEF, FEB,
are Similar; and therefore (by the 18th) AE: EF:: EF: EB ; where-
fore EF is a Mean Proportional. (Which is the 13th and 8th of the
6th.) W. W. D.
5. Hence we are taught how to divide
a Triangle ABC, as two Right Lines
AF, EF, by a Line drawn parallel to the
Side BC.
A
FD
B.
Conftr. Divide the Baſe AB in F, as EF
to AF (by the 3d of the 17th) betwixt
AF, AB, find a Mean Proportional AD,
and draw DE parallel to BC, and the
thing is done.
For
1
Euclide's Elements.
49
For by the Conſtr. AC : AE: :AB: AD: :AD: AF. Therefore, &c.
6. From the 4th Cor. it follows, that if a Right Line drawn in a
Circle, from any point of the Diameter Perpendicularly, and pro-
duced to the Circumference, is a Mean Proportional betwixt the
two Segments of the Diameter.
Propoſition XIX. A Theorem,
of Euclide, the 14th of the 6th.
D
IL
А.
B
G
E
F
Equal Parallelograms, BD, BF, having
one Angle ABC, cqual to one EBG,
have the sides which are about the equal
Angles: Reciprocal ; (AB: BG:: EB:
BC) and thoſe Parallelograms BD, EF;
which have one Angle ABC=to one EBG,
and the Sides which are about the equal
Angles Reciprocal, are equal.
For let the Sides AB, BG, about the equal Angles, make one
Right Line, wherefore EB, BC, ſhall do the ſame. (By the 6th of
the 3d.) Let FG, DC, be produced till they meet.
1. Hyp. AB: BG::Pgr. BD: Pgr BH (by the 16th.) :: Pgr. BF:
Pgr. BĦ (by the 11th of the 15th.): : BE: BC therefore, &c.
2. Hyp. Pgr. BD: Pgr. BH: : AB : BG (by the 16th) BE: BC (by
Hyp.) Pgr. BF: Pgr. BH; therefore, the Pgr, BD=BF (by the 5th
and 11th of the 15th.)
с
1. Hence we are taught, how upon a
given Right Line AE, to make a Pgr.
AEFG, equal to a given Pgr. ABCD.
Conſtr. Joyn the Points E and D, and draw BG parallel to DE,
(by tbe 5th of the 5th.) where this cuts AD as in G, will give the other
Side of the Parallelogram AGFE, required to be made.
The ſame Conſtruction holds good, if AE be leſs than AB:
2. Hence and from the 7th we are
C
tanght, how upon a given Right Line
H
AE, to make a Parallelogram AEIH,
equal to a given Parallelogram ABCD,
having a given Angle HĂE.
A
H
1. By
D
G
F
B
E
B
$
1
50
Euclide's Elements.
1. By making the Parallelogram AEFG, equal to the given Pa-
rallelogram ABCD, by the laſt, upon the given Line AE, and then.
making the Parallelogram AEIH=Pgr. AEFG (by the 7th)= Pgr
ABCD. W.W. D.
3. From (the 5th of the 7th, 17th and 191h) it follows, that equal
Triangles ABC, EBG, having one Angle ABC=to one Angle EBG,
their Sides which are about the equal Angles are Reciprocaly propor-
cional (ÅB: BG:: EB: BC) and thoſe Triangles that have one
Angle=ABC to one EBG, and have the sides that are about the
equal Angles Reciprocaly proportional (AB: BG::BE: BC) are
equal. (Which is the 15thof the 6th.)
4. Hence we are taught, how to make
a Triangle ADE=Triangle ABC, upon
a given Right Line AD.
E
B
D
Conſtr. Joyn DC, and draw BE parallel
to DC (by the 5th of the sth) where this
cuts AC as in E, draw ED, and AED is
the Triangle required.
For the Angle A is common, and AB: AE:: AD: AC (by the
15th of the 15th and 17th) wherefore, &c.
5. Hence, and from (the 10th of the 15th) we are taught, how to
make a Triangle equal to another, having a given Bale, and given
Angle.
6. Hence, and from (the 8th of the 7th) we are taught, how
up-
on a Right Line given, to make a Parallelogram at a Right Lined
Angle, equal to a Triangle given. (Which is the 44th of the ift.) and
the contrary. Alſo,
7. Upon a Right Line given, to make a Parallelogram, equal to
a Right Lined Figure given, at a Right Lined Angle given. (Which
is the 45th of the ift.)
In ſhort, all Problems whatſoever, relating to the Transformation
of Planes in general, reſolved after the moſt caſie manner, by the
help of theſe Properties.
8. If
..
- ܕ
Euclide's Elements.
51
H
D
A
BE
F
TAHU
B
8. If four Right Lines be proportion-
al (AB: FG:: EF: CB) the Rectangle
AC, comprehended under the Extreams
AF, CB, is equal to the Rectangle EG,
comprehended under the Means,
EG, EF, and if the Rectangle AC,
comprehended under the Extreams
AF, CB, be equal to the Rectangle
EG, comprehended under the Means
EG, EF, then are the four Lines proportiogal ; AB: EG::EF:CB.
(which is the 16th of the 6th.)
1. Hyp. The Angles B and F are Right, and conſequently equal,
and by Hyp. AB: FG:: EF: CB, therefore the Rectangles AC, and
EG, are equal (by the 19th.)
2. Hyp. The Ređangles AC, EG, are equal, and the <B=F
(by Hyp.) therefore, AB: FG::EF:CB (by the 19th.) W. W. D.
9. Hence it is eaſy to apply a Rectangle given EG, to a Right
Line given AB, viz. by making AB: EF::FG: BC.
10. If the ſecond and third Lines EF, EG, had been equal, the
Rectangle EG, would have been a Square. Whence we ſee the Rea-
ſon of the 17th of the 6th, viz. that if three Right Lines be pro-
portional, the Re&angle made of the Extreams, is equal to the
Square made of the Mean; and the contrary.
11. Hence we arc taught, a more elegant
way of demonſtrating the 35th of the 3d, viz.
that the Rectangle made of the Segments CE,
ED, AE, EB, of two Lines drawn within the
Circle are equal, joyn AC and DB, and becauſe
D
the Angles CEA, DEB, alſo C, B, are equal,
the Triangles are ſimilar. Wheretore, CE: EA
::EB: ED; wherefore, CEXED=AEXEB.
12. In like manner, from the roth we are
taught, an eaſier way of demonſtrating the 36th of B
the 3d, viz. that DAXDC=DBq.
For draw BA and BC, and becauſe the Angles
A, and DBC, are equal, and <D common to
both, therefore the Triangles BDC and ADB are
ſimilar, wherefore AD:
DB:: DB : CD; and
conſequently, ADxDC=DBq.
H2
А
B В
A A
D
C
13. From
52
Euclide's Elements.
А
D
13. From the 3d Cor. aforegoing, we
ſee the Reaſon why like Triangles ABC,
DEF, are in a duplicate Ratio, or as the
Squares of their Homologus Sides, BC,
EF. (Which is the 19th of the 6th.)
B
G C E
Make BC:EF::EF:BG; and let AG be drawn.
Becauſe, AB: DE :: BC:EF::EF: BG; and the <B equal
to the <E, therefore is the Triangle ABG=DEF (by the 3d of the
19th.) but the Triangle ABC: ABG :: BC:BG; and the Ratio of
BC to BG=Ratio of BC to EF twice, therefore the Ratio of ABC
to ABG, that is, of ABC to DEF=to the Ratio of BC to EF twice.
WV.W.D.
14. Hence if three Right Lines be proportional, then as the firſt
is to the third, ſo is a Triangle made upon the firſt, to a Triangle
made upon the ſecond, like and alike deſcribed; and ſo is a Triangle
deſcribed on the ſecond, to a Triangle deſcribed upon the third,
like and alike deſcribed.
15. And ſince like Poligons may be divided into equal Triangles,
both equal in Number, and Holomogus to the Wholes, it follows,
that like Poligons have a duplicate Ratio one to the other; what
one Homologus Side of the one, has to the other Homologus Side
(which is the 20th of the 6th.)
16. Hence if there be three Right Lines proportional, then as the
firſt is to the third, ſo is a Poligon made upon the firſt, to Poligon
made on the ſecond ; like and alike deſcribed, or ſo is a Poligon de-
ſcribed upon the ſecond, to a Poligon made on the third, like and
alike deſcribed.
17. By the help of this property we are taught, how to enlarge or
diminish'any Right Lined Figure, in any given Ratio, For Example,
Suppole AB, the side of any Triangle,
A
B
Square, or Poligon, Regular or Irregu.
lar, and it were required to make another Figure, as a Triangle, .
ſimilar to the former, and to be five times as large
Between AB and 5 AB find out a Mean Proportional, (by the 4th of
the 18th.) which ſhall be one side of the Figure required, upon which
if a Figure be deſcribed, like to the Figure given, the thing is donc.
18. On
Euclide's Elements.
53
18. On the contrary, Suppoſe ir were required to make another
Triangle, Square, &c. that ſhall be but į of the given Figure, then find
a Mean Proportion between the Line AB and its }, and you have
the Homologus Side required; upon which if a ſimilar Figure be
deſcribed, the thing is done.
19. Hence alſo, if the Homologus Sides of like Figures be known,
then will the Proportion of the Figures themſelves be eaſily found,
viz. by finding out a third proportional, (by the 6th of the 17th.) and
that compared with the firit, gives the Ratio of the Figures them-
ſelves.
20. From (the 13th and 15th of the 19th) it follows, that if four
Right Lines be proportional, the Right Lined Figures alſo deſcribed
upon them, being like and in like ſort ſcituated, ſhall be proportion-
al; and the contrary, (which is the 22d of the 6th)
21. Hence is deduced the manner and reaſon of multiplying Surd
Quantitys:
22. Hence if a Right Line AB (See the Figure of the 4th of the 18th)
be cut any wiſe in E, the Rectangle comprehended under the parts
AE, EB, is a Mean Proportional betwixt their Squares; likewiſe tlie
Rectangle comprehended under the Whole AB, and one Part AE,
or EB, is a Mean Proportional betwixt the Square of the whole
AB, and the Square of the ſaid Part AE or EB.
Propoſition XX. A Theorem,
of Euclide, the 31ſt of the 6th.
GA
A
L
In Right Angled Triangles BAC,
any Figure BF deſcribed upon the Side
BC, fubtending the Right Angle BAC,
is equal to the Figures BG, AL, de-
Icribed
upon
the Side BA, AC con-
taining the Right Angle, like and a-
like ſcituated to the former BE.
H
B
D
B
Conftr:
54
Trigonometry.
Conſtr. From the Right Angle BAC, let fall the perpendicular AD
(by the 14th of the 2d.) Becauſe DC : CA:: CA: CB (by the 4th of
the 18th) therefore, AL:BF::DC: CB (by the 15th of the 19th.)
alſo, becauſe DB: BA::BA:BC (by the 4th of the 18th) therefore,
BG: BF:: DB:BC (by the 15th of the 19th) therefore, AL+BG:
BF:: DC+DB (=BC): BC; therefore, AL+BG=BF.
Or thus,
For BG: BF::BAq: BCq (by the 20th of the 19th) and AL :
"BF:: ACq:BCą (by the ſame) wherefore, BC-+AL: BF :: BAq
- ACq: BCq; therefore, whereas BAq+Acq=BCq (by the 8th.)
then is BG + AL-BF. W.W. D.
1. Hence we are taught, how to Add or Subtra& any like Figures,
by the ſame Method as is aſed in adding and ſubtra&ing of Squares.
PART II.
Of TRIGONOMETRY.
SECT. I.
I.
T
.........
.
RIGONOMETRI, is that part of Geometry, which
is imployed more immediately in the Meaſuring of
Triangles.
A Plain Triangle, the Subjeđ of Plain Trigonometry, is a Plain
Figure contained under three Right Lines, any two of which being
taken together are greater than the third, that remains (See Cor. 2d.
of Prop. 4th, Part .)
In every Triangle there are ſix Parts, viz. three sides and three
Angles, any three of which being known or given; (except the three
Angles, for then the Proportion of the Sides one to another can only be
deter-
Trigonometry.
559
B
H
S
determined) the other three may be readily found, which to do is
the proper buſineſs of Irigonometry.
And becauſe every Angle of a Triangle, is meaſured by an Arch
of a Circle deſcribed about the angular Point as a Center, and con-
tained between the Lines which form the Angle; and the exact Pro-
portion of the Radius to the Circumference, and conſequently to any
part thereof, being not to be exprefled exactly by Numbers, Mathe-
maticians have been neceffitated to ſubſtitute certain Right Lines,
in the room of the Arches or Angles, the Nature and Conſtruction
of which, together with their agreement with the Arches, comes
next to be ſhewn.
Lines uſually applied to the Circle, are Chords, Sines, Tangents, and
Secants.
Definitions.
1. The Radius of a Circle, is a Right Line
drawn from the Center to the Circumference,
or it is equal to half the Diameter, thus, CA
is the Radius of the Circle ABDE
2. The Chord or Subtence of an Arch, is a.
Right Line connecting the Extremities of the
Arch together, thus, SP is the Chord of the
Arches SAP and SDP.
3. The Right Sine of an Arch, and which
is called the Sine ſimply, is a Right Line
drawn from one End or Termination of an Arch, Perpendicular to
the Radius drawn to the other Termination of the Arch, or its equal
to half the Chord of twice the Arch, thus,SR is the Right Sine of the
Arches SA and SD.
Hence it is evident, that the Radins or Semidiameter, the greateſt of all
the Sines, is the Sine of 90.00; for the Sine of an Arch greater than a
Quadrant, is leſs than the Radiss.
4. The Verſed Sine, (by the Ancients called the Sagitta) is that
part of the Radius, which is contained between the Sine and the
Arch, thus RA is the Verſed Sine of the Arch SA, and RD is the
Verſed Sine of the Arch SD.
$. The Tangent of an Arch, is a Right Line drawn Perpendicu-
larly from the end of the Diameter, paſſing to one end of the Arch,
and limited by a Right Line drawn from the Center, thro' the other
end of the Arch, thus AT is the Tangent of the Arches SA and
6. The
D
A:
C
R
P р
E
SD.
***
56
Trigonometry.
6. The Secant of an Arch, is a Right Line drawn from the Cen
rer, to the Extremity of the Tangent, thus, CT is the Secant of the
Arches SA and SD.
7. The Difference of an Arch, from a Quadrant, whether it be
greater or leſs than a Quadrant, is called its Complement; BS is the
Complement of the Arches SA, SD, whence SH is the Sine Comple-
ment, or Co-Sine of the Arch SA ; alſo BG, the Tangent Complement
or Co-Tangent, and CG the Secant-Complement, or Co-Secant of the
ſame Arch SA.
8. The Difference of an Arch from a Semicircle, is called its
Supplement. Hence and from Def. the 3d, 4th, and 5th, every Arch
and its Supplement, have the ſame Right Sine, Tangent, and Secant,
common to them both.
9. From Def. the 4th it is evident, that RD the Verſed Sine of the
Arch SD, greater than a Quadrant, is greater than the Radius CD O;
but R A the Verfed Sine of the Arch SA, leſs than a Quadrant, is leſs
than the Radius CA.
10. And becauſe CR, the part of the Diameter contained between
the Right Sine SR, and the Center C, is equal to SH, the Co-Sine of
the Arch SA; it is evident that the Sum and Difference of the Ra-
dius and Co-Sine of an Arch, will give the Verled Sines of that Arch
and its Supplement.
For, DC+CR=DR, the Verſed Sine of DS.
And, CA-CRERA, the Verſed Sine of AS.
Which Arches together make a Semi-circle.
But how to determine the length of the Sine, Tangent, Secant, &c.
Correſponding to any Arch, and conſequently to every Degree, Minute, &c.
of the Quadrant, in equal Parts of the Radius firſt given, ſhall be next
Jhewn both Geometrically and Arithmetically, or by Calculation. And
firſt,
Of the Geometrical Conſtruction of the Natural Sines, Tangents,
Secants, &c.
Eſcribe a Semicircle, abcd, and divide it into two equal
parts by the Radius bc.
2. Divide the Arch of the Quadrant ab, into nine equal parts
the Points 10, 20, 30, 60, then will each Diviſion anſwer to 10
Degrees
D
1.
in
1
Trigonometry.
57
Degrees; and if
you imagine Lines drawn from a to the ſeveral
Points 10, 20, 30, &c. in the Arch ab, they will be the Chords
of their reſpective Arches, by the ad Def. of the 1ſt Se&. and if trans-
ferred to the Chord Line cb, will form a Line of Chords; and if
each Diviſion of the Quadrant be ſub-divided into ten equal parts,
and the Diſtances of the ſeveral Diviſions from the Point , be
transferred to the Line ab, you will have by this means, a Line of
Chords to every ſingle Degree.
3. For the Line of Right Sines, from the Points of the ſeveral Di-
viſions 10, 20, 30, c. of the Quadrant ba, draw Lines parallel to
bc, by the 5th or 8th of the 5th, till they cut the Radius ca; ſo
ſhall c'á be divided into a Line of Sines : And if, as we ſuppoſed
before, the Arch ba be divided into 9o equal Parts, and from the
ſeveral Points of Diviſion, Lines parallel to the former be drawn, the
Line c a will be divided into a Line of Sines, anſwering to every ſingle
Degree of the Quadrant, by the 3d Def. of the ift Set. which muſt
be numbred from c towards a.
4. If the ſame Line of Right Sines be numbred from a towards c,
you will have a Line of Verſcd Sines, by the 4th Def. of the ift Set.
which may be continued to 180 Degrees, if the fame Diviſions be
transferred on the other ſide of the Center c.
5. From a, the extream Point of the Diameter, raiſe the Perpen-
dicular at: and from the Center c, thro'the ſeveral Diviſions of the
Quadrant ab, draw Lines till they cut the Tangent at; ſo will the
Line at become a Line of Tangents, by Def. the sth of Se&t. the
iſt, which muſt be numbred from a towards t.
6. If inſtead of laying the Ruler from c, the Center, you lay it from
d to the ſeveral Diviſions of the Quadrant ab, and obſerve where it
cuts the Line cb, and there make Marks, the Line cb will become
a Line of Semi-Tangents, by Prop. the 11th, and Def. ths sh of
Sect. the it!
7. And becauſe Lines drawn from the Center, thro' the ſeveral
Diviſions of the Quadrant ab, to form the Tangents, are the Secants
of the reſpe&ive Arches by the 6th Def. of the iſt Sext. If their
Lengtbs be transferred to the Line cb, continued to s, the Line os
will be the Line of Secants ; which muſt be numbred from c towards s.
After the ſame manner may the Chords, Sines, Tangents, and Se-
cants to every Minute of the Quadrant, be conſtru&ed, if the Circle
will admit of it, or to any particular Arch.
I
Again
***
:
Trigonometry.
58
70
1.
Teart.
5721???
1
t
Taident's
2130, 14.4546
60
ſtrument formed, called the Plain Scale; and ſince we are ſpeaking of
you
Fiy.1
Fig. 2.
30t
d
Again, if the feveral Diviſions of the Line ab in Fig. the 1st, be
transferred to the Line C in Fig. the 2d; alſo the Diviſions of
the Tangent Line a t in Fig. the ift, to Line T in Fig. the 2d.
In like manner, the Diviſions
of the other Lines of Right and Verſed
Sines and Secants in Fig. the ift. to Fig. the 2d, you will have the In-
a zo 41 212 213 214 215 216 47 48 49 20
+
ge
olt CEO
001
30.
th
610
810
Chords
707
60
60
210 400
5
012
man
So
...
f
sot
30.40.50.600
Rumos
2
em. Ting
2
or
20
012
god
210
011
1
20
02
10
OC 01
como
to
27
Eq. Parts
to
20 30 40 50 60 70. go
go C С
R. Seles
za zlo 40 50
60 50 40 30
7. Sines
a
8 70
C.
R:S
V.S.
R.
S.T.
E.P.
T
this Inſtrument, it may not be amiſs to ſhew the Conſtrudion of a-
nother Line uted in Sailing, called Rumbs, which is as follows:
Divide
. Auk
Trigonometry.
59
Divide the Quadrant db in Fig. the iſt, into eight equal parts in
the Points 1, 2, 3, c. and ſetting one foot of the Compafles in the
Point d, transfer the ſeveral Diſtances dı, d2, dz, from the Arch db
to the Line db; fo will you have formed Lines of Rumbs, cach
of which anſwer to 11° 15', which may be transferred to the Line R
in Fig. the 2d. To theſe may be added a Line of equal Parts, and
then the Scale will be compleat.
Now if the Numbers anſwering to the Lengths of the ſeveral Sines,
Tangents, and Secants proper to the ſeveral Arches of the Quadrant,
in cqual parts of the Radius, be ſet down in a Table, each in its
proper Column; you will by this means, have a Triangular Canon
of Numbers, by which the ſeveral Caſes of 'Trigonometry may be done ;
but theſe being not accurate enough for Arithmetical Calculations,
it is therefore abſolutely neceſſary, that the Lengths of thoſe right
Lines (eſpecially ſuch as are uſed in the Calculation of Triangles) be
determined to greater Accuracy in Numbers to an aſſigned Radius
which to do, ſhall be ſhewn at large in the next Section,
i
SECT. II.
Containing Methods for Calculating the Tables of Natural
Sines, Tangents, and Secants.
o
NE End propoſed in this Treatiſe, being to gratifie both the
Theoretical and Practical Reader, I have
determined with my
ſelf to put this Part in a Section by it ſelf, that fo the Praftical Reader,
who will content himſelf with bare Rules only, may paſs it over, and
haſten to the actual Solution of the ſeveral Cales in Plain Trigonome-
try, contained in the 4th and sth Se&tions ; whilſt the Inquiſitive and
Intelligent Reader, may indulge himſelf in the Contemplation, and
become a perfeá Maſter of the Grounds and Reaſons of one of
the moſt uſeful Branches of the Mathematicks.
:
Prob. I.
The Sine SR of an Arch SA, being given, to find SH, the Sine
of its Complement SB.
I 2
Becauſe
6o
Trigonometry.
Becauſe (by the 8th Prop.) CSq=SRq+ CRq, CSq-SRq=CRO
(by the ad Ax.) therefore, VCSq-SRq=CR (by the 3d of the 8th)
FSH (by the 7th of the sth.) which was to be found. That is, if
from the Square of the Radius, be taken the Square of the Right
Sine, the Square Root of the Remainder will be the Co-Sine re-
quired.
Prob. II.
B
The Sinc SR, of an Arch SA being given,
to find AN, or SN, the Sine of half the
Arch.
H
S
e
N
The Sine SR being given, CR the Co-
Sine may be found by the ift Prob.) which
taken from CA the Radius, leaves RA the
Verled Sine of the ſame Arch, but SAq
=SRq+RAq, (by the 8th Theo. of Part. Ift.)
therefore, (by the 3d Cor. of the ſame) VSR9 FRAq=SA=2SN=
2 AN, therefore half SASN=NA.
R
А
Prob. III.
The Sine AN, of an Arch AQ being given, to find SR the Sine
of AS, the double of the Arch AQ.
The Sine AN being given, the Co-Sine CN may be found by the
If Problem.) and becauſe the Triangles ACN and ASR are fimilar,
(for the Angles N and R are Right, and Angle A common, there-
fore the Angles S and C are equal) it will be (by the 18th Theo.) as
CA:CN::(AS -) 2 AN: SR; that is, as the Radius, to the Co-
Sine, fo is the double Sine, to the Sine of twice the Arch required.
Cor.
Becauſe, CA: CN :: AS: SR, therefore AC: 2CN:: AS: 2SR,
that is, the Radius is to twice the Co-fine of half the Arch, as the
Chord of the Arch to the Chord of twice the Arch ; alſo, becauſe
AC: 2CN:: (AS=) 2AN: 29R :: AN: SR, it will be AN:
SR :: half AC: CN, that is, as AN the Sine of any Arch AQ,
is to SR the Sine of the double Arch AS; fo is half the Radius to
CN the Co-Sine of the Arch firſt given:
Thc
.
Trigonometry
61
B
S
H
G
P
C
Prob. IV.
The Sines QD and SH, of two Arches
В
AQ and QS, being given, to find sf the
Sine of SA, the Sum of the Arches.
The Sines QD and SH, being given, their
Co-fines CD and CH, are alſo given (by the
1/7.) and becauſe the Triangles COD and
CHE are ſimilar, it will be cQ QD::CH
:HE - FG ; again, becauſe the Triangles
F E DA
CQD and SHG, are fimilar (for Angles D and
G are Right) and Angle Q=CPF=GPH=GHS; therefore, An-
gles S and C are equal) it will be cQ: CD :: SH: SG ; therefore,
FG+SG=SF, which was to be found.
Prob. V.
The Sines SF and QD, of two Arches SA and QA being given,
to find SH the Sine of se, the difference of the Arches.
The Sines QD and SF being given, their Co-ſines CD and CF
may be found by the ift.) and becauſe the Triangles CQD, CPF,
are fimilar, it will be CD :D2::CF:FP.; but SF-FP=SP.
Again,
becauſe the Triangles C2D and SHP are ſimilar (for the
Angles D and SHP, are right, and Angle Q=CPF-SPH, there-
fore, Angle S and Ć are equal) it will be cQ:CD:: PS (before
found): SH, the Sine of se required.
Prob. VI.
BD
х
E
lo
G
H
F
The Sine DK of any Arch DB, leſs than
30 deg. added to DG, the Sine of its Defeat
DE, multiplied by the V3 will give FH,
the Sine of FB, or of an Arch as much ex-
ceeding 30 deg. as the former Arch DB
wanted of 30 deg
Let BE be an Arch of 30 deg. BD. any
Arch leſs than BE, then will ED be the Arch
of Defe&; make EF-ED, and BF will be
an Arch as much exceeding BE the Arch
P
QA
of
+
sar
6 2
Trigonometry.
of 30 deg. as BD wanted of 30 deg. Becauſe the Angles O and G
of the Triangles FOD and ID are right, and the Angle D common,
therefore the Angle F=Angle I (by the ad of the 6th.) =Angle
(by the 5th.) = 30 deg, by Conſtruction; therefore Do= half DF
EDG, and becauſe FOq:- FD9-D09, therefore FOq=4DGI--
DGq=3DGI, therefore FO=GDxV3; again, becauſe DK=OH,
there'ore FHS10- KD-KD---DGY V 3.
Hence the Sines to 30 deg. being found, the Sine to 60 deg: are
hrad by the Addition of the Defect multiplied by the V3, for the
Sine of 40 deg, is equal to the Sine of 20 deg. added to the Sine
of 10 deg. inultiplied by the V3=1.732050807.
Prob. VII.
The Sine FQ of any Arch AF, leſs than 60 deg. added to
FG the Sine of the Arch FE it's Defe&, will give DP, the Sine of
the Arch DA, as much exceeding AE the Arch of 60 deg. as the
Arch AF wanted of 60 degrees.
For becauſe FG=DG=DO, and FQ =PO, therefore DP=DO
+OP=FG +-FQ; that is, DP-FG-+ FQ.
Hence the Sines to 60 Degrees being known, the Sines to 90 De-
grees may be had by the Addition of thoſe already found, for the
Sine of 40 Degrees added to the Sine of 20, will give the Sine of
80 Degrees.
Hence it is abundantly manifeſt, that the Sine of any one Arch in
the Quadrant being known, the reſt may eaſily be found; for if the
Sine of one Minute were given, the Sine of two Minutes may be
found by the 3d Prob. the Sine of three Minutes by the 4th Prob. &c.
and their Co-Sines by the ift. &c.
How therefore the Sine of one Minute, or of any ſmaller Arch in
equal parts of the Radius firſt given, may be found, comes next to
be ſhewn.
The ancient Method, and which is very eaſy to be underſtood,
was to find continually the Sine of the half Arch. For the Sine of
30 Degrees being equal to half the Radius, the Sine of 15 Degrees
may be obtained by the ad Prob. and thence the Sine of 7° 30', the
half Arch by the fame, and fo on continually, till the Sine of the
Arch laſt found, its Tangent, and conſequently the Length of the
"Arch it' ſelf, were expreſſed by the fame Decimal Part of the Ra-
dius.
Trigonometry.
63
T
S
R r
the Tangent.
dius. The Number of which Operations were more or leſs, accord-
ing to the Number of equal Parts the Radius was ſuppoſed to be
divided into
And that in very ſmall Arches, the Sines
and Tangents of the ſame Arches, are ve-
ry nearly in a Ratio of Equality, may be
thus demonſtrated.
Let SR be the Sine, and AT the Tan-
gent of any Arch, ſuppoſe AS,
Becauſe the Triangles CRS and CAT are
ſimilar, it will be as CR:CA::RS: AT
is
C
but the ncarer the Point S approaches to A,
or the ſmaller the Arch SA, the lefler will be the Verſed Sine RA, till
at lait the Point S returning to A, the Verſed Sine RA vaniſhes in
reſpect to the Arch AS: wherefore, CR becomes nearly equal to CA,
and conſequently SR the Right Sine, will be nearly equal to Aſ
Again, ſince the Arch is leſs than the Tangent, but greater than
the Sine, and becauſe the Sines and Tangents of infinitely ſmall Archos
are nearly equal, the Atch it ſelf will be nearly equal to the Sine or
Tangent; and conſequently in infinitely ſmall Arches, the Arches
will be to each other, as their Correſpondent Sines.
To Compute therefore, from the Sine of 30 Degrees firſt given,
the Sines of the Half Arches by the help of the 2d Prob, the Operation
muſt be repeated till a Sine be found, that ſhall be equal to half the
former, which to make a Table of Sines to ten places, will require
twelve Repetitions, or till the Sine of 52", 44"", 03''"', 45
be found; which being 0002.556634, it will be,
As the Arch of 52, 44",03"", "45"'", to the Arch of 1 Min.
so is the Sine laſt found 0002.5566.34+, to 0002.9088.82 the Sine
of 1 Minute required.
The Trouble and Difficulty attending the Repetition of theſe
Operations, ariſing from the great and frequent Involutions, and
Extractions of Roots of very large Numbers, (as any one will ſoon be
ſenſible of, who will try but two or three Operations) has put Man-
upon contriving eaſier and ſhorter ways of atraining the End,
as well to Examine the Tables already made, and carry them on to
greater Degrees of Exactneſs if required, as to convey this uſeful
Knowledge to Pofterity, with greater Eafe and Certainty; and a-
mong
kind
64
Trigonometry.
"
", be-
mong the reſt, the eaſy
way of inveſtigating the Proportion of the
Circumference of the Circle to its Diameter, with much greater
Exactneſs and infinitely leſs Trouble than the Ancients could do, has
not a little contributed to it.
For the Ratio or Proportion, of the Circumference of the Circle
to its Diameter, being known or given, the Length of any Arch in
equal Parts of the Radius firſt given, is eaſily had, thus if it were
required to find the Length of the Arch of 1 Minute, the Propor-
tion will be as 21600 the Minutes in the whole Circumference, is to
1 Minute, ſo is the Length of the Circumference in equal parts of the
Radius, to the Length of che Arch of 1. Minute in the ſame parts.
The Ancient Way of finding the Length of the Circumference, in
equal Parts of the Radius, firſt Invented by Archimedies, and af-
terwards Improv'd by the Doctrine of Angular Sections, was to find
the Length of the Chord of a ſmall Arch of the Periphery, which
being taken for the Length of the Arch it felf, that maltiplyed by
the Number denominating what part of the Circumference it was,
gave the Length of the Circumference it felt.
Thus in the laſt Example, the Sine of 52", 44", 03
03"'", 45
ing taken for the Length of the Arch (for it was there ſhewn, that the
ſame Number expreſſed nearly the Length of the Tangent of the
Arch, and conſequently of the Chord) and that Length multiplyed
by 24576, the number of times the Arch of 52" 07'", &c. is con-
tained in the Circumference, will give the Length of the Circumfe-
rence, in the fame parts the Radius or Diameter was ſuppoſed to be
divided into.
But ſince the new invented Methods for the Re&ification of Curves,
by the help of the infinite Series, much more eaſy, certain and di-
red Methods have been diſcovered for the Quadrature of the Circle,
to any poffible Degree of Exa&neſs required; aswill evidently appear
from what follows.
Let C be the Center of a Circle, CA=CS
=[ the Radius, As=a any Arch thereof
Sr=s its Right Sine, then will Cr the Co-
Sine, be equal to VI-ss by the 2d Prob.
Let CM be another Radius of the Circle,
infinitely near to Cs, then will Mn=; be
the Fluxion of the Sine Sr, and the infinite-
А
Rr
ly
B
M
n
S
C
Trigonometry.
65
ly ſmall Arch MS=a, the Fluxion of the Arch 1S. And becauſe the
Triangles.CSr and Mns are ſimilar, it will be Cr: CS:: Mn: Sn; that
S
:a; wherefore, i=Viss
is VI-ass:1::sia; wherefore, a= the Fluxion of the
Arch AS; that is, if the Fluxion of the Right Sine be divided by
the Square Root of s--ss, the integral or flowing Quantity of the
Quotient, will be the Arch it ſelf.
The principal End of this Section, being to convey this Knowledge
in the eaſieſt manner poſſible, and ſo as that it may be underſtood by
Youths, and Perſons of a middle Capacity. Inſtead of Extracting the
Root of 1–ss=î by the help of the Binomial Series, and multiplying
that Root bys, I ſhall find the Value of a, by the Simple Laws for
Extraction of Roots and Diviſion of Algebra, which tho' it be a
little tedious, yet it will ſufficiently Recompence the pains of the
Prađitioner, by letting him into the Reaſon of every ſtep he takes,
and opening a way for him to purſue the Knowledge of the infinite
Series farther.
The Square Root of I-ss may be found
after the following man-
ner, by the help of the Rules delivered for Extracting of Roots.
I=rssís-is?—šs4-sb- this, &c.
2-15?)-SS
ss +15+
2-55-14)—s+
454
2-ss4s+, C.)55
gistikos
1st ti stotis
6
18
រ
1.8
64
The Square Root of 1-ss, being 1-1,2 s 4-73so, &c. if the
Fluxion of s, be divided by this Root, we fall have the Fluxion
of the Arch SA, the Operation is as follows.
K
$
66
54
.
3 4
oc
$
3
is
3.6
1
?
Trigonometry
1-52-554-záss c.)(s+isas+is+stitsos
s- - sºs— 5s4srősos
sós Ec.
tisést sustissos c.
+4gºs—s*; 15sos, &c.
,
tissos, O C.
This laſt Quotient s+is+s+is+strip's, c. is equal to a the
Fluxion of the Arch, its integral or Aowing Quantity equal to st
öss +435) tifas?, &c.
Wherefore, a=stösi tiistifas?, c.
Op, a=sts IX 1,3 + 1X1X3X3, +1X1X3X3x5x5,
IX1X3?, &c.
2X3 2X3X4X5 2*345*6x7
s3 +
IX35
-s, 6C
2X3 2X45 2X4x6x7
Hence the Length of any Sine or Ordinate in the Circle being given,
the correſponding
Arch may be readily found.
Let it be required to find the Length of the Arch of 30 Degrees
to 10 Places, of Decimals, the Radius being ſuppoſed 1.
Becauſe the Radius is cqual to the Side of the Hexagon inſcribed
in the Circle, or Chord of 60 Degrees, therefore its half or the Sine
of 30 Degrees, will be =s. Wherefore, becauſe a=s+ós +4+5
&c. If the ſeveral odd Powers of s=1, bé multiply'd by 1,5, 46,
the Sum of the Products will be the Length of the Arch.
Now becauſe s=1, 'tis evident its Square is, its Cube
OC.
and becauſe to divide by 4 and multiply by 4 is the ſame, if.5=be
continually divided by 4, and the ſeveral Quotients multiply'd by
1.7, 4:, &c. the Sum of the laſt Products will be the Length of the
Arch, as may be ſeen by the following Example.
Or, a=st1X1,3 + 1x3
16.
I
i
1
Trigonometry.
67
ll
1
6
*
55
S
s
40
I 1 2
S
9
2
se
IIS 2
62
5000 0000.000
208.3333.333+
23.4375.000
3.4877.232 +
15933.974-
.1092.3914
211.826
42.617+
8.813 +-
1.863---
2816
5000.0000.000
1250.0000.000 its
312.5000.000
78.1250.000
19 5312.500
4.8828.125
1.2207031
3051.758
762.939
190.735
47.684
II.921
2.980
.745
I I
S
s']
IS
S
I 3 3 1 2
143
IO 2 of 0
17
S
s
19
$57058
1 2 1S
I 245 i 85
4618
$
BIS 9
928314
doc.
OC.
$21
$23
•400+
3521
$25
oc.
Oc.
&c.
87+
19+
4+
3 5 I08,
2 2 44
391090)
27
5235.9877.559
Whence the Length of the Arch of 30 deg. is .5235.9877.55 +-
in luch Parts as the Radius 1.0000.0000 00. Now, if this Arch be
multiplied by 6, we ſhall have the Length of the Arch of the Se-
micircle in ſuch Parts as the Radius is 1, or of the whole Circum.
ference in ſuch Parts as the Diameter is i.
Example
5235.9877.55.9
6
3.1415.9265.35.4
Whence, if the Diameter of a Circle be put equal to 1, the
Circumference will be 3.1415926535 + more than ſufficiently ex-
act, for any Uſes to which it is generally applied and capable of
raiſing a Table of Sines, &c. to 15 Places of Decimals.
The Semi-periphery in ſuch Parts as the Radius is 1, or the whole
Circumference in ſuch Parts as the Radius is 2, or Djameter , may
be directly inveſtigated; it inſtead of multiplying ·5 by the ſeveral
Powers afs, and the ſeveral Produas by 15,6 c. we multiply 6x.rs
=3. by the ſeveral Powers of S, and the ſeveral Products by 15, &c.
as in the following Example.
K2
3
.
68
Trigonometry.
I
6
3
40
3.0000.0000.00
• 7500 oCOo oo its
.1875 0000.00
468.7500.00
GE.
117.1875 00
29296875
7 3242 19
1 8310 55-
.4577.64-
1144.41--
2 86.10-1-
7153
17.88-1-
4.47-1-
1. I2 -
3 0000 0000.00
.1250 0000.00
140 6250 00
20 9263.39+
3.5603.84 -|-
.6554 34 +
•1270 95 +
*255.70+
52.88
I1.17-1-
2,40.-/
:52-4
12
3
I
3.1415.9265.354-
Agreeing exa&ly with the former.
If it were required to find the Circumference to s Places in De-
cimals only, which is exact enough for all common Uſes, and ſuffi-
cient to form a Table of Sines, &c. to 7 Places, the Operation
will be wonderful eaſy, and may be performed in a very few Minutes,
as the following Example will ſhew.
3.00000
3,00000
75000
•12500
.18750, OC.
14064
4687-+-
209+
1172
293
73 +
1-|-
I
al
.
364
.
3.14159
Agreeing exaâly with the former.
To pronounce the Proportion of the Circumference to the Di.
ameter by the ancient Methods to that degree of exactneſs,
done by the 2 firſt Examples was in a manner altogether Impra&i-
cable; and to obtain it to s Places, as in the 3d Example, (which
may
as is
Trigonometry.
6.9
(l,
I
as
may be performed in 5 or 6 Minutes at the moſt) by the help of the
2d Prob. would require 12 Repetitions of that Problem to a Radius
of 10000.0000.00, which alone is an Inſtance what vaſt Advantage
Arithmetical Calculation receives from the Modern Algebra, as now
Improved, and in things of ſo great Benefit.
Now, If the Semi-Circumference 3.1415926535+ in ſuch Parts
as the Radius is i, be divided by 10800 the Minutes in a Semicircle,
the Quotient.00029.0888208665 + will be the Length of the Arch
of 1 Minute exact to 15 places.
Again, If the Root of the infinite Equation star?, &c. be Ex-
tracted, we ſhall have the Value of s expreſſed in the terms of
and conſequently a direct Method for finding the Sine of any Arch,
its Length being firſt given.
If a=stis? +4633-4-728?, &c.
Then, s=8-10 tizace a sátort,&c.
Or, s=a-La't
a", &c.
2x3 2X3X4X5 2X3 X4X5 X6X7
For put s=A2+ Bas + Ca', &c.
Thenwill {s} = + A?q?+ A'Ba', &c.
And, 485=
+48 A'a ', &c.
And conſequently Aa=a, and A=1, alſo B+A=0,andB=-A
=-. Alſo, C+AB+4+A'=o. And C=AB-48 A=
--=+1-48=tio; wherefore A=1, B=-, C=x, a:
Or. and conſequently s=a-La trata', &c. W.W.D.
Wherefore, If a be put for the Length of any Arch, the Right
Sine will be a-šatizza', &cº
Let it therefore be required to find the Sine of one Minute, to
10 places, from the Length of the Arch before found, =0002.9088.
820 +=a, now becauſe in Cubeing of this Quantity you will have 9
Cyphers before the firſt Significant Figure which is 2, thercfore the
Series vanilhes, and 0002.9088.82 is the Sine required.
Hence it is manifeſt, that to a Radius of 1.0000.0000.00, the
Right Sine, the Arch and conſequently the Chord of 1 Minute are
equal in as much as each are exprefled by the ſame Number.
Again, Let it be required to find the Natural Sine of 10 Degrees
to 10 places
.
The
}=a
3
X
.,
Trigonometry.
yo
The Length of the Arch of 1 Min. = 0002.9088.2808.66
this multiply'd by the Min. in 10 Deg. =
600
Gives the Length of the Arch of 10 Degrees= 1745.3292.5196
Wherefore put a=
1745.3292.5 1.9
Then will
ta
-8.8609.61.5
3
And
ta?
1736 4682.90.4
Again
tizza' =
1134.96.0
And
4^{2}-4-1
a-isa'
1736.4817.86.4
Again
504
5470"=
- 9.7
And
123-1-12 on?
— 377ön?
1736.48 17.76.7
Wherefore becauſe the next Term, and conſequently all the other
Terms in the Series vaniſhes, S=1736.4817.76 = Sine of 10 De-
grees.
I ſhall give another Example for the Şine of 20 Degrees.
The Arch of 20 Degrees=a=
**3490.6585.03.9
-70.8876.924
I
3
11.'
3419.7708.115
7.4318.725
504302
3420.20.26.840
12.527
3420.2014.313
+, zirroa'=
---21
The Natural Sine of 20 Degrees= 3420.2014.334
The difficulty attending the Raiſing the ſeveral Powers of the Arch
a, according to the common Rules of Multiplycation, is enough to
thock the Pra&itioner, and make him conclude that the putting of
thefc Series in Pra&ice in ſome Caſes, is in a manner impoflible ;:
but if he will only Invert the Order of Figures in the Multiplier,
as in the following Example, he will find the Work wonderful Eaſy.
Let
Trigonometry.
71
Ler it be rquired to raiſe the ſeveral Powers requiſite to find the
Sine of 10 Degrees, from the Length of the Arch of 10 Degrees
equal to 17453 292,519.
1745.3292.519=
9152.9235.4715
0304.6174.197-a?
9152.9235.471=a
1745.3292,519
1321.7304.763
69.8131.700
8.7266.462
5235.988
349.066
157.079
3.491
872
17
0304.6174.197
213.2321.938
12.1846.968
1.5230.871
913.852
60.923
27.415
609
152
3
16
03046.1741.973=a
00531.6576.93.0=a?
0053.1657.693 =a?
914.7164.030=a?
-0001.6195.219-a
4.7164.030=a?
15.9497-308
2126 630
318.994
5.317
3:721
2 12
.4858.566
64.781
9.717
162
113
6
5
4
00000.4933:34:5=a
00016.1952.19.1=a?
The Number of Significant Figures after the next Repitition, or
of the 9th Power amounting but to fivė, viz. 15027, and this being
to be divided by 362880 the Uncia of the oth Power, thews that that
Term and confequently all the relt waniſh, and the Series terminates.
Every
.2
Trigonometry.
Every Figure of the ſeveral Powers of a are true except the laſt, and
therefore in every Operation, the Practitioner muſt take one Figure
more than the Number of Places he deſigns to Calculate to.
Beſides the benifite of having to do with no more Figures of the
Power than are requiſitc, thic vait Advantage of railing of Powers by
this way above the common way is very Obvious, tor as by the com-
mon way, the number of Figures in the Product are increaſed after
every Repitition by as many Figures nearly, as are equal to thoſe in
the Multiplier or Root ; till at laſt the Operation becomes almoſt Im-
practicable. So here on the contrary, the Number of ſignificantFigures
continually decreaſe after every Involution, till at laſt the Operation
vanilhes, and ſhews where the Series terininates.
The Sine of any Arch being given, its Co-Sine may be obtained
by the ad Problem.
By the ſame Problem may the Series for finding the Co-Sine, di-
rectly from the Arch it ſelf, be Inveſtigated.
For if from the Square of the Radius=1, be taken the Square of
she Right Sine=amatizza', &c. the ſquare Root of the Re.
mainder will be the Co-Sine, =I-a-1-248--72ca', &c. thus,
tila', &c.
s=ama tiżon', &c.
s=0
sa? +
a*---*+ion", &c.
za*-4-za', &c.
tizal', &c.
ss=n_$a4 +4ša", &c. which taken from the
Square of the Radius 1, leaves. I~-aa+ja+-43&º, &c. the Square
Root of which will be the Co-Sine.
1)1-aa-H-* a4 -#46&c. (I_ja?-124-44726, &c.
I
2-4a)--aa-
tat
aa-1 124
124
2-aa, &c.fi
4.94
&c.
6. &c.
2-aa, &c.)-, &c.
3701', &c.
And
.
Trigonometry.
73
.:
1
I
I
04
IX2x3x475489", &c.
COM
Wherefore, Putting C for the Co-line.
C=I- 10* +24a4 tot tota', &c. .
Orc=ritat
IX2X3 X4
Let it be required to find the Sine of 89 deg. 59 min. from the Archa
of 1 min. = .000 2.9088.82.
From
1.0000.0000.00
Take ja’=
4.23
Remains the Sine 89 deg: 59 min.
9999 9995 77
Let it be required to find the Co-fine of 10 Degrees, or Sine of
80 Degrees, from the Arch of 10 Degrees =1745-3292.51.9=1.
From
1.00.00.0000.000
152.3097.098
9847.6912.902
Add 22+ =
+3866.324.
9848,0779.226
Take 2:
3.926
The Natural Sine of 80 Degrees=
9848.0775.30.0
Let it be required to find the Co-line of 20 Degrees, or Sine of
70 Degrees, from the Arch of 20 Degrees 3490.6585.039.38
From
1.0000.0000.000
Take taas
-609.2348.395
93.90.765 1.605
Add 1945
+6.1861.181
9396.9512.786
Take ziba'=
251.253
9396.9261.533
Add 46 120228
546
9396.9262.079
Take žań
zzo
ol
+
Whence the Sine of 70 Degrees or Co-line of 20 Degrees is
9396.9262:08
L
The
74
Trigonometry.
:
The Sines of 20 and 8o being found, the Sine of 40 may be had
by the 7th Prob. for,
From the Sine of 80 Degrees
9848.0775.300
Take the Sine of 20 Degrees=
3420.2014.33.4
Remains the Sine of 40 Degrees
6427.8760.96,6
Again, if
From the Sine of 70 Degrees=-
Be taken the Sine of 10 Degrees=
There remains the Sine of 50 Degrees
9396.9262.079
1736.4817.76.7
7660.4444.312
Hence the Sine of 1 Minute is 0002.9088.82 +; of 10 Degrees,
1736.4187.76+ ; of 20 Degrees, 3420.2014.33; of 30 Degrees,
5000.0000.00 ; of
40 Degrees, 6427.8760.96+; of so Degrees,
7660.4444:31; of 60 Degrees, 8660.2540.38 ; of 70 Degrees,
9396.9263.08 ; of 80 Degrees, 9848.0775:30 ; of '89 Degrees
59 Minutes, 9999.9995.77—; of 90 Degrees, 1.0000.0000.00
In all the Examples hitherto given, the Sines have been Calculated
to 10 Places, but if it be required to carry them to 7 Places only,
which is as far as the Tables in uſe generally extend, one half of the
Labour would be faved.
Again, becaufe the Radius leſſened by the Co-fine, gives the
Verſed-line of the ſame Arch.
If from i be taken 1-antrat-ta', &c. there will remain
an,at tiiva', &c. the Series for finding the Verſed-line from
the Arch firſt given.
Alſo, becaliſe the Radius added to the Co-fine, gives the Verſed-
fine of the Supplement, therefore, if to i be added in to 24
itorno, &c. the Sum 2*** +za+Ta', &c. will be the Series
for finding the Verſed-line of the Supplement from the Arch firſt
given
i
But
7
Trigonometry.
75
*
r st
u
n
But to find the Sines, doc. of the intermediate Arches to every
Minute or part of a Minute if required, the following Theorem will
very much
contribute.
In a Rank of Equidiffering Arches, ſuch 9
as gz, sy, gx, &c. the Sine of any Arch
in that Rank as o (ſuppoſe of 20 deg.) will
be to the Sum of the Sines of any two
Arches,equally remote from it on each, viz.
rab
у
n-tp (ſuppoſe of 10 and 30) as the Sine
cla
of any other Arch in the ſame Rank n (ſup-
pole of 30 deg.) to the Sum of the Sines of
two Arches next to it on each Side, viz. f
m to (ſuppoſe of 20 and 40 deg.) having
the ſame common Difference, and this holds good whether the Provi
portion be continued or diſcontinued.
X
N
యం
In order to prove which, it is neceſſary to premiſe the following
Lemma,
C
A
If any Arch of the Circumference as ABD,
be bifected in the Point B, and from any
Point in the Periphery without the Arch as
C, the Chords or Subtences CA, CB and CD,
be drawn, CD being produced till BP be e-
qual to CB, I ſay the external Segment DP of
the Line CP, will be equal to CA.
B
D
Conſtruction. Let the Lines AB and BD
be drawn.
P
Dem. Becauſe the Angle ACB=BCP (by the ift of the 11th )=CPB,
and the AnglePDB=CĂB (by the 2d, and 3d of the i Ith.) and the Side
CB=BP by Conſtruction, therefore CA=DP (by the 8th of the 2d.)
W.W. D. Whence it follows,
La
That
76
Trigonometry.
A
B
D
K
That if from any Point C, in
the Circumference of a Circle,
any number of equal Parts be
fet off from C to A, to B, to D,
to E, &c, and the ſeveral Chords
or Subtences CA, CB, CD, CE,
&c. be drawn, CD being pro-
duced till BK be equal to CB,
allo CE till DL be equal to CD,
&c.
E
.
L
M
I ſay it follows, that CK ſhall
be equal to CA+CD, alſo CL
CE-+CB, &c, and becauſe
the ſeveral Triangles CBK,
CDL, CEMare ſimilar, it will
be,
N
any
CB:(CK=) CA+CD::CD:(CZE) CBI CE. Or,
CB :(CK=)CAICD::CF:(CN=) CE-I-CH, &c.
That is in a Rank of Equidiffering Arches, the Chord of
Arch in that Rank, is to the Sum of the Chords of any two Arches
equally remote from iton each side, as the Cliord of any other
Arch in the ſame Rank, to the Sum of the Chords of two other
Archs next to it on cach Side, having the ſame common Difference
whether the Progreſſion be continued or interrupted.
And fince the Chords are the double Sines of half the Arches
they ſubtend, and that the Halves have the fame Proportion as their
Wholes, it follows,
That in a Rank of Equidiffering Arches, the Sine of any Arch in
that Rank will be c. W. W. D.
Now if the Radius or Sine of 90 Degrees, be taken for the Sine
of the Middle of three Equidiffering Arches, it will be,
As the Radius, to twice the Sine of one of the Arches, or as
half the Radius, to the Sine of one of the Arches, ſo is the Sine of
any other Arch, to the Sum of the Sines of two Arches equally re-
more.
Hence
Trigonometry.
77
nutes, cc.
naturally from the ,
of the
Hence the Sine of 89 Degrees 59 Minutes being known or given,
it will be,
As the Radius, to the Sum of the Sines of 90 Degrees or Minute,
and 89 Degrees 59 Minutes, or which is the ſame thing, to twice
the Sine of 89 Degrees 59 Minutes; fo is the Sine of 89 Degrees
59 Minutes, to the Sum of the Sines of 90 Degrees oo Minutes, and
89 Degrees 58 Minutes, from which taking the Radius or Sine of
90 Degrees oo Minutes, tlic Remainder will be the Sine of 89 De-
grecs 58 Minutes; and ſo is the Sinc of 89 Degrees 58 Minutes, to
the Sum of the Sines of 89 Degrees 59 Minutes, and 89 Degrees
57 Minutes, from which taking away the ine of 89 Degrees 59 Mi-
nutes before found, there will remain the Sinc of 89 Degrecs 57 Mi-
Again, from the Sine and Co-ſine of 1 Minute the whole Cannon
may be formed, beginning at one Minute.
As the Radius, to twice the Sine of Minute, fo is the Sine of
1 Minute, to the Sine of 2 Minutes; and fo is the Sine of 2 Mi-
nutes, to the Sum of the Sines of i and 3 Min. from which taking
away
the Sine of 1 Minute, there will remain the Sine of 3 Minutes.
How troubleſome ſoever this Method may appear, yet if it be well
confidered, it will be found very eaſy; for in every Proportion, the
firſt Term being the Radius, or i with Cyphers, there is no need of
Diviſion : and ſince the ſecond Term remains the ſame, if the ſeveral
Multiples of it by the 9 Digits be made at firſt, and ſet down in a
Table orderly, the whole Cannon may be formed by Addition and
Subſtradion only.
The Sine of the Arch of Difference of three Equidiffering Arches
being known, the Sine of two Arches equally diſtant from any other
Arch given, may be found by the two following Proportions, wliich
General Theorem:
1. As the Radius, is to the Co-fine of the Arch of Difference, ſo
is the Sine of the middle Arch, to half the Sum of the Sines of the
Extream Arches.
2. As the Radius, is to the Sine of the Arch of Difference, fo is
the Co-fine of the middle Arch, to the half Difference of the Sines
Now the half Difference added to the half Sum, gives the Sine of
the Greater, but fubtracted leaves the Sine of the Lefler: 0
the
*8
Trigonometry
the whole Difference added to the Sinc of the Leſſer, gives the Sinc
of the Grearer, or ſubtracted from the Sine of the Greater, leaves
the Sine of the Leſler.
Whence the Sum and Difference of the Products, made by the
Side of the Mean Arch and Co-line of the Arch of Difference, alſo
the Co-line of the middle Arch, and the Sine of the Arch of Diffe-
rence divided by the Radius, gives the greater and leſſer of the Sines
of the two Extream Arches; whence the Interpolation of Sines in
the Quadrant is eaſy.
The Sines being thus obtained, the Tangents and Secants are eaſily
had
For becauſe the Triangles CRS, CAI,
are ſimilar, it will be,
T
B
H
C
А.
24
As CR the Co-fine, to RS the Right-
ſine ; fo is CA the Radius, to AT the
Tangent.
Wherefore, becauſe the Radius is ſupa
poled Unity, the Quotient of the Right-
fine divided by the Co-line, will give the
Tangent of the ſame Arch.
Hence the Tangent Series is eaſily found, for if a-ni tiiba',
&c. the Right Sine be divided by i--aatant, &c. the Co-line,
the Quotient a ta'tiša, &c. will be the Tangent, as in the
following Example.
1-aataat, &c.).—12+iiba', &c.(a tja' tiša', &c.
2-in} +14a', &c.
+a-a', &c.
+ža-a', &c.
tiša', &c.
Hence if a be put for the Length of any Arch, the Correſpon-
dent Tangent will be atta’ tišastija tiesa tiltja!
Again
trots
I 3
10.14
81-75
0
&c.
Trigonometry.
79
1
2
1
2
I
3
a
45
os
3
Again, becauſe the Triangles ACT, GBC, are ſimilar, it will be
as AT the Tangent, to AC the Radius ; ſo is CB the Radius, to BG
the Co-tangent.
Hence the Radius is a Mean Proportional, between the Tangent
and Co-tangent of an Arch, wherefore becauſe the Radius is 1.
If I be divided by the Tangent Series, viz atja tija', &c.
the Quotient
a?
n', &c.
945 4725
93555
will be the Series for finding the Co-tangent, from the Arch firſt
given
Again, becauſe the Triangles CRS, CAT, are ſimilar, it will be
as CR the Co-ſine, to CS the Radius, ſo is CA the Radius, to CI
the Secant.
Hence the Radius is a Mean Proportional, between the Secant
and Co-line of an Arch.
Wherefore if I be divided by ima tz^34-7X TQ", G. (the
Series for the Co-ſine) the Quotient 1 + ja’taat tiba + post
of thingbra'', &c. will be a Series for finding the Secant, from
the Length of the Arch firſt given.
Again, becauſe the Triangles RCS, CBG, are ſimilar, it will be
as RS the Right Sine, to SC the Radius ; So is CB the Radius, to
CG the Secant of the Complement.
Hence the Radius is a Mean Proportional, between the Righita
fine and Co-ſecant of an Arch. Wherefore, if I be divided by ----
3
a'tižsa', &c. the Series for the Right-line, the Quotientit
zation that come to zare? +4zidana', &c. will be the Se-
ries for the Co-ſecant of the fame Arch
I have omitted the inveſtigation of the three laſt Series, as alſo the
application of the Tangent, Secant, &c. Series to the adual finding
the Length of the Tangent, &c. from the Arch it ſelf for brevity
fake, it being very Ealy to any one who underſtands what went
By Extracting the Root of the ſeveral infinite Series, for finding
the Verſed-line, Tangent, &c. from the Arch firſt given, we ſhali
have
ſo many different Series for finding the Lepgth of the Cir-
1
1
before.
cumference.
But
6.
.
...*
}=
ز
80
Trigonometry.
But amongſt all, none is ſo eaſy to be retained in the Memory,
and ſo readily put in Pra&ice, as that for finding the Length of the
Circumference from the Tangent firll given ; as will evidently appear
from the Example.
If t=a+ja tija! +3*}a', &c.
Then a=t-*t + ft' -*++$t', &c.
For put a=Att Bt? + Ct', &c,
Then will ža’=tAt}+ABt', &c. =
And iša'=
tośAt', &c.
And conſequently, At=t, and A=1; alſo, B+}A=0,
And B=--A'=-*; alſo, C+AB+ i A' =0,
And C=AB-IŽA=-1}=.
Wherefore, A=I, B-, C=}, &c.
And conſequently, a=t-t} +}t', &c.
Now the Radius being Unity, the Sine of 30 Degrees =* ; and
confequently the Co-ſine=v1–4=V(by the 1ſt Prob.) and becauſe
the Co-ſine is to the Right-line, as the Radius to the Tangent ; it
will be, vå: V$::Vi:vį, the Tangent of 30 Degrees=t; whence
t=í, wherefore, if the Root of į be divided continually by 3, and
the ſeveral Quotients again by all the odd Numbers ſucceſſively,
viž. the firſt by the threc, the ſecond by five, &c. the Sum of all
theſe laſt Quotients (regard being
had to the Negative and Poſitive
Terms) will give the Arch of 30 Degrees.
And becauſe the Arch of 30 Degrees is part of the Semi-Cir-
cumference, if inſtead of V} be taken 6V =V=V12, we ſhall
have the Semi-Circumference in ſuch Parts as the Radius is Unity;
or the whole Circumference the Diameter being Unity, as will
appear from the following Example:
The
Trigonometry:
81
427 6608.06+
Succeſſive Quotients arifing from the Divi-
ſion of each preceding Number by 3.
The Square Root of 12 is 3.464160161.51.
3-4641 0161.50-
+3.4641.0161.51 to
1.1547.0053.84-
-0,3849.0017.95
.3849.001795
+769.8003.59-
.1283 0005.98 +
-183.2858.00-
+475185.414
142.5556.22+
- 129596.02+
47.5185:41
+36552.72+
15.8.95.14-
-1.0559677
5.279838
+.3105 79-
1.7599.46–
-926.29-
•5860 49
+279.36-
1955:49+
-85.02+
65 1.83 to
+26.07+
217.28
-8.05 .
72.43-
24.14
8.07
+24t
2.68 +
-8-
.89-
tat
Succeſſive Quotients ariſing from the Divi-
fion of the former Quotients by the odd Num-
bers, viz. the firſt by i, the ſecond by 3,
OC.
the 3d by 5,
+2.50-
78mm
+3:5452.3317 21
--4046.4051 86
3.141509265.35
The Impoſſibility of expreſfing the exa& Proportion of the Diame-
ter of a Circle, to its Circumference by any received way of Notation;
and the abſolute necetlity of as near a Quadrature as pffi oble, it be-
ing the very Bafis -upon which the uſeful Branches of the Mathematics
in general are founded ; has put the moſt Celebrated Men in all
Ages upon endeavouring to approximate as near as may be to the
Truth, and after the famous Van Ceulen who carried it to 3: Deci-
mal Places, which he ordered to be Engraven on bis Tombstone,
thinking he had ſet Bounds to farther Improvemenis, the firſt that
attempted it with Succeſs was the moſt indefatigable Mr. Abraham
Sharp, who by a double Computation, viz. from the Sine of o De
grees one way, and from the Sine and Co-line of 12 Degrees another
way, carried it to twice the Number of Places chartau Ceais had
M
dona,
82
Trigonometry.
grees one way, and from the Sine and Co-line of 12 Degrees another
way, carried it to twice the Number of Places that Van Ceulen had
done, viz. 72, which he ſent to me in a Létter, in the Year 1904;
which Lecter being Communicated to the Royal Society, at their re-
queſt Dr. Halley promiſed to ſee it Printed in the Philoſophical Iran-
ſactions ; but as yet by what Means I know not, it has never ſeen the
Light.
Ir deed mention is made by Dr. Halley, in Mr. Sherwin's Mathema-
fical Tables, publiſhed in the Year 1706, of a Quadrature of the Cir-
cle, done by Mr. Abraham Sharp, to 72 Places, from the Tangent
of 30 Degrees; after the manner taught in the former Page as if the
compendium of that Method had put him upon it ; when in reality
Mr. Sharp did that at the deſire of Dr. Halley, and it was no other
than a triple Computation of the ſame Quadrature; or rather a Con-
firmation of what he had done ſome Years before, as Dr. Halley very
well knew.
This I mention, as a piece of Juſtice due from my ſelf, to that in-
comparable Maftei of Numbers, for his free and generous Commu-
nication of it to me; having no other way left at preſent to make
him amends, for the long, if not intire Suppreſſion of his Letter.
The above mentioned Quadrature of Mr. Abr. Sharp, is as follows.
If the Diameter be 1.000000, c. the Circumference will be
3.141592.653589.793238.462643.383279.502884.197169.399375.
105820.974944.592207.816405.2-1-
It has been lince carried on by the curious Hand of the Inge-
nious Mr. John Machin, the preſent Profeſſor of Aſtronomy in Gre-
jham College, to 100 Places, as follows.
The Diameter being 1.00000, c. the Circumference will be
3.14159.26535.89793:23846.26433.87'279.50288 41971.69399.37510
58209.74944.59230 78164.05286.20899.86280.34825.34211.70679.
--
Which is a Degree of exactneſs far furplaffing all Belief, the former
Quadrature being more than fufficient to Calculate the Number of
Grains of Sand that may be comprehended within the Sphere of the
Fixed Stars.
I believe it will not be unacceptable to the Reader, if before I
conclude this Se&ion, I fhew how the Length of the Circumference
in equal Parts of the Radius, may be applied to the Menluration of
the Circle, and of ſuch Superficies and Solids as depend upon it.
It
Trigonometry.
83
multiplied by its Reciprocal :3 1830.98862--, the former Quotient
It has been already determined, that if the Diameter of any Cir-
cle conſiſt of 1:00000.00000, the Circunference will be 3.14159
.26536- of the fame Parts; and ſince all Circles are ſimilar Figures,
it will be (by the 18th.)
As
I, to 3.14159.26536-, ſo is the Diameter of a Circle,
to its Circumference. Wherefore, if the Diameter of any Circle be
multiplied by 3.14159.26536—, the Produd will be the Length of
the Circumference in the ſame Parts.
2. As :354159 26536– to 1, or as 1, to .31830,98862—, ro
is the Circumference of a Circle, to its Diameter. Wherefore, if
the Circumference of any Circle be divided by 3.14159.265364, or
I
or the latter Product, will give the Diameter.
E
B
G
H
2
2
3. In the Circle BEGH, put BG=d, OG=
r, then will d=ar. Let A ſtand for the Area,
and c for the Circumference, and ſuppoſe the
Arch FG=%, to be infinitely ſmall, then will
be the Area of the infinitely ſmall Sector
OFG, this therefore multiplied by the whole
Circumference c, that is, putting c in the room of X, we ſhall have
=A, equal to the intire Area of the Circle.
4. Hence the Area of any Circle is found, by multiplying half
the Circumference by half the Diameter or Radius, .
For rx = =1.
ş. Hence and (from the 6th of the 7th) every Circle is equal to a
Triangle whoſe Baſe is equal to the Circumference, and Perpendicu-
lar Height the Radius.
6. Again, Let the Proportion of the Diameter to the Circum-
ference, be as n to m.
M 2
Then
TC
.
esit
84
Trigonometry.
N
2
2n
q
md
Then as n:m::d: its Circumference. .
d. md mdd
Wherefore, X
=A, the Area.
40
Wherefore, 4n :m::dd: A.
7. And fince d may ſtand for the Diameter of any Circle, it fol-
lows, that Circles are to each otlier, as the Squares of their reſpective
Diameters
8. Hence as 4 times the Diameter, to the Circumference; ſo is
the Square of the Diameter, to the Area. Wherefore,
9. As 4, to 3,14159.26536-, or as i, to .78539.81634-«; fo
is the Square of the Diameter of any Circle, to its Area. Wherefore,
if the Square of the Diameter of any Circle be multiplied by
-78539.81634mm, the Product will be the Content.
And becauſe dd=4rr, it will be, as 4n:m:: 4rr : A. Therefore,
as n:m::98: A.
10. Hence Circles are to each other, as the Squares of their Ra-
dii.
11. Hence as the Diameter of a Circle, is to its Circumference, ſo
is the Square of the Radius, to the Area. Wherefore,
to 3.14159.26536mm, lo is the Square of the Radius, to the
Area of the fame Circle.
12. Wherefore, if the Square of the Radius of any Circle be mul-
tiplied by 3.14159.26536~-, the Product will be the Area.
ܕܐ A5
пс
Again, as m:n::6:
=d the Diameter.
712
ncc
C
nic
х
2 272
Wherefore,
= Area,
4m
Therefore, 4m: n::cc: A.
13. Hence Circles are to each other, as the Squares of their Pe-
ripherys or Circumferences..
14. Hence as 4 times the Circumference, to the Diameter, ſo is
the Square of the Circumference, to the Area. That is,
As I 2.56637.06144—, to i, or as I to .07957.74715; ſo is the
Square of the Periphery of any Circle, to its Area. Wherefore, if
the
T
Trigonometry.
85
D N
F
D
B
C
G
L
R
i
the Square of the Periphery be divided by 12.56637.06144-, or
multiplied by .07957.74715, the former Quotient or the latter Pro-
duet, will give the Content.
15. Let AB be the tranſverſe Diameter of
the Elipfis AFBG, FG its conjugate Diameter,
ADBE the Circumſcribing Circle.
Л
Becauſe in the Circle ACxCB: APxPB ::
CD: PNq; and in the Elipſes, ACxCB: AP
XPB :: CF4: PMq, it will be, CD: CF :: PN
:PM; or as (ED=) AB: FG::NR: MQ; but all the NRs
make the Circle, and the ſame number of MQs make the Elipſi.s.
Hence (by the 5th of the 15th.) as AB to FG, or as the Longeſt Dia-
meter of the Elipſis to its ſhorteſt, ſo is the Area of the Circumſcri-
bing Circle, to the Area of the Inſcribed Elipfis. That is, as AB:
FG:: Area of the Circle, to the Area of the Eliffis. Wherefore,
ás (ABXAB - ) ABq:ABXFG :: Circles Arca,to the Eliptical Arca.
16. Hence as the Circle is to its Circumſcribed Square, or to the
Square of its Diameter ; ſo is the Elipſis, to its Circumſcribing Pa-
rallelogram, or to the Rectangle of its Diameters.
17. Hence the Quadrature of the Elipſis, depends upon the Qua-
drature of the Circle.
18. Hence alſo, the Area of any Segment of the Elipſis, is to the
Area of the Correſpondent Circular Segment, as the ſhorteſt Diame-
ter of the Elipſis to the longeſt.
19. Hence as 4, to 3.14159.26536-, or as i to .78539.81634,
fo is the Product of the Diameters of the Elipſis, to its Area. Where-
fore, if the Product of the Diameters of an Elipſis be multiplyed by
78539.81634, the laſt Product will give its Area
20. How the Area or Superficial Content of any Right-Lined
Superficies may be found, has been taught in the 7th Prop. of the
1ſt Part, whence we may learn how to find the Superficies of any
Priſm.
21. For fince each Face of a Priſm is a Parallelogram, the Sum
of the ſeveral Faces will be the whole Surface.
Hence the Surface of every Priſm is equal to a Parallelogram,
whoſe Baſe is the Perimeter of the Priſm, and Height equal to
the Priſm's Altitude : wherefore, if the Perimeter of any Priſm be
multiplied by the Altitude, the Product will give the Superficial
22. Again,
Content.
86
Trigonometry.
be by , the .
22. Again, ſince the Face of every Pyramid is a Triangle, the
Surface of every Pyramid is equal to a Triangle, whoſe Baſe is equal
to the Perimeter of the Priſm, and Height equal to the Height of
the Priſm; wherefore, half the Product of the Pyramid's Circumfe-
rence multiplyed into its Height, is equal to its Superficial Content.
23. Hence it is evidert, that the Superficies of a Pyramid, is to
the Superficies of the Circumſcribing Priſm, as i to 2.
24. Again, ſince every Surface of a Cylinder, is generated by the
Motion of one side of a Parallelogram, the other Side remaining
fixed as an Axis, it follows, that every Cylindrical Surface is equal to
a Parallelogram, whoſe Baſe is equal to the Cylindrical Circumference,
and Height equal to the Height of the Cylinder, Hence,
25. As i, to 3.14159.265366, ſo is the Diameter of any Cylinder
multiplyed by the Height, to the Superficies of the Cylinder: Where-
fore, if the Product of the Height of the Cylinder into the Diameter,
multiplyed 3.14159.26536Product will give the Su-
perficies
26. If the Height of the Cylinder be equal to the Diameter, then
willits Superficies be quadruple of the Area of the Baſe, for the Area of
the Baſe is, and the Superficies of the Cylinder 2cr=3 7*X4.
27. Again, ſince the Surface of every Right-Cone is generated by
the revolution of the Hypothenuſe of a Right Angled Plain Trian-
gle, the Perpendicular remaining fixed as an Axis, it follows, that
every Conical Surface is equal to a Triangle, whoſe Baſe is equal to
the Circumference of the Baſe of the Cone, and Height equal to the
Side of the Cone or Line, reaching from the Baſe to the Apex. Where-
fore, half the Product of the Baſe of the Cone, multiplyed into the Side,
is equal to the Superficial Content of the ſame Conė.
28. Wherefore, as I to 3.14159.26536%, ſo is half the Produ& of
the Diameter of the Baſe of the Cone, multiplyed into the Side, to
the Superficies of the ſame Cone.
29. Hence it is manifeſt, that the Superficies of every Cone, is half
the Superficies of the Circumſcribing Cylinder.
Gr.
cr
30. IC
oli
Trigonometry.
87
D
PP C
CY
18
1
Square of the Circumference of the Sphere, to its Superficies. Where-
fore, if the Square of the Periphery be divided by 3.14159, Oc. or
30. If the Semicircle ABD, be turned
B
about
upon
its Diameter AD as an Axis,
M
R
the Semi-Periphery ABD, will deſcribe the
Surface of a Sphere ; and each Point as M,
will deſcribe the Çircumference of a Cir A
cle, whoſe Radius will be MP, the Right-
line of the Arch AM. Put AP=%, PM=), AC=r, and the whole
Circumference =c, then as r:0::y: theCircumference generated
by the Point M. Let mM be an infinitely ſmall. Arch, and draw mp
parallel to MP, then will MR=x, and becauſe the Triangles CMP
and MMR are ſimilar, it will be (PM=),:(CM)=r:: MR=%
су
: Mm
=
this multiplyed by the Circumference formed by the
Ý
cyrx
Point M, will give = cx the Area of the Increment formed by
ry
Mm, this multiplyed by 2r, or which is the ſame thing, putting 2r in
the room of x, will give 2cr for the intire Surface of the Sphere;
for when Mm-becomes equal to ABD, x becomes equal to ar.
31. Hence it is manifeſt, that the Surface of the Sphere is
to 4 times the Area of the greateſt Circle, for 2cr the Surface of the
Sphere, is equal to 4 times the Area of the greateſt Circle.
It has been already proved in Art. 6, that a:m:: 4dd: Area of
the Circle. Therefore, as n:m::dd:Surface of the Sphere ; that is, as
the Diameter of any Circle, is to its Circumference, ſo is the Square
of the fame Diameter, to the Surface of the generated Sphere.
32. Wherefore, as 1, to 3.14159.26536—, fo is the Square of
the Diameter of a Sphere, to its Superficies.
Hence the Square of the Diameter being multiplyed by 3.14159
26536m, or the Square of the Radius by 12.56637.06144, the
Produ& will be the Superficial Content.
33. Again, becauſe (by Art, 12.) 4m ::: cc to the Circles Area.
Therefore, as m : *:: cc to the Superficies of the Sphere ; that is,
3.14.159.26536, to i, or as 1, to .31830.98862, ſo is the
multi-
equal
CY
2
As
88
Trigonometry.
multiplyed by 31830, &c. the former Quotient or the latter Product,
will give the Content.
34. Hence and from the 26th Art. the Surface of any Globe and
its Circumſcribing Cylinder are equal, as being each quadruple of the
greateſt inſcribed Circle.
35. Hence it follows, that the Surface of any Portion of a Sphere
cut by a Plane or Planes, as MP is to the intire Surface of the Sphere,
as the part of the Axis AP cut off by the Plane or Planes, to the
whole Diameter of the Sphere AD. Whence it is Eaſy to find the
Surface of a Portion of a Sphere if the Correſpondent Portion of the
Diameter be kuown, and conſequently the Portion of the Surface of
a Sphere cut by two parallel Planes.
36. Hence alſo, the Portion of any Surface of the Sphere contain
od between any two Great Circles, is to the intire Surface of the
Sphere, as the Angle of Inclination of the two Great Circles, to the
intire Circumference, or four Right Angles.
37. From Art. the 34th and 35th, it follows, that the Correſpon-
dent Surfaces of the Globe and its Circumſcribing Cylinder, cut off
by parallel Flanes, are equal.
38. Again, ſince every Solid Priſm or Parallilepepedon, is generated
by the uniform Motion of a Plane Surface, and may therefore
be conſidered as compoſed of an infinite number of Plane Surfaces,
equal and ſimilar to each other, and conſequently to the Baſe ; its
Solidity will be found by multiplying the Area or Superficies of the
Baſe, into the Perpendicular Height, or Sum of the Component
Parts.
39. Hence the Solidity of a Cylinder, (which is formed by thế Re-
volution of a Parallelogram about one side, , remaining fixed as an
Axis) is found by multipiying the Area of the Circular Baſe, by its
Height or Altitude; putting therefore r for the Radius of the Baſe,
c for the Circumference, and a for the Altitude, we ſhall have
Solidity:
But if the Altitude be equal to the Diameter, then the Solidity
will be- x2r=crr.
But it has already been ſhewn in the ath Art. that, as 1, to .78539
.81634—; fo is the Square of the Diameter of any Circle, to its Area.
Where-
CY
2
CT
2
Trigonometry.
89
Wherefore, as I, to 78539.81634“, ſo is the Square of the Dia-
meter of the Baſe of the Cylinder multiplyed into its Height, to its
Solid Content.
41. Becauſe by the rith Art, as I to 3.14159.26536--, ſo is the
Square of the Radius, to the Area of the Circle ; it will be, as 1,
to 3.14159.26536-, ſo is the Square of the Radius of the Baſe of
the Cylinder, multiplyed into the Height, to the Solid Content.
42 Again, becauſe by the 14th Art. the Square of the Circumfc-
rence of any Circle is to its Area, as to i to 07957.74715, there-
fore, as I to 07957.74715, ſo is the Square of the Circumference
of any Cylinder, multiplýed by its Altitude, to its Solidity.
43. Let ABDC be a Cone, formed by the Revo-
A А.
lution of the Triangle ABC, about the Side AC,
remaining fixed as an Axis; it is manifeſt, that
every Line as MP, drawn parallel to BC, will de-
Scribe a Circle, of which that formed by BC will be
the greateſt, and that the Sum of all theſe Circles
will be the whole Cone. Put CD=r, AC=1, and
the Circumference formed by the Point B=c,
AP=x, and MP=y; then, asr : c::y: the circumference of
суу
the Circle formed by the Point M, this multiplyed by gives
the Area of the Circle it ſelf ; this multiplyed again by x=Pp, will
суух
give
the Conical Increment. Again, becauſe of the Similar
Triangles APM, ACB, it will be, x:y::a:r; wherefore, x=
and x="" putting this therefore in the room of x in the for-
me
to
P
P
B
D
C
y
21
2r
r
cyyx acyyy
mer Equation we ſhall have
for the Conical Increment,
27
2rr
and conſequently 6rr
acyyy
for the Value of the Cone, deſcribed by
acrry
act
brr
6
APM; and conſequently
for the whole Cone ; for when
* becomes equal to a, , becomes equal to r.
.
N
But
Trumani
90
Trigonometry.
ст
Car
аст
аст:
2
6.
or as
3
But when the Height of the Cone is equal to the Diameter of the
Baſe, then the Solidity will be ő * 27 =
3
44. Hence it follows, that every Cone is of the circumſcribing
Cylinder, having the ſame Baſe and Altitude ; for the Solidity of the
Cylinder by Article the 39th was this multiplyed by š, gives
the Solidity of the inſcribed Cone.
45. Hence and from Art. the 40th, as I to
78539.81634
3
1, to .26179.93878, fo is the Square of the Diameter of the Baſe
of the Cone, multiplyed by its Altitude, to the Solid Content.
3.14159.26536
46. Again, from Art. the 41ſt as I to
or as s', to
1.04?19.75512, ſo is the Square of the Radius of the Conical Baſe,
multiplyed into its Altitude, to its Solidity.
47. Alſo, from Art. the 42d it follows, as I, to .of.0795.7.74715,
or as i, .02652.58238-; fo is the Square of the Periphery of the
Baſe of the Cone, multiplyed into its Altitude, to its Solid Content.
48. By reaſoning after the ſame manner as in the 43d Art, it will
appear, that every Pyramid is of the circumſcribing Prifin, and
conſequently the Solidity of the circumſcribing Priſm being found by
Art. the 38th, the Solidy of the Pyramid is had by taking off its
49. As a Sphere may be couſidered as a Polyedron of an infinite
number of Surfaces; and that each of theſe Surfaces are the Baſes of
ſo many Pyramids, which have for their common Vertex the Center
of the Sphere, and for their common Altitude the Radius of the
fame Sphere ; it follows, that if the Surface of the Sphere equal tozcr
(by Art. the 39th) be multiplyed by Ź of , the common Altitude,
the Product will be the Solidity of the ſame Sphere.
3
$o: Hence it follows, that the Sphere is of the circumferibing.
Cylinder, for the Solidity of the Cylinder whoſe Height is equal to
the Diameter of its Baſe, is crr by Art. the 39th; that multiplýcd
by gives the Solidity of the inferibed Sphere.
3
part.
ز
2017
2.677
3
51. Hence
Pay
.
crr
2 crr
2, gives
3
;
Trigonometry.
91
51. Hence the Globe is double to the Cone, having for its Baſe a
great Circle of the Sphere, and for its Height the Diameter ; for the
Solidity of ſuch a Cone is by Art. the 44th, this multiplyed by
3
for the Solidity of the Sphere.
52. Hence it follows, that the Cylinder, the Sphere, and the Cone,
having the ſame Height and Diamcter, are as the Numbers 3, 2, 1.
53. Whence the Cone is equal to the Exceſs of the Cylinder, above
thie Globe.
54. Hence alſo it is evident, that the Sphere is to the Cube of its
Diameter, as į of its Circumference, is to four times its Diameter
for tcrr:Srrr :: C:8r; and conſequently the Sphere is to į of
the Cube of its Diameter, as the Circumference, is to 4 times its
Diameter.
55. Hence and from Art, the sos7 and 45th, it will be, as i, to
52359.87756, ſo is thc Cube of the Diameter of any Sphere, to its
Solid Content.
56. Alſo, from Art. the sist, and 46th it follows, that as r,
to 4.18879.02048, ſo is the Cube of the Radius of any Sphere, to its
Solid Content.
Again, from Art. the 5 iſt and 47th it will be, as i, to .01688
68639, ſo is the Cube of the Circumference of any Sphere, to its
Solid Content,
57. Conceive the adjacent Figure to be
turned about upon its Axis AB, then will
NMP
the Semicircle ADB generate a Sphere,
whoſe Diameter is AB, and the Semi-
Elipfis AMFB will generate a Spheroid,
whofe longeſt Diameter is AB, and ſhort-
eſt FF; and every Ordinate in the Circle
as PN, will generate a Circle, whoſe Di-
ameter is NN, while the correſpondent
Ordinate in the Elipfis generates a Circle,
whoſe Diameter is MM; and theſe Circles being to each other as
the Square of NN to the Square of MM, (by the 6th Art.) and the
Square of NN being to the Square of MM, as the Square of DD,
to the Square of FF; (by the 15th Art.) it follows, (by the 5th of the
1514
A
MN
D
F
B
N 2
92
Trigonometry.
15th) that all the Circles generated by PN, will be to the like num-
ber of Circles generated by PM, as the Square of DD, to the
Square of FF; but all the Circles formed by the Revolution of PN,
make the Sphere and the like number of Circles formed by the Re-
volution of PM, make the Spheroid ; wherefore, the Sphere is to
the Sphieroid, as the Square of DD, to the Square of FF; or as the
Square of the longeſt Diameter of the Spheriod, to the Square of the
forteſt.
58. Hence the Spheroid is to its circumſcribing Cylinder, as the
Sphere to its circumſcribing Cylinder; but the Sphere is į of its cir-
cumſcribing Cylinder, (by Art. the soth) therefore, the Spheroid
is } of its circumſcribing Cylinder.
59. Hence and from Art. the 45th it follows, that as ļ, to 52359.
87756, ſo is the Square of the ſhorteſt Diameter of the Spheroid,
multiplyed into the longeſt, to the Solid Content.
60. Again, from Art. the 46th it follows, that as I, to 2.09439
:5 1024, lo is the Square of the Seini-conjugate Diameter, multiplyed
into the tranſverſe or longeſt Diameter, to the Solidity of the Sphe-
roid.
61. Alſo, by Art. the 47th it will be, as i, to .05305.16476, fo
is the Square of the Circumference of the Spheroid, multiplyed into
the longeſt Diameter, to the Solid Content.
62. The Rules delivered in the ſeveral Articles for the Menſura-
tion of Cylinders and Cones, having circular Baſes, will hold good if
the Baſes are Eliptical, &c. as alſo in the Spheroid, as the inteli-
gent Reader will eaſily perceive.
63. Again, froni Art. the 58th it follows, that any Segment of
the Spheroid, as MPMPA, is to the correſpondent Segment of the
Sphere NPNPA, as the Square of the longeſt Diameter of the Sphe-
roid, (or Diameter of the circumſcribing Sphere) to the Square of
the ſhorteſt Diameter; for of as many Circles whoſe Diameter is MM,
as make the Segment of the Spheroid, of the like number of Circles
whoſe Diameter is NN, is formed the correſpondent Segment of
the Sphere, and theſe being every where as the Square of FF, to
the Square of DD, ir follows, &c.
64. Hence it will be eaſy to find the Value of any Portion of a
Spheroid, cut by two parallel Planes.
.
65. Hence
Trigonometry.
23.
RE:
65. Hence it is evident, that the circumſcribing Cylinder, the
Spheroid, and in cribed Cone are in the ſame Ratio as the Cylinder,
the Sphere, and its intcribed Cone, viz. as the Numbers 3, 2, and 1.
66. From thefe Premiſes duly conſidered, a great variety of uſeful
and pleaſant Theorems might be drawn, but having already exceeded
the Bounds I at firit laid down, I ſhall deſiſt till a more convenient
Opportunity offers,
67. In the Application of theſe Rules to a&ual Menſuration, the
Practitioner is at liberty to make uſe of what Number of Decimal
Places he judges convenient for his purpoſe, being always fure of ha-
ving his Anſwer true to what Degree of exa&tneſs he pleaſes; which
is an Advantage (beſides the facility of the Operation) that he can-
not reap from making Uſe of the Numbers 7 to 22, or 113 to 355)
oc, commonly in Uſe.
Section III.
Containing the Demonstration of ſome ufeful Theorems, common,
ly cald Axioms, neceſſary for the Solution of the ſeveral .
Caſes of Right and Oblique- Angled Plain Triangles.
Theorem the ift, uſually called the ift Axiom
I Now
N every Right-Angled Plain Triangle,
When the Hypotenuſe or longeſt Side is made the Radius of a Cir-
cle, the other Sides or Legs, will be the Sines of their oppoſite An-
gles; but if either of the sides or Legs, containing the
Right Angle
be made the Radius, the other Side or Leg will be the Tangent of its
oppoſite Angle, and the Hypothenuſe the Secant of the fame Angle.
:B D
Tlus,
Wien
94
Trigonometry.
Thus, in Fig. the iſt AC being made the Radius, BC is the Sine
of the Arch CD, (by the 3d Def. Set 1.) the Meaſure of the Angle
A, and AB the Sine of the Arch AE, (by the 3d Def. Se&t. 1.) the
Meaſure of the Angle C.
Again, in Fig. the 2d. AB being made the Radius, BC is the Tan-
gent of the Arch BF, by the 5 def. Sečt: 1.) the Meaſure of the An-
gle A, and AC the Secant of the ſame Arch or Angle, (by the 6th
def. Sect. 1.)
In like manner, in Fig. the 3d. BC being made the Radius, AB is
the Tangent of the Arch BG or Angle C, (by the 5th def. Sect. 1.)
and CA the Secant of the ſame Arch or Angle, (by the 6th Sect. 1.)
This Axiom is uſeful in the Solution of all the caſes of Right-angled
Plain Triangles.
Theorem the 2d, or Axiom the 2d.
In all Plain Triangles,
The Sides are to each other, as are the Sines of their oppoſite
Angles ; that is,
B
B
D
D
AG
E
A
CF H
Е ЕНС
In the Triangle ADE it will be, AD: DE::S. AngleE : S.Angle A.
Constr. Produce ED to B (by the iſt Poft.) make EB = AD, and
ler fall the perpendiculars BC and DF, (by the 14th of the 2d.) which
will be the Sines of their oppoſite Angles E and A, (by the 1ſt Theo-
rem) to the Radius EB=AD. Wherefore, (by the 18th Prop.) it
will be
As EB:ED:: BC: DF. That is,
As AD: ED :: S,<E:S, A W. W. D.
This Axiom is uſeful in the Solution of the three firſt Cafes of Oblique
Angled Plain Triangles, and may be applyed to all the caſes in Plain Tri-
gonometry, except that wherein the two Sides and the Angle contained are
given, and the three Sides to find the reſt.
Theorem
Beweguh
Trigonometry.
25
Theorem the 3d, or Axiom the 3d.
In all Plain Triangles it will be,
As the Sum of any two Sides, is to their Difference, ſo is the
Tangent of half the Sum of their oppoſite Angles, to the Tangent of
half their Difference. Or, of the Ditterence of either of them from
the half Sum
Now the Arch of halt Difference, added to the half Sum, will
give the greater of the two Angles, and ſubtracted from the half
Sum, will leave the lefler.
That is, in the Triangle ADE it will be,
ift, As AD+DE: AD-DE:: T. <Et::T. <$E-A..
Z
D
D
X
B
B.
X
А
A А
E
E''
Conſtr.. On D as a Center, with the Diſtance of DE deſcribe the:
Circle DEZ (by the 3d Poſt.) produce AD to Z (by the 1st Poſt.)
joyn the Points X and E (by the 2d Poft.) and draw XB parallel to
AE (by 5th of the 5th:).
Demonſtr. Becauſe D is the Center of the Circle XEZ, therefore
are XD, DE, DZ, equal (by the 15th def.) whence AZ is equal to
AD #DE (by the 2d Axiom) and AX equal to AD-DE (by the 3d
Axium.)
Again, becaufe XZ is the Diameter of the Circle XEZ, therefore
the Angle XEZ is Right (by the 3d of the 11th) and EZ the Tangent
of the Angle EXZ (by the 1ſt Theorem)=XED (by the 4th of the
2d.)=1 EDZ (by the N Prop.) =<SATE (by the 6th Prop.) to
the Radius XE, and EB isthe Tangent of the < $ EXB (by the 1st
Theorem) =EXB (by the 5th Prop) to the ſame Radius.
And becauſe XB is parallel AE, it will (by the 17th Prop)
As AZ: AX;: ZE:BE. That is,
As
.
..........
96
Trigonometry.
G
AS AD+DE: AD-DE:: T. KSA+E:T. <EXB= (by
the 5th Prop.) < XEA, the Arch of Difference of either, from the
half Sum.
Now, DXE-EXB=(by the 3d Ax.) DXB=(by the sth Prop.)
DAE. And DEX XEA =(by the 2d Ax.) AED. W. W. D.
In ſuch caſes where the logarithms of the two containing Sides are gi-
ven, which
frequently happens, and particularly in the Caſe for finding the
Bearing and Diſtance between any two inaccej able Places) the following
Theorem is very applicable, being eaſier for Practice, and producing a truer
ufwer than the former.
Theorem the 4th, or Axiom the 4th.
in all Plain Triangles it will be,
As the letter of the two Sides, to the greater, ſo is the Radius to
the Tangent of an Arch, from which taking away 45 Degrees, it will
be,
As the Radius, to the Tangent of the remaining Arch; fo is the
Tangent of half the Sum of the oppoſite Angles, to the Tangent of
half their Difference. Thar.is, in the Triangle ADE, it will be,
As DE: AD ::R: T. of an Arch. Again,
As R :'T. Arch-45 Deg. ::T. <sA+E: T.;<$EA.
X
D
B
B
A
E
Conſtr.
Trigonometry.
97
Conſtr.On D as a Center, with the Diſtance of DE, deſcribe the
Semicircle EBX (by the 3d Poft.), ED being firſt produced to Z, till
DZ=DA. Let DB be Perpendicular to XE (by the 13th of the 2 d.)
and draw the Lines BE, BX, (by the ad Post.) wliich will be equal
(by the 2d. Prop.); from X draw Xe, Perpendicular to BX (by the
13th of the 2d, then will the Triangles BZE, CZX be ſimilar, (by the
5th Prop. and 2d of the 6th.)
. And becauſe the Triangle ZBD is Right-angled at D, it will be
(by the 1ſt Theorem) (DB=) DE:(D=) DĂ::R:T.ZBD.
Now ŹBD-BD (= 45 Degrees) (by the 4th of the 6th.)=
CBX Wherefore,
As R:T. <CBX::(BX=) BE:CX (by the ift Theorem) :: EZ
:X2 (by the 17th Prop.) :: T. KE+A: T. į Ks E-A (by the
precedent Theorem.) W. W. D.
Theſe two Axioms are uſeful in the Solution of the 4th and sth Caſes.
of Oblique-angled Plain Triangles, where two Sides and the Angle
contained are given to find the reſt.
e
Theorem the 5th, or Axiom the 5th.
-
In every Plain Triangle it will be ,
As the Baſe, to the Sum of the other two Sides, ſo is their Diffe-
rence, to the Difference of the Segments of the Bale ; made by a
Perpendicular let fall from the oppoſite Angle when it falls within,
or to the alternate Baſe when it falls without.
Z
Z
D D
D
X
X
Α.
P
B В
A
E
E...P...B
That
1
:
98
Trigonometry.
That is, in the Triangle ADE it will be,
As AE: AD-HDE:: AD-DE: AB.
Conſtruction. On D as a Center, with the Diſtance of DE, de-
ſcribe the Circle XBEZ (by me 3d Post.) produce AD to Z, (by
the 2d Poft.) joyn the Points D and B (by the ift Poft.) and let fall
the Perpendicular DP (by the 14th of the 2d.)
Demonſtr. Becauſe D is the Center of the Circle XEZ, therefore,
are DX, DE, DZ, equal, and AZ=AD DE (by the ad Axiom.)
and AX=AD-DE (by the 3d Axiom.) and becauſe the Point A is
without the Circle, AEXAB=AZxAX (by the iſt of the 13th.)
Wherefore, (by the 8th of the 19th.) it will be, ÅE : AZ :: AX
: AB.
That is, AE : AD-DE:: AD-DE: AB, the Difference of
the Segments AP and PE in the firſt Caſe, and the alternate Baſe
AB in the ſecond. W.W.D.
Now, AE+ AB=2 {AP } in the firt Cafe.
And ABI AE=2 {Ap}
in the ſecond Caſe.
EP
This and the following Axiom are of Uſe in finding the Angles in the 6th
and laſt Caſe of Oblique Angled Plain Triangles, where the three Sides are
given.
Theorem the 6th, or Axiom the 6th.
In all Plain Triangles it will be,
As the Rectangle, or Product of any two containing Sides, is to
the Rectangle, or Product of half the Sum of the three Sides, mul-
tiplyed into the Difference between half the Sum of the three sides,
and the Side oppoſite to the Angle contained, ſo is the Square
of
the Radius, to the Square of the Co-line of half the contained An-
gle: That is,
In the Triangle ADE it will be,
AE+AD+DE AE+AD+DE
As AEXED:
X
DE: :Rq:0579
1 the Angle DAE
AS
2
2
*
5
Trigonometry.
99
:
.D
G
G
JE
H
A
H
A
LB B F
L
BF
M
M
Conſtruction.
I. Let the Angle D AE be biſected by the Line AC (by the 11th
of the 2d.) and ler DCB be drawn Perpendicularly to AC, (by the
14th of the 2d.) then will DC=CB, and AD=AB, (by the 8th of
the ad) and conſequently BE is equal to the Difference of the Sides
AE, and AD.
2. From C draw CM parallel and equal to DE, (by the sth of
the 5th.) then will CF=iDE (by the 17th Prop.) the Side oppoſite
= Fm,and BF=FE, equal to half the Difference of the Sides AE,
AD, and conſequently Af=half their Sum.
3. Joyn the Points M and E, and produce ME till it meet AC
produced in G, then will MEG be parallel to DCB (by the 6th of
the 5th.) and conſequently the Angle at G right, (by the sth Prop)
4. On Fas a Center, with the Diſtance of CE=FM=DE the
oppoſite Side, draw the Circle LCHM, which will paſs thro' the
Point G, by the 3d of the 11th.) then will AH be equal to half the
Sum of the Sides AE, AD, and the oppoſite Side DE ; and AL e-
qual to their Difference. Wherefore, by the ift Theorem or iſ Axiom
of Irigonometry it will be,
As ÁR: AC :: R:cs. <BAC=;DAE (by Conſtr.)
And AE: AG:: R:cs. <BAC=;DAÉ (by the fame.)
Wherefore, (by the 8th of the 15th) it will be,
ABXAE : ACX AG :: Rq:cs,q · <DAE.
But, ACxAG=AHXAL (by the iſt of the 13th.) and AB=AD
by Conſtruction. Wherefore,
As ADXAE: AHXAL :: Rq:05,9 <DAE. That is,
As
O 2
ܪ .
.
*
ng
hii
X
2
2
-DE it will be,
2
2
X
DE :: RO
2
2
DE and X=
2
2
.
x
100
Trigonometry.
AD + AE+ DE AD FAE-DE
AS ADXAE:
:: Rq:cs,q.-DAE
AD + AE--DE AD+ AE + DE
And becauſe
AD" AEDE AD + AE-+DE
As ADX AE :
Rq
cs:. <DAE. W. W. D.
AD 4 AE | DE
AD+AE+DE
Put Z
--DE.
Then it will be, ADXAE: Rq:: 2xX:cs,q. * <A Wherefore,
R R
x ZxX=(!,9 <A. That is,
AEDA
Becauſe the Difference between the Logarithmic Radius, and Logn-
rithm of any other Number, is called the Complement Atithmetical.
1. To the Arithmetical Complement of one Side AE, add the
Arithmetical Complement of the other Side AD.
Alſo, the Logarithm of half the Sum of the three Sides, and the
Logarithm of half the Sum lefſen'd by the Side oppoſite, half the Sum
of theſe 4 Logarithms will be the Co-line of half the Angle required.
2. Becauſe ADXAE : ZxX :: Rq. : cs,g. <A.
ADXAE
cs,q. Ź <A
Therefore,
X::R:
Z
R
As half the Sum of the Sides, is to one of the containing Sides, fo
is the other containing Side to a fourth Number.
As that fourth Number, is to the Difference of the half Sum of the
three sides and Side oppoſite, ſo is the Radius, to a ſeventh Sine ;
which vadded to the Radius, gives the double Co-line of half the
Angle contained
ADXAE
cs,9. A.
3. Again,
:Z::R;
X
R
As the Difference of the half Sum of the three Sides and Side
poſits, is to one of the containing Sides, fo is the other containing
Side, to a fourth Number.
As this fourth Number, is to halfche Sum of the three Sides, ſo is.
the Radius, to a ſeventh Sine, &c. as in the former.
Again,
that is,
op-
***
Trigonometry.
101
cs,q. } <A
4. Again, AD:
::R:
that is,
cs,q. KA
-
AEXAD *Z*X
ZxX
R:
that is,
AE
R
As the greater of the two containing Sides, is to the half Sum of
the three Sides, ſo is the half Sum leflen'd by the Side oppoſite, to
a fourth Number.
As the leſſer of the two Sides, is to the fourth Number, ſo is the
Radius, to a ſeventh Sine, &c. as in the 2d Cor.
ZxX
cs,9 < A.
5. Again, AE:
DA
R
As the leflir of the two Sides, is to the half Sum of the three Sides,
ſo is the half Sum leflen'd by the Side oppoſite, to a fourth Number
As the greater of the two Sides, is to the fourth Number, ſo is the
Radius, to a ſeventh Sine, &c. as in the ad Cor.
R
6. Again, becauſe
it will be,
R
As the Radius, is to one of the containing Sides, fo is the other
containing Side to a fourth Number.
As that fourth Number, isto the half Sum, ſo is the half Sun
leſſen'd by the Side oppofite to a ſeventh Sine, &c. as in the 2d Cor.
The Rules given in theſe five lait Corroliarys are all applicable to the
Gunters Scale.
Various other Axioms might be produced for the Solution of the
Caſes in Trigonometry, but theſe that are already Exhibited are
more than ſufficient for the preſent Deſign.
Before I conclude this Sedion, I ſhall ſhew the Reaſon of one
Rule made Uſe of in the 6th Caſe of Right-angled Plain Triangles,
for finding the Hypothenuſe from the two given Legs by the Lo-
garithms, viz.
From twice the Logarithm of the greater Side, fubtract the Lo-
garithm of the lefler Side Seek for the abſolute Number anſwering
to that Sum, and to it add the leſſer Side. Seek for the Logarithm
of this laſt Sum, and to it add the Logarithm of the lefler Side, halt
the Sum of theſe two Logarithms will be the Logarithm of the Hy-
pothenuſe required; and the fame will hold good if the lefler be ta-
ken inſtead of the greater, 66.
For,
102
Trigonometry.
For, put H for the Hypothenuſe,b for one of the Sides, and p for the
other, then HH=pp+bb
xb= - ppt bxb.
Whence, H=2+bxb. W. W. D.
PP - 66
b
::
Section IV.
Containing the application of the three former Sections, to
the Solution of the ſeven Cafes of Right-angled Plain Triangles.
T
HE whole Science of Trigonometry is contained in this one
fingle Problem, viz. from three Conſtituent Parts of a Triangle
to find the reſt, and the various Queſtions ariſing from changing the
things given and required, are called the ſeveral Caſes, and in Plain
Trigonometry are uſually reduced to 13, viz. to 7 in Right, and o in
Oblique-angled Plain Triangles.
And to render thele Sections as eaſy as poſſible, it will be neceſla-
ry to call to mind ſome few things already prov’d.
1. That any two sides of a Plain Triangle taken together, are
greater than the third that remains. (See Prop. 4. Cor. 2.)
2. That the greateſt Side of every Triangle is oppoſite to the great-
eſt Angle ; and on the contrary, that the greateſt Angle is oppoſite
to the greateſt Side. (See Prop. 4. and if Cor.)
3. "That the Suin of the Angles of every Plain Triangle is equal to
a Semicircle, or 180 Degrees. Wherefore it follows,
4. If two Angles of a Plain Triangle are known, the third is alſo
known, being found by ſubtra&ing their Sum from 180 Degrees.
5. If one Angle of a Triangle be Obtuſe, that is greater than a
Quadrant or 90 Degrees, each of the other two ſhall be Acute; that
is each fall be leſs than 90 Degrees.
6. If one Angle of a Triangle be Right, or 90 Degrees, ithe o-
ther two together ſhall be equal to one Right Angle, orgo Degrees;
(and theſe fort of Triangles are called Right-angled, Triangles.)
Wherefore,
In
9. IN
Trigonometry.
103
K
E
7. In a Right-angled Plain Triangle, if one of the Acute Angles
be given, the other is alſo known ; being found by taking the given
Angle from 90 Degrees or a Quadrant, and this defećt is called
its Complement.
For the Reaſon of the s laſt Cor. (ſee Prop. the 6th and its Cor.)
8. That the Tables of Natural Sines, Tangents, and Secants, are
Tables of Triangles readily Calculated to all the variety of Angles :
So that ſuppoſing AD of the Triangle A DE
to be 10000, c. equal Parts, the Natural
or Tabular Tangent ſtanding againſt or anſwer-
ing to the Degree and Minute, &c. of the Au-
gle A, is the Length of DE in the ſame Parts.
A
Alſo, the Correſpondent Tabular Secant is the
Length of AE.
Again, if AK of the Triangle AKL be=100003&c. equal Parts,
the Tabular Sine anſwering to the Quantity of the Angle A, will be
the Length of KL in the Same Parts; and the Natural Sine Corre-
fponding to the Meaſure of the Angle C, the Complement of the
Angle A, will be the Length of AL in the ſame Parts.
Hence it is abundantly manifeſt, that no Triangle can be propoſed,
but that a Triangle fimilar to it may be found in the Tables of
Natural Sines, Tangents, and Secants ; all whoſe Sides being
known or given, the Correſpondent Sides of the Triangle firſt pro-
poſed may be readily found, by the help of the 17th and 18th Prop.
of the iſi Part, as will more evidently appear from the Solutions
themſelves of the ſeveral Caſes, which are as follow.
I D
Caſe I.
*
The Angles and one of the Legs of a Right-angled Plain Triangle
being given, to find the other Leg; or
The Baſe and Angle at the Bare being given, to find the Perpen-
dicular.
Example.
In the Triangle ABC, Right-angled at B,
AB
64
Are given, {CA=*3!!* }required BC:
Geome
not
Trigonometry.
Geometrically.
B
.
.
'F
:E 1. Make AB equal to 64
K.
by a Line of equal Parts.
2. Ered the Perpendicu-
lar BC.
3. Make the Angle at A
A,
.:D equal to 36° 52', by the
L
H.::
Line of Chords, and pro-
;I
duce the Line AC, till it
meet the Perpendicular BC
P...
in C, then is the Triangle
Conſtru&ed, and the Length
of BC may be found, by applying it to the Line of equal Parts
But if a more exact Anſwer be required in Numbers, produce AB
to D, till AD be equal to the Tabular Radius 10000000, c. then
will DE the Tangent of the Angle A be 7499119, &c. of the ſame
Parts.
And becauſe the Triangles ABC, ADE, are ſimilar, it will be,
(by the 17th Prop.)
As AD the Radius, to DE the Tabular Tangent of the Angle
A, ſo are the Parts in AB, to the Parts in BC. That is,
As roc00000 :: 7499119::64:47.99=BC.
64
29996476
44994714
47.9943616=BC.
2. Produce CB to F, till CF be equal to the Tabular Radius
10000000, &c. then will FP be the Tangent of the Angle C=53°
08', 13334900 of the ſame Parts, and becaufe the Triangles
CBA, CFP, are ſimilar, it will be, (by the 17th Prop.)
As PF, the Tabular Tangent of the Angle C, to FC the Ra-
dius, ſo are the Parts in AB, to the Parts in BC. That is,
As
7
Trigonometry.
105
As 13334900: 10000000 :: 64: 47.99=BC.
64
133349.00)6400000.00(47.99, &c=BC.
1066040
1325970
1258290
58149
3. Produce AC both ways to K and H, till AK and CH be each
equal to the Radius; then will KL the Sine of the Angle A be 5999549,
and HI equal to AL, the Sine of the Angle C equal to 8000338, and
it will be (by the 17th Prop.)
As (HI =) AL the Tabular Sine of the Angle C, is to LK the
Tabular Sine of the Angle A, ſo are the Parts in AB, to the Parts
in BC, That is,
As 800038:5999549 ::64: 47.99, &c. =BC.
64
23998196
35997294
8000338)383971136147.99, &c. =BC.
63957616
79552500
75494580
3491538
P
But
I.
106
Trigonometry.
&c.
But ſince the Invention of Logarithms by the Lord Naper, (of
which I intend to Treat largely in a Section by its ſelf) the Artificial
Sines, Tangents, and Secants, (which are no other than the Logarithms
of the Natural-Sines, &c.) have been Introduced in the room of the
Natural Sines, &c. to avoid the trouble of Multiplycation and Di-
viſion ; ſince the Addition of Logarithms anſwers to the Multiply-
cation of the Natural Numbers they repreſent, and their Subtraction
to the Diviſion, as will evidently appear from the Solution of the
former Caſe by the Logarithmic Sines, &c.
N. B. The Tables of Natural Sines, Tangents, and Secants, of which
the Artificial are the Logarithms, are Calculated to a Radius of
1.00090.00000, whence the Logarithmic Radins, or Sine of 90 De-
grees, according to Brigs's Cannon, which is now in Uje is, 10.0000.
0000.,
The Solution of the fame Cafe by the Logarithms.
The Logarithm.of. AD the Radius is 10.0000, 6c. and the Lo-
garithm of DE the Natural Tangent is 9.8750102 ; wherefore, be-
cauſe it is, as AD the Radius is to DE the Tangent of the Angle
A, fo is AB to BC; it will be,
As the Logarithm of the Radius 1000, 6C. 10.0000000
To the Log. of 7499119, &c. the Nat. Sine
of the Angle A =36° 52'.
-9.8750102
So is the Logarithm of AB=64
1.806 1800
To the Log.of BC =47.99 in the ſame Parts=
1.68 11902
In this Caſe the Multiplycation of 7449, &c. by 64, is perform-
ed by adding their reſpective Logarithms together.
2. The Logarithm of 1334900 the Natural Tangent of 53° 08'=
Angle C, is 10.1249898, the Logarithm of the Radius as before, is
30.0000, c. wherefore, becauſe it is as PF the Tangent of the
Angle C, is to FC the Radius, ſo iš AB to BC; it will be,
As the Log. of 13334900, c. the Nat.
} 10.1249898
Tangent of Angle C=53° 8'
To the Log. of 1000, &c. the Radius
So is the Log. of AB - 64
1.806 1800
10.0000000
To the Log. of BC=47.99
1.681 1902
In
*
Trigonometry.
107
In this caſe the Diviſion of 64000, &c. by 13334900, is per:
formed by ſubtracting their reſpective Logarithms from each other.
3. The Logarithm of HI 8000338, the Natura Sine of 53° 08'=
Angle C, is 9.9031084, the Log. ot 599,95 49, the Natural Sine of 36°
52'=Angle A, is 9.7781186, and the Log. of the Number 04,
1.8061800. And becauſe it is as HI the Sine of Angle C, is to LK
the Sine of Angle A, fo is AB, to BC; it will be,
As the Log. of 8000338,.Nat. Sine of <C=539.08' 9.9031084
To the Log. of 5999549, Nat. Sine of <A=36°52' 9.7781186
So is the Log. of AB=64
1.806 1800
To the Log. of BC=4799, &c.
1.68 11902
In this caſe the Multiplycation of 5999, c. by 64, is perform-
ed by adding their reſpective Logarithms together, and the Diviſion
of the Product 3839, c. by 8003, &c. by ſubtraging the Log. of
80003, &c. from the Sum of the former.
Whence we have this General Rule for working by the Logarithms,
viz. to the Logarithm of the ſecond Term, add the Logarithm of
the third Terin, from that Sum ſubtract the Logarithm of the firſt
Term, and the Remainder will be the Logarithm of tlie fourth Term
required; whether it be a Side or an Angle.
ز
To Condu&t the Reader after the moſt eaſy Manner, into the Uſe and
Application of the Logarithmic Sines, &c. to the Solutions of the Caſes
of Plain Triangles; ar well as to let him ſee the reaſon of every Step, and
at the ſame time to convince him of the neceſſity of being Maſter of the fe-
cond Section, was the Reaſon that induced me to reſolve the firſt Caſe by
the Natural Sines, &c. which I ſhall omit to do in tbe following Cafes for-
Brevity Sake ; and I would adviſe every Mafter that intends his Pupil
ſhould underſtand what he is doing, to follow this Method.
*
Hence it is manifeſt, that the Angles, and one Side of a Right-angled
Plain Triangle being given, each of the other Sides may be found three diffe-
rent ways, viz. by comparing each Side with the Radius of the Cannon,
as will more plainly appear from the Solution of the two following Caſes.
P 2
Caft
108
Trigonometry.
Caſe II.
The Angles and one side of a Right-angled Plain Triangle being
given, to find the Hypothenuſe. Or,
The Baſe and Angle at the Baſe being given, to find the Hypo-
thenuſe
Example.
In the Triangle ABC Right-angled at B,
<A=30°.52' }req. AC:
Are given AB =64--
E
This Caſe is Conſtructed like
the former, and AC may be
meaſured by the Line of equal
Parts.
A
B....D
But to find the Length Lo-
garithmically, or by Calcula-
H
I
tion.
P.
..F
1. Produce AB to D, till
AD be equal to the Tabular
Radius, and draw the Tangent ĐE, and Secant AE, and becauſe
the Triangles ABC, and ADE, are ſimilar, it will be (by the 17tb
Prop.)
As AD:AE: : AB: AC ; that is,
As the Radius
10.0000000
}
To the Tangent of the Angle A=36° 52'
So is the given Leg AB=64-
10.0968916
aim 1.8061800
To the Hypothenuſe AC=80-
-1.9030716
2. Produce CA to H, till CH be equal to the Radius, then will
HI be the Sine of the Angle C, equal to the Complement of the Angle
A, and becauſe the Triangles CAB, CHI are ſimilar, it will be,
As
2
Trigonometry.
109
As IH: HC :: BA: AC. That is,
As the Sine of the Angle C=53° 08'
9,903108
To the Radius
10.0000000
So is the Bale, or given Leg AB=64
1.806 1800
To the Hypothenule AC=-80
1.9030716
3. Produce CB to F, till CH be equal to the Radius, then will FP
be the Tangent, and PC the Secant of the Angle C, and becauſe
the Triangles CAB, CPF, are ſimilar, it will be,
As FP : PC :: AB: AC. That is.
As the Tangent of the Angle C=53° 08'-
10.1249898
To the Secant of the Angle C=53° 08'
10.2 2 18814
So is the given Leg AB=64
I 806 1800
To the Hypothenuſe AC=80 -
1.9030716
Caſe III.
>
The Hypothenuſe and Angles of a Right-angled Plain Triangle
being given, to find either of the Legs. Or,
The Hypothenuſe and Angle at the Baſe being given, to find the
Baſe.
Example
In the Triangle ABC Right-angled at B,
Are given,
{
AC 80
Geometrically.
1. Draw the Line AB at plea-
fure.
2. Make the Angle ar A e-
qual to 36° 52' by the Line of
Chords, and continue the Line
AC, till it be equal to 80, up-
А
B.
on the Line of equal Parts.
ө
H
3. Let fall the Perpendicular
CB, and the Triangle is Con-
ſtructed; and the Baſe AB may
P.......si
F
be meaſured by the Line of e-
qual Parts,
Lug
3
I
..
110
Trigonometry
Logarithmically, or by Calculation.
1. Make CH equal to the Radius, then will HI be the Sine of the
Angle at C, and becauſe the 'Triangles CAB, CHI, are ſimilar, it
will by the 17th Prop.
As CH: HI :: CA: AB. That is,
As the Radius
vern comments on you 10.00000DO
To the Sine of the Angle C=53° 08
So is the hypothenuſe AC-80
9.903 1084
1.9030900
To thic Baſe or Leg AB=64
1.800 1984
2. Make AD equal to the Radits, then will DE be the Tangent,
and AE the Secant of the Angle A, and it will be (by the 17th Prop.)
As EA:DA::CA: AB. That is,
As the Secant of the Angle 4 - 36952
'
10.0968916
To the Radius
So is the Hypothenuſe AC=80
10.000.000
1.9030900
To the Baſe or Leg AB=64
1.8061984
3. Make CF equal to the Radius, then will FP be the Tangent,
and PC the Secant of the Angle C, and it will be (by the 17th Prop.)
As CP: PF::CA: AB. That is,
As the Secant of the Angle C =53° 08' -- 10.2218814
To the Tangent of the Angle C=53°08'
So is the Hypothenuſe AC=80
10.1249898
1.903 0900
To the Baſe or Leg =
1.8061984
:
Caſe IV.
The Legs of a Right-angled Plain Triangle being given, to find
the Angles: Or,
The Baſe and Perpendicular being given, to find the Angles.
Example.
Trigonometry.
111
Are given < AB=64
E
B
:D
ret
0
F
C=53° 08' (by the 3d of the 6th.)
from the given Tangent, which call the Exceſs.
Example.
In the Triangle ABC, Right-angled at B,
>req. the Angles A and C.
Geometrically.
1. Ler AB be made equal to
64, from the Line of equal Parts.
2. From the Point B, erect
the Perpendicular BC equal to
48 equal Parts from the ſame
Line, and joyn the Points Cand
A
A, by the Right Line AC, and
the thing is done : And the
Quantities of the Angles A and
Pi..
C may be meaſured by the Line
of Chords.
Logarithmically, or by Calculation.
1. Make AD equal to the Radius, then will be be the Tangent
of the Angle A, and it will be by the 17th Prop.
AS AB : BC :: AD: DE. That is,
As the Bale or given Leg AB=64
1.806 1800
To the Perpendicular, or given Leg BC=48 1.6812412
So is the Radius
10.0000000
To the Tangent of the Angle A =36° 52' 9.8750612
Which taken from 90° co' or a Quadrant, will leave the Angle
But if greater exa&tneſs be required, that is, if the anſwer be de-
ſired to Seconds, proceed thus S;
1. Take the Tangent of 36° 52' the next leſs Tabular Tangent,
from the Tangent of 36°53' the next greater Tabular Tangent,
which call the Tabular Difference.
2. Take the Tangent of 36° 52', the next leſs Tabalar Tangent,
3. Then
1 1 2
AR
Trigonometry.
Then ſay, as the Tabular Difference, is to the Exceſs, ſo is 60
Seconds, to the Proportional Augment.
Example.
The Tabular Tangent of 36° 52' is
9.8750102
The Tabular Tangent of
36° 53' is
9.8752734
The Tabular Difference is
2632
The given Tangent is
The next leſs Tabular Tangent of 36° 52' is
9.8750612
9.8750102
The Exceſs is
510
Nov, as 2623 : 510::60" : 11"; which added to 36° 52',
the Number anſivering to the next leſs Tabular Tangent, gives 36°
52' 11" the true Quantity of the Angle required,
After the ſame manner, may the length of the Arch correſponding
to any Sire, Tangent, &c. be found to Seconds.
By the Reverſe of this Method, may the Sine, &c. of any Arch
given in Seconds, be found.
2. Make CF equal to the Radius, then will FP be the Tangent
of the Angle C, and it will be (by the 17th Prop.)
As CB : BA:: CF: FP. That is,
As the Perpendicular or given Leg BC =48-
1.6812412
To the Baſe, or given Leg AB=64
So is the Radius
1.806 1800
10.0000000
To the Tangent of the Angle C, 53º.08
10.1 249388
Whoſe Complement to a Quadrant, will give the Angle A =36*
52', as before.
Hence it is obſervable, that when the two Sides are given, the Angles
ma be found two different ways, viz. by comparing each of the given Sides
with the Rndins, as will yet farther appear, by the Solution of the next
Cafe.
Trigonometry:
113
Caſe V.
The Hypothenuſe and one of the Legs of a Right-angled Plain
Triangle being given, to find the Angles. Or,
The Baſe and Hypothenuſe being given, to find the Angles.
Example
In the Triangle ABC, Right-angled at B,
80.
Are given; {AB=84:}req
req. s A and C.
. <
E
Geometrically.
1. Let AB be made equal to 64, by a Line of cqual Parts.
2. Ere& the Perpendicular
BC.
3. With the Diſtance of 80
equal Parts, between the Points
of the Compalles, ſetting one
foot in A, with the other croſs
the Perpendicular in C, and
A
B
joyn the Points A and C,
then is the Triangle projected,
and the Qantity of the Angles
may be found bythe Line of P
Chords,
Logarithmically, or by Calculation.
1. Make AD equal to the Radius, then will AE be the Secant of
the Angle at A, and it will be (by the 17th Prop.)
As BA : AC :: DA: AE; That is,
As the Bale or given Leg AB=64
-1.8061800
:
H
1
☆
To the Hypothenuſe AC=80
So is the Radius
1.9030900
10.0000000
To the Secant of the Angle A =36° 52'.
10.0969100
Q
Whorc
114
Trigonometry.
Whoſe Complement to a Quadrant is the Angle at C, =53° 08'
(by the 3d of the 6th.)
2. Make CH equal to the Radius, then will HI be the Sine of
the Angle at C, and it will be (by the 17th Prop.)
As CA: AB ::CH: HI. That is,
As the Hypothenuſe AC =80
1.9030900
To the Baſe or given Leg AB=64
So is the Radius
1.806 1800
10.0000000
To Sine of the Angle at C=53° 08'
9.9030900
Which taken from 90° co' leaves the Angle at A, equal to 36"
sa' as before found.
Caſe VI.
The Legs of a Right-angled Plain Triangle being given, to find
the
Hypothenuſe. Or,
The Baſe and Perpendicular being given, to find the Hypothenuſe.
Example
In the Triangle ABC, Right-angled at B,
Are given BC ef}req. AC.
Geometrically.
E
K
20
The Conſtruction of this Cafe
is in all reſpects the fame with
the fourth, and the Length of
AC may be found by a Line of
equal Parts.
A-4
B
D
H
I
I
P
F
Loga:
Trigonometry.
115
This Caſe is a Compound of the ſecond and fourth Caſes, and the
Hypothenuſe or Side 4C may be thus found;
Logarithmically, or by Calculation.
Let AD and CF be made each equal to the Tabular Radius, and
let the ſeveral Correſpondent Sines, Tangents, and Secants be drawn
as in Caſe the firſt, and it will be,
1. As AB :BC :: AD: DE. as in Caſe the fourth, That is,
As the Baſe or given Leg AB=64
1.806 1800
1.68 12412
To the Perpendicular or given Leg BC=48
So is the Radius
10.000000
To the Tangent of the Angle A =30°52'.
-9.8750612
Which taken from 90° oo' or a Quadrant, leaves 53° 08', for
the Angle at C, (by the 3d of the 6th.)
2. The Angles A and C being known, their Correſpondent Sines,
Tangents, Gr. are given, and therefore to find the Hypothenuſe
AC, it will be
1. As AD: AE :: AB: AC. That is,
As the Radius
10.0000000
To the Secant of the Angle A=36° 52' 10.0968916
So is the given Leg AB=64
1.8061800
To the Hypothenuſe AC=80
1.9030716
2. As IH: HC: : AB : AC. That is,
As the Sine of the Angle C=53° 08'.
9.9031084
To the Radius
So is the Baſe or given Leg AB=64-
-10.0000000
1.806 1800
To the Hypothenuſe AC=80
1.9030716
Qa
3. As
23
116
Trigonometry.
3As FP: PC :: AB: AC. That is,
As the Tangent of the Angle C=53° 08'
To the Secant of the Angle C=53° 08'
So is the given Leg AB=64
10.1249898
10.2218854
1.806 1800
To the Hypothenuſe AC=80
1.9030716
4. As FC : PC :: BC: CA. That is,
As the Radius
10.0000000
To the Secant of the Angle C=53° 08' .
So is thc Perpendicular or given Leg BC=48
10.2218814
1.6812412
To the Hypothenuſe AC=80.01
1.9031226
5. LK: KA::BC: CA. That is,
As the Sine of the Angle at A 36° 52'
9.7781186
To the Radius
So is tlie Perpendicular or given Leg BC=48
10.0000000
1.681.2412
To the Hypothenuſe AC=80.01 mammas
1.903 1226
6. As DE: EA:: BC:CA. That is,
As the Tangent of the Angle A=36° 52
9.8750102
To the Secant of the Angle 4= 30° 52'
So is the perpendicular or given Leg BC = 48
10.0968916
1.6812412
To the Hypotienuſe AC=80.01-
1.9031226
4096
7. By the help of the (3d of the 8th.) the Hypothenuſe AC may
be found directly, without finding the Angles; thus,
To the Square of one of the given Sides AB=645
Add the Square of the other given Side BC=48=
2304
The Square Root of that Sum
6400
Will be 80 equal to the Side AC.
Qr,
Trigonometry.
117
Or,
8. From twice the Log. of the greater Side AB=6453.6123600
Subtract the Log. of the lefler Side BC=48=-
1.6812412
The abſolute Number anſwering to the Rem. is 85.3332=1.9311188
To this Number 85.3332, add the leffer Side 48, and the Sum
Will be 133.3332.
To the Log. of this laſt Sum 133.3332=
2.1 249379
Add the Log. of the lefler Side BC=48=
1.6812412
Half the Sum of theſe two Logarithms
3.8061791
will be the Log. of the Side required AC=80 1.9030895
For the Demonſtration of this laſt Method, I refer the Reader to
the End of the 3d Se&tion.
Hence it is manifest, that the two Legs of a Right-angled Plain Tri-
angle being given, the Hypothenuſe AC may be found 8 different ways.
Caſe VII.
The Hypothenuſe and one of the Legs of a Right-angled Plain
Triangle being given, to find the other Leg. Or,
The Baſe and Hypothenuſe being given, to find the Perpendi-
cular.
Example.
In the Triangle ABC Right-angled at B,
Are given, {AG 80} reqi BC:
AB=64
The Geometrical Con-
ſtruction of this caſe, is the
ſame with the fifth, and the
Length of BC may be mea-
ſured by a Line of equal
Parts
A
D
B
C
:I
H
This
18
Trigonometry.
This Caſe is likewiſe a Compound of the 5th of the 1ſt and of the
3d, for the Angles A and C being found by the 5th, the Perpendi-
cular BC may be found 6 different ways as in the laſt, viz. three
ways by the help of the Angles, and the Baſe or Leg AB, as in the
firſt Care; wid three other ways by the help of the Angles, and the
Hypothenuſe as in the third Caſe ; all which I ſhall omit för Brevity
fake, lince the inteligent Rcader may eaſily ſupply them himſelf.
The Perpendicular BC may likewiſe be found (by the third Cor.
of the Sih) Thus,
From the Square of the Hypothenuſe AC80=
6400
Take the Square of the Baſe AB 6+=
4096
The Square Root of the Remainder
2304
Will be 48 the Perpendicular
Or,
To the Logarithm of the Sum of the Hypothenuſe }2.1583625
and Bale or given Leg AB=144
Add the Log. of their Difference=16
1.2041200
Half the Sum of theſe 2 Log. will be the Log. 3:3624825
of the Perpendicular 48
1.6812412
N. B. Thoſe Proportions in which the Radius iš not concern'd, are moſt
eaſily worked by taking the Arithmetical Complement of the firſt or upper-
mort Number, which is what it wants of 10.0000000, for then the
Sum of the three Terms will be the Logarithm of the 4th required
To perform any of the Proportions in Trigonometry by Gunter's
Scale, take this one General Rule.
Exterd the Compaſſes from the firſt Term, or Number found
the Scale, to the ſecond, and the ſame Extent will reach from
the third to the fourth required.
For the Reaſon of theſe two laſt Rules, ſec the Logarithmic
Se&ion.
upon
Section
Trigonometry.
119
wao.Wiwa
bedded
reducedi web.edcedido
Section V.
Containing the Solution of the Six Cules of Oblique-angled
Plain Triangles.
Caſe I.
TWO
D
:
WO Sides and an Angle oppoſite to one them being giveny,
to find the other oppoſite Angle.
Example.
In the Oblique-angled Plain Triangle ADE,
AD=104–
Are given DE=70
req. <$ D and A.
2<E=4412
Geometrically.
1. Having drawn the Line AE, make
the Angle AED equal to 44° 12', by
the Line of Chords.
2. Make ED equal to 70, by the
Line of Equal Parts.
3. With the Diſtance of 104, between
the Points of the Compaſſes, ſetting one
foot in D, with the other croſs the Line AE in A, and joyn the Points
D'and A, and the Triangle is Conſtructed; and the Angles D and
A may be meaſured by the Line of Chords.
To find the Angies by Calculation, it will be by the 2d Tlie rem or
Axiom of Trigonometry, AD:DE::S. <E:S. A. That is,
As the Side oppoſite to the given Angle A=104 - 2.0170333
To the side oppoſite to the Angle ſought=701845.0980
So is the Sine of the given Angle E=44' 12' 9.8433356
A40
JE
.
To the Sine of the required Angle A=27° 59.-9.6714003
The
..
1 20
Trigonometry.
The Angles E and A being known, the Angle at D may be found
by the 6th Prop. thus ;
From the Sum of the Angles A, D and E,
180 00
Take the Sum of the Angles A and E,
72 II
Remains the Angle at D=
107 49
In this caſe, if inſtead of the Angle E the Angle A bad been given, that is,
if the Side oppoſile 10 tbe given Angle had been the leler of the two given
Sides, then the caſe had been doubtful; and the Angle at D either
Acute or Obtuſe, as shall be ſhewn at large in the third Caſe.
Caſe II.
The Angles and one Side of an Oblique-angled Plain Triangle
being given, to find either of the other Sides.
Example.
In the Oblique-angled Plain Triangle ADE,
KA= 27° 59
Are given E= 44° 12'
1078
AD E104
Geometrically.
1. Make AD equal to 104, by a Line
of Equal Parts.
=
req. AE.
44
2. Make the Angle A equal to 44°
12', and the Angle D equal to 107° 49'
by the Line of Chords, and continue the
Lines AE, DE, till they interfect each other in the Point E, and
the Triangle is formed ; and AE may be meaſured by the Line of
Equal Parts
To determine its Length by Calculation it will be, by the ſecond
Theorem or Axium of Trigonometry, As S. <E:S. <D :: AD: AE:
That is,
As
->
Trigonometry.
121
As the Sine of < E 44° 12' oppoſite to the given Side 9.8433356
To the Sine of <D 107° 49'oppo. to the req. Side 9.9786554
So is the given Side AD=104
2.0170333
To the required Side AE=142.02
2.1523531
After the ſame manner may the Side DE be found.
Caſe III.
Two Sides and an Angle oppoſite to one of them being giveri, to
find the other Side.
In the Oblique-angled Plain Triangle ADE,
req. AE.
D
AE may
Are given, DE=70
<E=44° 12'
The Geometrical Conſtruction of this Caſe
is exactly the ſame with Caſe the firſt, and
be meaſured by a Line of Equal
Parts.
ALO
To find the Angle D it will be as in the
firſt Caſe, viz.
As AD: DE:: S. Angle E: S. Angle A. That is,
As the given Side DA=104
2.0170333
To the given Side DE=70
1.8450980
So is the Sine of the given Angle E = 44° 12' 9.8433356
To the Sine of the Angle at A=27° 59'
9.6714003
Which being obtained the Angle ar D will be found to be 107
49', by the 6th Prop.
Wherefore to find AE it will be,
As S. Angle E : S. Angle D :: AD: AE. That is,
R
AS
1 2 2
Trigonometry.
As the Sine of the <E=44° 12
9.8433356
To the Sine of <D=107° 49' its Supplem. 71° 11' 9.9786554
So is the given Side AD = 104
2.0170333
To the required Side AE=142,02
2.1523531
if inſtead of the Angle E, the Angle A had been given, the Sides AD,
DE, remaining the ſame, the Caſe had been doubtful; and the third
Side AE would have had two different Values, as the following
Example will ſufficiently Jhew.
Caſe III.
Example. 2.
In the Oblique-angled Plain Triangle ADE,
AD=104
}
req. AE.
OnE
I:
Are given, DE=70.
<A=27° 59
The Ambiguity of this caſe appears from
the Conſtruction it felf, for having tormed
the Angle A, and ſet off AD froin A to D.
If with the Diſtance of DE, one foot of the
Compaſtes being placed in D, the Arch EE
be drawn, it will cut AE in the Points E
and E, on the fame Side of the Point A; and conſequently the Angle
Das well as A will have two different Values, and to find each of
which it will be by the ad Theorem or 2d Axiom of Tiigonometry.
AS DE : AD: :S. KA:S. <E. That is,
As the given Side DE=70
1.8450980
To the given Side AD=104
2.0170333
So is the Sine of the Angle A=27° 59'
9.6713716
To the Sine of the Angle E=44° 12'
9.8433069
The meaſure of the Angle E in the great Triangle AED, but ta-
ken from 180° oo' (becauſe the Triangle EDE is Iſofceles) leaves
135° 48' for the meaſure of the Angle E in the lefler Triangle ADE;
whence
Trigonometry.
123
whence and from the 6th Prop, the Angle D of the great Triangle
ADE will be 107° 49', but the Angle D of the lefler Triangle ADE
will be only 16° 13'; whence firſt, to find the Side AE of the great
Triangle ADE it will be,
As S. <A:S. <D::DE: AE. That is,
As the Sine of the Angle A = 27°59'.
9.6713716
To the Sine of <D=107° 49' its ſupplement=71°119.9786554
So is the given Side ED=70-
1.8450980
To the required Side AE=142.03
2.1523818
2d. To find the Side AE of the leſſer Triangle ADE, it will be,
As S. <A: S. <D::DE: AE. That is,
As the Sine of the Angle A=27° 59'
9.6713716
To the Sine of the Angle D=16° 13'
9.4460250
So is the given Side ED=70–
1.8450980
To the Side AE required =41.66-
1.6197514
!
Hence,
0
If the <Ebe{Dostupe } the<D is{ 107:49 }and AE is 142,3}
If inſtead of DE, in the first Example of the third Caſe, the Side AE
had been given, the Caſe would have been likewiſe doubtful; and DE
would have had two different Values, as will appear from the following
Example.
Caſe III.
Example. 3.
In the Oblique-angled Plain Triangle ADE,
SAD=104
Are given, 3 AE=142.02
<E=44°12'
}
req.DE
R2
Cont
Po
124
Trigonometry.
.
A
E
Conſtruction.
1. Having drawn AE equal to 142.02 E-
qual Parts, and formed the Angle E equal
to 44 deg. 12 min. by the Line of Chords,
take the Diſtance of 104 the Length of AD,
and placing one Foot of the Compaffes in A,
with the other deſcribe the ſmall Arch DD,
which will cut the Side ED, in the Points D and D, on the ſame
Side of the Point E, whence the different Values of the third Side
DE may be found, by the Line of Equal Parts ;' but to determine the
ſame by Calculation it will be,
1. As AD: AE::S. <E:S. <D. That is,
As the given Side AD=104
20170333
To the given Side AE=142.02
2.1523495
So is the Sine of the Angle E=44° 12'
9.8433356
To the Sine of the Angle D=72° 11'-
9.9786518
'The meaſure of the Angle at D in the great Triangle ADE, but
taken from 180 deg. oo min. (becauſe the Triangle ADD is Iſoſce-
les) leaves 107 deg. 49 min. for the Angle D in the lefler Triangle
ADE, whence the Angle A in the great Triangle will be 64 deg.
37 min. and the Angle A in the lefler 27 deg. 59 min. wherefore to
find the Side ED, in the great Triangle ADE, it will be by the 2d
Theorem or 28 Axiom of Trigonometry.
As S. <E: AD:: S. <A: DE. That is,
As the Sine of the <E=44° 12'
.
9.8433356
To the Sine of the Angle A=64° 37'
So is the side oppoſite to E, viz. AD=104
9,9559089
- 2.0170333
To the Side DE req. = 134.77
2.1296066
2. To find the Side DE in the lefler Triangle ADE it will be,
As S. <E:S. <A::AD: DE. That is,
AS
!
Trigonometry.
125
As the Sine of the Angle E=44° 12'
To the Sine of the Angle A = 27° 59'
So is the Side AD=104
-9.8433356
9.6713716
2.0170333
To the Side req. DE=70
1.8450693
1
ŞObtufe }the Ai
}the <A is{64.37
Hence,
264.375
527.59 Sand DE is
If the <Dbe{ Obtuſe?
2 Acute
and {
S 70.-?
134.77
S
In like manner, If the Sides AE, DE and the Angle A, were given, then
would AD be found to be of two different l'alues; as is manifeſt from
the former Conſtruction, by changing the Letters A and E.
Caſe IV.
Two Sides and the contained Angle of an Oblique-angled Plain
Triangle being given, to find the other oppoſite Angles.
Example
In the Oblique-angled Plain Triangle ADE,
SAE=142.022
Are given, ZAD=104, Sreq. Angles E and D.
KA=27° 59'
Geometrically.
D
ܓܢܐ
1. Draw the Line AE, and ſet off upo:
it 142.02 equal Parts.
2. Make the Angle at A equal to 27 deg.
59 min. by the Line of Chords.
3. Let AD be made equal to 104 Equal Parts,
4. Joyn the Points D and E, by the Right Line DE, and the Tri-
angle is made ; and the Angles ac D and E, may be meaſured by the
Line of Chords as uſually,
The
126
**
Trigonometry.
AE=142.02
AD=104
Sum = 246.02
Differ.= 38.02
The Numerical Solution of this Caſe,
depends upon the 3d Theo. or 3d Ax. of
Trigonomeiry, for having found the Sum
and Difference of the iwo Sides, ſub-
ſtract the given Angle from 180 deg.
and take half the Remainder, which
will be the half Sum of the unknown
Angles (by the 6th Prop.) and it will
be by zd Axiom of Trigonometry,
SA+D+E=180 00
KA= 27. 52
Sum <D and E=152 OI
half Sum <s= 76 oo
As the Sum of the two Sides AE and AD=246.02-2.3909704
To their Difference = 38.02
1.5800121
So is the Tangent of half the Sum of the Angles at } 10.6034981
D and E=76° 00' 30''
To the Tangent of į their Difference or the Diff.
of either of them from the Sum=31° 48 28"
39.7925398
Now,
o
To the half Sum = 76 00
Add the half Diff.=31 48
30 From the half Sum=76 00 30
28 Take the half Diff.=31 48 28
Sum greater<D=107 48 52
Rem. leller < A=44 12
02
Caſe V.
Given any two Sides and the Angle contained, to find the third
Side.
Example.
In the Oblique-angled Plain Triangle ADE,
Are given S4D3104
Are given; <D=107° 49' req. AE.
DE=70 masin
Breg
Make
Trigonometry.
127
D
fas
Make AD equal 104, the Angle D equal
to 107 deg. 49 min, the Side DE equal to 70,
and draw AE and tlie thing is done ; and AE
E
may be meaſured as uſual.
This Caſe depends upon the 4th and 3d
for its Solution, and for varietys fake, I ſhall apply the 4th Axion for
finding the Angles,
As the leſſer of the two Sides DE=70
1.8450980
neo
To the greater AD=104
So is the Radius
-2.0170333
10.0000000
To the Tangent of an Arch = 56°03' 23"
10.1719353
From which taking away 45 deg. oo min. there will remain 11°
93' 23'', and it will be,
As the Radius
10.0000000
To the Tangent of 11° 03' 23"
So is the Tangent of the half Sum=36°05' 30".
9.2909285
9.86272 14
To the Tangent of half the Difference=8° 06' 25" 9.1536499
Whence the greater Angle E will be 44° 11' 55", and the leſſer An-
gle A equal to 27° 59' 05"; whence to find the Side EA, it will be
by the ad Axiom of Trigonometry,
Ass. <A:S: <D:: DE: AE. That is,
As the Sine of the Angle A=27° 50' 05"'.
9.6713914
To the Sine of the <D.= 107° 49'its Sup.72° 11'=9.9786554
So is the Side DE=70
1.8450980
To the Side AE required=142.02
2.1523620
Care
128
Trigonometry.
Cafe VI.
Given the three Sides, to find the Angles.
2
SCE
s req.
SA
AL
E
р
Example.
Ir the Oblique-angled Plain Triangle ADE,
AE=142.02
<
D7
Are givenS AD=104–
DE= 70-
Geometrically.
1. Make AE equal to 142, from a Line
of Equal Parts
2. With 104 the Length of AD, between
the Points of the Compafles, ſetting one
Foot in A, with the other ſweep a ſmall
Arch at D.
3. With 70 the Length of DE, ſetting one Foot in E, with the
other croſs the former Arch in D.
4. From the Interſection D, to the Points A and E, draw the
Lines AD DE, and the Triangle is Conſtructed ; and the Angles
may be meaſured by the Line of Chords as uſual.
But to deterinine their Quantities by Calculation.
From D let fall the Perpendicular DP, then will the Triangle
ADE be reduced into two Right-angled Triangles ADP, EDP, in
each of which the Hypothenules AD, DE, are given, and the Baſes
AP, PE, or Segments of the Baſe AE, may be found by the 5th
Axiom of Trigonometry: For,
As the Baſe AE=142.02
2.1523495
To the Sum of the Sides AD, DE, =174
2.2405492
So is the Difference of the Sides AD, DE, =34-1.5314789
TO AB the Diff. of the Segments AP, PE, =41.56 1.6196786
Which taken from AE=14202, leaves BE=100.364, the half
of which is 50.182=PE the leſler Segment.
This
NAM
Trigonometry.
129
This again taken from AE the whole Baſe=142.02, leaves 91.
838=AP, the greater Seginent.
The Segments AP, PE, or Baſes of the Triangles ADP, EDP,
being thus found, the Angles may be eaſily had, by the 5th Caſe of
Right-angled Plain Iriangles
. For,
In the Triangle ADB it will be,
AS AP: AD::R:Sec. A. That is,
As the Segment AP=91.838
1.9630224
To the Side AD=104
So is the Radius
2.0170333
10.0000000
To the Secant of the <A=27° 59' 11"
100540109
Again, in the Triangle EDP it will be,
As EP:ED::R: Sec. <E. That is,
As the Segiment PE=50.182
1.7005480
To the Side DE=70
So is the Radius
1.8450980
10.0000000
To the Secant of the Angle E=44° 52' 07''
10.1445500
Now if to the Complement of the Angle Ą= 62° OO' 49"=ADP
Be added the Complement of the Angle E=
45 47 53-EDP
The Sum will be
107 48 42=ADE.
B
P
D
2. The Angles A, D, E, of the Tri-
angle ADE may be found, if the Perpen-
dicular be let fall from any other Angle,
ſuppoſe A of the Triangle ADE, upon the
oppofite Side ED produced if need be,
for then the Hypothenuſes AE, AD, of
the two Right-angled Triangles AEP
ADP, are given, and the Baſes EP, DP, may be found by the 5th
Axiom of Trigonometry. Saying,
S
As
A
130
Trigonometry.
As the Dale ED=70
1,8450980
To the Sum of the two Sides AE, AD, =240.02 2.3909704
So is the Difference of the Sides AE, AD,=38.02-1.580012 i
TO EB, the alternate Baſe=133.624
2.1258845
From which taking away ED=70, the Remainder 63.624, will
be DB, the half of which is DP=31.812, the Baſe of the Triangle
ADP ; whence to find the Angle D it will be, by the 5th Caſe of
Riglit-angled Plain Triang'es.
As PD: R:: AD: Sec. <D. That is,
As the External Baſe DP=31.812
1.5025910
To the Radius
So is the Side AD-=104
10.0000000
2.0170333
To the Secant of the Angle D=72° 11' 19' 10.5 144423
Which taken from 180° 00', leaves the great Angle D=107 deg.
48 min. 41 ſec.
Again, If to the External Segment DP=31.812, be added the
Side DE-70, the Sum 101.812 will be PE, the Baſe of the great
Triangle APE; whence to find the Angle at E, it will be by the
5th Care, &c.
As EP : EA:: R. : Sec. E. That is,
As the great Baſe EP=101.812 -
2.0077990
To the Side AE=142.02
So is the Radius-
2.1523495
10.0000000
To the Secant of the Angle E=44° 12'07" TI0.1445505
Now, if from the Complement of the <E=45° 47' 53"=<EA!
Da taken the Complement of the <D=17 48 41 DAP
The Remainder will be mm
27 59 12DAE
3. After the ſame manner, and by the help of the fifth Axiom and fifth
Caſe the Angles may be found by letting the Perpendicular fall from the
Angle E, upon AD produced, if need be, but this the inteligent Render
will tafily Supply himſelf
.
4. By
Trigonometry.
131
4. By the help of the firſt Cor. of the fixth Trigonometrical Axiom,
may any one Angle as D, of the Triangle ADE, be more readily
Inveitigated, after the following manner.
From half the Sum of the three Sides 158.01, ſubſtract AE the Side
oppolite to the Angle required, and keep the Remainder 15.99.
Then,
To the Arithmetical Complement of 4D=104, one 7.982908
of the Sides containing the Angle required
Add the Arithmetical Complement of DE=70, the o-
$ 8.1549020
ther containing Side
Alſo the Logarithm of the half Sum 158.01
2.1986840
And the Logarithm of the Remainder 15.99
1.2038485
Half the Sum of theſe four Logarithms
19.5404018
Will be the Co-line of 53° 54' 20"
--9.7702009
}7.9
Which being doubled gives 107° 48' 41", thc Angle D required.
The Angle D being obtained, the Angles A and E, may be
found by the firſt Caſe and ſecond Cor.
5. By the help of the 2d Cor. of the ſame Axiom, may the Angle
D be Inveſtigated after the following manner.
As the Log. of half the Sum of the three Sides 158.01 2.1986846
To the Log. of one of the containing Sides AD 104 2.0170333
So is the Log. of the other containing Side DE=70--1.8450980
To a fourth Logarithm
1.6634467
As the fourth Logarithm
1.6634467
To the Logarithm of the Remainder 15.99 1.2038485
So is the Radius
10.0000000
To a ſeventh Logarithm
To which add the Radius
9.5404018
10.000000O
Half that Sum, viz.
Will be the Co-fine of 53° 54' 20" =<D
therefore, <D=107° 48'41"
S2
19.5404018
-9.7702009
6. By
132
Trigonometry.
6. By the help of the 4th Cor. of the fame Axiom, may any one
Angle fuppoſe A, of the Triangle ADE, be readily found after the
following manner.
As the Log. of the greater of the 2 cont. Sides AE 142.02 2.1523495
To the Log. of the kalf Sum of the three Side 158.01 2.1986846
So is the Log. of half the Sum, leſſen’d by BC=88.01—-1.94453 20
To a fourth Logarithm-
1.9908671
Again,
As the leffer of the two cont. Sides AD =104 2.0170333
To the fourth Logarithm
1.9908671
So is the Radius
10.0000000
To a ſeventh Log, to which add the Radius
19.9738338
Half that Sum will be the Co-ſine of 13° 59' 35"-99869169
which being doubled gives 27° 50' 11", the <at A
7. By the help of the 6th Cor. of the ſame 6th Axiom, may any
one Angle ſuppoſe E, of the Triangle ADE be Inveſtigated, after
the following manner.
As the Radius
10.0000000
To the Log. of one of the cont. Sides AE=142.02 2.1523495
So is the Log. of the other contained Side DE=70--1.8450980
To a fourth Logarithm
Neg. 6.0025525
As that fourth Logarithm
Neg. 6.0025525
To the Logarithm of the half Sum 158.01
2.1986846
So is the Logarithm of the Remainder
1.7324742
To a 72h Logarithm, to which add the Radius 19.9337113
Half this Sum will be the Co-line of 22° 06'03"},
9.9668556
the double of which 44° 12' 07'', will be the <at E reg.
Tliere
54.01
3
Trigonometry.
133
There are various other ways for the Reſolving this as well as the
former Caſes, but theſe that I have thewn, are more than ſufficient
for our purpoſe.
Each of theſe three laſt ways may be made Uſe of, for finding the
Angle by Gunter's Scale, for if you extend the Compailes from the
firſt to the ſecond, c. againſt the ſeventh will ſtand a Sine in the
Verſed. Sines, which gives the Angle required.
Suppoſe it were required to find the Angle D, by Extention ac-
cording to the fifth Rule proceed thus; Extend the Compaſles from
the half Sum (upon the Line of Numbers) to one of the containing
Sides (ſuppoſe AD) the ſame Extent will reach from the other cone
taining Side (DE) to a fourth Number.
Extend the Compaſſes from this fourth Number, to the Remain-
der upon the ſame Line; the ſame Extent will reach from the Verſed
Sine of oo deg. to the Angle required, upon the ſame Line of Verſed:
Sines, and ſo for any other.
N. B. Before the Render enters upon the Navigation, I would adviſe
him to read over, and be well acquainted with the Circles of the
Sphere ; the Definitions of which he will meet with in the fifth part:
of the Book..
PART
134
Navigation.
అంతంతమంతతమయendardian.తు...
తంతత. ఆ.తంతతకు
PART. III.
Of NAVIGATIO N.
M
SECT. I.
T
HE great End and Buſineſs of Navigation, is to inſtru& the
Induſtrious Mariner how he may Conduct his Ship the
ſhorteſt Way, and in the ſhorteſt Time poſſible, between any
two Placęs aſſigned in Paſſages Navigable.
If the Places propoſed be at no great Diſtance, båt ſo that a Ship
may
fail within fight of Land or Sounding's, as between England,
Holland, France, &c. it is more properly called Coaſting ; for the
Performance of which, there is required only the Compaſs, Lead, o.
Sounding-Line, and a competent Knowledge of the Nature of the
Coaits.
But how to Conduct a Ship through the wide and pathleſs Ocean,
where nothing is to be ſeen for a conſiderable time but Sky and Wa-
ter, is the proper Buſineſs of Navigation, and ſhall be the Subject of
the enſuing Treatiſe.
For the due and regular Performance of which, are requiſite,
1. A true and perfect Knowledge of the Diſtance and Scituation
of the Places, as to the Latitude, Longitude, Bearing, &c:.
2. Inſtruments for Directing and Meaſuring the Ship's Way and
Courſe ſteered, ſuch as the Log-Line, Compaſs, Lead-Line, Half-
Minute Glaſs, &c.
3. Inſtruments for making Cæleſtial Obſervations, ſuch as the
Quadrant, Fore-ſtaff, Azimuth-Compaſs, &c. to meaſure the Heights,
Distances, and Azimuths of the Sun and Stars, or to find the Hour
of
*
}
Navigation..
135
of the Day and Night, alteration of Latitude and correcting the Courſe,
together with Charts or Maps, for pricking down the Ship's Way.
4. A competent Knowledge of thic Nature of the Currents, of the
Mould and Trim of the Ship, and the Sail ſhe bears, that ſo due al-
lowance may be made for Lee-way.
5. Skill in the Navigator, ſo that by the help of theſe he may be able
to know at all times the place the ship is in, how far flie hath to
fail, and which way, or upon what Courſe, to gain her intended
Port.
For the obtaining of which there are generally given,
1. The Latitude and Longitude, both of the Port you are to fail
from, and of the Port you are to fail to; by the help of which may
be found,
2. Upon what Courſe, and how far the Ship muſt fail.
3. The Account of the Ship’s Way obtained, by the Log and
Compaſs.
4. The Latitude of the Place the Ship is in by Obſervation.
How to order, and manage theſe, ſo as to produce the neceſary Quæſita,
Jhall be hewn at large hereafter ; for the better underſtanding of which
it is very convenient to Explain fome Terms already made Uſe of.
Def.
1. The Latitude of any Place, is the ncareſt Diſtance of that Place,
from the Equator, and is meaſured by an Arch of the Meridian, in-
tercepted between the place and the Equator, and therefore can ne-
ver exceed 90 deg. or a Quadrant, taking its Denomination accorda
ing as the place is ſcituated, either to the Northward or Southward of
the Equator.
Hence all Places that lie at the ſame Diſtance from, and on the
ſame Side of the Equator, are ſaid to lie under the fame Parallel of
Latitude.
Coroll.
Whence it follows,
1. That if a Ship fail from a South
Ş North} Latitude dire&tly
North
}
2 South
which is failing under the fame Meridian, the increaſes her Latitude
juſt ſo much as is her Diſtance fail'd.
Wherefore,
If the Diſtance fail'd be reduced into Degrees, by allowing 60
Miles to a Degree, (the reaſon of which ſhall be ſhewn hereafter)
and added to the Latitude the Ship fail'd from, the Sum will be the
Latitude the Ship is in.
Example
]
9
1.1.4
136
O
Navigation.
Example. 1.
Suppoſe a Ship at Sea in the Latitude of 46 deg. 30 min. North,
fils directly North 192 Miles, and it be required to find the Latitude
the Ship is in.
To the Latitude fail'd from
46 30N.
Add the Diſtance fail'd recuced
3 12 Nly.
The Sum is the Latitude the Ship is in=
49 42 N.
Example. 2.
Suppoſe a Ship in the Latitude of 35 deg. 40 min. South, fails di-
realy South 200 Miles, and it be required to find the Latitude the
Ship is in.
To the Latitude ſail'd from
Add the Diſtance fail'd reduced
-3 20 Sly.
The Sum is the Latitude the Ship is in
35 40 s.
39 Oo S.
Cor.
2. If a Ship fail from a { North} Latitude , dire&ly {Sorteo}
ſhe decreaſes her Latitude, juſt ſo much as is her Diſtance failid:
Wherefore,
If it be firſt reduced into Degrees, and ſubtracted from the Latitude
the Ship fail'd from, the Remainder will be the Latitude the Ship is in.
Example 1.
Suppoſe-a Ship at Sea, in the Intitude of 49 deg. 42 min. Northin
fails directly South 192 Miles, or 3 deg. 12 min. and it be required
to find the Latitude the Ship is in.
From the Latitude fail'd from
Take the Diſtance fail'd reduced-
12 Sly.
There remains the Latitude the Ship is in
46 30 N.
Example 2:
If a Ship from the Latitude of 39 deg. oo min. South, fails direaly
North 200 Miles, or 3 deg. 20 min. and it be required to find the
Latitude the Ship is in.
From
O
1
49 42 N.
3
Navigation.
137
oo S.
Froin the Latitude fail'd from
Take the Diſtance fail'd --
39
3
20 Nly.
35 40 S.
Remains the Latitude the Ship is in
If in either of theſe two laſt Examples, the Diſtance fail'd be more
than the Latitude the Ship fail'd from, the Ship has crofled the Equa-
tor; and as the Sailors uſually term it, deprefled one Pole and raiſed the
other, and in this Caſe from the Diſtance fail'd, take the Latitude
the Ship fail'd from, and the Remainder will be the Latitude th;
Ship is in, the contrary way to what it was before.
Example
Suppoſe a Ship in the Latitude of i deg. 30 min. North, fails di
re&ly South 192 Miles, or 3 deg. 12 min. and it be required to find
the Latitude the Ship is in.
From the Diſtance fail'd
Take the Latitude the Ship fail'd from
3
I
12 Sly.
30 N.
Rcmains the Latitude the Ship is in
I
42 S.
Def.
2. Difference of Latitude, is the neareſt Diſtance of two places lying
under the ſame Mertaran, or the Diſtance between the Parallels pal-
fing thro' any two places lying under different Meridians, and is
found,
If the places lye on the ſame ſide of the Equator, that is, both in
North or both in South Lalitud., by ſubftra&ing the leffer from the
greater.
Example.
Let it be required to find the Difference of Latitude between the
Lizard, in the Latitude of so deg. oo min. North, and Barbadoes, in
the Latitudeof 13 deg. 30 min. North.
1
From the Latitude of the Lizard
Take the Latitude of Barbados
so
oo N.
-13 30 N.
Remains the Difference of Latitude
36 30
+
T
138
Navigation.
If the places propoſed lie on contrarý Sides of the Equator, that
is, if one be in Northern Latitude and the other in Southern Latitude,
then the sum of the Latitudes will be the Difference of Latitude.
Example
Let it be required to find the Difference of Latitude between
Barbadoes, in the Latitude of 13 deg. 30 min. North, and the Cape
of Good Hope in the Latitude of 34 deg, 15 min. South.
Å
To the Latitude of Barbadoes
Add the Latitude of the Cape of Good Hope
13 30 N.
-34: Is $.
܂
Their ſum makes the Difference of Latitude
47 45
If the Latitude of one place be given, and the Difference of La-
titude between it and a ſecond place, to find the Latitude of the
ſecond Place.
If the Latitude and Difference of Latitude are both the ſame way,
that is, both Northerly or both Southerly.
To the Latitude given, add the Difference of Latitude made, and
the Sum will be the Latitude required.
Example.
Suppoſe a Ship from Cape Verd, in the Latitude of 14 deg.
43 min. North, fails Northerly, until her Difference of Latitude
be 4 deg. 40 min.
To the Latitude of Cape Verd
Add the Difference of Latitude made
14 43 N.
4 40 Nly.
Their ſum is the Latitude the Ship is in
19 23 N.
If the Latitude and Difference of Latitude are contrary, that is,
if one be Northerly and the other Southerly. Then,
From the Latitude fail'd from, take the Difference of Latitude
made, and the Remainder will be the Latitude the Ship is in.
Example
1
Suppoſe a Ship from Cape Verd, in the Latitude of 14 deg.
43-
min. North, ſails Southerly until her Difference of Latitude be
4 deg. 40 min. Then,
int
From
.
Navigation .
1
139
-14 43 N.
From the Latitude of Cape Verd
Take the Difference of Latitude made.
4 40 Sly.
10
Remains the Latitude the Ship is in
03 N.
If in this laſt Caſe, the Difference of Latitude exceed the Lari-
tude fail'd from, then from the Difference of Latitude made, take
the Latitude the Ship fail'd from, and the Remainder is the Latitude
the Ship is in; the contrary way to what it was before.
Example.
Suppoſe a Ship from the Latitude of i deg. 20 min. North, ſails
Southerly, until her Difference of Latitude be 190 Miles, or 3 deg.
10 min. and it be required to find the Latitude the Ship is in.
3
10 Sly.
From the Difference of Latitude reduced
Take the Latitude the Ship fail'd from
1
Star
20 N.
Remains the Latitude the Ship is in
I
50 S
:
Def.
3. The Longitude of any Place upon the Earth, is an Arch of the
Equator, intercepted between the firſt Meridian, and the Meridian
pafling thro' the Place propoſed, or it is equal to the Angle formed
by the two Meridians.
So that if a Ship fails direaly on the Equator, ſhe alters her Lon-
gitude juſt as much as is her. Diſtance faild.
Def.
4. Difference of Longitude, is an Arch of the Equator, intercepted
between the Meridians paſſing thro' the places propoſed, and is the
ſame with the Angle at the Pole, formed by the Meridians themſelves.
Cor.
Wherefore,
1. If to or from the Longitude of the Place the Ship fail'd from,
be added or ſubſtrated (according as the Caſe requires) the Diſtance
fail'd (reduced as for the Latitudes) the Sum or Remainder, will
be the Longitude of the Place the Ship is in.
2. But if the Ship fail under any Parallel of Latitude, ſhie alters
not her Longitude according to her Diſtance fail'd, but more ; and
that in the Proportion of the whole Equator, to the Circumference
of the Parallel under which the fails. Which is,
T2
As
)
140
Navigation.
As the Co-ſine of the Latitude, to the Radius. (As ſhall be
made appear hereafter) and the Diſtance fail'd in this caſe, is uſually
called the Meridional-Diſtance, or Departure. So that,
Def.
4. Meridional-Distance, or Departure, is the Diſtance of any two
Places lying under the ſame Parallel, or the Diſtance of the Meri-
dians paſſing thro' two Places lying under different Parallels, num-
bred in Degrees or Miles of the Equator.
Def.
s. For determining the Courſe of the Winds, and to diſcover their
various Alterations or Shiftings, the Horizon is uſually divided into
32 equal Parts, by Lines drawn from the Place whereon the Obfer-
ver ſtandeth, (forming Angles with each other of Ti of 360 deg. or
11 deg. Is min.) the four Principal of which, take their Names from
the Places to which they tend, viz. that which extends it ſelf under
North
the Meridian towards the North
3 is called
}
2 South
)
is towards the Right-hand, when the Face is dire&ed North, and at
Right-angles to the Meridian, Eaſt; and that which extends it felf
the contrary way Weſt. The others having their Names compound-
ed of the principal Lines which are on each ſide of it, as may be ſeen
in the following Figure ; and over which ſoever of theſe Lines the
Courle of the Wind is directed, that Wind takes it Name accord-
ingly. And ſince it we move never lo little to the Eastward or Weft-
ward, we come under a new Meridian; and foraſmuch as all the
Meridians do of necelity meet in the Poles, (according to their
Nature) it follows, that the Eaſt aud Weſt Line in the laſt Place,
will not be Parallel to the Eaſt and Weſt Line in the firſt Place, but
a little inclined to it, and conſequently the other intermediate Rumb-
Lines in each Station will be inclined to each other, (ſince they al-
ways form equal Angles with the Meridian.)
Now, If the Alterations or Diſtances of the Meridians be infinite-
ly Imail, the infinite ſmall Portions of theſe Rumb-Lines, (proper
to each Alteration of Latitude) when Connected, together will form
Curve Lines upon the Globe, which Lines will form equal Angles
with all the Meridians, and are the Rumb Lines deſcribed upon the
Surface of the Globe ; and the ſeveral Paths of a Ship that lails ac-
cording to the Dire&ion of the Compaſs.
Def.
6. The Compaſs, is a Circle made to reprefent the Horizon,
and
Navigation.
141
and divided into 32 equal Parts by Lines drawn from the Center,
to the ſeveral equal Diviſions of the Circumference, repreſenting the
ſeveral Rumb Lines upon the Globe, and by touching the Needle faſten'd
on the back ſide of the North and South Line, with the Magnet or
Loadſtone, a certain Vertue is infuſed into it, which makes it al-
ways tend to, and hang nearly Parallel with the Meridian, and there-
fore of great Uſe for direæing and determining the ship's Way, and
diſcovering the Courſe and Alterationof the Winds.
30
30
North
211
N.N.W
N. by W.
N.W by N.
N.by E.
N.N.E.
50
N. E. by N
N.w.by W.
N. W.
ht
W. N.W.
S N.E.
N.E.by E
W. by N.
E.N.E.
E. by N.
Welt
.
-K
- Eaſt
E. by S
W.by s.
S.
E.S.E.
W. S.W.
S. E.by E.
09
S.W by W.
S. W.
W by s.
S. E.
S.S.W
OL
S. by W.
S. E.by
S. by E.
Yino
S. S.E.
OK
18. The Courſe is the Angle which the Rumb Line upon which the
Ship has or muſt fail, makes with the Meridian; ſo that it the Ship
fail upon the ſecond Rumb, or N. N. E. &c. the Courſe is an Angle
of 22 deg. 30 min. and ſo for any other, as the following Table
will lew.
A
142
Navigation:
A Table of the Angles which every Rhomb (or Point
of the Compaſs) maketh with the Meridian.
D. 11
NORTHS OU I H
Points.
D. M.NORTHS OU I H.
1
I
I
I
2/16
3
I
2
2
4/25
2
3.
2
1
1
3
II
102 49
05 37
08
26
Northby Eaſt South by Eaſtı II
15 N. by W. S. by W.
414 04
52
419 41
N. N. E. S. S: E.
22 30 N. N. W.
S.S. W.
19
28 07
430 567
N. E. by N. S. E, by S. 3
33 45 N. W. by N. S. W. by S.
3
4/36 34
3
2139
3
4/42
North Eaſt. South Eaſt 14
4) - North Weſt South Weſt.
4' 47 49
4 3150 37
4 4153 26
N. E. by E. S. E. by E. 5 15 N. W. by W. S. W. by W.
5 159 04
61 52
5 64 42
E. N. E. E. S. E. 6
67 30 W. N. W.
W.S. W.
19
07
75 56
Eaſtby North
Eaſt by South'
78 45 W. by N.
W. by S.
7
7
7 87
Eaſt
Eaſt
Weſt
Weſt
Hence
1
56
1
4
1
2
arnab
ex frict
264
1
6
6
770
273
1
6
3
4
81 34
41
184
22
II
Ś 1996
oo
Navigation.
43
Hence it is plain, that if a Ship ſail between the Meridian and the
B
K
I
in
a.
r
f
Eaſt or Ut ejt Line, flie alters both her Latitude and Longitude, and
each of them more or leſs, according as the Angle the Rumb-Line
(tipon which the Ship fails) makes with the Meridian, is greater or
letler : And in order to determine the juſt Quantity of each, it is ne-
ceflary to enquire a little into the Nature of the Rumb-Line ic ſelf,
and the Triangles deſcribed upon the Surface of the Globe, by the
Rumb-Lines, Meridians, and Parallels of Latitude.
It being then the
EſſentialProperty of
the Rumb Line, to
P
form equal Angles
with every Meridi-
an thro' which it
paſſes, it muſt of
neceſſity be a Curve
or Heliſpherical Line
(ſince none but the
Equinoctial as
Great Circle, and A
the Parallels asſmay
Circles, can cut the
Q
Meridians at equal
Angles) always ten-
ding towards the
Pole, and approaching it by infinite Gyrations or Turnings, ſtill co-
ming nearer and nearer, bớt never falling into it; for if it were poſſi-
ble to ſail by any Rumb whatſoever (except under the Meridian)
directly under the Pole Point, it would follow,' that one and the
ſame Line, could cut infinite other Lines conſidered as ſuch, at e-
qual Angles in a meer Point ; which is contrary to the very Notion
we have of a Line and an Angle.
Hence it is, that no part of a Rumb-Line how ſmall ſoever, can
be a ſtreight Line tho' for all Nautical Uſes, a ſmall Portion is, and
may without ſenſible Error be conſidered as ſuch.
This being premiſed; Let P repreſent the Pole, AQ an Arch of
the Equator, PA a Quadrant of the Meridian, and the Arches ap,
cq, er, &c. Parallels of Latitude, at ſmall Diſtances from each
other, fuppoſe of i min. and let the Rumb-Line Abdf, &c. be
drawn
d
14 9
Z
22
.
m
n2
144
Navigation.
drawn. Thro' the Interſedions b, d, f, &c. of the Rumb-Line, with
the Parallels before deſcribed, draw the ſeveral Meridians Pl, Pm,
Pn, &c. whence will be formed leveral ſmall Triangles Aab, bod, def,
&c. which tho' ſtriąly ſpeaking are Curvilinear, yet becauſe of their
ſmallneſs, we here, and may without ſenſible Erro, conſider them as
Re&tilinear Triangles ; in each of which da, bc, de, &c. are the
Differences of Latitude, ab, cd, ef, &c. the Departures, Ab, bd,
df, &c. the Diſt.nces upon the Rumb, the Angles a Ab, cbd, edf,
&c. the Courſes, and the Angles at a, , e, &c. Right, and conſe-
quently in the whole Rumb Triangle ABC, AB is the whole Diffe-
rence of Latitude, AC the whole Diſtance, but the whole Depar-
ture from the Meridian, is Compounded of the ſeveral intermediate
Departures ab, cd, ef, &c.
1. Now becauſe the Triangles Aab, bod, def, &c. have each
one ſide An, bc, de, &c. equal, the Angles at a, c, e, &c. Right,
and the Angles a Ab, ob d, edf, &c equal among themſelves, it
follows (by the 2d I'r p.) that the other parts of the Triangle ab,
od, ef, &c alfo, Ab, bd, df, &c. ſhall be equal among themſelves.
Whence it follows,
1. That the Segments or Portions of any Rumb-Line, con-
tained between two Parallels having the ſame Difference of Latitude,
are equal to each other; for the Segments Ab, bd, af, &c. are e-
qual among themſelves.
Wherefore, If a Ship fail from the Latitude of 10 deg. North,
rll ihe be got into the Parallel of 20 deg. North, and then ſail froin
chence upon the ſame Rumb, till ſhe be in the Latitude of 30 deg.
N the latter Diſtance ſhall be equal to the former, and the like will
hold good from 30° to 40°, and thence to 50°, &c. tothe very Pole ; if
it were poſſible to ſail thither according to the Direction of any Rumb.
2. That to a like Diſtance upon the fame Rumb, there will be al-
ways the ſame Difference of Latitude and Departure; for the Sides
Ai, bi, de, &c. are equal among themſelves, as are the Sides a b,
c.d, ef, &c. ſo that if a Ship in ſailing 100 Leagues or 300 Miles,
upon the third Rumb from the Equator either Northward or South-
ward, make an alteration of 4° og' in her Latitude, and 166.68
Miles in her Departure, ſhe will make the ſame alteration of Lati-
tude and Departure, in failing Northerly from any other Point or
Place in the Northern Hemiſphere, or in failing Southerly, from any
Point or Place in the Southern Hemiſphere; and the contrary.
3. That
Navigation.
145
3. That the Differences of Longitude, or Arches of the Equator
Ai, im, mn, &c. correſponding to the equal Scparations, or De-
partures from the Meridian, ab, cd, ef, &c. in different Parallels
are unequal, increaſing as the Ship fails farther from the Equator, in
the Proportion of the Radius, to the Co fine of the Parallels Di-
ftance from the Equator; as ſhall be made more fully appear here-
after, when I come to Treat of the Mercator's Sailing.
4.
That if a Ship in failing from A, upon a given Rumb and a given
Diitance, arrive at b, in returning back by the oppoſite Rumb, ſhe
will not arrive at A, bur fall wide of it, nearer to l; for becauſe Al
is greater than ab, the Angle Ablis greater than a Ab, and con-
fequently the Ship in returning, muſt lail upon a different Rumb,
making a greater Angle with the Meridian, to arrive at A.
And this is one Notorious Error that attends the Plain Sailing, and is the
Reaſon why thoſe who keep their Reckonings by the Plain Chart are
forced to make more Eaſting or U eſting as they call it, to arrive at
the ſame place when they ſail towards the Equator, than when they ſail
from it, tho' they know not from whence the Cauſe proceeds.
s. Hence it is manifeſt
, that the Difference of Latitude, Diſtance
faiid and Departure, form a Right-angled Plain Triangle, the
Courſe being the Angle oppoſite to the Departure, and therefore
from any two of theſe with the Right Angle, the reſt may be eaſily
found, by the help of the Do&rine of Plain Triangles ; as will more
evidently appear in the next section.
Section II.
Containing the Application of the Doctrine of Right-angled
Plain Triangles, to the Solittion of the ſeveral Cafes in
Plain Sailing, relating to a ſingle Courſe.
Caſe 1.
O
NE Latitude, Courſe, and Diſtance fail'd being giren, to find
the other Latitude and Departure from the Meridian.
U
Example.
*
}
140
Navigation.
Example.
A Ship at Sea in the Latitude of 46 deg. 30 min. North, fails 96
Miles upon the third Rumb, or N. E by N. I demand the Latitude
the Ship is in, and how much he has departed from her former
Meridian.
1. Geometrically.
B
с
Com:Cours
Diſtance saild
A
Departure
2. Let AB repreſent the Meridian, and
for Methods ſake the fore part towards B, the
North part, the part neareſt to you towards
A, the South, then will that part towards the
Right-hand be the Eaſt, and that towards the
Left hand the Weſt; which Order will be Ob-
Served thro' the whole Courſe of this Work :
And let A be the place the Ship departed from.
2. Draw the Line AC, forming an An-
gle of 33 deg. 45 min. or three Points,
with the Meridian AB, upon which ſet oft
96 Miles from A to C, then will AC repre-
ſent the Rumb-Line upon which the Ship has faiid, and the Place
ſhe is arrived at.
3. From C let fall the perpendicular CB, then will AB be the
Difference or Alteration of Latitude, and ſhews how much the Ship
has got to the Northward of the firſt Place ; alſo CB the Departure
or Meridional Diſtance, and ſhews how much ſhe has got to the
Eaſtward of her former Meridian, the juſt Quantity of each of
which, may be found by the Rules given in Plain Triangles.
And here Note, that altho' each may be found three different ways, by the
Trigonometrical Rules, yet in the Application we never make Uſe but
of one way, and that uſually where the Radius becomes the fill Term
in the Proportion, as being the moſt apt for Practice.
And firft to find the Departure or Meridional Diſtance, it will be,
by Caſe the 3d of Se&t. the 4th of Trigonometry.
As R : AC:: S. BAC: BC. That is,
As
Navigation.
147
•10.0000000
As the Radius
-1.9822712
2.74-17390
To the Diſtance fail'd 96
So is the Sine of the Courſe 33 deg. 45 min.
To the Departure 53.34 Miles
1.7270102
And ſo much is the Ship to the Eaſtward of her former Meridiaa:
If the Courſe had been Weſterly, then the former Number would have
jheron how much the Ship was got to the Weſtward, and the faine Meri-
dional Diſtance willcbtain in failing froin any Point upon Globe, 96 Mile;
upon the third Rumb.
2. To find the Alteration or Difference of Latitude, it will be by
Cafe thic 3d of Sect. the 4th of Trigonometry.
As R: AC::cs, BAC: AE. That is,
As the Radius
10.0000000
To the Diſtance fail'd 96
So is the Co-fine of the Courſe 33 deg. 45 min.
1.9822712
9.9198464
To the Difference of Latitude 79.82 Miles
1.9021176
And ſo much in this caſe is the Ship got to the Northward of her
laſt Place, wherefore becauſe he is got farther from the Equator.
To the Latitude fail'd from
46 2ON.
Add the Difference of Latitude made
1978 Nly
The Sum is the Latitude the Ship is in
47 49 50 N.
I
S?
If thcCourſe had been S.erly, the Ship would then have been gotten
79.82 Miles to the Southward of her former Place, and therefore in
this Caſe to find the Latitude the Ship is in,
From the Latitude fail from
46 30 N.
Take the Difference of Latitude made
O
1
I
39.82Sly.
There Remains the Latitude the Ship is in
44 50.18 N.
U 2
Again
148
Navigation.
Again becauſe the ſame Alteration of Latitude, will always happen
in ſailing 96 Miles upon the third Rumb, if the Latitude propoſed had
been oo deg. 30 min. South, and the Courſe had been Northerly,
as in the firſt Example, to have found the Latitude the Ship is in,
becauſe the Difference of Latitude is greater than the Latitude fail'd
from.
From the Difference of Latitude made
39.82 Nly.
Take the Latirude ſail'd from
O,
I
30 S.
Remains the Latitude the Ship is in
1
09.82 N.
Contrary to what it was before, and ſo for any other.
:
By Reaſon ſometimes of the unſteadineſs of the Winds, ſwelling of the
Sear, unknown Currents, ſetting with or againſt the Courſe of the Ships
ad ſeveral other Accidents, the ejtimate Diſtance by the Log is viciated;
and comfiquently the Difference
. of Latitude and Departare thus found is
Niiht, a:id not to be relied or, the Navigator muſt be ſure therefore, so
foon as the Sun or Scars appear, to find the Latitude by Obſervation, ac-
cording to the Dire&tions in the latter part of this Treatiſe, which it found
to diſagree with the Latitude deduced by this caſe, the Difference between
the Latitude you departed from, and the laſt Obſerved Latitude muſt be
taken, and with this true Difference of Latitude and the Courſe, find the
Diſtance and Departure, by the following Cafe.
1
Caſe II.
Both Latitudes and Courſe being given, to find the Diſtance fail'd
and Departure from the Meridian.
Example.
A Ship at Sea in the Latitude of 46 deg. 30 min. North, afrer ha-
ring failed for ſome time upon the third Rumb, or N.E. by N. is found
by Obſervation to be in the Latitude of 47 deg. so min. North, I
demand the true Diſtance fail'd and Departure from the Meridian.
1. To find the Difference of Latitude.
Becauſe the Latitudes are both North, and the Ship’s Courſe is
Northerly
From
Navigation.
149
From the Latitude found by Obſervation
Take the Latitude faiļd from
47 50 N.
46 30 N.
Remains the Difference of Latitude made
1
20 Nly.
Geometrically.
1. Make AB equal to the Difference of
Latitude 80, and erect the Perpendicular BC.
B
53.45
Sais
7.96
NEBN
1
Draw the Rumb-Line AC, making an
Angle of 33 deg. 45 min, with the Meridi-
an, and produce it till it cut the Perpendi-
cular in C, which is the Place the Ship is
in, then will AC be the Diſtance run, and
BC the Departure, to find each of which by
Calculation, it will be,
1. By Caſe the firſt of Se&t. the fourth of
Trigonometry.
As R : AB :: t. BAC: BC. That is,
As the Radius
A
10.0000000
To the Difference of Latitude made 8
1.9030900
So is the Tangent of the Courſe 33 deg. 45 min. 9,8248926
To the Departure 53.45
1.7279826
And ſo much is the Ship got to the Eaſtward, doc.
For the Diſtance faild it will be by Caſe the ad of Se&t. the 411
of Trigonometry.
As R: AB :: Sec. BAC : AC. That is,
As the Radius
10.0000000
To the Difference of Latitude made 80
So is the Secant of the Courſe 33 deg. 45. min.
1.9030900
10.0801536.
Ta the direct Diſtance 96.21
1.9832436
:
But
150
Navigaton.
thwilt Currents; then with the true Difference of Lacitad., and
But if having no Reaſonto ſuſpect the Diſtance faild, for the former or any
other Reaſons, and are aſſured in your ſelf of having made a good Eſti-
mate thereof, but miſtruſt the Care of bin that Stoors, that he has let
her yaw or fäll off, or that you have nezleted the Variation, or met with
:
the Diſtance faiļd, find the Courſe and Departure, by the jülowing
Caſe.
Caſe III.
Potl Latitudes and Diſtance Sail'd being given, to find thc truc
Courfe and Departure from the Meridian:
.6
Example
A Ship at Sea in the Latitude of 46 deg. 30 min. North, ſails
upon Come Rumb between the North and Eaſi 96 Miles, and then by
Obſervation is found to be in the Latitude of 47 deg: 50 min. North,
I demand the true Courſe ſteered and Departure froin the Meridian.
To find the Difference of Latitude.
Becauſe the Latitudes are both North,
O
1
From the Latitude found by Obſervation
Take the Latitude fail'd from
47 SON.
45 30 N.
Remains the Difference of Latitude
-1
20 Nly.
Geometrically.
B
53.06
с
W33.33€
56
1. Set the Difference of Latitude made
80, from A the Place departed from to B.
2. Erect the Perpendicular BC, to the
right from the Meridian, becauſe the Courſe
is Eaſterly.
3. With the Diſtance fail'd between the
Points of the Compaſſes, ſetting one foot
in A, with the other croſs the Perpendicu-
lar in C, which gives the Place the Ship is
-C
A А
arrived
Navigation.
151
arrived at, and conſequently the Angle BAC will be the true Courſe
ſteerd, to find which by Calculation it will be, by Caſe the 5th
of Set. the 4th of Trigonometry.
As AC:R:: AB : S. ACR. That is,
As the Diſtance faild 96
1.9822712
To the Radius
So is the Difference of Latitude 80
10.0000000
1.9030900
To the Co-line of the Courſe 33 deg. 33 min.
9.9208188
Which becauſe ſhe fail'd between the North and the Eaſt, is North
33 deg. 33 min. Eaſt or N. E. by N. nearly; whence to find the
Departure it will be, by Caſe the 3d of Sect. the 41h of Trigonometry,
As R: AC:: S. BAG: BC. That is,
As the Radius
10.0000000
To the Diſtance fail'd 96
So is the Sine of the Courſe 33 deg. 33 min.
1.9822712
9.74.2016
To the Departure 53.06 Eaſtwardly
1.7247328
The Latitudes of any two Places being given, and their Meridional Di-
ſtance, the following Caſe is of Uſé in determining the true Rumb the
Ship muſt ſail upon, and how far.
Caſe IV.
Both Latitudes and Departure being given, to find the direct
Courſe and Diſtance.
Example
A Ship at Sea in the Latitude of 12 deg. 10 min. North is bound
to Barbadoes, in the Latitude of 13 deg. 30 min. North, the Meri-
dional Diſtance by Eſtimation being 53 Miles Weſt, I demand the
direct Courſe and Drtance, from the Ship to her Port.
TO
1
152
Navigation.
To find the Difference of Latitude.
From the Latitude of Barbadoes
Take tlie Latitude the Snip is in
13
12
30 N.
10 N.
Remains the Difference of Latitude
I
20 Nly:
Geometrically.
53
R
1. Set off the Difference of Latitude
from A to B.
2. From B erect the Perpendicular
BC, equal to the Departure 53 Miles,
to the Left-hand of the Meridian, and
draw the Line AC, then will A repre-
ſent the Place of the Ship, C the Port
bound to, AC the direct Diſtance, and
the Angle A the truc Courſe.
To find which by Calculation it will
be, by Caſe the 4th of Sect. the 4th of
Trigonometry:
As AB: R:: BC:t. BAC. That is,
As the Difference of Latitude 80
1.9030900
95:35
133.37 w
A
To the Radius
10.0000000
So is the Departure 53
-1.7242759
To the Tangent of the Courſe 33 deg. 31 min. -9.8211859
Which becauſe the Ship is bound from a leſs North Latitude to a
greater, and the Meridional Diſtance is Weft, is N. 33 deg 3 1 min.
Weſt, or N. W. by N nearly : whence to find the direa Diſtance
it will be, by Caſe the ad of Se&t. the 4th of Trigonometry.
As R: AB: : Sec. BAC:AC. That is,
As the Radius
10.0000000
To the Difference of Latitude 80
So is the Secant of the Courſe 33 deg. 31 min.
1.9030 900
10.0789771
To the dire& Diſtance 95.95
1.9820671
If
MW
Navigation
153
If for want of a perfect Knowledge of the Latitude fail'd from,
or thro' the carelelneſs of him that kept the Helm, you miſs the
Place you were to fail to, but made an exad Eſtimate of the Diſtance
fail'd by the Log, then may the Latitude of the Place the Ship is in,
and the true Courſe ſhe has fail'd, be exa&ly known by the follow-
ing Cafe,
Caſe V.
B
One Latitude, Diſtance Taild, and Departure being given, to find
the other Latitude and direa Courſe.
Example.
A Ship at Sea in the Latitude of 12 deg. 10 min. North, having
fail'd between the North and Weſt 95.95 Miles, and having made
53 Miles of Weſting, I demand the dire& Courſe ſteer'd, and
Latitude the Ship is in.
Geometrically.
1. Having drawn DC parallel to the Meridi-
an, at the Diſtance of the Departure 53 Miles; 53
with the Diſtance fail'd between the Points
of the Compaſſes, ſetting one foot in A, the
Place the Ship departed from, with the other
croſs the former Parallel in C, which will be
the Place at which the Ship is arrived; from
whence having let fall the Perpendicular CB,
and drawn the Line CA, BA will be the Al-
teration of Latitude, and the Angle at A
the Courſe, to find which by Calculation it
will be, by Caſe the sth of Se&t. the 5th of
A
Trigonometry.
As AC:R::BC:S. BAC. That is, -
As the Diſtance fail'd 95.95
1.9820450
98;95
8664
ON33.32w
:
To the Radius
So is the Departure 53
10.0000000
1.7242759
To the Sine of the Courſe 33° 32'.
9.7422309
X
Whence
7.
154
Navigation.
Whence the true Courſe is North 33 deg. 32 min. Weſt, or North
Weft by North nearly : Wherefore, to find the Difference of La-
ticuce it will be, by Caſe the 3d of Seet. the 5th of Trigonometry.
AsR: AC::cs, BAC : AB. That is,
As the Radius
10.0000000
To the dire& Diſtance faild 95.95
So is the Co-line of the Courſe 33° 32'
1.9820450
-9.9209393
To the Difference of Latitude 79.98
1.9029843
Or 80 Miles nearly, equal to i deg. 20 min, wherefore, to find the
Latitude the Ship is in, becauſe ſhe fail'd from a North Latitude
Northerly.
To the Latitude fail'd from
Add the Difference of Latitude made-
I 2
10 N.
20 Nly.
I
Their ſum is the Latitude the Ship is in
-133
13
30 N.
If the Diſtance Run estimated by the Log, be not ſufficiently exa&t for the
Reaſons given in Caſe the firſt, but you are well affured of your Courſe,
then by the following Cafe may be found the true Diſtance faild, as alfo
the Latitude the Ship is in.
Cafe VI.
One Latitude, Courſe and Departure, being given, to find the
Latitude the Ship is in, and her dire&t Diſtance fail'd.
Example
A Ship at Sea in the Latitude of 46 deg. 30 min. North, fails
upon the third Rumb or North Eaſt by North, antil her Meridional
Diſtance be 53.34 Miles, I demand the Latitude the Ship is in, and
her dire& Diftance fail'd.
Geome
Navigation.
155
B
55.18
79:22
cr
Geometrically:
1. Having drawn the Meridian AB: from
A, the place of the Ship, draw the Rumb-
53.34
Line AC, forming an Angle with AB, equal
to 33 deg. 45 min.
2. With the Meridional Diſtance draw
CD parallel to AB, where this interſects the
Rumb-Line as at C, will give the place the
Ship is arrived at; from whence having let
fall the Perpendicular BC, we ſhall have
AB for the Difference of Latitude, and
AC for the direct Diſtance fail'd, to find
A
which, by Calculation it will be, by Cafe
the ist of Set. the 5th of Trigonometry.
As S. BAC : BC ::R: AC. That is,
As the Sine of the Courſe 33 deg. 45 min.
9.7447390
To the Departure 53:34 Miles
1.7270530
So is the Radius
NEBN
10.0000000
To the direct Diſtance 96.01
1.9823140
To find the Difference of Latitude it will be, by Caſe the 1st of
Se&t. the 5th of Trigonometry.
As t, BAC : BC :: R: AB. That is,
As the Tangent of the Courſe 33° 45'
9.8248926
To the Departure 53.34
So is the Radius
1.7270530
10.0DO0000
To the Difference of Latitude 79.83
1.9021604
Whence to find the Latitude the Ship is in, becauſe ſhe ſails from a
North Latitude Northerly.
To the Latitude fail'd from
46 zó N.
Add the Difference of Latitude made
19.83 Nly
The Sum is the Latitude the Ship is in
47 49.83 N
01
X 2
And
156
Navigation
And that the Reader may the better underſtand the Nature of Com-
pound Traverſes, I thought it neceſary to incert the following Problem.
A Ship at A, (which for Methods ſake we call the firſt Port) in
the Latitude of 15 deg. 30 min. North, (See Fig. the firſt, Plate the
firſt) is bound to P a ſecond Port, in the Latitude of 23 deg. 10 min.
North, the Departure between them being 390 Miles Eaſt, and ha-
ving lail'd from A to S, is found by Dead Reckoning to be in the
Latitude of 21 deg, 15 min. North, and to have departed from her
former Meridian 180 Mites Eaſt; I demand the dire&t Courſe and
Diſtance from the Ship at S, to the ſecond Port P.
1. To find the Difference of Latitude between the Ship and her
fecond Port, becauſe the Laçitudes are both Northerly,
Froin the Latitude of the fecond Port-
23 ION.
Take the Latitude of the Place the Ship is in 2.1. 15 N.
Remains the Difference of Latitude
55 Nly.
2. To find the Departure between the Ship and her ſecond Port.
From the Departure between the two Ports CP=-390 Eaſt.
Take the Departure the Ship has made BS=CD= 180 Eaſt.
Remains the Dep. between the Ship and the ad Port DP=210 Eaft:
Wherefore to find the Courſe it will be, by Caſe the 4th of Plain
Sailing.
As the Difference of Latitude 115.
2.0606978
To the Departure 210
2.3222193
So is the Radius
10.0000000
A
To the Tangent of the Courſe 01.18.
19.2615215
Which becauſe the Difference of Latitude is Northerly, and De-
parture Eaſterly, is North 61° 18' Eaſt, or North Eaſt by Eaſt, sdeg.
03 min. Eaſterly. And,
To find the direct Diſtance it will be, by the ſame fourth Caſe of
Plain Sailing
As
Navigation.
157
As the Radius
10.0000000
To the Difference of Latitude 115
So is the Secant of the Courſe 61° 18'
2.0606978
10.3185566
To the direct Diſtance 239.5 Miles
2.379254+
If the Courſe and Diſtance upon which the Ship has fail'd, or be-
tween the iſt Port and the Ship be required, then in the Triangle BAC
are given, AB the Difference of Latitude 340 Miles, in this Cafe
Northerly, and BS the Departure 180 Miles Eaſt, whence by the
fourth Cafe of Plain Sailing, the Courſe will be found to be North
62 deg. 06 min. Eaft, or North Eaſt by Eaſt 5 deg: 51 min. Eaſt,
and the diſtance fail'd 384.7
2. Let it be required to find upon what Courſe ſhe muſt ſail, and
how far, to arrive 100 Miles dire&ly to the Weſtward, of the ſecond
Port.
Becauſe in this caſe the Difference of Latitude is the ſame, and the
Departure equal to 110 Miles, it will be by the fourth Caſe of
Plain Sailing
As the Difference of Latitude 115--
2.0606978
To the Departure 110
So is the Radius
2.0413927
10.0000000
To the Tangent of the Courſe 43° 44'
9.9806949
Which becauſe the Difference of Latitude is Northerly, and the
Departure Eaſterly, is North 43 deg. 44 min. Eaſt, or North Eaſt by
North-9 deg: 55min Eaſt, and to find the direct Diſtance, it will
be by the ſame Caſe,
As the Radius
10.0000000
To the Difference of Latitude 115
2.0606979
So is the Secant of the Courſe 43 deg. 44 min. 10.1411230
To the dire& Diſtance 159.2 Miles
2.2018208
3. If it had been required to have found upon what Courſe lhe
muſt have fail'd, and how far, to have brought the ſecond Port to
bear
158
Navigation.
bear 100 Miles Weſt, then in this Caſe the Difference of Latitude
would have been 118 Miles as in the former, and the Meridional
Diſtance 310 Miles, and conſequently by Caſe the 4th ot Plain Sail-
ing, the dire& Courfe would have been North 69 deg. 39 min. Eaſt,
or Eaſt North Eaſt 2 deg. 9 min. Eaſt, and the direct Diſtance 330.6
4. In like manner in ſailing North s i deg. si min. Eaſt, or North
Eat 6 deg. 51 min. Eaſt, Dillance 339.9 Miles, the ſecond Port
will be brought to bear due South Miles from the Ship.
5. Alfo in failing North 72 dég. 48 min. Ealt, or Eaſt North Eaſt
5 deg. 18 min. Eaſt, Diſtance 219.8 Miles, the Ship will be brought
to bear from the ſecond Port, South 50 Miles, and conſequently
the fecond Port will bear North 50 Miles from the Ship.
6. Ler it be required tomknow what Courſe and Diſtance the
Ship muſt ſail, to bring her ſecond Port to bear North Eaſt by
North, diſtance 100 Miles, which place for diſtinction lake, we will
call the Ship's Stacion. (See Fig. the ift, Plate the ift.)
1. In the Triangle m Po are given, the Angle m Po the Bearing,
and Pothe Diſtance, whence to find the Difference of Latitude m P
it will be by the firſt Caſe of Plain Sailing.
As the Radius
10.0000000
To the Diſtance given 100 Miles
So is the Co-ſine of the Courſe 33° 45'
2.0000000
-9.9198464
&
To the Difference of Latitude 83.15
1.9198464
Which taken from SD 115 Miles, the Difference of Latitude
between the Ship and her ſecond Port, leaves SE 31.75 Miles, the
Difference of Latitude between the Ship and her Station, which in
this Caſe is Northerly.
Again, to find the Departure mo, between the ſecond Port and
the Station, it will be by the firſt Cale of Plain Sailing.
As the Radius
10.0000000
To the Diſtance given 100
- 2.0000000
So is the Sine of the Courſe 33° 45'
9.7447390
To the Departure 55.56 Miles
1.7447390
Which
Navigation.
159
Which in this Cale added to Emequal to DP, equal to 210 Miles,
the Departure between the Siaip and her ſecond Port, gives EO -
qual to 265.56 Miles, the Departure between the Ship and her
Station, in this Caſe Easterly, whence to find the true Courſe and
Diſtance it will be, by Care the fourth of Plain Sailing.
As the Ditterence of Latitude 31.75----
1.5017437
To the Departure 265.56.
So is the Radius
2.4241626
10,0000000
To the Tangent of the Courſe 83° 11'
10,92 24189
Which becauſe the Difference of Latitude is Northerly, and De-
parture Eaſterly is North 83 deg. I min. Eaſt, or Eaſt by North
4 deg. 26 min. Eaſt, and for the direct Diſtance it will be,
As the Sine of the Courſe 83° 11'
9.9969191
To the Departure 265.56
So is the Radius
2.4241626
10.0000000
To the direct Diſtance 267.4.-
2.4272435
So that in Sailing N. 83 deg. II min E. Diſtance 267.4 Miles,
the Ship will arrive at her Station, and the ſecond Port will then bear
North Eaſt by North from the Ship, Diſtance 100 Leagues.
Hence it will be eaſy to find upon what Courſe and how far, the
Ship muft fail, ſo as to bring the ſecond Port to have any given or
determinate Bearing and Diſtance from the Ship: But to returi,
Ler A, (in Fig. the 7th, Plate the ift) repreſent the firſt Port,
in the Latitude of 23 deg. oo min South, P the ſecond Por, in the
Latitude of 16 deg. 30 min. Sourh, the Meridional Diſtance CP,
between the two Ports being 180 Miles Eaſt, and let us ſuppoſe the
Ship arrvied at s, in the Latitude of 14 deg. 20 min. South, and to
have made 390 Miles of Departure Eaſt, and let it be required to
find upon what Courſe the Ship muſt fail, ſo as to arrive at the fe-
cond Port P.
To find the Difference of Latitude, between the Ship and her fe-
cond Port.
Be-
G
1
160
Navigation
I
Timer
Becauſe the Latitude ihe is bound to, is greater than the Latitude
he is in, an iboth on the ſame ſide of the Equator. Therefore,
From the Latitude of the ſecond Port
16.30 South.
Take the Latitude the Ship is in
14.20 South.
Remains the Difference of Latitude
2.IO
Or 130 Miles, which becauſe the Ship is to the Northward of
her Port, is Southerly.
Again, to find the Departure, becauſe the Departure made is
greater than the Departure between the two Ports.
From the Departure the Ship has made
390 Miles E.
Take the Departure between the two Ports
180 E.
Rem. the Departure between the Ship and her 2d Port 210
Which is Weſt, becauſe the Departure made is greater than the
Departure between the two Ports.
Wherefore, to find the direct Courſe it will be, by Caſe the fourth
of Plain Sailing.
As the Difference of Latitude 130
2.8139434
To the Radius
10.0000000
So is the Departure 210
2.3222193
To the Tangent of the Courſe 58° 14'
10.2082759
Which becauſe the Difference of Latitude is Southerly, and De-
parture Weſterly, is South 58 deg. 14 min. 1 Weſt, or South Weſt
by Weſt i deg. 59 min. Weſt, whence to find the dire& Diſtance it
will be, by the fourth Caſe of Plain Sailing.
As the Sine of the Courſe 58° 14'
9.9295598
To the Departure 210
So is the Radius
2.3222193
10.0000000
To the dire& Diſtance 247
2.3926595
Whence the Courſe from the Ship to her ſecond Port, is S. 58°
14' | Weſt, or S. W, by W. rºse'i, W. Diſtance 247 Miles.
From
Navigation.
161
From a due Conſideration of what has been done, and a View of
the 2d, 3d, 4th, 5th, 6th, and 8th Figures in Plate the 1st, the Reader
will eaſily account for all the Varieties, that can attend Problems of
this kind, and readily give Solutions to any of them.
And inaſmuch as Plate the ist, is ſuppoſed to repreſent a ſmall
Part of the Earths Surface, in which, beſides the Figures already
made Uſe of, the 9th, soth, 11th, 12th, &c. repreſent the ſeveral
Caſes in Plain Sailing, it will not be difficult by a diligent Inſpedion,
to ſee the Reaſon of every ſtep that has been taken,
For as the Latitude of a Place is no other than the neareſt Diſtance
of that place from the Equator, it is manifeſt that a Ship at A, in
Fig. 1516, 17th, 197h, on the South-fide of the Equator, in failing a
Northerly Courle from A to C, approaches nearer to the Equator, by
the Diſtance of AB, and conſequently makes her Latitude ſo much
leſs than before ; while on the contrary, the ſame Ship at A, in Fig.
the 13th rith, and 9th, on the Northern ſide of the Equator, in
failing the ſame Courſe, continually goes farther from the Equator
by the ſame ſpace, and conſequently makes her Latitude ſo much
the greater.
In like manner, the fame Ship at A, in Fig. 1oth, 12th, and 14th,
on the North ſide of the Equator in failing Southerly towards C, ap-
proaches the Equator, and therefore diminiſheth her Latitude; while
the Ship at A, in Fig. the 16th, 18th, 20th, on the South ſide of the
Equator, keeping the ſame Courſe, continually increales her Latitude.
Again, as the Departure is nothing elſe but the Diſtance between
the Meridian fail'd trom, and the Meridian ſaiļd into, it is manifeft,
that a Ship in Fig. 9th, 11th, 13th, 15th, &c. in failing Weſter-
ly from A to C, is got to the Weitward of her former Meridian, by
the Diſtance of BC; while the ſame Ship in Fig. 10th, 12th, 14th,
16th, &c. in failing Eaſterly from A to C, gets to the Eaſtward of
her firſt Meridian, by the Diſtance of BC.
Again, as a Ship in failing from A to S, (See Fig. the 5th) is got to
the Eaſtward of her 1st Port, by the Diſtance of BS, and likewiſe to
the Eaſtward of her ſecond Port, by the ſpace of PD; 1o the fame
Ship in Fig. 3d, in failing from A to S, is got to the Eaſtward of her
1/ Port, by the Diſtance of BS; and to the Weſtward of her żd Port,
by the Diſtance of PD, the difference between the Meridional Di-
ſtances CP, and CD, equal to BD, and confequently her Courſe to her
ſecond Port muſt be South Eaſterly. Theſe things being premiſed, I
proceed to
Y
SECT;
0
162
Navigation.
>
Boone YoYou
hora
28 Miles, then NW
Section III.
Containing the Application of the Doctrine of Right-angled
Plain "Triangles, to the Solution of a Traverſe or Com-
pound Courſe.
A
Traverſe or Compound Courſe, is when a Ship upon the ſhifting
of the Winds, fails upon ſeveral Courſes, in any given ſpace
of time.
To reſolve a Traverſe, is ſo to Order and Compound the ſeveral
Courſes and Diſtances, obtained by the Log and Compaſs, that the In-
duſtrious Mariner may know at all times, the Place the Ship is in,
and be able to determine how far, and upon what Courſe he muſt
fail to gain his incended Port.
And to this End it is neceſſary to let the Learner know, that the
Seamen after the manner of the Aſtronomers, reckon their Days from
Noon to Noon; and 'tis their uſual Cuſtom, to take from the Log-
board every Day at Noon, the moſt notable Tranſactions of the pre-
ceding Day, relating to the Wind, Weather, &c. and if the Ship in
that time has gon upon ſeveral Courſes, that Compound Courſe beomes
2 Traverſe; of which I fhall give ſome Examples.
Example, I.
A Ship takes her Departure from a Head-Land, in the Latitude
of so deg. oo min. North, it bearing then E. N. E. diſtance by Eri-
mation 6 Leagues, and ſails $. by E. 36 Miles, then w. by S.
the Latitude the Ship is in, and her Bearing and Diſtance from the
Head-Land.
The Solution of this problem, depends intirely upon the firſt and
fourth Caſes of Plain Sailing, and the ſeveral Departures and Diffe-
rences of Latitude anſwering to each Courſe and Diſtance, muſt be
found according to the Rules given in Cafe the firk, after the manner
following
I. Course
है
T
:
Navigation.
163
T.
I. Courſe W. S. W. Diſtance, 18 Miles.
10.0000000 As Radius
As Radius
10.0000000
To the Dift.ſaild 18 1.2552725 To the Diſt. ſail'd 18 1.2552725
So is theS.of C.67.30 9.9656153 So is thecs,ofC.67.30 9.5828397
To the Dep. 16.63-12208878 To Diff.of Lat. 6.89 0.8381122
2. Courſe, S. by E. Diitance 36 Miles.
10.0000000 As Radius
As Radius
10.0000000
To the Diſt. fail'd 36 1.5563025 To the Diſt.fail'd36 1.5563025
So is the S.ofC. 11.15 9.2902357 So is the cs,ofC.11.15 9.9915739
To the Depart. 7.02 0.840538 To Diff. of Lat.35.31 1.5478764
After this manner muſt the Differences of Latitude and Departure
for every Courſe and Diſtance be found; but I ſhall omit the Opera-
stions for Brevity fake.
Now from th: feveral Departures and Differences of Latitude, thus
deduced from the Courſes and Diſtances, obtained by the Log and
Compaſs, may be found the Latitude of the Place the ship is ini, and
how far the is to the Eaſtward or Weſtward of her firft Meridian, by
difpofing of them as in the following Table, called the Traverſe
Table; which conſiſts of five Colums, in the firſt of which muſt be
placed the ſeveral Courſes and Diſtances, in the other four the Dif-
ferences of Latitude and Departures, deduced from each Courſe and
Dittance, placing each according as the Courſes are ; that is, if the
Courſe be N. {Welterly} the Difference of Latitude muſt be placed
under the Column N, or Column for the North, and the Depar-
Eaſt
ture under the Column Welt } but if the Courſe be's.
then the Departure muſt be placed as before, and the Difference of
Latitude under the South'; and being thus difpoled, it is manifeft
that the Fotals of the ſeveral Columns ſhew the Northings, Southings,
Eaftings, and Weſtings, that.the.Ship has made.
Wherefore, if the Sum of the Northings, exceed the Sum of the
Southings, it follows, that the Ship is to the Northward of her airft
Y2
Place
Eaſterly
Weſterly
통
​PLATE the 1
25
C
D
P
B
25
А
Tropick of Cancer
E
B
m
9
20
B
20
10
B
B
C
P
15
А
A
с
B В
D
15
:11
23
B
12
10
10
P
A B
D
B
3
I 3
р
5
B
5
14
B
A
B
Equa
tor
Meridian
Meridian
C
22
P
15
22
5
A
B
16
5
o
B В
C
10
6
10
B
B
S
B
17
A
P
--
15
211
18
115
B
S
B
cܠ
F
D
B В
7
20
19
D
20
A
D
C
20
Z
P
SK
K TropickofCapricorn
B В
25
B
125
Place this between Page 16% and 163 •
164
Navigation.
Place, and that juſt as much as is the Exceſs, and the contrary. In
fike manner, if the Sum of the Eaſtings, exceed the Sum of the
Weſtings, the Ship is to the Eaſtward of her firſt Meridian ; other-
wiſe to the Weſtward, as will appear by the following Table.
The Traverſe Table.
Dif: of Lat. Departure
N. S. E. W..
Courſes.
Diſtances.
E
16.6.3.
--18
36
W. S. W.
S. by E.
W. by S.-
N. W.
South
6.89
35.31 702
5.46
28
=
27.47
1.21,
30/ 21.21
41
41.00
11
21.21. 188.66
7.02 65.31
7.02
2 1.2 1
Dill of Lat. -67.45
Dep. 38.39
Now becauſe the Sum of the Weſtings exceedsthe Sum of the Eaſt-
ings, by 58:29 Miles, it is manifeſt, that the Ship is gotten 58.29
Miles to the Weſtward of her firſt Meridian. Again,
Becauſe the Sum of the Southings exceeds the Sum of the North-
ings, by 67.45 Miles, it follows, that the Ship is 67:45 Miles, or
1° 7'6", to the Southward of the Port departed from; wherefore,
to find the Latitude ſhe is in, becauſe the Latitude failed from is
Northerly, and the Difference of Latitude Southeriy:
so oo N..
From the Latitude fail'd from
Take the Difference of Latitude made
1 077** Sly:
$3
Remains the Latitude the Ship is come into
-49 52766
Now, to find the direa Courſe the Ship has Sail'd, or its Bearing
from the Head-Land, it will be by the fourth Caſe of Plain Sailing.
Navigation
165
As the Difference of Latitude 67.45
1.8289820
To the Departure 58.29
So is the Radius
1.7655941
10.0000000
To the Tangent of the Courſe 40 deg. so min, 9.9366121
Which becauſe the Difference of Latitude is Southerly and De-
parture Weſterly, is S. 40° sol Wi; or S. W.by S. 7° 05' W.
Again, for the direct Diſtance it will be, by the fame Caſe.
As thc Sine of the Courſe 40 deg. 50 min.
9.8154854
sܕ
To the Departure
So is the Radiús
58.29-
1.7655948
10.0000000.
To the direct Diſtance 89.15
1.9501087
***
***
Example 2.
A Ship at Sea in the Latitude of 36° 10' North, is bound to a
Port in the Latitude of 32° 00' North, the Departure between the
Ship and the Place being 180 Miles Weſt, and therefore the direct
Courſe by Caſe the fourth of Plain Sailing is, S. 35° 45'. W. or S. W.
by S. 2° 00' Weſterly, and the direct Diſtance 308.1 Miles, but
finding the Wind variable from the S. by E. to the S. S. W. a ſmall
Gale, and ſmooth Water plies upon theſe ſeveral Courſes, with the
Diſtance on each Courſe obtained by the Log as follows.
Larboard Tacks on Board, Wind variable from S. by. E. to S. S.
W. Courſes S. W. by W. 27 Miles, W. S. W. W. 30 Miles, W.
by S. 25 Miles, W. by N. 18 Miles.
Starboard Tacks on Board, Wind ſhifting from S. S. W. to S. W.
and W. S.W.
Courſes S. S. E. 32 Miles, S. S. E. * E. 27 Miles, S. by E. 25
Miles, S. 31 Miles, S.S. E. 39 Miles.
I demand the Latitude the Ship is in, her Departure from the Me-
ridian, and upon what Courſe, and how far. fhe muſt fail, to gain her
intended Port:
Geometrically.
Let A repreſent the Place of the Ship at Sea, AHD the propes
Meridian, make AD cqual to the Difference of Latitude, 250 Miles
between
166
Navigation.
"
tion, KAH the direct Courſe, KH the Departure fhe has made Weſt,
between the firſt Station and the Port, and Perpendicular to it, draw
PD equal to the Departure 180 Miles, then is P the Port the Ship
is bound to; AP the direct Diſtance, and the Angle PAD the diret
Courſe, (See Plate the ad.)
And to find the Place of the Ship at the end of the
ſeveral Courſe's,
or which is the ſame thing, to lay down the ſeveral Runnings of the
Ship, proceed thus.
1. From A the Place of the Ship firft given, draw Ab, the S. W.
by W. Line, and ſet off upon it 27 Miles, from A to b, then will
Áb repreſent the firſt Courſe, and b the Place of the Ship at the
end of it.
2. From b draw the Line bc, equal to 30 Miles, the ſecond
Diſtance, and parallel to 2n the W. S. W. W. Line, then will
be the Place of the Ship at the end of the ſecond Courſe.
3. From c draw cd, parallel to go, the W. by S. Line, and fet
0.4'upon it 25 Miles from c to d, then will d be the Place of the Ship
ar tie end of the third Courſe.
And proceeding in like manner, by drawing Lines from the Place
laſt found, parallel to the ſeveral Rumb-Lines drawn in the Qua-
drants, anſwering to the ſeveral Courſes, and equal in Length to the
feveral Diſtances, upon each Courſe we ſhall have the feveral Courles
de, ef, fg,gh, bi, and ik, upon which the Ship has fail'd, and
k the place where the Ship is arrived at the end of them.
After this manner may the ſeveral Runnings of the Ship, in the former
Traverſe be laid down Geometrically, if required.
Laſtly, having drawn the Lines K A, KH, Perpendicular to AD,
KR parallel to AD, or Perpendicular to PD, alſo the Line KP, then
will 'K A be the neareſt Diſtance between the Ship and her fir Sta-
KR equal to DH, the Exceſs of AD above AH, the Difference of
Latitude between the Ship and her Port, in this Caſe Southerly;
PR the Difference between PD and DR, equal to KH the Depar-
ture Weſt, alſo KP the direct Diſtance between the Ship and her
Port, and the Angle PKR the direct Courſe, upon which the muſt
fail, if poſible, to arrive at her intended Port,
To find which by Calculation, the ſeveral Differences of Latitude
and Departure, anſwering to each particular Courſe and Diſtance,
mult firft be found and diſpoſed of as in the following Table, after
the manner taught in the former Example.
The
Navigation.
167
The Traverſe Table.
Diff. of Lat. | Departure.
Diſtances. N. SI E.
Courſes.
W.
15.00
8.71
22.44
28.70
lu
4.88
24:51
17.65
3:51
1. S. W. by W.
2. W. S. W. W.
3. W. by S.
4. W. by N.
5. S. S. E.
6. S. S. E. E.
7. S. by E.
8. South
9. S. S. E.
27
30
* 25
18
32
27
25
31
39
29.56 12.24
23.15 13.88
24.51 4.88
31.00
36.03 14.92
3.51 172.8445.92 93.30
3:51 45.92
Diff of Lat=169.33. Dep.=47:38
The two S. S. E. Courſes might have been compounded together
and made one, and the anfwer would have been exactly the fame.
Hence it is manifeſt, that the Ship is 47.38 Miles to the Weſt-
ward, and 169.33 Miles to the Southward of her firſt Station, and
therefore the whole Difference of Latitude is 2. 49' 7 Southerly.
Wherefore, to find the Latitude the Ship is in;
Becauſe ſhe fail'd from a North Latitude Southerly.
1
From the Latitude fail'd from
Take the Difference of Latitude made-
36
IO N.
49.7*? Sly.
Remains the Latitude the Ship is in
33
20 767 N.
To find the dire& Courſe the Ship has fail'd it will be, by the
4th Caſe of Plain Sailing.
As
168
Navigation
2.2287339
As the Difference of Latitude 169.33
To the Departure 47.38
So is the Radius
- 1 6755950
I0.0000000
To the Tangent of the Courſe 15 deg. 38 min. 9.4468611
Which becauſe the Difference of Latitude is Southerly, and De-
parture Weſterly is S. 15° 38' W. or S. by W. 4° 23' W. and for che
Diſtance it will be by the ſame Caſe.
As the Sine of the Courſe 15 deg. 38 min.
9.4305267
To the Departure 47.38
1.6755950
So is the Radius
10.0000000
i
To the dire& Diſtance-fail'd 175.8
2.2450683
The next thing wanting to compleat the Traverſe, is to find the
diſtance of the Ship from her intended Port, and upon what Courſe
ſhe muſt ſail, if poſſible to gain it.
And becauſe the Difference of Latitude between the Place of the
Ship at her firſt Station, and her intended Port, is greater than the
Diference of Latitude made, and both the ſame way.
From the Difference.of. Latitude firſt given
Take the Difference of Latitude made
Miles.
250.00 Sly.
169:33 Sly.
Remainsthe Difference of Latitude Themuſt make 80.67 Sly.
Again, becauſe the Departure between the Place of the Ship in
her firſt Station and the Port, exceeds the Departure made, and
both are Weſterly.
Miles
From the Departure firſt given
180.00 Wiy.
Take the Departure made
47:38 Wly:
Remains the Departure the Ship muſt make 132.62 Wiy.
Hence
Navigation
169
Hence and from the fourth Caſe of Plain Sailing, the Courſe and
Diſtance may be eaſily had. For,
As the Difference of Latitude 80.67
1.9067121
To the Departure 132.62
2.1 226090
So is the Radius
To the Tangent of the Courſe 58° 41' .
10.2158969
10.0000000
Which becauſe the Difference of Latitude is Southerly and Departure
Weſterly, is S. 58 deg, 41 min. W.or S. W. by W. 2 deg. 26 min. W.
Whence for the direct Diſtance it will be, by the ſame Caſe,
As the Sine of the Courſe 58 deg. 41 min.
9.9316143
To the Departure 132 62
So is the Radius
2.1 226090
10.0000000
To the direct Diſtance 155.2
2.1909947
So that the Ship in failing 155.2 Miles, S. 58 deg: 41 min. W. or
S. W. by W. 2 deg. 26 W. will arrive at her intended Port.
In the two former Examples, the Courſes given are luppoſed to
be true, that is, Corrected according to the Allowance made for the
Leeway and Variation; and how that may be done, the Reader
may ſee in Section the 6th Part the 5th, and therefore I ſhall conclude
this uſeful Sečtion with ſome few Examples adapted to the Pra&ice,
and give their Solution, leaving the Operation as an Exerciſe for
the Young Practitioner
Example 3.
A Ship takes her Departure from a Cape or Head-Land, in the
Latitude of 5 1 deg. Io niin. North, the Head-Land then Bearing by
the Compaſs N. N E. Diſtance by Eſtimation 7 Leagues, and fails
S, by E. 30 Miles, then S. E. by Ś. 37, then South 34, then S. W.
32, then S. by W. 25.; the Variation of the Compaſs being rPoint
to the Weſtward : I demand the Latitude the Ship is in, her true
Bearing and Diſtance from the Cape, and how far the has departed
from her former Meridian,
Z
Anſwers
N
.
170
Navigation.
1
oo E.
Anſwer.
The Latitude the Ship is in, is
48 3076N.
Her true Bearing from the Cape .
S. 8
And direct Diſtance
161 Miles
Example 4.
Being at Sea in the Latitude of 32° oo' North, 10 Leagues to
the Southward of the Iſand I departed from, I found by a good Ob-
fervation that there was no Variation, and being bound to the South-
Weſtward, I fail'd upon the ſeveral Courſes and Diſtances following,
the Wind as per Column, and allowing one Point Leeway; I de-
mand the Latitude the Ship is in, and her Bearing and Diſtance from
the Iſland.
The Traverſe Table.
Couries. Courſes Diff. of Lat. Departure.
Winds. Steer'd.
Steer'd. Correct. Diſ. N. 1 S. E. W.
S.W byS.S. E. by S.
S. E. 16
11.3111.31
W. by N. W.N. W. 38 14.54
35.11
S. S. W. S.E. S. E. by E. 20 IIII 16.63
Weſt. w. by N. 40
39.22
S. W. S.S. E. S. E. by S. 24 19.96 13:33
W.by N. S.W.by S. S. S. Wil 48 44:34
18.37
7.80
=
22.34 86.72 41.27 92.70
22.34 41:27
Diff
. of Lat. 64.38) Dep. 55.43
Anſwer,
Latitude the Ship is in
30° 55' North
Bearing from the Iſland
South 28° 26' Weſt,
Direct Diſtance from the Iſland
107.34 Miles.
Example. 5.
A Ship from the Latitude of 46° 40' North, fails the ſeveral
Courſes and Diſtances following upon a Wind, the Wind as per Co-
lumn, the Leeway 1 Point and Variation 1 $ Weſt ; I demand the
Latitude the Ship is in, and her Bearing and Diſtance from her firſt
Station
The
1
7
Navigation
171
The Traverſe Table.
Courſes
Dift. of Lat. Departure.
Correct. Dift. N.
S. E. W.
Courſes
Winds. | Steer'd.
1.18
-
34.84
12.47
17 18
W.N.W. North. N, 2 W.
24 23.97
S. W. S by W.
S by W. W. 37
N. N.W. N. E. N.E.byNE 10 7.41
6.72
Weft. S.Wby V IV; 19
8.13
S.W. S. S. E. SF. E. 43 28.87 31.86
W. N. W.W.N.W.V.
E. by N. N. by E. N. byw. W. 19 18.43
S. E by Sis. E.by S 4E. 12
9.641 7.15
N. E. N. N. W.IN. W. 4 W., 30 20.15
South E. S. E. E. by N. . E. 37
36.95
2 I
7.08
19.77
4.62
22.23
22.23
1.81
78.85
78.85 81.48 82.68 77.45
78.85 177.45
Diff. of Lat. =2.631 5.23 | Dep.
409
Anſwer.
The Ship is in the Latitude of
37'7. North
Her bearing from her firſt Station
S. 63° 18 E.
Her direct Diſtance
5.85 Miles.
Example 6.
A Ship in the Latitude of 40 deg. 30 min. North, is bound to a
Port in the Latitude of so deg. oo min. North, the Departure to the
Eaſtward 158 Miles, the Variation i$ Weſt, ſhe fail'd on the direct
Courſe 30 Miles, and then the Wind came up at North, and ſhe
hauling up, went E. N. E.56 Miles, and then the Wind being N.
by E. ſhe went away E. by N. 48 Miles, then the Wind was N. E.
and ſhe Tack'd and ſtood away N N. W. 44 Miles ſhe making on
each Courle i Point Leeway, then the Wind ſprung up fair at S. E:
ſo ſhe ſail'd on her direct Courſe towards the Port 56 Miles, then the
Wind coming to the N. E, by E, ſhe went away N. by W. 40 Miles,
ſhe ſtill making i Point Leeway, then the Wind came up fair. I
demand the Latitude the Ship is in, her Bearing and Diſtance from
her former Station, and her dire&t Courſe and Diſtance to the Port.
Anſwer
.
172
Navigation
.
Anſwer.
Latitude come into
49° 18' North
Bearing from her firſt Station
N. 27° 39' Eaſt.
Diſtance from her firſt Station
190.51 Miles.
Direct Courſe to her Port
N 59° 21' Eaſt
Dire&t Diſtance
80.89 Miles
Example 7:
Being at Sea, and by a good Obſervation at Noon, I find the Ship
to be in the Latitude of 47 deg. 30 min. North, having then run by
Account 210 Miles to the Weſtward of the Port I failed from, (which
was in the Latitude of so deg. oo min. North) and being then bound
to a Port in che Latitude of 43 deg. 10 min. N. and likewiſe 180 Miles
to the Weſtward of the Port I failed from, and having the Wind
at S. S. E, S. and S. W. was obliged to ſteer the following Courſes,
Eaſt so Miles, then S. W. 54 Miles, then E.S E. 67 Miles, then
W.S.W. 80 Miles, then S-SE. 50 Miles, ſhe making on each Courſe
Iż Point Leeway, the Variation being one Point Weſtward, and then
the Wind (prung up fair at N.N.W. I demand the Latitude the Ship
is in, her Bearing and Diſtance from her firſt Station to the Port the
fail'd from, her direct Courſe and Diſtance from her firſt Station, to
the Place ſhe is now in ; alſo, her Bearing and Diſtance from the
Station ſhe is now in, to the Port the firſt fail'd from, and upon what
Courſe ſhe muſt fail, and how far, to arrive at the Port ſhe is bound to.
Anſwer.
The Ship is in the Latitude of
46° 30': North
Bearing from her firſt Station to the Port She} s. 54° 28' Weſt.
fail'd from
Diſtance
from her firſt Station to the Porr ſhe}
258.1 Miles
Her Courſe from her firſt Station to the Place S. 27° 47° Eaſt
ſhe is now in
Her Diſtance from her firſt Station to the
}
Place ſhe is now in
66.76 Miles
Her Bearing from the Port She ſaid from, to} s. 40° 33' Welt
the Place the is now in
Her Diſtance from the Port ſhe ſail'd from, to
to}
the Place ſhe is now in
Her direct Courle from the Place ſhe is now ? S. 00° 19' Weſt or
in, to the Port the is bound to
due South fere.
Her direct Diſtance from the Place ſhe is now 2
in, to the Port ſhe is bound tom
Š
A
.
ol
275.13 Miles
200.943 Miles
A Large and very Uſeful
T A B L E
1.
OF
DIFFERENCE
OF
LATITUDE and DEPARTURE
IN
Minutes and Tenth Parts.
to every Degree and Quarter Point
OF THE
C O M PASS:
For the Exact Working
OF A
TRAVERSE
London, Printed for Rich. Mount, at the Poſtern on
Tower-Hill.
별
​174
A Table of DIFFERENCE
Dilt.
1
01.0
1
OI.O02.0
2.0 100.1
Diſt. annanol
2
6106.0 0.1
IO
II
16.9 OL.S
21.000.4
004
8 ܐܐ
22.0
23
42 42.000.7 42.001.5 41.902.5 141.902,2 41.902.9 418 03:7
45 145.0 00:8|45.001.6 44.902,244.902.4 44.9 P3.1. | 44.8
46 46.000,8 46.0 016 45.902,2 145.902.445.9 3.2 45.8
47 47 000.8 47.001.6 46.902.3 46.902.5 46 9"3.3 46.8
Depl Lat Depl Lat Dell Lat Dep 16 49.9 103.5 49:8 24.4
I Deg. 2 *
2 Deg. 1 Point
3
3 Deg. 4 Deg.
5 Deg.
Lata Dep Lat | Dep Lat| Dep Lat, Dep Lat Dep Lat Dep
101.000.0
O1.000.0
01.] 10.0
00.1
. I 01.0 100.1
202.000.0
-)2.000.1
O2.0 0.1
2.0 100.1 0 2.0 OO. 2
3103.000.103.000.1 03.01001
03.09.203.000.2 03.05.3 3
404.000. I 04.00. I
34.000.2 04.010.2 04.000.3 04.0 100.3
S05.0 100.1 05.000,2
05.0 10.2 05.05.305.000.3 105.005.4
06.01 00.2 06.000.3 06.000.3 06.000.4 06.000.5
6
707.000.107.000.2 07.000 3 07.000.4 07.000.5 107.000.6 7
308,000.1
08.009.3 108.000.4 03.003.4 108.000.608.000.7 8
909.05.209.000.3 09.05.4 09.00u.s 09.000.6 09.000.8 9
IO 10.000.2
10.000.4 10.000.5 10.0 0.5 10.000.7
10.000.9
II11.000.2
11.000.4 I1.000.5
11.005.6 II.000.8 II.OTOI.O
121 2.000.2 I 2.000.4 12.000.6 13.0 10.1,6 12,0 100.8 I 2.0 101.o I 2
13 13.0 05.213.000.5 13.009.6 13.009.7 13.0 100.9 12.901.I 13
14 14.002.2 14.0 Ocis
14.0 CO.7
14.0|09.7 14.0010 13.901.2 14
15115.000.3 15.0 100.5
15.09.7
15.0 103.815.001.0 14.901.3 15
16 16.0 09.3 16.000.6 16.0oo.s 16.0 100.816.00I.I
15.9 01.4
16
17117.000.3 17.0 90.6 17.000.8 17.000. 17.001.2
17
IS118.000.3 18.09.6. 18.003.9 13.000.917.9 01.3 17.901.6 18
1919.000.3 19.0 09.7 19.000.9 19.0010 18.901.3 18.9 01.71 19
20.000.4 20.000.7 20.0 1.0 20.0 01.019.9 914 19.9 01.7 20
21
21.003.7 21.0 OI.O 21.601.120.901.5 20.901.8 21
2
22.0 03.8
22.01011 22.0 101.1 21.901.5 21.901.9 22
23 23.009.4123 000.8 23.0101.I 23.0 01.222.9101.6 122.902.0
24 24.0 20.4.24.0 20.8 24.001.? 24.0 21.3/23.9101.7 123.902. I
251 25.000:4 25.0 CO 9
25.0 0.2 24.0101.3 124.901.7 24.9|02.2
25
26 26.0109.5 | 26.000.9 26.0 101.3 26.0 91.425.9 01.8 25.902.3 26
26.9
27 127.0 03.5 27.100.9 127.0 01.3 27.301.4 01.9
27
28 128.000.5 28.0 01.0 23.001.4 29.0 101.5 27.902.6 27.902.4 28
28.9
29 29.000.5 29.0 1.0 29.0 21.4 29.0 101.5 02.028.902.5
02.1
30.011 30.0 101.5 33.0 101,6 29.9
30 30.000.5 1 30.01 OLI
29.902.6
30
31 310 00.5 31.0 01.131.0 101.5 31.0 21.6 30.9 02.2 30.902.7
31
32 32.000 6 32.001.1 31.9101.6 31.9101.7 31.902. 2 31.902.5
32
33133.000*6 33.001.2 32.901.6 32.901.732.9 22.3 32.902.9
33
34 34.000.6 34.01.233-90107 33.9 o1.8 33.9 02.4 33.9 03.0
34
35 35.000.6 35.9912 349 01.7 34.901.8 34.9 02.4 34.903. 35
36 36.000.6 36.001:3 35.9 01.8 35.901.935.9 102.5 35.9 103.1 36
37 37.000.737.001:3 36.901.8 36.901.936.902.6 36.903.2
37
38 38.000.7 38.001.337.901937.902.37 202.7 37.903.3
38
39 39.000.239.01.4 38.901.9 33.902.938.9 02.7 138.903.4
39
40 40.0 10.7
39.902.039.902.39.9102.5 139.8 03.5
40
41 41.000.7.41.0 01.4 40.902.0 40.9 22.1 40.9 2.9 40.8 03.6
42
43 43.0 20.8 43.0 101.5 42.9 22.1 42.992.2 42.9 103.6 42.8 03.81 43
44 44.000.8 44.0/01.54309 02.243.902.343.9 103.1 43.303.8
44
03.9
45
04.0
46
|
04.1
8.000.8 48.0 01.7 47.702.347.9 oz. 47.9 33.4 47.
48
49 49.000.949.0017 48.9 22.4 48.9 102.6 48.9 53.4 48.8 04.3
49
50
Lat Dep. Lat Dep|Lat
189 Deg. 138 Deg. '7Point./87 Deg. 186 Deg 185 Deg.
24
26.902.4
:29
40.0 01.4
41
47
&
04.2
48
50 50.000.9 50.0 01.8
9|Diſt.
ToDict
Y
of LATITUDE and DEPARTURE 17:
Diſt
Liit.
S
.....
61
ser
62.2305.5
65.7 5.8 6
95.001.24.203.394.904.7194.91050 194.806.6 94.6 08.
01 Deg. 2 Deg.
1 Point 3 Deg
4 Des. 5 Deg.
Laty Dep Lat, Dep Lat| Dep Lat Dep Lat| Dep Lat Def
SI 51.000.951.0 101.8 50.902.5 50.9 02.7
50,902.5 50.9102.7 50.923.6 50.8 04.4 5
52 52.000.9152.0 101.8 51.902.5 51.9 02.7 51.9 103.6 51.8 04.5 5
53 53.000.9 53.0 01.852.902.652.902.8 52.3 103.7 52.8 04.6
54 54.000.9 54.001.953.902.653.902.8 153.903.3 53.9 24.71 5.
55
55.0 01.055.0 101.9 54.902.7 54.) 02.9 54.9 103.3 54.8 04.815
5656.00L0|56.002.5 559|22.755.5| 02.) [559 103.9 155.8104.9 56
57 57.01 01.057.002.056.902.9 56.903.156.904.0 56.825.05
58158.01 01.0158.002.0 57.902.357.903.0 57.904. I 57.995.1
59
59.0 01.059.002. I 58.902.9 58.903.158.8 04.I 55.825.2
3
60 60.0 01.0 60.002.1 59.902.959.90z.159.8 04.2 15.05.21 :
61 61.0 1.1 161.002. I 60.7|23.0609103.2 60.8 04.3 70.3 15.3
62 62.01.162.002.2 61.903.061.9 03.3 61.8 04:3 161.7 25.4
63 | 63.001.163.002.2 62 903.162.903.3 62.9
03.3 1 04.4
6:
64 64.001.164.002.263.903.1|63.9 103.4 63.8 04.5 163.8256
6565.001.1165.0 02.3 1649103.2 1649103.4 64. 04.5 64.7 125.76
65.9 65.9 03.5
67 67.001.267.. 02.3 66.9 103.3 66.9 03.5 66.8 04.7 56.705.9 67
68 68.01.2670202.4 67.9193.3 67.903.6 67.8 04.8 67.7 25.9.6
69 69.401.2 168.902.4 68.903.4 68.9 03.6 68.8 04.8 68.7 06 0
70 70.01.2169.902.469.903.469 9103.7 69.8 04.9 69.7 06.117
7171.001.270.902.5 70.903.5 709 03.7 70.8 05.0 70.7 06.27
72 72.101.371.902.5 71.9103.5 71.9103.8 71.805.0 171.706.31 72
73 73. 01.3172.902.5.1 72.903.672.903.8 72.8 1051 72.7 56.4 73
74 74.01.373.902.6 73.2 3.6|73.903.9 73.8 05.2 173.7
73,7 106.5) 74
75 75.0 01.3174.902.6 74.9103.7 74.903 9 74.8 O5.2 74.7 06.6 75
70 76.0 01.375.902.775.903.775.904.0 75.805.3 75.706.6 76
77) 77.003 76.902.776.903.8 76.9 04.0 76.805.4 176.7 26.7 77
78 780001.4177.9 02.7 77.903.8 77.904.1 77.805.5 177.7 6.3 78
79 79.0 01.4178.902.8 78.903.9 78.904.1 78.805.5 78.7 06.9 77
80 80.0 01.479.9 02.879.903.9 79.9 04.2 79.805.6 79.7 07.080
81 81.001.4 30*
902.8 80.9 04.0 80.904.2 80.905.7 80.7 07.1
07.1) 81
820 82.0 01.4 31.9102.9 $1.9 04.0 81.9104.3 81.805.7 81.7 107.2 82
83
83. 01.4
32.902.9 82.9 041 32.904.4 82805.8 92.7 107.31 83
84
84.fol.s $3.902.5 83.9 4.1 33.904.4 83.9 05.9 83.7|07.3 84
85
85.Cor.5 14.9 03.084.9104.2 34.904.5 34.805.9 $4.707.4 8s
86
$5.9 85.904.2 35.9104.5 89.8 06.0 35. 07.5 86
87.0 01:5|86.9 03.0869104.3 36.904.6 86.8 i 06.1 86.7 07.6 87
88 88.0 91.5 87 9103.1 87.910.1.3 37.9 54.6 87.8 06.2 87.797.7 89
89 89.0 91.5 38.9 103.188.9 34.4 38.904.7 88 8 06.2 83.707.8 89
90 90.001.689.9.03.189.904.4 134.91 24.789.8 06.3 89.707.91 90
91 91.001.6 90.9.03.2 90904.5 190904.8 90.8 05.4 90.7 08.0 91
92 92.001.c.1.9103.2 91.9104.591.904.8 91.8 06.4 916108.0 92
931 93.001.6929103.2 92.9 04.692.904.9 92.8 16.5 926 08.11 93
94 94.001.-3.903.393.914.693.904.993.806,6 93.6 08 94
95
95
96 96.0 01.795.903.495.904.7195.905.0 195.805.7 1956 28.2 96
97 97.091.796.903.4 16.904.596.9 05.106.806.8 96.603.3) 97
981 98.0 01.727.9 3.4 97.4 34.8197.995.197.8 069 97.6 8.98
99 99.0 01.728.9 23.5 23.9 54.943.9|05.203.805.9 98.6 58.77 99
100 100.0 01.99.903.5 99.9 31.997 45.2 1198 07.3 193282100
Dep Lai Depl Lat Dep Lat Dell Lat Dell Lat Depl Lat
89 Deg. 88 Deg.17 Point.184 Deg. 136 Deg. 185 Deg.
А а 2
1
CO CO CO CO CO CO CO CO CO
86.0 01.5
87
Diſt
176
A Table of DIFFERENCE
Diſt
Diſt.
OI.O DI
2.0 0,2
1
2
6
8
IO
II
12
. We
w
12.8
02.0
8
16
20
21
22
04 22.8 102.8
34.8 3.7 | 34.7 04:3
39 38. P3.8 38.804.130.704.8 | 38.6
4039.5 3.9
4241.8104."141.894.4 41.70S. I
43 42.8104.2 42.8 04:5 42.7|05.242.606.042.5|06.3 42.506.7
44 43.8 04:3143.7 104.64347 05.4 43.066.143.806.5| 43.5 106.9
Point. 6 Deg. 7 Deg' 1.8 Deg. Point, 9 Deg.
Lat, Dep. Lat DepLat, Dep Lat Dep Lat Dep) Latſ Dep
I 01.ooo. i 01.0 DO.I
OI.OOO.101.000.1 01.000.21
2
02.000.2 02.03.2
02.0 no.3 02.000.3 02.000.3
302.000.3 03.090.33.0 534
03.000.4 03.009.4 03.000.5
4104.000.4 04.010.4 37.0 23.5 04.000.604.000.6
03.900.6
4
Slos.o 00.5 65.0 0.5 105.00.6
04.9037 04.903.7 | 04.900.8
606,000.6 106.0 30.6 36.0 05.705.900.8 05:9 00.9 05.900.9
7
07.6 (0.737.0 10.7 105.909.806.9 01.0|06.901.006.901.I 7
8 08.c 100.8 5.0 33.8 07.2010 07.901.107.9 0.21 07.901.2
9 09.000.9 0.105.9 108.901.1
08.901.108.901.208.901.3| 08.9 01.4 9
10 09.9101.0 109.901.0|09.901.209.901.4 09.901.509.9 01.6
II 10.9 OLI 10.901.1
10.901.3 10.901.5 10.901.6 10.901.7
12|11.901.201.9111.2 11.901.5 11.9101.711.9 01.8 11.8 01.9
13 12.01.312.901.4 12.401.6 12.901.8 12.901.9 12.
13
1413.901.413.9 101.5 13.901.7 13.901.9 13.8 oz. I
13.02.2
14
15 14.901.514.901.6 14.9|018 14.8102.1 14.8 02.2 14. 02.3 IS
16 15.9 01.6 15.9 01.7 15.9 01.9 15.8 02.2 15.8 02.3 15.8 02.5
17 16.01.7 16.901.8 16.902. I 16.8 02.4 16.8 2.5 | 16.802.7 17
1817 OL.8 17.9 21.9 17.402.2 17.8 02.5 17.8 02.6 17.8 02.8 18
19 18.9019118.902.0 18.902.3 / 18.8 02.618.8 02.8 | 18.8 03.0 19
26 19.9-2.0.17.9 22.119.802.4. 19.8 102.8 198 02.9 19.7 03.1
21 | 2014 02.120.9106.2 20.802.6 20.8 02.9
20.8 02.620.802.9 20.8 03.120.7|03.3
22121. 102.2 21.702.3 21.802.7 21.8.03.1 21.8 03.2 21.7 02.4
22.9
22.303.2 22.7 03-4 22.703.6 23
24/23.> C23 23.902.5 23.8|02. 23.03.3 23.7 23.5 23.703.8 24
25 24.502.4 24.902.6 24.8103.0 24.8|03.5 | 24.7 03:7 24.703.9 25
26 25.9 2.5 25 902.7 25.8103.2 25.7703.625.7 03.8 25.7 04. I 26
27
02.8 26.8103.; 26.703.7 25.7|04.0 26.7.04.2
26.9
27
28271907 27.8 02.9 27.803. 27.703.927.704.127,6 04.4 28
29
23 y 102.8 26.8 03028.803.5 28.704.028.704.2 28.6 04.5 29
30 29.8102.9 29:8 03.12488103.7 23.704.2|22.7 04.4
30
31 | 30.8 03.0 30.8103.2 30.8 03.8) 397 04.3 307 04.5 30.6 04.9 31
32 31.8 03:131.8 03.3 31.8 -3.9 31:7 04.4 31.6 04.7 31.6 05.0 32
33
32.803.4 32.7 04.0 327 04.6 32.6 04.332.605.233
34 33,8 p.3.3 33.8 103.5 33.7 04.133.7 04.7 33.605.0 33.605.3 34
34.7
|
35
303.003. 35.8. 03.8 35:7 04.4 35.605.0 35.6 35.3 35.5 05.6 36
37 36.8123.6 36.8 103.9 36.7104.5 36.605.136.61 25.4 36.5|05.8 37
38 37.8 337 37.8 04.0 37.7040 37.505.3137 6105.637. 06.0 38
|
39
39.8
05.6
40
40 8 104.0 40.8 04.3 70.705.6 40.6 05.7 40.606.040.5|06.41 41
41
05.8 41.506.2141.5 406.6
42
43
44
45 44.8 04:4 44.7 04.7 44? 05.144.606.3 44.5 6. 44.4 07.0 45
46 45.5 104:45.7 04.8 45.7105.645:206.4 45:206.7 45:4 07:
246
476.8 104.646.7 04.9 40.0 0.7 46.505.5 46 06.9 46.4107.3 47
48.478 104.7 47.7 105 476 105.147506.7 47.5 07.0 47.407.5 48
8.8 04.8 46.105.148.6106.6 48.5 106.8 48.5 107.2 18.407.7
49
49.5 1049 149.71 05.249.606. 42:5 07.0 49.5|07.349.407.8
Deflat Del Lat Dell Lai Dep. Lai Dep. Lat Depi Lat
17{Point.
Point. 84 Deg, 83 Deg.I 82 Deg Point.) 81 Deg
26.9 102,5
29.6 04.7
O VOO W
35 34.8103ng
416
49
So.
Diſt.
11
of LATITUDB and DEPARTURE
177
Diſt.
Lat Dep
Dift.
66.2 10.5 67
IO.O
10.1
11,1)
11.41 73
Point 6 Deg.
1 Deg. 8Deg Point, 9 Deg.
Latq Dep Lat Dep Lat Dep Lat Dep Latej
51 sayl 05.053.7 05.3 50.6.05.250.5 07.050.4 27.5 50.4 08.0 SI
52
$1.7 35.15 1.7 105.4 51.605.3 51.5 57.2 11.4 27.6 51.4 08.1: 521
53 52.7 05.252.705.5 52.6 105.5 52.5 27.4 52:
407.8 52.30%.3 53
54 53:7 35.3153.705.5 53.6 05.6 53.5 27.5 $3.407.9 53.3 | 08.4 541
55
54.7 05.4 54.105.7 154.5 25.754.51 07.654.4 101.I 54.3 08.655
$6
55.705.555.705.8 55.6 06.8 55.507.855.4 03.2 55.308.71 56
57. 56.7 25.6|56.7 06.0 56.606.9 56.407.) 56.4 35.4 56.3 | 08.91 57
58
57.705.7 7.7. 06.1 57.6 07.1 $7.401., 57.403.5 57,3 09.1 58
59 58.7105.853.706.258.607,258.41 08.255.403.7 58.3 09.24 59
60 59.7 05.957.706.3 59.5 107.3159.4 08.359.3 03.8 59.3 | 04.41 60
61 60.4106.0 50.7 06.4 160.5 07.4 50 4 08. 169.3
60.3 03.9 60.2 09:51 61
62 61.7106.161.6 06.5 61.507.661.4 03.661.3 07.1 61.209.7 62
63 62.706.252.6 06.6 62.507.762.4 68.8 62.3.09.2
62. 2 09.9 63
64 63.7 06.353.6 06.7 163.5 07.363.403.963.3 07.4 63.2 iod 64
65 64.706.464.6 06.8 64.5 107.7 64.4 0.1 64.3 0.9.5 64.210.265
651:
66 65.706.3165.6 06.9 65.5 23.0 65. 109.2 65.307.7 65.2 10.366
67 66.7 56.666.507.0 66.5 103.266.3 09.3 66.309.8
68 67.7 16.7 67.607.167.5 38.367-309.567.3
67.2 10.6.68
69 68.705.8 13.6 107.2 63.501.4 68.3 0 1.6 63.2
68.1 10.81 691
70 69.706.9667.6 07.3 69.5 93.5169.3|07.769.2 10.3
69.1 10.9 70
71 70.6 07.070.607.4 70.5 8.7 70.3 0.7 70.2 10.4 70-1
71
72 71.907.1171.607.5 171.5 08.871.319.371.2 10.6
71.1 11.31 72
73 72.5 07.172.607.6 72.4103.972.3 19.272.2 10.7 721
74 73.6 07.273.6 07.7 73.405173-3 10.3 73.- 10.2 73.1 11.6 74
75 74.607.3174.6, 07.3 74.40).1 74.3 10.4 74.2
74.1
75
71 75.607 4175.6107.9 175.40931753 10.6 75.211.1
75.1
77 76.607.5176.6 08.0 76.4 07.1176.215.776.211.3 176.0 12.0
77
78 77.607.6 77.6 103.1 77.4 01.5177.2 10.9 7701 11.4
77•1 1 2.2
78
79 78.607.778.6 103.3 173.4 94.678.2 11.078.1 11.6 78.0 12.4
79
10 79.607 8179.6 08.4 79.41 34.3 179.2 11. I 7). 1
79.0 12.5 80
31 80.607.93905 03.5 10.4 97.9 80.211.380.1|11.9
32 81.608.081.501.6 85. 10.381.211.4 31.1
12.5
81.0 12.8
13
82.4103 iliz.s 03.7 82.17.1822 11.5 82. I
83..233.5 03.3 83.4 19.233.211.7 83.112.3 83.0 3.1 84
84. 08.45 2. 84.110.334.11.887.1 [ 2.5 83.9'3:31 85
85.03.4 5.5 0). 35.4 12.45 12.0 1351 | 12.5 34.7 13.4 86
86.608 5136.5 0.1 36.310.53. 35.0 12.8
sál 37.08.147.50.).2 87.215717., 12., 37.0 12.9 86.4 13.8
88
y $8.00.3.7 38.5 21.3 33.3 10.) 33.5 124 330 13.1 .
90 89.008.67.510 1.4 97.3 11.0 11.2.5 89.0 13.2 88.4 14.1
90
91 9 16 08.990.5 10.5 70.3 11.1 1 2.7 7.0 13.4
89.9 14.2
91
92 91.09.15.6 (1.2191.12.8
11.7 13.5 90.) 14.4
92
93 92.60).13.5 10.3.7 2.3 1.3
12.0 13.6 2018 14.5
_93
94 93.5109.2 +3.509.3 73.7 11.5 3. 13.193,0 13.8-192.8 14.7
94
os 94. 09.334.5 109.9 1.311.5)!. 13.294.0 13.9
13.8 14.995
96145.5095.5 110.3.11.7950113.4 5.3 14. I 94.8 15.0
96
27 96.509.5.5 10.1 -16.7 11,31 15.0 13. 95.714.2 95.3 15.2
97
981 97.5| 09.077.5 13.2 97.3 1 2.0 17.0|13.6 196.) 14.4 96.9 15.31 98
98.5107.8.5 10.3 13.312.800 13.397.9 14.5 97.8 15.5
99.5|09.8 95.4 10.499.2 12:2
99
17. | 1.13.) 14.7
OS.S 15.6
Dep. Lai Dep. Lat Dept Depl Lat Dep. Lar DepLat
71 Point 34 Deg: 18; Deg. 82 Deg. 74 Point
81 Deg.
II.
11.7
11.9 76
11.7
80.0 12.71 81
82
$2.0 13.083
I 2.2
I 2.1
85.9113.687
7
70
2.1
( 2.
15.6 100
*8 Diſt.)
178
A Table of DIFFERENCE
Dift.
Diſt.
OI.Ooo.2
ol'o 00.2
OI.COo.2
I
2
1
2
00.4
coulanow
6
6
1o
10
II
I2
16
036
20
20
22
22
27.3106.3 1276206.81
1o Deg. 11 Deg i Point 12 l'eg 13 Deg. 14 Deg.
Lat Def Lat. Dy Lat| Dep Lai Dep Lat, Dep (Lat. Det
01.000.2
010 00.2 01.000.2
02.0co.3 02.
02.0 00.4 C2.000.4 01.900.4 01.9 00.5
3
02.9co.502.9oc. 02.9 100.6 02.900.6 02.900.7 02.900.7 3
03900.7
03.9 co.& 03.9oo.8 02.00.8
4
03.900.9 103.901.6 4
5
04.900.9 04.900.9 04:9 01.004.101.o 04.901J 04.8 01.2 S
05.9CJ.O 05.9 01.) 05.9 01.2 05.01.2 07. 01.3105.SOT.4
901.206.9012
7
06.9 01.4 06.101.5 106.8 101.6 CC.8101.7 7
8
07.901.4107.8 01.5 07.8101.6 07.01.107.801.8 07.801.9 8
9
08.901.6 08.801.7 08.8101.8 08.& 101.9 | 08.802.008.7 | 02.2 9
09.801.709.801.509.800.9 09.02. I 09.7 02.209.702.4
10.8 01.9 | 10.8 02.) 10.8 102.1
II
10.102.3
10.7 02.5 10.7 1.02.7
II.S 02.)
11.8 02.3
I 2
02.3 11.8 02.3 II.; 02.5 11.7 02.7 11.602.9
13
12.802.3 12.8 |02.5 12.7 102.5 12.7 02.7 12.7 02.9 12.6 03.11 13
14
13.8 02.4 137 102.7 13.702.7 13.7 02.913.6 oz.1 13.603.41 14
15
14.8 102.6
14.7 029
147 02.9 14.703.114.603.4 (14.5 3.0 15
16
15.702.8 15.7 03.015.703. 1 15.03.3 | 15.6 03.6 15.503.9
17
16.702.9 16.7 -3.2 16.703.3 103.5 166 03.8 16.5 104.1
17
18 17.7 03.1 17.7 103.4 17.702.6 17.03.7 17.5 104.0 17.5 04.4
18
19
18.703.3 186
18.603-7186 03.9 18.5 04.2 18.4 04. 19
19.703.5 19.6 03.8 19,6 103.9 19.004.219.504.5 19.4 04.8
21
20.703.6
20 604.0 20.6 04.1 20.5 04.4 20.504.7 20.4. 05. I
21
21703.8 21.6 04.2 21.6.04 3 21.504.6 21.4. 04.9 21.305.3
22.604.0
22.6
04.4 22.6 | 04.5 22.504.8 22.4.05.2 22.3105.6 23
24
23.6.04.2 23.6 04.623.504.7 23.5 05.0 23.4.05.4 23-305.8 24
25
24.04.3 24.504.8 24.504.9.24.405.224.3 105.6 24.3 06.0 25
26 25.04.5 25.5105.0 25.5105.1 25.4 05.4 25.3 195.8 25.2 06:3
27
26.6 047 26.55.1. 26.5 105.3 26.405.6 26.3.06.1 26.206.5 27
28
27.04.9
|
28
29
28.605.0 28.505.5 28.405.7 28.4 06.0 28.3 06.5 28.107.0 29
30
29.5 05.2 29.495.7 29.405.8 29.306.2 29.2 06.7 29.107.3)_30
31
30.5|05-4 30.4 05.9 30 4 06.030.306.4
30 4 06.030.306.4 | 30.2 07.0 30.107.5 31
ži sosis 31.406.1 31.4 06.2 31.306.6 31.207.2 31.0107.7
32
32
33
32.5 05.7 32.4 06.3 32.406.4 32.3 069 32.107.4 22.01 08.01 33
34
33.505.933.406.5 33.3 06.6 133:2 07.1 33.3|07.6 33.008.2
34
35 34:5 06.1
34.4 06.71 34.3 06.8 34.207.3 34.1 07.9 34.0108.5 35
36
35.3 06.9 35-3 07.0 35.2 07.5 35.1 08.1
71 36
36.406.4 36.3107.1 36.3.07.2 36.207.736.0108.335.909.0
37
37
37.4 06.6-37-307.2 37.3.07.4 37.2 07.9 37.0.08.5 36.9109.2 38
38-406.8 38.307.41 38.2 07.6 38.1 08.1 38.0 08.8 37.8 09.4) 39
39.4 06.9 39.307.6 39.2 07.8 39.1 08.3 39.009.0 38.8|09.7 40
40.4 07.1 40.2 07.8 40.2 8.0 40.1 08.5 39.9 09.239.8|69.9
41
41.4 07.31 41.2 08.0 41.208.241.108.7 40.909.4 40.7 10.2
42
42.3107.5 42.2 08.2 42.2 08.4 42.] 08.9 41.909.7 41.7 10.4 43
43.307.71 43.2 08.4 43.1 08.6 43.009.142.9 09.9 42.7 10.6
44
44
44.3 oz.8 44.208.6 44.1 08.8 44.009 4 43.8 10.1 437 11.0
45
45.3108.0 45.208.845.159.0 45.009.6 44.8 10.3. 44.6 11.1
46.3|08.146.109.46.109.2 46.01 0938 45.8 10.6
47
47:3 08.347d09.2| 47.109.4 47.0 10.0 46.8 10.8 46.6 11.6
48.3 8.5 48.109.: 48.1 09.6 47.9 10.2
47.5 11.9
49.2 08.7 49.1 09.5 49.99.8 48.9 10.4 48.7 11.2 48.5 12.1
so
Dep Lat Dep. Lal Dep|Lat Depl Lat Dep Lat Dep Lat
8o Deg. 79 Deg 7 Point 78 Deç. 177 Deg.176 Deg.
23
26
35.4[06.2
34.908
38
39
40
41
42
43
45
46
47
46
45.6
I1.4
* 1 ****** 01 Diſt
49
49
47.7 11.0
48
49
Diſt:
179
I
Dift.
niit.
91.6 16,1 91.3 17.7 91.218.1 191.0 19.3 90.6 20.990 222.5
2.2 23.0
of LATITUDE und DEPARTURE.
10 Deg. 11 Deg. Point 12. Deg. 13 Deg. 14 Deg.
Lat. Dep Lat. Dep. Lat, Dep. Lar. Dep. Lat Depi Lat Dep
SI 10.2 OS.80.109.7 150.0 0.0 49.9 10.5 49.7 11.5 49.5 12.3 51
5251.2 09.051.0 09.9 51.0 TO.I 50.9 19.8 90.7 11.7 50.5 12.6 52
5352.203.2 520 10.1 52.0 10.3 51.8 11.051.611.9 51.4112.8 53
54153'209 453.0 10.35 3.0 10.5 52.8 11.2 52.6 12. I 52.4 13. I 54
5554.209.5154'0 10.5 53.210.7 53.811.4 53.612.453.4 13.3 55
SE55.109.755.0 10.7 54.9 10.9 54.8 11.6 54.9 12.6 54.3 13.5 56
57 56.109.956.0 10.8 55 911.I
55.8 11.8
55.5 12.5 155.3 13.8 57
5857.1 10.1156.9 11.1 56.911.3 56.71 12.1 56.5 13.0 156.3 14.0
58
5958.1 10.2 57.9 11.3 57.9 11.5 1 57.7 | 12.3 57'5 13.3 57.2 14.3 59
6059.1 10.458.915.4 58.8 11.7 58.7 12.5 158.5 13.5 153.2 14.5 60
6160.1 10.6 59.9 11.6598 11.9 59.7 12.7 59.413.7 59.2 14.3 61
6261:1 10.8 60.9|11.8 60.8 12. I 60.6 12.9 60.4 13.960.2 15.0
62
6362.0 10.961.8 12.061.812.3 61.6 13.1 61.4 14.2 61.1115.2 63
64/630 11.1 62.8 12.262.8 12.5 162.6 13:3162.3 14.4 62.1 15.5 64
65 64.4 11.3 63.8 12.4 63.7 12,7 63.6 13.5 163.3 14.6 63.1 15.7 65
66165.0 11564.8 126 64.7 12.9 64.0 13.7 64.3 14.864.0 16.0 66
67 66.0 11.665.8 12.905.7113.1 165.5 13.9 165.3115.1165.0 16.2 67
68 67.0 11.8 66.7 130 66.7 13.3 66.5 14.1 66.2 15:31 66.0 16.4 68
6968.0 12.067.7 13.2 167.713.5 167°5114.3 67.2 15.5| 66.9 16.7 69
70168.912 263.7 13.4 68.7 13.7 68.514.5 68.215.7167.9 16.9 70
71 69.9 12.369.713569.6 13.9 169.4 14.8 69.216.0 68.9 17.2
71
72 170.9 12.5 70.7 13.7 70.6 14.0 70-4, 15.0 170.116.269.917.4, 72
7371.912.7 71.7.3.9 71.6 14.2 171.4 15.2 71.1 16.4 70.8 17.6 73
74 72.9 12.8 72.6 14.1 72.6 14.4 72.4.15.4 72.116.671.8 7.9. 74
75 73.9| 13.0173.6 14°3 173,6 14.6 73.4 15.6 73-116972.8 18.1 75
76 74.5 13.2 74.6 14.5 74.5 14.8 74.3 15.8 74.0 17'1 73.7 18.4
76
77 75.8 13.4 75.6 14.7.175.515.0 75.3 16.0 175 0 17.3 74.7 18.6 77
78 76.8 13.5 76.6 14.9 76.515.2 1763 16.2 76.0 17.5 15.7 18.9 78
79177.8 13.7 77.5 15.1 77.5 15.4 77.3 16.4 77.0 17.8 76.6 19. I 79
80 78.8 13.9|78.5 15.3 78.5 15.6 78.2 16.6 77.2 18.0 77.6 19.3 80
81 79.8 14.1 79.5 15.5 79.4 15.8 79.2 16.8 78.9 18.2 78.6 19.4 81
8280.8 14.280.5 15.6 80.4 16.0 80.217.0 179.9 18.4 79.6 19.8 82
83 81.7 14.4 81.5158 81.4 16.2 181.2 17.2 80.918.7 80.5 20.1
84 82.7 146 :2.5 16.9 82.4 16.4 82.2 17.5 81.818.9 81.5 20.3 84
85 83:7| 14.8 83.4 16.2 83.4 16.6 83.1 17.7 82.8 19.1 82.5 20.6
85
86 84:7 14.9 84.4 16.4 84.3 16.8 84.1 17.9 83.8 19:3 83-4 20.8 86
87 85.7 1501 85.4 16.6 85.3 17.0 85.1 18.1 84.8. 19.6 84.421.0
87
88 86.7 15.386.4 16.8 86.3 17.2 86.1 18.3 85.7 19.8 854 21.3 88
89 87.6 15.487 117.0 8.317.4 87.1 18.5 86.7 20.0 86.4 21.5
90 88.6 15.5 88.3 17.2 88.3 17.6 88.0 18.7 87.2 20.287.321.8 90
9189.6 15.8189.3 17.4 89.2 17.8 89.0 18.9 88.7 20.5 88.3 22.0 91
92 90.6 16.0 90:3 17.6 90.2 17.990.0 19.1 89.6 20.789.3 92
93
93
94 92.6 1663 92.3 17.9 922 18.391.9 19.5 91.6 21.1 97.2 22,7 94
95
96 94.5 16.7 94.2 18.3 94.2 18.7 93.91200
935 21.6193.1 23.2 96
9795.516.8.195.218.5 91 18.9 94.920*2
94.5 21.894.1 23.5
97
98 96,5 17:096,2:8.7 96.1 19.1 95.9 20.4. 95.5 22.0 95.1 237 98
99 97.5 17.2 57.2 18.9 97.1 19.3 96.8. 20.6 96.5 22.3 96.1 23.9 99
100 98.5 17.4 98.2 19.1 98.1 19.5 97.8 2098
97.4 22.5 97.0 24.2.
Dep 'Lat. Dep. Lat Dep. Lat. Depl Lat Dep. Lat Dep. Lat
80 Dg. 79 Deg.
7 Point! 78 Deg. 177 Deg, 76 Deg.
83
CO CO ICOCO
89
222
g|Diſt
og Dift
180
A Table of DIFFERENCE
Diſt.
Diſt.
1o lo.2
01.
1
2
2
2.9
3
4
S
6
OI.5
7
02.2
025
7
8
9
10
II
12
II
I 2
13
14
15
16
17
18
19
20
21
*22
16 15.5 104.0 15.504.1 15.4 104 41 15.! 1.5
31 609.6 31.5 1.51
41.31 2.7 47.1
13 Point, as weg 7 10 Deg IzPoint 17 Deg: 10 Deg.
Lat|Dep Lat | Dep Lat; Dep Lat Dep Lat, DepLar | Dep
29.31 01.0)3101.727.3 21.799.31 13.709.3
01.909.5 01.30.31 olya7.501.9 3.6 01.999.61 11.7
1.00.6
02:9 29.7 22.9 338 02.12.31 0.125.4 90.) 02.300.9
4
03 9 01. 03. 91.0 03.321.1 23.8 21.2 03.01. 03.01.2
04.8 01.2 04.8 21.304314104.8101.5 04.901.5 07.3
05.801.505.8. PL. 05.7 11.505.3 317 05.7 01.7105.7 OI.S
06.8 101.706.891.8 06.101.9 25.7 22.5 05.7 | 02.0) 35.)
Ś07.8 01.907.732:11 07:7-2.2 27.7 52.3 07.602.307.6
08.702.208.7) 12.30},5 22.1. 03.6122.5 03.5 52.6 103.6 02.8
9
Jo 09.7 02.407.72:6 09. 03.502.901.602.909.5 23.1
10.702.5 10.612 81.0.13.0 13.5 23.3 15.5 3.210.1 93.4
I1.6 049 11.6 03.1'11.5 3.3 11.503.5 11.5 23.5 11.193.1
13
12.6 23.2 12.033.41 12.5 103.612 1.23.3 I 2.403.8 12.404.0
13.6103.4 135.93.6 13.5/53.) 13:10 13.49.1 1.3 04:3
15 14.503.5 14.8103: 14.4 241 14.4 144 14:31 94• * 14:3 | 04.5
11.347 15.104.9
17 16.5 104.1 16 424.4) 16.3217 16.706.316.3°5.0 16.205.2
18 17.5 04:4 17:4 24.7) 17:315.2172 23.2 17.215.3 17.10;.6
18.41046 18 4 94:9 18.395.213.295.5
19
13.205.5 15.125.4
20 19:494.919-35.2 19.01.19.105.3 19.105.319.0062
21 20.405.120 335.4 29.105.820.106.1 29.106.12.206.5
2221.305 31 21:21 057 21.105.1 21.0 25.4 21.106.4 27.7 6.8
22.206.0 22.1 105.3 22.616.7 12.2 105.721.707.1
24 23.305.8 23.2 96.2 23.005 623.0 05.3
22.9°7.? 22.307.4
25 24.234.006.5 24.06423.207.3 23.7 27.3 23.3 107.7
26 25.206.3125.106.7 24.07.2 24.2 27.5 24.91 $7.5 24.7 08.0
26.2 96.6 26.107.025 2 07.425.3 97.3 25.3 27.7 25.7 103.3
28 27.256.8 27.07.2 257 107.7 25.8. 03.1 25.1 18.2 25.5 10.5
29
28.107.3
07.5278 105.3 27.8 93.437 7103.5 27.0 09.3
30. 29.1. 07.3 290 27.8) 23.3 33.3 23.7 38.7 231 23.3 2.5 103.3
31 35.107.5 39.903.) 29808.5 27.7 09,3
29 808.5127.7 09,027.693.127.5
0.6
32 310 07.130:9 08:3327 33 33.6993 33.6 23.3 33.4 10.3
34 33.008.3) 32.808 8 32.7 9.4 32.5 3.9 32.5 33.932.3 10.5
33.903.3 33.8) 09.033.6 29.5 37.5 17.2
33.5 10 2
33.3
10.8
36 34.9 08.1 34-8) 09.3 34 99.9 34.41.427110.1342 11.1
37 35.9 103.0 35.709.6 35.0
10.2 35.4 19.7 35.+ 10.8 35.2 11.4
38 36.907.2 36.7 09.8 35.5
10.5 36.4 11.036.3 11.1 35.111.7
39 37.8 459.5 37.7 10 1 37.5
10.7 37.3 1.3 37.3 11.4 37.112.3
40 33.8 9.7
110 38.3 11.6 38.2 11.7 38.5 12.4
11.3 39.211.) 372 12.5 39.0 12.7
11,645.212.2 45.3 12.3 39.9 130
43 41.7 10.4 41.3 11.1 41.3 1.8 41.1 12.5 41.4 12.6
43.9 133
42.7 10.7 42.5 11.3 42;12.142.112.842.112.9
44
41.8
45 43.610.9 43.5 11.0 43.2
12.4 43.1 13.143.113.142.8 13.9
46 44.6 11.2 44.411.944.2
12.7440 1391 47.3 13.4 43.7 14.2
47 145.611.445.4 12.2 45.2
12.245.0 13.6 44.9 13.744.7 14.5
48 46.6 11.7 46.4 12.4
46.1 13.245.9|13.9 45.714.0 45.6 14.8
|
13.546.9 14.2 46.714.3 46.615.İ
48.512.1 48 3 12.9 43.1 13.8 47 314.547.3 14.6 47.1 15.4
Depl Lat Depl Lar Depl Lat Depl Lat Dep. Lat Dep'Lat
i Point 75 Deg.d 74 Deg. Io Point 73 Dez.1 72 Deg.
23 22.305.6
23
24
25
28.0|
10.2
35
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
38.6
41 39.8 10.0 39.6
42 43.7 10.2 40.6
10.3) 38.4
10,6 32.4
IO.9 40.4
13.6
44
more Dift
45
46
47
48
49
of LATITUDE und DEPARTURE. 181
Mift.
SI
52
63
៨ដដដដ៩ala។
66
و)
Point 15 Deg. 16 Deg. I Point, 17 Deg. 118 Deg.
Lat. Deb Lar. Dep. Lat Dep. Lat. Dep. Lat Dep Lati Dep
51 42.5 12.4 49.; 13.2 19.0 14.0 48.3 14.8 143.8 14.9 48.5 15.8
52 5.3.4 12.6 50.2 13.5 49.9 14,3 49.7 15.1 47.7 15.2 49.4 16.1
531:14 12.951.2 3.7 0.914.6 50.715.3 59.715.5 50.4 16.4 53.
54 12.4 13.152.214.0 51.914.9 151.7 15.7 51.615.851.3 16.7
54
55. 53.3 13.4 3.1 14.2 52.9|15.2152:6 16.0 52.6.16.1 52.3 17.0 55.
56 4:3 13.6 4.114.5 53.815.4 53.6 16.2 53.5 16.4 53.3 17.3 56
5755.3 13.855.1 14.8 54.8 15.754.5 16.5 54.5 157 54.217.6 57
58 16.14.156.0 15.0 55.7.16.0 55.5 16.3 55.517054.2 17.9
58
59 7.2 14:357.0 15.3 56.716.3 56.5 17.156.4 17.2 56.1 18.2
59
60 158.2 14.658.0 15.5 57.7 16.5 57.4 17.4 574 17.5 157.1 18. 60
61 59.214,8 58.9 158 58.6 16.5 18.4 17.758.377.8 58.0 18.8. 61
62160.1 15.159.9 16.1 596 17.1 19.3 18.0 159.3 18.1 19.0 19.2 62
63 61.1 15.360.8 16.3 60.5 17.4 60.3 18.3 160.2 18.459.9 19.5
64 62.1 15.5 61.8 16.661.5 17.6 61.2 18.6 61.2 18.7 60.9 19.8 64
65 630 15:8 162.8 16.8 62.5 17.9 62.2 18.9 62.219.0 61.8 20.1
65
66.64.0 16.0 63.7 17.1 63.4 18.2 53.2 19.2 63.119.3162.8 20.4
67165.0 16.3164.7 17.4 64.4 18.5 64.1 19.4 64.1 19.6 63.7 20.1 67
68 66.0 16.5 65.7 17.6 65.4 18.7 65.1 19.7 65.0 19.9 64.7 21.0 68
69 66.9 16.8 66.6 17.9 166.3 19.0 66.0 200 65.0 20.265.6 27.3
70 67.9 17.067.6 18.167.319.3 67.0 20.3 66.9 20.5 66.6 27.6
70
7168.9 17.268.6 18.3 68.2 19.6 678920.6 167.9 20.8 67.5 | 21.9
72 69.8 17.5 69.5 18.6 69.2 19.8 (8.3 20.9 68.8
68.8 21.0 68.5 22.2,
72
73 70.3 17.7 70.5 18.9. 70.2 20.1 69.821.2 169.8 21.3 69.4 22.6
73
74 71.8 18.071.5 19.171.1 20.470.8 21.5 70.8 21.6 70.4 22.9 24
7572.7 18.272.4 19.4 72.120.7 71.8 21.871.721.9 71.3 23.2
75
7673.7 18.5 73.4 19.7 73.0 20.9 72.7 22. I 72.7 22.2 72.3 23'5
76
77 | 74.7 18.7 74.4 19.9 74.021.273.7 122.3 73.6 22.5 73.2 23.8
77
78175.7 18.975:3 20.2 75.0 21.5 74.6 22.6 74.6 22.8 74.2 24.118
79176.6 19.276.3 20.4. 75.925.8 75.622.9 175 523,1 175.124.4
79
8077.6 19.4 27.3 20.7 176.922.0 76.6 23.2 76.5 234 76.1 24.7 8a
81 178.6 19.7178.2 21.0 77.9 22.3 177.5 23.5 77.523.7 77.0 25.0 81
82179.5 19.9179.225.2 78.8 22.6 78.5 / 23.5 78.4 24.0 78.0 25.3 82
83180.5 20.380.220.5 79.8 28.9 79 4 24.1 79 4 24,3 78.925.6
84 81.5 20.4 81.1 -1.7: 80.8 23.1 80.4 24.4 80.3 24.5 79.9 26.0 84
85.182.4 20.7 621
1200 $1.7 23.4 813 24.7 81.3 24.8 80.8 25.3
85
86 83.4 20.983.1. -2,3 32.7 23.7 82.3 25.5 82.225.18 1.8 26.6 86
87 84.4 211 84.022.5 33:
624.0 83.3 25.2 83.225.4 82.7, 26,987
8885.4 21.4 85.0 22.8 34.6 24.284.225.5 84.125.783-727.2 88
89 86.321.686.0235 85.6 24.5 35.2 25.5 85.1 26.0 84.6 27.5 89
90 87.3 21.9 86.9 23.3 86.5 24.8 86.226.186.126.3 85.6 27.8 90
91 88.3 / 23.1 87.9 23.5 87.5 25.1 87.1 26.7 87.0 25.6 86.5 28.1 91
9289.2 22.4 88.9 23.888.4125.3 88.226.7 88.0 26.987.5 2.8.492
93 90.2 22. 89.8 24.1 89.4 25.6 89.0 27.0 88.4 27.3 88.4 28.793
94191.223.8 90.8 24.3.190.4 25.9 90.0 27.3 899 27.5 89.4 29.0 94
95 92.1 23.191.8 24.6 91:3 26.290.9 27.6 9018 27.390 3 29.3 95
98 93.1 23.3 92.7 24.8923126.4 91.9 27.9 91.928,191.3 29.7
96
9794.1 23.693.7 25.193.2 26.7 92.8 28.2 92.8 23.492.3 30.0
97
98 95.1 23.8 94.7 25.4 24.2 27.0 93.5 28.4.1937 28.693.2 30.3 98
99 96,0 24.1195.6125.695.2 27.3 947 28.7 94.7 28.9194.2 30.6 99
100 97.0 24.3 96.6 25,9 96.1 27.6 95:7 29.0 95.5 29.2 95.1 30.9 100
Dep Lat, Dep. Lat Dep Lat. Dep. Lat Depl Lar Depl Lat
slo
16 a Point 75 Dog! 74 Deg's
Poing 73 Deg. 72 Deg.
Bb
71
83
Dilt
182
A Table of DIFFERENCE
Diſt.
Diſt.
!
1
2
nanono
10
11
12.204.4
14
20
21
22
1
.
19 Deg.Point 20 Deg: 21 Deg 22 Deg.12 Points
Lat|Dep LatDet Lat Dep Lat| Dep Lat Dep Lat Dep
Io0.9 09.31 00.900.3 00.900.300.900.4 00.900.4 100.900.4
2 0..9 103.601.900.71 01.900.7 01.900.7 01.800.701.800.8
3 02.8 los.o02.8 |0.0 02.801.002.8 OL. I 02.8 OLI 102.8 01.I 3
4 03 801.3 03.8 01.3 03.801.403.7 01.4 03.7 | 01.5 103.701.5 4
04.7 01.04.701.7 047 01.704.701.7 04.601.9 04.601.9 $
6
05.701.905.6 02.01 05.6 02.005.6 02.1 os.6 02.205.502.3
6
706.6 92.3 06.622.4 06.6 02.4 06.5 102.5 | 06.502.6 06.502.7 7
07.6 02.807,502.707.5 02:7 07.502.907.4 03.0 07.4 03. I
8
908.5102.908.5|03.0 03.503.108.403.208.3 03.4 08.3 03.4 9
10 09.503.307.4103.409.4 03.4 09.303.6 09.3103.7 09.2 oz.:
10.4 03.10.4103.7 10.3 23.8 10.3 03.9 10.2 04.1 10.2 04.2
12 11.303.9 11.304.0 11.3 04.111.2 04.3 11.104.5 11.104.6
13 12.3 (04.212.2 12:204.4 12.104.7 12.0 04.9 12.0|05.0 13
14 13.2(04.6 13.204.71 13.204.8 13.105.0 13.008.2 12.9|05.4
15 14.204.9 14.105.1 14.105.1 14.0 105.4 13.905.6 13.9105.7 15
16 15.105.2 15.105.4 15.0 05.4 14.905.7 14.8 06.07 14.8106 1
16
17 16.105.5 16.005 2 16.0 05.8 15.9 06.115.8 06.4 15.706.5 57
18 17.0 05.9 16.906.1 16.9 06.3 16.806.4 16.70607 16.606.8 18
19 18.006.1 17.906.4 17.9 06.5 17.706.8 17.6 07.1 17.6 07.3 19
2018.9 106.5 18.806. | 18.8 106.8 18.7.07.218.5 07.5 18.507.6
2119.906.8 19.8) 07.1 19.7|07.2 19.6 07.5 19.5 07.9 19.408.0
22 20.8 107.2 20.707.4.2007 07.5 20.5 07.9 20.408.2 20.3 08:4
23 21.707.5 21.707.7 21.6 07.9 21.5 08.2 21.308.6 21.208.8 13
24 22.707.8 22.603.1 22.5 08.222.4 08.6 22.2, 09.0 2 2.2.09.2
25 23.6 08.1 23.508.4 23.508.5 23.3 09.0
23.3 09.023.2 09.4 23. 109.6
25
26 24.608.5 24.5 08.8 24.4 08.9 24.3109.3 24.1|097 204.009.9 26
27 25.5 68.8 25.409.1 25.4 09.225.209.7 25.010.1 24.9 10.3 27
28 26.5 (09.126.409.4 26.3.09.6 25.110.0 26.0 10.5 125.9 10.7
28
29 27.4109.4 27.3 09.8 27.2 09.9 27.1 10.4 26.9 10.9 26.8 11.I 29
30 /
28.4 09.8 28.2 10.1 28.2 10.3
28.2 10.3 28.0 10.7 27.8 11.2 27.71 11.5 30
31 29.3 10.1 | 29.2 10.4 29.110.6 28.9 11,1 28.7 11.6 28.6 11.91 31
32 30.3/10.430.1 10.8 30.1 10.9 29.9 11.5 29.7 12.0 29.6 12.2*:
32
33 31:2 10.7 31.1 11.1 31.0 11.3/ 30:8 11.8 30.6 1204 30.5 12.6
33
34 32.1 11.1
3250 11.5| 31.9 11.6 31.7122 31.5 12.7 31.4 130
34
35 33.1|11.4) 33.0 11.8) 32.9 12.0 32.7 12.5 32.4 13.1 323 134
35
36 34.0 11.7 33.9 12.1 33.8 12.3 33.6 12.9 33.4 13.5 33.3 13.8 36
37 |
35.0 12.1 34.8 12.5 34.8 12.6 34.513.3 34.3 13.934.2 14.2 37
38 35.9 12.4) 35.8|12.8) 35.7.13.0/ 35.5 13.6
357 13.01 35.53.6 35.2 14.23841 14.5 38
36.9412.536.7|13.1 36.6 13.3 36.4, 14.0 36.2 14.6 36:0 14.9
39
40 37.8 13.0 37.7 13.5| 37.6 13.7 37.3 14.3 37.1 15.0 36.9 15.3 40
41 38*8 13.3 38.6 13.8 38.5 14.0 38.3 14.7 38.0 15.3 37.21157
41
42 39.7|13.7 39.5 14.
39.5 14.1 39.5 14.4 39.2 15.1 38.91507 38.5 16.1
42
43 40.7 14.0 10.5 14.5 40.4 14.7 40.1 15.4, 39.9 16.139.7 16.543
44 41.6 14.3) 41.4 14.8) 41:3 15.0 41.1|1568 40.8 16.5 40.6 116.8
44
45 426 14.6 42.4 15.21 423115,442-9164 41.7 16.8 41.6 17.2
45
46 43.5.15.0 43.3 15.5 43.2 15.7 42.9 16.5 42.6 17.2 42.5 17.6
46
47 444 15.3 44.1 15.6 44.216.14 43.916.8 43.6 17.6 13.4 18.
47
48 45.4 15.6 45.216.245.1 16.4) 44.817.2 44.5 18.0 44:3 18.45
46.1 16.5 16.0 16.8 45.7 19.6 45-4 18.3 45.3 18.749
47-316.3 47.1 16.8 47.0 17.1 46.71 17.9 46 4 18.7 46.2 19.1
Dep|Lat Depl Lat Depl Lat Depl Lat Dep. Lat Dep Lat
70 . 16
Points
.
50
ಇಟ Diſt
Dift
:
hi Deg. of Point) 10 Deg. og Deg. 168
Deg.
E...
of LATITUDE und DEPARTURE.
183
Dift.
Dift.
53
ور
61
.
85180.4 27.7 80.128.6 79.9 29.1179 330.5 178.8 31.878.5 32.5
19 Deg. T Point; 20 Deg. 27 Deg. 22 Deg. 2 Points
Lat. Dep Lat., Dep. Lat Dep. Lar. Dep. Lat | Dep Lat Dep
S148.2 16.6 48.0 17.2 47.9 17.4 47.618.3 47.3 19.1 47.1 19.5
$1
52.49.2 16.91 49.0 1705 43.9 17.8 48.5 18.6 48.2 194 48.619.9
52
$350.1 17.3) 49.9 17.9 49.8 18.1 19.5 19.0 49.1 19 8 490 20,3
54151.117.6 50.8 18.2 50.7 18.5 10.4 19.3 150.1 20.2 49.9 20.1
$4
5515200 17.95 1.8 ! 18.5 151.7 18.851.3 19.1 151.0 20.5 308 21.0
55
56152.9 18.2 52.7 18.9 52.6 19.252-3 20.1 51.) 21.051.7 -1.4
57153.9 18.6 53.7 19.2 5 3.6 19.553.2 20.452.8 21.3, 52.7 21.8
57
5854.818.9 54.6 19.5 54.5 19.8 54.1 20.8 53.821.75 3.622.6
58
59155.8 19.255.5 19.9 55.4 20.255.121.1 54.722. I 54.5 22.2
60 56.7 19.5156.5 20.2 56.4 20.5 56.0 21.5 55.622.5 55.4 23.0
60
61 57.7 19.9157.4 20.5 57.3 20.956.9 21.9 56.5 22.8 54.3 23 3
6258.6 20.258.4 20.9 58.3 21.2 57.922.2 57.523.2 57:323.7
62
63 59.6 20.5 59.321.259 221.558.8 22.6 58.4 23.658.2 24.1
63
44 60.5 20.8 60.3 21.6 60.1 21.9 59.7 22.9 59.3 24.059.1 24.5
64
65 61.5 21.2161.2 21.9 161. 22.2 63.7 23:3 60.3 24.360.0 24.9
65
66|62.4 21.5| 62.122.262.0 22.6 65.623.6 61.2 24.7 61.025.3
66
67 63.3 21.8 63.1 | 22•6 63.0 22.9 62.5 24.0 62.125.061.9 25.6.
67
68 64.3) 22.1 64.0 22.963.9 23.3 163.5 24.4 63.025.5|62.84 26,0
68
69 65.2 32.4 65.0 23.2 54.8 23.6 164.4 24.7 64.0 25.8163-725.4
69
70.166.2 22.8165.91 23.6 65.8 239965.3 25.1 64.9 26.264.7 26.8
70
71167.123.166.8 23.9 66.7 243 663 25.4 65.8 | 26.6 65.6 27.2
71
72 68.1 23.4 67.8 24.2 167.724.6 167,325.8 66.7 27.0166.5 27.6
72
1369.0 23.8 68.7 24.6 68.6 25.068.126.2 67.227.3167.4 27.9
73
74 70.2 24.169.7 24.9 64.5 25.3 169.126.5 68.6 27.7 68.3 28.3
74
75 70.9 24.4170.61*5.3 70.5 25.6 70.0 26.9 169.5
28.1 169.31 28.7
75
76171.9247171.6 25.6 71.4 26.01739187.2 70.5
28.5 70.2 29.1
7772.8 25.1 72005125.9 172.4 26.3 71.9 27.671.4
28.8 71.129.5
77
78 73.7 25.4 73.426.3 73.3 25.7 72-8 27.9 723129.2 72.0 29.8
79 74.7 25.7 74.4 266 74.2 27.0 73.7 28.3 173.2
29.6 73.030.2
79
8075-6 26.0 75.3 26.9 75.2 27.4174.7 28.7 74.2
30.0 73.9 30.6
80
8176.6 26.4 76.3 27.3 76.1 27.7 75.6 290 75 130.3 24.8 31.0
81
82 77.5 26.777.2 27.6 77.1 28.0 76.5129.4 176.0 30.775 7 31.4
82
8378.5 27.0 78.1 28.0 78.0 28.4 97.5 29.7 116.9 31.176.731.8
84 79.4 27.3 79.128.3 70.228.7 78.4 30.1 77.9 36.5 77.6 32. I
86 81.3 28.0 81.0 29.0 80.8 29.4 83.3 30.8 19.7 32.2 794 32.9 86
87 82.3 28.3 81.9 29.3 31.8 29.7131.2 31.2 80.732.6 80.4 33.3
88 83.2 28.6 82.8 29.6 82.7 30.182.1 31.5 81.6 33.0 81.3 33.7
88
89 84.1 29.0 83.8 30.0 33.6 30.4183.1131.5 182.5 33.3 82.2 34.1
89
90 85.1 29.3 84.7 303 84.6 30.8.84.0 32.3 83.4 33.7 S3.1 34.4
90
91 86.0 29.6 85.7 30.7 85.5 31.0 84.9 132.6 34.4 34.184.1 34.8
91
92 87.0 29.986.6 31.0 864 131.585.9, 33.9 85.3 345 185.0 35.2
92
93 88.0 30.387.6 31:3 187.431.8 868 33.386.134.885.935.6
93
94 98.9 30.6 88.5 37.7 88.3 32.187.7337 87.2 35.2 86.8 36.0
95 89 830.9189.4/320 89.332.588.7 34.0 83.135.6 87.8 36.3
95
96 90.8 31.3 90.4 32.3 90.2 32.889.6 34.4 89.0 35.9 88.7 36.7
97 91.7 31.8 91.3 32.5 91.1 33.290.5 34.8 89.9 36.389637.1
97
98 92.7 31.9 92-333.0 92•1 133.5 91.5 35.190.939.7 905 37.5
99 93.6 32.2 193.3 33.3 193.0 33.992.4 35.5 91.8 37.196.5 37.9
99
94.5 326 94.233.7 940 34.2 93.4 35.8 192.7 37-592.4 38.3
Dep Lat. Dep. Lat. Dep Lat. Dep Lat Dep Lat Dep. Lat
71 Deg. 6 * Point70 Deg 69 Deg.1 68 Deg: 1 o Point
76
78
83
84
85
87
100
въ 2
*It:
Diſt.
Dift.
2
5
7
10
II
I2
14
13.5 05.5
13.105.3
20
21
22
18.7 41.7 19.441.6
184
A Table of DIFFERENCE
3 Deg: 24 Deg) 25 Deg, Point 25 Degal 27 Deg.
Lat Dep Lat. Dep Lat Dep Lat Dep Lari Dep
Lat Depl Lat | Deb
I 09.929.400.919.4) 25.903.409.9.25.4 20.933.1 33.903.4
zlor.80.801.805. 01.327.301.8 5).; 01.30.) 01.303.7
302.8 25.202.701. 02.7 21.302.701.3 22.7 1.3 0 2.7 1.4 3
403.7 13 03.901.603.5 21.7 03.501.7 03.5 21.71 03.501.8 4
$ 104.601.904.602.004.5 22.104.5 10.5 01.102.3 1.5 02.3
6 05.5 12.305.57 22 41 05.1 22.505.402.6 01.122.505.302.7
6
79614102.706.4102.8) 05.3103.706.31 03.025.3 23.105.3 23.2
807.4 03.107.3 03.207.2 3.407,203.4 07.2"}.5 0.1 03.5 8
9f08.310315 08.2 03.7 03.2423.303.133.8
03.1|23.8 03.103.)oj. 01. I 9
10 09.203.909.104.1 09.101209.71913 01.091.4 33.907.5
II 10.124.3 10.0 04.5 10.0 101.6 02.901.7 0).)| 04.3 0.3 05.
1211.0 24.7 11.004.2) 10.995.110,3 05.1 15.3 11.3 10.1 05.4
13 12.005.111.905.3) 11.805.5 11.721.6 11.715.7 11.5 0.9 13
14 12.905:5 12.8/ 05.71 12.721:9 12:7|0s. 12.65.1 12.5 25.4
15 13.805.9 13.7 06.1 13.505.313.6105.4 13.5
15
16|14.7 06.2 14.6 36.5 14.5 105.3 14.5 35.5
14.4 97.5 14.307.3 16
1715.606.6| 15.5.06.9 15.407.215.107.3 15.307.4 15.107.7 17
18. 16.6 07.0 16.4 07.3) 16.307.6 16.3 27.7 16.209.9 15., 08.2 18
19 17.5 1074 17.407.717.208.3 17.21631
17.1103.31 16.10.6 19
2018.4.127.8 18.3103.1 18.03.418.130.5 18.303.5 17.3 03.I
2119.31-52 19.2 08.5 19.003.9 19.2 07.0 18.7 57.2 19.702.5
22 20.2 98.6 20.103.9 19.910).319.9. 27.4 19.805.9 19.5 10.0
23 121.2 59.0 21.0109.323. 03.7 20.8/).S
25.7 10.1 25.5 10.4 23
24 22.1. 09.4 21.909.8 21.7 10.1 2.1.733 21.5 10.5
21.4 10.9
25 123.0 09.8 22.8 10.2 22.7 10.61 21.6. 19.7 22.5 11.3
25
36 123.9 192 23.7 10.6 23.6|11.0 23.5.11.1. 234 11.7 23.211.8 26
27 24.810.524.7 11.0 24.5 11.4 24.411.5 24.311.8
27.1 12.3
27
28 25.8101925.611.4) 25.4111,825.3112.6
25.212.31 2.49|12.7 28
29 26.7 11:3 26.5 11.8 26.3 12.3 25.2 12.4 25.112.7 25.9 13.2 29
30 27.611.7 27.4 12.2 2.7.2 12.7 27.1. 1 2.8 27.3 13.1 25.7136 30
31 28.512-1 28.3 12.6 28.1 13.4 23.013.3 37.713.6 27.6. 14.1 31
32 29.5 12.5 29.213.0 29.0 13.5| 38.21:3.7 | 15.3: 14.0 23.5 14.5 32
33 30.4 12.9 30.113.429.9 13.9 29.3 14.1 29.7|14.4 23.445.0
34 31:313:3) 31.113.81 338 14.4 30.7 14.5 33.5 14.9 30.3 15.4
34
35 322 13.7 22.01.14.2 31.7 14.8 31.6 15.0 31.5 15.3 35.2115-9
35
36 33.1.14.1 32.9 14.6 32.6 15.2 32.5 15.4 32.4 15.3
37 34.1 14.4 33.8 15.0 33.5 15.6 33.4.15.8 33.216.4 33.9 16.8
37
38 35.0 14.8 34.7|1504) 34.4 16.2 34.3 16.236.1 16.6 33.9.117.2
38
39 35.9.15.2 35.615.4 35.3 16.5 35.3 16.7 35.7|17.1 34.7 17.7
39
40 36.815.6 36.5 16.3 36.216.91 36.2 19.135.9 17.5 35.6 18.2
41 377 16.0 37.5 | 16.7 37.2 17.337.1 17.5 36.8. 13.) 36.5 18.6
42 2807 16:41 38.4 17.1 33.1 17.7 38.0 13.3 37.7 18.4 37.4 19.1
43 39.6 16.5 32-3 17:
53:18.2 38.9 18.438.6 18.8 35.3 19.5 43
44 40.517.240.217.9 37.91 18.5 39.8118.3
395 17:31 39.2 20.0
44
45 41.4 17.641011 18.347.8 19.3 45.7 19.2 40.4:19.74.125.4
45
41.3
46
47 43.3) 18.4 42.9.19.1426 19.9 42.5 23.1
42.2 27.5| 41.21.347
48 44.2 18.8 43.8 19.5| 43.5 20.343.4 27.5
42.82108
48
49 45.1 19.2 44.8 19.944642017 443 25.9 44.0121.5 43-7122.2
49
46.0 19.5| 45.7 20.3 45.3 21.145.2141.4 41.921.9) 44.5227
Depl Lat Depl Lat Dep. Lat Dep Lar Dep. Lat Dep'Lat
167 Deg. 1 66 Degos Deg. s} Point) 04 Deg. 63 Deg.
24
22.3 11.3
33
32.116.3
36
40
41
42
2
43.0 21.0
Dilt
Dift.
Dift.
64
67.
65.230.4 65.1 30.3 64.7 | 31.667.232.7
41.0 87.741.5 87.242.86.4 44.0971
41.9183.143.0873144.5) 98]
of LATITUDE and DEPARTURE 185
23 Deg. 24 Deg: 125 Deg. 24 Point. 26 Degl 27. Deg.
Lat, Dep Lat Dep Lat| Dep Lati Dep Lat | Dep Lat Dep
SI 46.9 19.9 46.5 25.7 46.2 21.5 46.0 21.844.3 23 3 45.4 23.2 SI
5247.9 23.3 17.5 21.5 47.122.0 47.0 22.2 46.7 22.3 40.3 23.64 521
5348.6 20.749.4 21.5 48.5 22.447.722.7 47.6 23.3 47.24.53
5449.7 21.147.3 22.0 48.9 22.5 43.8 23.143.5 23.7 18.114.594
5550.6 21.550.2 122.4 49.82 3.249.7 23.5 41.4 24. I 19.5 25.055
56151.521.951.2 2208 50.7 23.7 53.5 23.250 3 24.5 492 25.4 56
57 152.5 122.352.11 23.251.7 21.151.524.4 51.225.01 so.S 25.44 57
58
53.422.75 3.7 23.652.5 24.5 52.4 24.5 52.125.451.7 26.31 18
59154.3 23.0 53.9 24.0 53.5 24.9153.3 25.253.125.9 52.6 26.81 59
60 55.2 23.4 54.8 24.4 54.4 25.4 54.225.653.326.31 53.5 27.21 60
6156.123.855.7 24.8 55.325.355.126.1 54.8 26.7 54.427.7
61
62 57.124.2 156.6 25.2 56.2 26.2 56.7 25.5.55.727.21 55.2 28.1 62
6358.0 24.6 57.5 25.6 57.1 26.6 56.91 26.757.6 27.61 56.1 28.663
64 58.9 25.0158.5 26.5 58.5 27.0 57.927.4 57.5 23.0 57.729,1
65 59.6 25.4 59.4 26.4 53.927.5 58.5 27.8 58.428.5 57.9 29.5 65
6660.7 25.8160.3 26.8 59.327.959.728.2 59.3 28.9 51.3 30.0
66
67161.4 26.261.2 27.260.7 28.3160.6 28.6 50.2 29.4 59.7 30.4
68 62.6 26.662.1 27.7 61.6 28.7 61.51 29.1 29.8
60.6
30.91
68
69 63.5 27.0163.0 28.1 62.529.262.429.5 62.0 30.2 61.5 31.3 69
70 64.4 27.3 53.9 28.5 63.429.6 63.3 29.9 62.9 30.71 6204 31.8 70
11165.427.764.928.9 64.3 30.0 64.2 39.4 63.8 31.163.332.2 71
72
2367.3 28.506.729.7 66.2 30.8166.031.255.6 32.0 65.033.1 73
74 68.1 28.967.6 33.167.1 31.3 66.7 31.6 66.5 32. 165.3 33.6 14
75 69.0 29.368.5 30.5 68.5 31.7 67.832.167.432965.8 34 75
7670.0 29.7 169.4.30.9 68.9. 32.1 68.7 32.5 68.3 133-3! 67.7 34.51 76
77 70.9 20.170.3 31:3 69.8.32.5 164,632.9 69.2 33.7
7821.8. 20.5171.23167 70.7 33.370.5 3313 70.1 34.267.5 35.4 78
19 22.7 30.9172.2 32.1. 71.633.471.4 33.871.0 34.5 70.4 35.79
80 73.6 31.3 73.1 32.5 72.5 33.872. 3. 34.271.935.1 71.3 36.380
81 74.6 31.674.0 32.9.73.4 34.2 73.2 34.6 78.8 35.5 | 72-2. 36.8 81
82 15.5 132.0 74.933.374.3 34.7 74.135.1 73.7 35.91 73.137.2 82
83176.4 32.475.8 33.8.75.2 35.475.035.5 74.6 36.4 740 37.7 83
84 77.332.8176.7 34.2 76.135.575.235.4 75.536.8 74.8 38.1 84
851782 33.2177.6 34.6 77.0 35.9.76.8 36.3 76.4 37.3 75.7 38.6 85
86|19.2 33.678.6 35.0 77.9 36.3 77:7 36.8 71.3 37.2 76.6 39.0 86
87 80 134.0 79.5 35.4 73.8 35.5 78.6 37.2 78.2 38.1 77.5:1 39.5 87
88 | 81.034.4 30 4.35.8.79.7 37.2 79.5 37.6.79.133.6. 78.4 400 88
8981.9 34.8 31.3 36.2 80.737.6 80.538.80.0 39.0 79.340.4 89
90 82.8 35 2 82.2 36.6 81.0 33.581.4 38.5 80.939.4
80.2
40.9) 90
91 $3.7 35.6 33.4 37.9 82.5 8.5 82.3138.9 31.8 37.3. 81241.31 91
92 84.2 35.9 34.0 37-4 83.41 38.9 832 39.382.7 40.31 82.41.8) 192
93 85.6.36.335.378 643 39.3 841 353 83. 428. 82.7 42.2 93
94 86.5 36.735.9 38.2 185.21 39.7185.0 40.3 84.541.21831842.71 94
95 87.4 37.1 86.8|38.6 36.1 40.185.943.6. 8.441.6 84.6 43.1 95
96 88.4: 37-5187.7 39.5 87.0 40.5 86.8 41.086.342.185.543.696
99 91.1 38.7 0.4 40.3 89.7 40.3 89.5 42.3 89.9.43.4 88.2 44.999
100 92.0 39.1 31.4 407 90.6 423 90 4 42.7 89,9 431887145-4100
DeplLar Dep Lat Dep Lat Depl Lat Dep. Lat Depl Lat
167 Deg. 66Deg. 165 Deg. 5, Point 64 Deg 63 Deg.
68.6135.0 77
DIA
186
A Table of DIFFERENCE
Diſt
Diſt.
1
1
2
4
S
6
6
7
IO
II
I 2
.
.
20
21
22
28 Deg.2 Point. 29 Deg. 30 Deg. 124Point. 31 Deg.
Lat Dep Latf Dep Lal Dep Lat, Dep Latı Dep Latſ Dep
00.90o 5 b5.9 05.59.9 5.5 03.919.5 00.900.5 00.9 0.5
201.800.9 br.8 03.901.7010 01.7 01.0 101.71010 01.7|OI.O
302.6 101.4 2.6 01.4 52.6 01.4 0.6 01.s 22.6. or.s
02.601.5
3
403.5 01.9 $3.501.903.5 1.9 103.502. 03.402.1
03.40 2. I
04.4. 02.34.402.4 54.4 102.4 04:302.5 043 02.6 04.31 02.6
OS:3 02.8 5.3 02.8 05.202.9
05.203.005.103.1 05.I 03.1
7106.2 03.346.203.3 05.103.4 05.103.5 06.003.6 06.0103.6
8 07.1 fo3.8 27.1103.8 07.0|03.9 06.9|04.0 06.904.106.9 04.1 8
9 07.904.2 27.9.04.207.904.407.8 04.5 107.7104.607.7104.6 9
10 08.8 104.798.8 04.708.7 04.808.705.00$.6 05.105.6 05.I
109.7los.z 109.7 05.209.6105.302.5 25.509.4 105.5 | 09.405.7
12 10.6 los.6 10.6 05.6 10.5 05.8 10.4 06.0 10.3 06.2
10.3.06.2
13 11.5 106.1 11.506.1 11.406.3 11.306.5 11.106.7 III|05.7 13
14 12.4 106.612.306.6
12. 206.8 1.127.0 12.0 07.2 I 2.0 07.2 14
15 13.2(07.0 13.2|07. I 13.107.3 13.0|07.5 12.9077 12.907.7 15
16|14.107.5 14.107.5 14.0|07.7 13.9 08.0 13:7|08.2 13.7 08.2
16
1715.0 108.615.000.0 14.9108.2 14.7 08.5 14.603.7 14.6 08.8 17
18 15.9 08.4115.9 08.5 15.708.7 15.6 09.0 15.407.2 15.409.3 18
1916.8 08.9 16.808.9 16.609. - 16.409.5 16.3 69.8 16.309.8 19
20 17.769.4 17.6109.4 17.5409.7 17.3 10.0 17.1 10.3 17.1 10.3
21 18.5|99.9 18.5| 09.9 18.410.2 18.2 10.5 18.0 10.8 18 0 10.8
22 19.4 10.3 19.4 10.3 19.210.7 19.0 11.0 18.9 11.3 18.9 11.3
23 20.3 10.8 20.3 10.8
20.1 II.1 19.9 11.5 19.7 11.8 19.7 11.8 23
24 21.211.3 21.2 11.3
21.0 11.6 20.8 12.0 20.6 12.3 20.6 12.4 24
25 22.111.7 22:0|11.8
22.0/ 11.821.912. I
21.6
1 205 21.4 12.8 21.4 12.9
26 23.012.2 22.9 123 22.7 12.6 22.5 13.0 22.3 13.4 22.3 13.4) 26
27 23.8 12.7 238 12.7 23.6 23.1 23.4 13.5.23.1 13.9 23.1 13.9 27
.28 24.7|13.1 24.7 13.2 24.5 13.6 24.2 14.0 24.0 14.4 24.0 14.4 28
24 25.6 13.6 25.6 13.7 25.4 14.125.1 145 24.9 14.9 24.9 14.9 29
30 26.5 14.126.5 14.1 | 26.2 14.5 26.0 15.0 25.7 15.4 25.7 15.4) 30
312714414.5 27.3.14.6 | 27.1 15.0 26.8 75.5 26.615.9 126.6 16.0
žž 28.215.0 28.2 15 1 28.0 15:5 27.7 16.6 27.4 16.4 27.4 16.5
32
33 29.1. 14.6129.1 15.5 28.9 16.0 28.6. :6.5 28.3 17.0 25.3 17.0
33
34 30.0 16.0 30.0 16.0 29:2 16.5 27.5 17.0 29.2 17.5
34
35 30.9 16.4 30.9 16.5 30,6 17.0 33.3 17.5 30.0 18.0 30.0 18.0 35
36 31.8 16.9 31.7 17.0 31.5 17.4 35.2 18.6 309 18.5 30.9 18.5 36
3732.7 17.4 32.6 17.4 32.4 17.9 32.0 18.5 31.7 19.5 31.719. I 37
38 33.5 17.0 33.5 17.9 33-218.4 32.9 19.0 32.5 19.5 32.5 19.6 38
39 34.4 18.334.4 18.4 341 18.9 33.8 19.5 33.4 20.3 33-4 20.1 39
40 35-3 18.8 35.3 18.9 35.0 19.4 34.6 20.5 34.3 20.6
34.3 20.6
40
41 36.1 19 2 36.1 19.3 35.8 19.9 35.5 20.5 35.2 21.1
35.1 21.1
41
42 37.1 19.7 37.0 19.8 36.7 20.4 86.4 27.6 360 21.6 36.021.6
42
4338.0 20.1 37.9 20.3 37.620.8 37.2 21.5 (35.9 22.1 36.922.1 43
44 38.8 20.6 38.8 20.7 38.5 21.3 38.1 22.0 377) 22.6. 37.7 22.6 44
45. 39.7 21.1 39.7|21.2. 39.321.8 39.2 22.5 33.6 23.1
38.6 23.2
46 40.6 25.6 40.625.7 40.222.3 39-8123.0 39.5 | 23.6 39.4) 23.7
47/41.522. 1 41.4 22.241.1 22.8 40.7 23.5 40.324.2 40.324.2
47
48 42.4 22.5 42.322.6 42.0 23.341.6 24.0 41.224.7 41.1 24:7
49143.3 23.1 43.2 23.2 42.87 23.7 42.4 24.5 42.0 25.2 42.0 25.2 49
44.1 23.5 144.1 23.6 43.7 24.2 43.3 25.642.925.7
42.9 25.7
DeplLar Depl Lat Depl Lat Dep. Lat Dep. Lat Dep! Lat
62 Degl s{Point' 61 Deg,' 60 Deg. 's Point.
5 Point. 159 Deg.
31
29.1 17.5
com F********** Diſt
45
46
48
Dile.
of LATITUDE and DEPARTURE
187
DA
Dift.
Dirt
43.7 26.31 SI
51.2
60
53.0
2
55.6/29,6
64
7' 167.135.7 67.0 35.8 66.5 1 36.8 65.838.0 65.2 39.1 165.132.11 701
74.2 39.4174.139.6 73.5 40.7 727142.0 72.1. 43.2 72.0 43.31 84
28 Deg.12 Point 29 Deg: 30 Deg. 2. Point. 31 Deg.
Lat Dip Lat|Dep. Lat Dep Lat Dep Lat Dep | Lat Det
51145.0/23.9 45.0 24.0 44.6 24.744.225.5| 43.7 26.2
5245.9 24.4 145.924.5 45.5 25.2 45.0126.0 44.6 26.7
44.6 26.8 52
53146.8 24.9 46.725.0 46.3 25.745.9125.5 49.5 27.2 45.427 53
54 47.7 25.3 47.6 25.5| 47.2 26.2 46.8 27.0 46.327.8 46.3 27
463 27 8 54
55.48.6 25.8 48.5 25.9 48.1 26.7 +7.6 27.5 47.223.3 17.128.3
56 49.4 26.3 49.4 26.4 49.0 27.1 4:5|28.6 48.0 28.8
28.8 48.0 28.8 56
57
26.8
$0.3
50.326.9) 49.8 27.649.4 28.5 48.9 29.3 45.9 29.4/ ?
58 27.25 1.227.31 50.728.150.129.0 19.7 29.8 49 229.9 58
59 52.127.752.0 27.81 $1.6 28.6 511 29.5 5066 36.3 50.6 30.4 59
60
52.928.31 52.5 29.1 52.0 30.0 51.5 30.8 51.4 30.9
61
61
53.9
28.653.6 28.753.329.652.830.552331.4 52.3 31.4
62
62
54.7 29.154.7 29.2 54.2 30.1 33.7 31.0 53.2 31.9 33.1 31.9
63
55.6 29.755.130.5 54.6 31.5 54.0 32.4 54.0 32.4
63
64136.5 30.0 56.43.2 56.31.0 55.4 32.0 54.932.9 54.9 33 0
6 157.4 30.5|57.330.5 56.8 31.5 56.3 32.555.7
33.4 155.7 33.51 65
66
6€ 158.3 31.058.2 311 57.7 | 32.0 57.2 33.0 56.633.9 56.634.0
67
67
59.2
31.459.1 31.6 58.6 32.558.0 33.5 57.5 34.4 57.4 34.5
68
68 60.0 31.9160.0 32.0 59.5 33.0 58.9 34.0 58.31 35.0 58.3 35.
69 50.9 324160.832.5 60.333.459.7 34.5 | 59.2 35.5 39.1 35 51 69
70 618 32.9161.7 33.0 61.233.9 30.6 35.0 60.0 36.0 60.0 36.0 70
7162.7 33.3162.6 33.5 62 1 34.4 61.5| 35.5 60.936.5 60.9 36.6 71
72 63.6 33.8635133.9163.034.962.3 36.0 61.8 37.0 161.737.11 72
7: 64.4 34 364.4 34.463.8 35.4632 36.562.6 37.5 62.6 | 37.6 73
74 65.3 34.7 65.3 34.9 64.7
64.735.9 64.137.0 63.5 38.6 63.4 38.1 74
75 66.2 35.2166.135.465.5 36.4 34.2 37.5 64.3 38.6 64.3 33.6 75
7768.0 36.167.9 36.367.3 37.3166.7 38.5 66.0 139.6 66.0 39.7 77
98. (8.9136,663.8 36.81 68.2 37.8 67 539.0 56.9 40.1
66.9 40 2 78
1 69.1 37:1167.737.2 69.1 38.3 68.4 35.5 67.1 406 6727 40.7
8273.6 37 6705 39.770.0 38.8 169.340.0 68.6
88.641.2
8171.5 38.0 71.4 38.2 708 39.3 70.1 40.5 69.541.6 69.441.7
82 72.4 33.5172.338.671.7 39.7 702 41.070.3 42.2 110.3 42.2
85175.0 39.9175.0 40.1 74:34 41.2 73.642.5 72.943.7 72.943.8) 85
86 75.9 40.475.840575.2 41.7 74.5 43.073.8 44.2 73.744.3
86
89176.8 40.8 76.741.0 76.142.2 75.343.574.6 | 44.7 74.644.887
8877.741.3.177.6 41.5 770 4207 76.2 44.0 75.545.2 74.445.31 88
8917806 41.878.5 41.9 77.8 43.177.1445 76.3
76.345.8 76.3 45.8 89
90 79.5 42.2 29.4 42.4 78.7 43.6 712145.0 77.2 463 77.146.3 90
91 80.342.7 30°2 429 79.6 44.1 78.8
45.5 78.146.8 78.0 46.9 91
92
81.2 47.2 51.1143.4 80.5 44.6179.7 46.0 78.9 47.3 78.947.41 92
93
82.1 43.6 32.0 43.3 81.3 45.1805 46.5 79.8 47.8
47.8 79.7 47.9 93
94 83.0 44.) 32.9 443 82.2 45.6 81.4 47.0 80.6 48.3 806 48.44 94
95 83.9 44.6 33.844.883.1 45.182.3 47.5 81.5 48.8 81.4 48.9
96 84.8 45.1 84.7 45.2 84.0 46.583.1 48.6 82.349.3 81.3 49.496
97
85.6 45.5 85.5145.7 84.8.47.0 84.0 48.5 | 83.249.9 83.1 50.0 97
9$ 46.5 46.0 36.4 46.2 85.7 47.5 84.949.6 84.1 10.4 84.0 50.5 98
99
87.4 46.5 37.346.7 86.6 48.085.7149.584.950.9 84.951.0 99
88.3 46.438.2 47.1 87.5 48.5 86.6 50.0 85.851.4 85.756.5
Dep Lar. Depl Lat Dep. Lat Depl Lat Depil Lat Depl Lat
62 Deg.'s Point 61 Deg. 60 Degl 5* Point 159. Deg.
41.1
83
95.
100
Ulalala
1
1
188
A Table of DIFFERENCE
Diſt.
Lat, Dep
Diſt.
I
2
2
6
olant
10
II
12
20
21
22
19.1|12.3
-3.3 134. 23.5 34.424.1
32 Deg.133 Deg.) 3 Point,
34 Deg. I 35 Deg. 1 36 Deg.
Lat Dep Lat Dep | Latop
Dep) Laty Dep Lat Dep
120.8oo.5 p0.3 00.5
10.31 23.6 05.8 59.600.800.6 0.8 100.6
01.70I,I PI.701.I 17.1.1 01.7 01.121.6 01.1
01.601.2
302.501.6 p2.5| 91.6 23.5 11.7 02.5 101.702.5 101.702.4 01.8 3
4 03:4 02.1 23.+ 22.2 103:30 3.2
03:3 02.203.302.303.202.3 4
S04.202.6 P4.202.7 1.202.)
04.102.8 04.102.904.002.9
S
05.103.2 5.303.3 05.293.3 01.03.106.903.4 04.8103.5 6
29.9.03.7 25.903.8 3.5 3.9 105.303.905.704.005.704. I 7
8 106.8 04.2 25.7 4.4 03.6 04.126.6 94.5 96.504.606.5 104.7 8
9107.6 04.8 67.57.07.1 P.) 27.505.07.105.2 07.305.3 9
10 oś.sos-
35:41 05.103.35. 03.305.6 108.2 05.7 08.i oss
II 09.3105.824.2 6.0 2.1 25.1
09.105.1 0.0 06.3 | 08.906.5
12 10.706.4 19.105.5
19.0125.7 19.496.707.8 06.909.707.0
13 11.0906.9 10.91011 10.99
[0.3 27.3/10.6 107.5 10.5 07.6 13
14 11.907.4 11.7) 07.6 11.6197.3
11.627.8 11.508.o 11.308.2 14
15 12.71 07.12.6 53.2
12.501.31-4108.4 12.3 03.6 I 2.1 08.8 15
16 13.608.5 13.408.7 13.303.9 13.3 03.9 13.107.2 12.909.4
16
17|14.409.7 14:3013 (14.10).4 14.193.5 13.909.8 13.7 10.0 17
18 15:3109.5115.10.15.0 10.0 15.9 10.1 14:7 10.3 14.6 10.6 18
19 161 10,1 15.9 19.3 15.8 19.6 15.7 10.615.4 10.9 15.4 11.2
19
2017 010.6 16.8 13.9 16.6 ILI
16.5 11.2 16.4 11.5
16.2 11.8
2119.81111 17.6 11.4 17.5 11.7 17:411.7 17.2 12.0
17.0 12.3
22 18.6 11.7 18.5 12.7 18.312. 2
1$..
12.3
18.3 12.6 17.8 12.9
23 19.5 12.2 1.3 12.5
19.312.8 18.8
13.5 23
24 20.3 12.7 ... 13. I
23.3 13.3 19.9 13.4 19.7 13.8 19.4 14.I 24
2521.213.2 21.0113.6 23.8 13.-5.714.9 29.5 14.3
25
26. 22.013.8 21.914.221.6 14.4 25.5|14.5 21.3
26
27 22.9 14.3
22.6
14.7 22.4 15.0 121 15.122.1 15.5
27
28 23.7 14.813.5 152 23.3 15.5 23.2 15.6 22.9 16.1
28
29 24.6 15.4 24.3115.8 24.1 16.124.0 16.2 23.9 16.6 23.5 17.6 29
3025-415.9 25.2 10.3 24.9 16.7 24-9:6.8 24.6 17.2 24.317.630
3126.316.4 26.0 16.9 29.8 17.2 25.7 17.3 25.4 17.3 25.1 18.2 31
32 27.1 17.0 26.3 17.4 26.6 17.8 26.5 17.9 26.2 18.3 25.9 18.8
32
28.0 17.5 127719.0 27.4 13.3 27.418.4 27.0 18.9 26.7 19.4 33
28.318.6 28.5 19.5 29.3 10.9 28.2 19.6 27.919.5 27.5 20.0 34
35 29.7 18.5 29.4.19.1 27.1 19.4 29.0119.6 28.7 29.1 29.3 20.5
35
36 30.5 19.130.2 19.6 29.9|23.0 27.5 25.127.5 20.5
29.125.21 36
37 31.4 19.6 31.0 20.1 30.8129.6 337 27.7 30.3 21.2
29.921.7 37
38 32.220.131.9 20.7 31.621.131.5121.2 31.121.3 30.7 22.3
39 33.1 20.732.721.2 32.4 21.7 3.2.3.21.332.3 22.3 31.5 22.9 39
40 33.921.2 33.6 21.833.3 22.2 33.2. 22.4 32.8 22.9 32.4 23.5 40
4134.821.7 34.4 22.3 34.1 22.8 34.0 22.033.6 23.5 33.2 24.1 41
34.0 24.7
43 36.5 12.36.1 23.4 35-2 -3.3 35.24.0 31.2 2706 34.025.3 43
74 37.3 23.3 36.2270 36.612*4 35.5 24.5 360 25.2 35.6 25.9
44
438.1 23.8 37.71245 37.4 25.03731251236.9 25.8 36.4 26.4 45
46 39.0 24.4 38:025.0 35.2 25.5 33.1 25:71 377 26.4 37.227.0
47 39.91 24.9 39:4 28.6 39.121.137.9 26.3133.5 26.9 38.0 27.6
48 40.7 25.4 49.3 26.5 37.9 26.7 39.325.3 353 27.5 38.8 28.2
49 41.5 26.0 41.1 26.7 40.7) 27.2 10.127.4 49.1 23. í 39.6 28.8
49
42.4 26.5 41.327,2 41.5 27.8 trit -3.014021 23.7 40.4 29.4
Depl Lar Depl Lat Def Luta De Lar DepLato
58 Deg. 157 Deg.' 's Point, 55 Dog's Deg. 154 Deg.
13.2 18.6
20.1 14.7
21.0 15.3
21.8 15.9
22.6 | 16.5
42
46
47
48
of LATITUDE and DEPARTURE 189
Dift.
Diſt.
Time
66
32 Deg.133 Deg: 3 Points 34 Deg: 35 Deg. ;Deg
Lat Dep Lat Depl Lat| Dep Lat|Dep Lat| Dep Lal Dep
5143.2 27.0 42 8:27.8 42.4.28.3 42.3 28.5 41.8 29.2 41.3 30.0 SI
52 44.1 27.6 13.6 28.3 43.21 28. | 43.129.142.6 29.8 42.1 30.6 32
$3.44.9 28.1 44.5 28.0 44.127.443.9 29.6 43.4 30.4 42.931.2 53
54 45.828.6 45.3 29.4 44.9 30.0 44.5 30.2 44.2 31.0437 31.71 54
5546.6 29.146.130.0 45.7 30.6 45.5 30.7 45.1 31.5 144.5 32.355 .
56 47.5 29.7 47.0 30.5 46.631.1 46.4 31.3 45.2 32.1 45.3 32.9 56
5748.3 30.2 17.8, 31.0 47.4131.7 47.3 31.9 46.732.7 46.133.61 37
5849.2 30.7 48.7 31661 48.232.243.1 32.41 47.5 33.3 46.934.158
59 50.0 31.314905 32.149.0 32.8) 48.933.5| 43.3 33.8 147.7 34.7 59
60 $0.9 31.8150.3 327 49.933.3 49.7 33.5 47.1 34.4 148.5 35.3 60
61 51.732 351.233.2 50.7 33.9 50.6 34.1 500 34.9 149 335$
62 52.6 32.9 52.0 33.56.5 34.4 51.4 34.2) 5o.33.6 30.2 36.4 62
63 153.4 33.4 $2.9 34 5.2.4 35.0 52.235.251.6 36.1 31.037.063
64 54.333.953.7 34.9 53.235-5.5 3.1 35.8. 524 36.7 51.8 37.6 67
651 55.1 34 4154.5 35.4 54.0 36.1 53.9) 36.3 53.237.3 152.6 38.2 65
66156.0 35.0*5.3 35.9 54.9 35.7.154.7 35.9 54.137.9 534 58.8
67156835:55 6.2 36.555.7 37.2 55.5 37.5 54.9 38.4 54.2 39.4
68 57.7 36.687.0 37.5 56.137.8 38.0 55.7 39.0 55.0 40 68
69) 53.5 36.67:937.6 57.433.337.238.6 56.5 39.6 55.8 40.6 69
701'59.4 37.15 8.7 38.1 8.2 38.958.0 39.1 $1.3 40.5 56.6
41.1 70
21) 60.937 69.6 38.7 52.037.458.939.7 58.2407 157.441.7 71
12161.0 38.60.4 39.2 19.8 40.0 59.740.3 59.041.3 58.2 42.3 77
73161.5 38.7161.2 39.8 60.7 40.6 69.5 40.8 59.81419 59.142.5 13
-74162.7 39.262.140.3 61.5 40.1 61.341.460.6 12.4 139.9 43.5 44
73163.6 39.762.9 40.8 62 4 41.7 612 41.9 61.4 43.9 160.7 441 75
164.4 40.3638 41.5 63.3 42.5 63.0 42.5 6.2.3. 43:6 61.5 447 76
77
27165.340$64.6 41.964.8428163.343.0 63.1 44.2 1623 45.3 72
78
66.1 41.365.4 42.564 943.364.7 143:6 63.91 44.7 63.1, 45
19 67.0 41.366.343.0 65.7 43.9 65.5 44.2 647 | 45
4-745 3 163.9 46.
80 67.8 42.4167.123 66.5 44466.3|447 45.96407 47
81 68.742.968.0 441 67.3 45.767.0 45
45.31667146.5 16.5
65.547,6
82 69.5474163.844.7) 68.2 45.5.168 A5.8
1.247.0 66.3
83
70.4 44.0357.645269.0 46.1 68.3146. 68.0 47.6 167.1
84
71.244.5 70.545.869.8.46.769.6 47.0 68.81 48.2 68.0
72.145.0713 46.3 70.7 47.2 70.5 47.5| 69.6 48.8 63.8
30.08
3272.945.621 46.8 11:547.8 7 1:348.1 70.5 49.3 169.6 Sos
87 73.8 46.
173.0 47.372.3148.3 72.148.6 71.3149,9 70.4
74.6 46.6 73.8 47.9|73,2 48.9 77.9 49.2 72.1 50.5 71.2
897505 47.2 747 48.5 740 49.4 23.8 49.8 72.9 51.0 72.0
90
761347-7 75.5 19.0 74.81 50.0 74.6 50.3 73.71 51.6 72.8
77.248.2 76.
.276.3149,6 75.750.6
75.750.67.4 150.9 74.5 1 52.2 73.6 53.5
91
78.0 48.7 ja 76.5151.1176.3151.4 75.41 52.8 74.4 34 1
92
78.9 49.378.0 50.677.3 51.7 7741
77-351.7 77.152.0 76.253.3 75.2 54%
93
79,7|49.8 78.931.2 78.252.277.95 2.6 77.0 53.9 176.0
76.0 55.3
94 89.6 50.3 79.7|51.7 79.0 52.8 78.8 153.1 17.8 54.5 16,9
95
95
81.4 50.90's $2.31 79.8 53:3 79.6 53.7 78.6 55.1 7707
96
99 82.351.431.4152.880.653.3130.434.2 29.5 55.6 48.5
78.5 57
97
97
98
83.1 31.952.2 53.4 81.5 34-4 81.2 54.8 80.356.2 79.3 57
84.0 52.5183-1 $3.9 82.3 55.0 8.1 55.4 81.1 56.880.1 58.
98
90 848 3.0 83.9 54.3 83.165.683,9155.9 81.9 37.4 809 58
De'La Depl Lat Da Lat Dep Lat Depl Lar Depl Las
58 Deg: 57 Deg. Points 56 Degl ss Deg. I 54 Deg!
C.
9.4 84
88 74.6
71 88
91 77.2
Dia
(190
A Table of DIFFERENCE
*
Diſt.
.}
2
aute
10
11
I 2
16
20
21
22
23 18.5 13.7 18.4 138) 18.1 14.217.2 14.5 17.8 14.6 17.6 14.8
24 19.3 14.3 19.214.4118.914.818.6 15.1
33.9 26.5 33:41 27.1 133-2 27-33269 276
47 37-728.0 37.5 28.5
48 38.51 28.6 38.28;51 37.8
3 Point 137 Deg. 38 Deg. | 39 Deg
Deg. 38 Değ: 39 Deg 13{Points 40 Deg.
Latj Dep Lat Depl Lat Dep Lar Dell Lat| Dep Lati Dep
100 8100.6 0.8 0.6 0.8 0.6 0.8 10.7 00.800.600.8 00.6
z 101.6lov.z 01.601.21 01.601.201.5101.3
01.201.5013 01.501.301.501.3
302.4 01.8 102.401.8 02.4 01.8 02.3101.9 02.3101.902.3 01.9 3
4 103.2 02.4 03.202.41 03.102.503.502.5 03.102.5 03.102.6
504.003.0 04.003.01 039 03.103.9 103.1 03.9.03.203.8/03.2
6 04.8 03.6 04.8 03.6 04.7 03:7 046103.9 04.603.8 04.603.9
705.6 04.205.604.205.5 04:31 05.401.4 05.404.4 105.4 04.5 7
8 6.404.8 6.4 04.81 063 049) 06.2105.0 06.2 os.I 106. Hlos.
9.07.205.4 07.205.407 i los.s07.0 105.7 07.0|05.7 06.105.8
10 08.006.008.01 06. 07.9106.2 07.81063 07.7 06.3 07.7 06.
II 08.806.6 108.806,608.706.808.5 106.9 08.507.008.4 07.1
12 109.6 07.1 109.6 07.2 09.4 07.4 09.307.5 09.307.6 09.207.7
13 10.407.7 10.4 07.81 10.2 OS. I rol | 08.2 10.0 08.2 10.0 08.4 13
14 11.2108:3 11.2 08.4 11.0 09.7 10.908.8 10.8 08 9 10.7 (09.0
14
15 12.0 08.9 12.0 09.0 11.8 09:3 11.6109.4 11.6 09.5 11.509.6 15
16 12.8109.5 12.809.612,6 09.8 12.4 10.! 12.4 10.1 12.3 10.3
17 13.6 101 13.6 10.2 13.4 10.5 13.2 10.7 13.1 10.8 13.0 10.9 17
18 14.5 10.7 14.4 10.8 14.211.1 13.9 11.3 13.9 11.4 13.8 11.6 18
19 15:3113 15.2 11.4 15.0 11.7 14.8 12.0 14.7 12.0 14.5 12. 2 19
20 16.1 11.9 16.0 12.0 15.8 12.3 15.5 12.6 15.5 12.7 15.312.9
21 16.912.s 16.8 12.6 16.5 12.9|163 13.2 162|13.3 16.1 135
22 17:7 13.1.17.6 13.2 17:313.5 17.1 13.8 17.0 14.0 16.8 14.1
23
19.2
24
25 20.114:9 200 15.0 19.7 15.4 19.4 15.7 19.3 15.9 19.1 16.1 25
26 20.9|15.5 208 15.6 20.5 16.0 20.2 16.4 20.116.5 19.9 16.7 26
27 21.1 16.1 21.6 162 21.3 16.6.210 17.0 20.9 17.1 20.9 174/27
28 22:5|16.7 22.4 16.8 22.1. 17.2 21.8 17.6 21.6 17.8 21.4 18.0 28
29. 23:3 17:3 23.2 17.4 22:8 1781 22.1 18.3 22.4 184 22.2186 29
30 24.1 17.9 24.0 18.0 23.6 18.5 23:3 18.7 23.2 19.0 23.0 19.330
31 24.9 18.5 24.8 18.6 14-4 19.1 24.1 19.5 24.0 19.7 23.7 19.2 31
32 25.7 19.125,6 19.3 25 2 1927 | 24.9 20.1 24.7 20.3 24.5 20.6 32
33 26.5|19.7. 26.
19.9 26.0 20.3 25.6 20.8 25.5 209 25:3 21.2 33
34 (27-320.2 27.120.5
27.1 20.5 26.8 20.925.4 21.4 16.3 21.6 26.021.9 34
35 28.1 2018 -7,9 21.127,6 21,5 27.2 22.0 37.0 22.126.8 22.5
35
36 18.9 2114 28.7 21.7 28.4 22.2 -77 227 -7.8 22.8 27.6 23.1 36
37 29.7 22.9 29.5 22:31 29.2 22.8 8.8 23.3 28.6 23.5 28.3/23,8 37
38 30.5|22.6 30,31 229 299 23.4 29.5 123 $9.4 24.1 29.124.4 38
39 31.3 23.2 31.123.5 30.7 24.030.3 24.5 30.1 24.7 29.9.25.139
40 32.1 23.8 31.9 24.1 31.5 24.6 31.125.2 30.925.4 30.6 25.7 40
41 32.9 34.4 32.7 24.7 31.3 25.7 31.9 25.8 317 26.0 31.4 26 441
42 33.7 25.0 33:5 125.3 33.1 2509132.6 26.4 32.5 26.6 32.2 27.0
5
43
7
44 35.326.. 35.26.5| 34 27:2 34.2 277 349 37.933.7 28.3
36.] 26.8 135.9 27.1 35.5
44
45
27:7 3.0 28.3 34.8 28.5 34.5 23.9
46 36.927.4 36.7 27.7 36.2
18.3 35.7 | 29.0386 29.2 35.2 29.6
18 $ 36.5 29.6 36.329:8 36.0 30.2
47
29.5 37.8 39.2 37.1 30:4 36.830948
39.3 29.2 39.129.5 38.6 30.2 38.1 30.8 37.9 31.9 37.5 31.5
40.2 29.8. 39.930.1 39.4 30.818.91 31.5 38.6 31.7 38.3 32.1
Depl Lar Defl Lat Dep. Lai Dep! Lat. Dep. Lat Dep Lat
4 Point | 53 Deg. 52 Deg.1st Deg. 14Point 50 Deg.
é
42
57.0
49
49
Of Latitude and Departure. 191
51
53 42.6
૨૨ ૨,
54
8
57 45.8
58 46.6
59 47.4
60 45.2
61 49.0
62 | 49.8
60
64
67 / 53.8
68 546
70
71
570
57.8
58.6
32.5142. 12.641.2 33.3
32.8 43.933.143.3 33.9 42.7 34.0 42.5 34.9 42,135.4
33:3 44.7 33.8 | 44.1 34.5 43.5 35.2 43.335.5 42.91 36.0
42.3 56.7 42.7 35.943.7 55.2 44.7 54.9145.0 54.4, 45.6
80 64 3
172.7 54.8 71.7 56,0 70.7 57.3 70.3 $7.769,7 58.5
| 57.2 74.9 158.5| 73.859.8 73.4 60.3 72.861.1
3 Point 37 Deg. 38 Deg.: 39 Deg: 3{Point. 40 Deg.
Lat Dep Lat Drep Lat Dep Lat Dep Lat Dep Lat Dep
51 41.0
30.4 40.7
30.7 40.2 31.4 39.6 32. 39.4 32.3 39.1 32.8
5241.8
31.041.5
31.3 41.0 32.0 40.4 32.7 40.2 33.0 39.8) 33.4
52
31.42.3
53
54 43.4
43.1
. 41.5
55 44.2
55
56 | 45.0
56
33.9 45.5 34.3 44.935. 44.335.9 47.136.243.7 36,6
SZ
34.5 46.3 34.91 45.7 35.
45.1 36.5 44.836.8 47.4 37.3
58
35.1 47.1 35.5 46.5 36.3 45.8137.145.637.4) 45.2 37.9
59
3507 47.9 136.1 47.33.246,6 37.8 46.438.1 46.0 38.6
36.3 48.7 36.7 48.1 371 47.4 38.4 47.238.7 46.7 39 2 61
36.9 49.5 37.3 48.938.2 48.2 39.0 47.939.3 47.5 39.9 62
37.5 50.3 37.9 49.6 38.8 49.0 39.6 48.7 40.0 48.3 40 5
63
50.6
63
64
51.4 38.1 51.1 38 550.4 39.4 49.7 40.349.5 40,6 49.0 41.2
65
52.2 38.7 31.9 39.151.2 40 050 540.9 50.2 41.2 40.8 41.8
65
66 53.0 39.352.7 39.7 52.0 40.651.341.5 51.0 41.9 So.s 42.4 66
39953.5 40.3 52.8 45.252.. 42.2 51.842.551,343.1 67
40.5 5413 40.7 33.6 11.9 52.8 42.8 52.643.1 32.1 13.7
68
69
55.4 41.41551 41.5 54.4 42.5) 53.6 43.453.343.852.944.4 79
56.2 41.7 | 55.9 42.1 55.2 43.154.4 44.0 54.4.4.4 53.645.0
70
9
42.9157.5 43.356 7 44.3 55.945.355 745.755 146.3
72
72
43.558.3 43.9 57.5 449 56.7 45.256.4 46.3 55.9 46.9
73
73
59.4 44.1 59.1 44.5 58.3 45.6 57.5 46.6 57.2 46.956.7 47.6
7+
74
60.2
44.7 59.9 45.159.2 46.2 58.3 47.258.0 47.6 57.4! 48.2
75
75
45.3 160.7 45.760.0 45.8 59.1 47,8 58.7 48.2) 3.3 48.9
76
76
61.8 45.961.5 46.3 60.7 47-41598 48.5 59.5 48.8 59.0 49.5
77
77
16.3 62.3 46.961.5 48.0 60.6 49.1 100.3 49.5 59.71 50.1
78
78
47.1 63.1 47.562.2 48.6 61.4 49.761.1 50.1 60.5 40.8
79
81
65.148.3617 48.763.8 49.9|62.9 51.0 62.5 51.4 62.0 52.1
81
82 65.9 48.865.5 49.357.6 50.5| 63.7 31.6 63.4 32.0 62.852.7
52.7 84
83
49.4 65.3 49.9165.4 54.1 64.5 52.264.252.6 63.6 53.4
83
84
67.5 50.0 67.5 30.5 66.2 51.7 65.3 52.964.953.3 64.3 54.08
68.3
50.667.9 51.1
51.1 67.0 52.3 66.1 53.5 65.7 33.9165.154.6
85
$1.2 168. 51.7 67.8 5.9 65.8 54.1 66,5 54.6 65.2 55.3
69.1
85
87 69.9 51.8.69.5 52.4 68.6 53.6 57 | 548 572 55.266.6 55.9
52.470.3 33.0 69334.2 68.4) 55.4 68.0 55.867.4 56.6 88
71.5 53.0 71.1 53.6 70.1 34.869.256.0 68.8 36.5 68.2 57.2 89
89
72:31 $3.6 171.9
53.6 121.9 154.2 79.9 58.4169.9156.6 69.6 57.1 68.9 57.4
90
73.1
91
739 54.8 73.5 35.4 72.5 56.671.557.971.1 58.470.5| 59.1
92
S5.4 1743
56.0 23.3573) 72.358.5 71.9159.0 71.2 598
93
94 75.556.0 75.1 36.6 74.1 57.973.0159.2 72,71 59.6 72.0 60.4
94
56.675.9
57.276.5 37.8) 75.6 59.1 746 60 4174.260.973,5 61.7
96
779 | 57,8 77.5
57.8 77.5 58.476.+ 19,7 75.4 61.0 75.0 6.5 1743
62.1
97
$8.4 78.3 59.0 77.2 60.376.2 61.775.762.2 75.2 63.0
98
79.5 59.0 79.1 39.678.0 60.9 76.9|62.376.5 162.875863.6
99
99
80.3 59.6 79.9
60.2|78.8 161,67747 62.9177.3.63.4 76.6 6403
Depl Lat Dep 'Lat Dep Lat Depl Lat Dep Lat Depl Lat
14 Point 53 Deg. 52 Deg. 5, Deg.143 Poinil so Deg
Сс 2
Dift. ln er om olho Release to be lã ã nomi magiDilt
61.0
62.7
63.5
79
80
66,7
85
87
83 707
90
91 173.1
92
74.7
93
I
95
76,3
77.1
95
96
97
98
78.7.
100
100
1192
A Table of DIFFERENCE
Ꭼ
4. Point
Diſt.
I
2
3
4
6
9
10
16
19
20
21.
22
0.700.7 20.7 0.7 | 03.7 23.703.7 007 00.700.700.700.7
|
1914.312.5 14.1171114.11 2.8 13.2.13.0 137 13.213.4 13.4
21 15.8 13.8 13.6 14.0 15.0 14.1 15.4 14:3|15.1 14.0 14.8 14.8
24 18.115.77.916.117.8 116.1 17:50 164 173 16.7 17-0|17.0
34 25.02203125 31227 25.222.9 24.9 23.2
+38 28.7 24.9
41 Deg.142 Deg: 33 Point, I 43 Degy 4A Deg.
Lat Dep LatD:P Lar Dep
Lat Dep Lat, Dep Lat Dep Lat Dep
01.501.3bis 01.301.5013 01.01.4 01.401.401.401.4
302.302.022.2 02.0 02.200 022102.0 02. 202. I O2 I OLI
4 03.0 02.643.002.703.0067 02.902.702.9 92.8 02.8 02.8
Slož.8 03-353.7 103.3 83.7 03-403.603.4 03.603.503.5 103.5
04 503.204.5 24.01 04:4104.0 04.4
4.4.04. 04.404.104.304.204.204.2
7105.304.6|05.2042708,2017 05.104.805.004.904.91049 7
8 06.0 05.205190593 0.405.4 OF 8105.505 705.6 05.708.7
9 106.8 05.9 6.7 9590 05.705.0
05.7105.0 106.606.106.506.206.4 106.4
107.5106,6 07 0307 074106.7 07:3106.807.206.907.1 107.1
11.08.31079203.2 07:4 03.10.7.4 08.0 9.5 07.9 7.6 078 07.8
12 09.07, 98.9 26.0 08.903. 1 08.8 08-2 08.6083 08.5 108.5
13. 09.81a8is 9.7 1087 09.609.7 109.5 108.9 09.309.009.2 109.2 13
14. 10,6 09.210.4 394 10.4094 10.2 109.5 | 10.1 29.1 029 09.9 14
15 H 3109.8 14.110.7
I10 10.2 :0.8 10.4 10.6 10.6 15
16. 12.11.11 10.7 1.7 10.9 11.710.6 11.5 11.1 11.3 11.3
17 Fiz.8 11.1 12.6 11.4 12.611.4 124 11.61 12.2 11:8 | 12.0 12.0 17
18 13.6 11.813.4 12:21 13:3|12.1 113.212.31 12.9 12.5 12.7 12.7
18
2015.113.1 149 13:47 148-1304 146 13.6 144 13.9 14.1 14.1
2416.614-4 116.3 14.7 16.3 14.8 16.115.0(15.8 15.3 15.5 15.5
2317 4.15.1 17.1 15.4 17.0|15.4 16.8 115.716.5 16.0 16.3 16.3 23
24
25 18.9416.4 18.6 16.7 18.81168 118-3 17.0 18.0 174 177 17-735
26 19.0 17.1 7.3 174 173 174 19.0127187 18.118.4 18,4
27 (20.427.7 25.1 18.1 23.0 18.1 19.718.4 19.4 18.819.1 19.1 27
28 (21.118.4 20.8 118.7 20.718.8 20.5 19. 1. 20.1 19.4 19.8 19,8
29 21.9 19 19 19.4 21.5 19.5 21.2 19.8 20.920.0 20.5 20.5 29
30 22.6 19.7 123
219 3.5 27.6 20821.2 21.230
31 23.4 (20.3 1139 207 230 23.8 23.6 21.0 21.3 21.511920.931
32 124.121.0238 21:45 237 21.5
21.4 23,7 11.5 23.4 21.8 23.0 23.2 23.6 22.6 32
22.1 24.4 24.2 24.11 225 123.7 12.9 23.3 23.30
3526.4 23.0 26,0 23:41 25-9|23.5 256 23.92501 2403 247 24,7 35
3627.2 23:6 26.7 24.4 26.7 24.2 263 24.5 25.9 25.0 25.42514
3727924-3 27.5 24.7 27.4 24 8 17.0 25.26.6 25.7 26.125.1 37
25.4
39 129.4 25.627.0126.1 28.9 26.2.1285 266 28.0 27.1 27.6 27.6 39
40 30.4.26.2 19.7 26.8 29.6 26.9 29.1 17.3.128.8 27.8 28,3 28.3
40
41 31.0 26.9 30.5 27.4 39.4 27.5 39.0 28.022.5 28.5 29.0 29,0
41
142 31.7 -7.5 3112 28.1 31.128.2 30.7. 28.6 30.2 29.2 297 29.7 42
43 32.5 18.2 31.928.8 31-928.) 1.4 293 30.9 29.9 30:4 304443
4433.428,933.7 29.4 32.6 29.5 32 2 30.031.6 30.6 31.131.1
45 34.0 29.5.133.4 30.11.33:3 30.2 32.9 30.7 34.4 31.3 31.8 11.8
31.845
46 34.7 30.2 34.2 30 8 341 309 33.6 31.4 33.1 320 32.5 32.5
47 39.5 30,8 349 31.4 34.8 316 344 32 1 33.8 3216 33.2 33.2
48 36.5 31.5
357 32.1 35.6 32.2 35.1 347 34.5 33:3 33.2|33.9 48
49 37.0 325 36.4 32.8136.3 32.9 358 334 33
34.6. 49
50 3771 32.8 37-2 33.5 137.0133.6 36.6
36.0 34.1 36
34.7 35.3 35.3
Der Lar Pepl Lat Dep. Lat Depl Lat Depl Lat Dep|Lat
49 Deg.48 Deg. 41. Point 47 Deg. 146 Deg: 4 Point.
28
20,12 2.2 2. I
33124.9 21.6
46
47
34,0 34,6
Dit.
* Diſt.
Dift.
Dift
SI
56
57
61
62
/|
of LATITUDE and DEPARTURE. 193
41 Deg 42 Deg. 3. Point. 43 Deg. 144 Deg. 4 Points
Lat Dep | Lat. Dep | Lat Pep Lat | Dep Lat pep Lat Dep
51:8.5 33.5 37.9 34.1 37.8 34.2 37.3 34.8 36.735.4 36.136. I
$239.2 34.1 38.6 34.8 38.5 34.9 38.0 35.5 37.436.1 36.8 36.8
52
53.40.0 34 8 39.4 35.5 39.335.6 38.8 36.1 38.1 35.8 37.5 37.5
53
54 49.8 35.440.136.1 40.0 36.39.5 36.8 338 37.5 38.2 38.2
54
55 41.536.0 409 36.8 40.736.9 40.2 37.5 13.1.6 38.2 38.933.9
55
56.12.31 3667 41.6. 37.5 41.5 37.6 1.0 38 2 42.333.9 39.6 39.0.
57 13.0 37 4142.4 35.142.2 38.3 +507) 38.9 41.037.5 47.340.3
58 13.8 38.1 113.6 33.8 43.0 33.9 42.4 39.5 41.2 110.341.0 41.0 58
59 14.5 38.7 4368 39.5 43.7 39.6 43.142.2 424 41.0 41.7 41.7
$
60 15:339.4 44.6 40.144.5 49.343.840. 43.241.7 424 42.4 66
61 46.6 40.0 45.340.8 45.241.0 14.641.7 43.9 42.4 43.143.1
62 6.8 40.7 46.1 41.5 452 41.645-3 43.3 44.5. 43.143.5 43.8
63 47.641.3 46.8 422 467.42.345.143.0 45 3113.8 44.5 44.5 62
04148.3 429 47.5. 448 47-4 43.246.8 43.6 16.0 14.5 45.3 45.3 64
65 19.142.6.48.3 43.5 48.
43.6.48.243.6. 47-5. 14.3 46.8145.1 46.0 46.0
65
66 49.8 13.349.0 44.2 48.9 44.3 48.3 45.0 47.5 145.8 467 46.7 66
67 $0.6 44.0 49.8 44.8 49.6445.0 1490 145.7 48.2 46.5 47.441.4 67
68 1.344.650.5 45.5 50.4 145.7 49.1 464 48.7 47.248.1. 48.5
69 52.1 45.351.346.2 51.146.350.5 47.1 49.6 47.948.848.8 69
70 52.8 4.9 52.0 46.851.27.0 51.2 47.7 50 343.6 47.5 47.5
70
21 $3.6 46.0 52.8 47.5 52.6 47.7 81.9 484 Slot 49.3 502 50.2
71
72 54:3 47.253.5 48.2 53.3 48.352.7149.1 151.850.0/509 50.9
72
7355.1 47.91542 48.8 54.1 49.0 53.4 49.8 52.5 50.7151.51.6
73
7455.9 48.555.0 149.5 54.8 49.7 541 So.5 53.3 51.4 52.35223 24
75 56.649,21557|50.2 556 50. 54.8 51.1 539132,153.05 3.0
73
76 57.4 49.9156.5 50.956.3. 51.0 55.6 50.8 54.715 2.3 53.7 53.7
76
58.150.5 57.1 53.5 57.1 51.7 56.3925 564 3.5 34.4 54.477
78 58.9 5 1 3 58.0 521 57.852.4 57.0 53.2 56.1 54.255.255.2
78
19 159.0 51.8158.7 52.8 18.5 $3.1 57.8 13.956.8 4.9 95955.9
80 60.4 32.5 19.4 $3.5593 53,7 k8.5 5406 575 55.6 566 36.6 8c
81 61.153.100.2 54.2 60.0 54.4 15.9.2 95.2 58.356.3573 57.3 81
82 61.9 53.8160.9 54.9 60.8 55.1 60.0 55.9. 154.0 57.0/58.0 18.0 82
83 62.6 54.5 161.7 35.3 61.5 55.760.7 56.6 69:7 57.6 58.7. $3.7
84 63.4.55.1 62.4 36.1 62,2 56.4 61.4 573 1464 58.359.459.4
64.2 55.963.2156.9 63.0 57.1 62.2 58.0 61.19.0160.1 60.1
86 4.9 56.4 63.9 57.5 63.7 57.763,0 58.6 161.9 59.760.8 60.8
87.65.7 57.1 64.7 58.2 64.5, 58463.6 59.3 62.6 50.4 61.561.5
88 166.4 87.7145.4 58.9 65.2 59.1.644 690 163.361.1|62.2 622 86
89
67.9159.0 66.96.2 66.460.4165.3 61.4 164.9182.863.263.68
901
9.1 68.7 59.7167.660.9167.4 1.1 66's 1621 65.5 63.264.3 64.3 91
92 991460,4684161.668.2 618 67.3 62.7. 66.2 63.9 65.0 16.092
93 70.261.0 69.1 62.2 68.9 62.4 68.0 63.4 66.9 64.6 65.805.8
93
94 71.0 61.7 169.9 62,9 69.6 63.5 68.7 64.1 67:6165.3 665 166.5.
94
9521.7|62.3 706 63.6 70.463.8 169.5,648 68:366067.2 67.3
96 7245 630 713 64.2 71.1 64.5 70.2 65.5 69.166.7 167.9 672
9723.2 63.6 72.1 64.9 71.965.1 70.9 66.1 69.8 67.4.168.668.6
97
98174.064.3 72.865.6 72.6 65.8 21.9 66.& 795 168.1 1693 169.3
99 14:7 65.0 73.666.2 73.4166.5 1924 67.5 171,2 168.8 70.0 70.999
75,5 65.6 74:3 66.9.741 67.2 73.1 682 21.9 69.5 70.7 2017
Dep|Lat. Dep Lat. Dep Lat. Dep Lat Dep|Lat Depl Lat.
49 Deg. 48 Deg. 4. Point 47 Deg. 45 Deg. 1 4 Point
79
83
84
85
8€
95
194
Navigation
Fih
Of the Nature and Ufe of the preceding Table.
OR the more ready working of a Traverſe, in order to find
the Place of the Ship at all times, the preceding Table has been
Inſerted made according to Caſe the iſt of Plain Sailing, fhewing the
Difference of Latitude and Departure, to every Degree, Point, and
Quarter Point of the Compaſs, the diſtance from 1 Mile to 100,
and may be made to ſerve for any other diſtance, provided it be firſt
divided into parts not exceeding the Limits of the Table.
At the Head that is in the uppermoſt Rank, are placed the
Courſes beginning from 1 degree and proceeding on through the
feveral Degrees, Points, and Quarter Points to 45 Degrees, and ar
the foot or in the lowermoſt Rank but one, are placed the Degrees,
Points, and Quarter Points of the remaining half of the Quadrant,
that is from 45 Degrees to 90.
In the two outmoſt Columns of each Page, are placed the
Diſtances, the Table on the Left-hand Page begining åt i Mile, and
continuing to so; and that on the Right-hand Page begining where
the other left off, viz. 50, and going on to 100.
And in the common Area, that is dire&ly under the Courſe, and
againſt the Diſtances are placed the Differences, or Alterations of
Latitude and Departure, correſponding with, or anſwering to the
reſpe&ive Courſes and Diſtances.
A few Examples will make it plainer.
Example. the ift.
If the Courſe were N. E. by N. { Eaſterly, and Diſtance 76, and
the Difference of Latitude and Departure were required.
On the Right-hand Page (becauſe the Diſtance is above 50)
find the Diſtance 76 ; and at the Top (becauſe the Courſe is leſs
than 4 Points, or 45 Degrees) look for the Courſe 3 | Points, and
right under it, and againſt 76, the Diſtance fail'd will be found in the
Column Lat. 58.7, for the Difference of Latitude and in the Column
Dep. 48.2, for the Departure.
If the Courſe had been N. E. * Eaſterly, and the Diſtance as be-
fore 76, then muſt the Courſe have been fought for at the bottom,
and the Difference of Latitude would have been found 48.2, and the
Departure 58.7
Example
Navigation.
195
1
:
Example, the 2d.
Suppoſe a Ships Courſe N.E.by N. Eaſterly, and the Diſtance 176.
On the Right-hand Page at the Top find out the Courſe 3 į Points,
and againſt the Diſtance 100, you will find the Difference of Lati-
tude 77.3, and Departure 63.4: Again, find out 76 the remaining
part of the Diſtance under the Column Dist. right againſt it, and in
the Column under 3 į Points you will find 58:7, for the Difference of
Latitude, and 48.2, for the Departure ; now the Sum of this Diffe-
rence of Latitude and the Difference of Latitude before found, 136
will be the Difference of Latitude required; alſo the Sum of the
Departures 111.6, is the Departure ſought.
Example the 3d.
Suppoſe a Ship fail South 36 deg. 30 min. Eaſterly, 45 Miles, and
the Difference of Latitude and Departure were required.
Having found the Diſtance failed in the Left-hand Page, becauſe
the Courſe cannot be found exa&ly, leek the Difference of Latitude
36.4 under the next lefs, viz. 36, alſo the Difference of Latitude
35.9, under the next greater, viz. 37 ; then ſay as the Difference of
the Tabular
Courſes 60 min. to the Difference of the Tabular Diffe-
rences of Latitude 5, ſo is the Exceſs of the Courle given 30 min. a-
bove the next lefs. Tabular Courſe, to 25 ; which ſubſtracted from
36.4 the Difference of Latitude belonging to 36 degrees, gives 36 15,
the Difference of Latitude ſought : After the ſame manner the De-
parture will be found to be 26.75, and to for any other.
In like manner, if the Diſtance were a Frađional Part, then the
Difference of Latitude and Departure under the fame Courſe muſt be
found, to the two next Extream Diſtances, and Proporcion made
accordingly.
Example
Let the Courſe be S.S. E. Eaſterly, and the Diſtance 44.75 Miles.
In the Left-hand Column againſt 44 Miles, the next leſs whole
Number to the Diſtance, and under 2 { Points, find the Difference
of Latitude 38.8; likewiſe in the fame Column againſt 45, the next
Integer find the Correſpondent Difference of Latitude 39:7: Then
fay as too to 9 the Difference of the Tabular Numbers, fo is 75 the
Exceſs of the given Diſtance above 44 Miles to $76, or 13 almoſt,
which therefore added to the Difference of Latitude 38.8, correſpond-
ing with 44 Miles gives 39.5 nearly, the true Difference of Latitude
required.
In
196
Navigation
againſt 5 in the ſame Column of Dep. you will have
In like manner for the Departure, ſay is too to s the Difference
of the Tabular Departures, anſwering to 44 and 45 Miles, ſo is 75
to 71, or nearly, which therefore added to the Departure 20.72
anſwering to 44 Miles gives 21.1 nearly, the Departure fought.
Or Having found the Difference of Latitude 38.8, anſwering to
the Diſtance 44, under 21 Points in the ſame Column againſt 7, you
will have. 62; alſo againſt s in the ſame Column, you will have .044,
the Sum therefore of theſe three will give 39.464, the true Difference
of Latitude, agreeing nearly with, but more exa& than the former.
Alſo, having found the Departure 2017, anſwering to the Diſtance
44 Miles, in the ſame Column right againſt 7 you will have 33,
and
.0243
the
Sum of theſe three Dep. will be 21.054, the true Departure, agrce-
ing nearly with, but more exact than the former.
After the ſame manner, may the Difference of Latitude and De-
parture be found, anſwering to any given "Courte and Diſtance, and
therefore I fall give only one ſmall Traverſe, with its Solution, and
Jeave: the Operation to exerciſe the Young Student.
"Admit a Ship at Sea in the Latitude of 34° 30' North, fail away
N. by E. 35, N. N. E. 3 Eaſterly 29, N. E. 92, N. E. by E. Eaſter-
ly 40, E. N. E. 4 Eaſterly so, and it be required to find what
Latitude the is in, and how much the has departed from her firkt
Meridian.
Having made a Table, and placed the Courſes and Diſtances in
the Sabfequent Manner; the Differences of Latitude and Departure
will be found as follows.
The Trauerle Fable.
Dift. of Lat. Departure.
DUTIN.
S.
W.
Courſes.
N. by E.
N.N.E.E-
N.E.
N. E. by E. E.
E. N. E. E.
35 34:3
27 2414
22.0
40 18.9
So 12.1
||||||
6.8
11.5
22.6
35.3
48.5
E
Diff. of Lat.112.31 Dep: 124.7
Hence
Navigation
197
Hence it appears that ſhe has departed from her firſt Meridian
124.7 Miles, and altered her Latitude 112.3 Miles, wherefore to
find the Latitude ſhe is in. Becauſe the failed from a North Latitude
Northerly.
0
To the Latitude ſaiid from
Add the Difference of Latitude reduced
34 : 36 N.
I: 525Nly.
The Sum is the Latitude ſhe is in
36 : 227N.
By the help of this Table, all the ſeveral Caſes of Plain Sailing
may be eaſily reſolved; but this being a little Forreign to the preſent
Buſineſs, avd not the deſign for which theſe Tables were Calcula-
ted, we ſhall paſs it by.
兴​兴头​央​决​选民​男​男​男​头​兴​兴​兴庆​兴​兴​头头​兴​兴头​头头​头头​头头​头头​头头​头
​Section IV .
Containing the Doctrine of Oblique-angled Plain Triangles
applied to Problems of Sailing.
HE varietyof Queſtions that may be propoſed under this Head,
being in a manner innumerable, I ſhall ſelect out ſuch only as
may be of Uſe in the Prađice of Navigation.
TH
- ore
Caſe I.
::
Coaſting along the ſhore I ſaw a Cape of Land, which bore from
me N. by E. then I ſtood away W. N. W. s Leagues or 15 Miles,
and the ſame Cape bore from me N. E. ŽE. I demand the Di-
ſtance from the Ship in her laſt Station to the Cape.
Dd
Geome-
198
Navigation.
B
NEKI
8.Ne
Geometrically.
1. Having drawn the Compaſs N,
E, S, W; let A repreſent the place
of the Ship in her firſt Station, and
draw the W. N. W. Line AC, equal to
15 Miles, then will C be the place of
the Ship in her ſecond Station.
N
CW NWT5
NE E
W
E
A А
S
2. From C draw the Line CB, pa-
rallel to the N. E. į E. Line, till it
meet the N. by E. Line AB in B, then
will B repreſent the Cape of Land, and CB the diſtance of the Ship
from the Cape, at her ſecond Station.
To find which by Calculation, in the Triangle ABC are given,
AC equal to 15 Miles, the Angle A equal to 78 deg. 45 min.
equal
to 7 Points, equal to the diſtance between the N. by É. and W. N.
W. the Angle B equal to 39 deg. 22 min. , equal to 3i Points, e-
qual to the diſtance between the N. by E. and the N. E. į E. and
the Angle C equal to 61 deg. 52 min. , equal to si Point, equal to
the diſtance between the W. N. W. and the S. W. 1 W. whence
to find the Diſtance CB it will be, by the ad Caſe of Se&t. the 5th of
I rigonometry.
S. ABC : AC :: S. BAC:CB. That is,
As the Sine of the Angle at B=39° 22'1
-9.8023585
To the Diſtance Run ACS15
So is the Sine of the Angle at A=78.45
1.1760913
-9.9915739
To the diſtance between the Ship and the Cape 23.19-1.3653067
If the diſtance of the Ship in her firſt Station from the Cape be
required, it will be by the ad Caſe of Se&t. che 5th of Trigonometry.
S. ABC : AC :: S. ACB: AB. That is,
As the Sine of the Angle at B=39° 22' ;
9.8023585
To the Diſtance Run AC=15
1.1760913
So is the Sine of the Angle at C=oI° 52'1-
9.9454298
To the Diſtance required AB=20.85 Miles
74
1.3191626
This
*
**
Navigation.
199
This Solution will ſerve for various other Queſtions, dreſſed up
after different manners, of which I fhall give a few Examples to in-
large the Learners way of thinking.
1. Being at Sea I ſaw two Headlands, whoſe bearing from one another
I find by the Chart to be S. S. E. and W. N. W. diſtunce s Leagues or
15 Miles, the Northermost bore from me S. W. Ź W. the Southermoſt
S, by W. I demand my Diſtance to each of the Headlands.
Anſwer.
My Diſtance from the Northermoſt Headland, is 23.19 Miles or
8 Leagues nearly, and from the Southermoſt, 20.85 Miles or 7
Leagues nearly,
This laſt Queſtion and the firſt Caſe, are of Uſe for finding
the Diſtance from a Headland, &c. when a Ship is about to take her
Departure from the Land.
2. Two Ships ſail from the Same Port, the one fails W. N. W.5
Leagues or 15 Miles, the other ſails N. by E. So far until she finds the
firſt Ship to bear S. W. W. I demand the ſecond Ships Diſtance from
the Port, and the Diſtance between the two Ships.
Anſwer.
The ſecond Ships Diſtance from the Port is 20.85 Miles or 7
Leagues, nearly and the Diſtance between the two Ships is 23.19
Miles or 8 Leagues nearly.
3. A Ship under fail diſcovers a dangerous Shole, and finds it to bear
from a Mark on the shore W. by S. then Sailing W. N. W. 5 Leagues,
finds the Mark to bear N. E. E. I demand the diſtance from the
the Mark on the Shore, to the dangerous Shole
Anſwer.
The Diſtance between the Mark on Shore and the dangerous
Shole, is 20.85 Miles or 7 nearly Leagues.
4.
There are three Iſlands in light of each other, a Ship at A finds B
to bear N. by E. and C, W. N. W. then ſailing 5 Leagues or 15 Miles
to C, finds B to bear N. E. E. I demand the Distance from A to B,
and from B to C.
Dd 2
An-
.
200
Navigation.
Anſwer.
The Diſtance from A to B 20.85 Miles or 7 Leagues, nearly and
the Diſtance from B to C, is 23.19 Miles or 8 Leagues, nearly.
5. There are two ports that lie N. by E. and S. by W. a Ship from the
Southermoſt fuils W. N. W. 5 Leagues or 15 Miles, another ſails from
the Northermojt S. W.; W. and then meets the firſt. Ship. I. demand
the Distance between theſe two ports, and the ſecond Ships diſtance faildo
Answer.
The Diſtance between the two Ports is 20.85 Miles or 7 Leagues,
and the ſecond Ships Diſtance fail'd is 23.19 Miles or 8. Leagues, ,
nearly
Caſe II.
Coaſting along the Shore I ſee two Headlands, the firſt bears
from me N. W. by N. diſtance by Eſtimation 5 Leagues or is Miles,
the fecond bears from me S.W. Diſtance 6 Leagues or 19 Miles.
I demand the Bearing and Diſtance between the two Headlands.
Geometrically.
N-
ANWBNIS
W
A
1. Having drawn the Compaſs N, E, S;
W. Let A repreſent the place of the Ship,
and draw the N. W. by N. Line AB, cqual
E to 15 Miles, alſo the S. W. Line AC, equal
to 19 Miles, and joyn the Points B and C.
then will EC be the mutual Diitance, and
the Angle B the Bearing from the N. W.
by N. Line.
S
SW 19
To find which by. Calculation, in the Triangle ABC are given;
AB=15, AC=19, and the Angle at A equal to the Diſtance be-
tween the N. W. by N. and the S. W.equal to 9 Points, or 101 deg.
15 min. whence to find the Angles at B, &c. it will be by the 4th
Cafe of Oblique-angled Plain Triangles. .
As
Navigation.
201
As the Sum of the Sides AC and AB=34
1.5314789
To their Difference 4
0.6020600
So is the Tangent of half the Sum of the Angles B
and C=39,221
> 9.9141732
To the Tangent of half their Difference=5° 31' 48.9847543
Which added to 39° 22'!, gives the Angle at B=44°53', whence
the Bearing of B, the firſt Headland, from the ſecond Headland,
will be N. 1° 08' E or N. by E. nearly, and conſequently the
Bearing of the ſecond from the firſt, will be S. 11° 8' W. or S.
by W. nearly.
Whence to find the mutual Diſtance BC it will be, by the ad Caſe
of Oblique-angled Plain Triangles.
Asthe Sine of the Angle at B=44.53
9 8485989
To the Diſtance AC-19
So is the Sine of the Angle at A= 101° 15'
1.2787536
9.9915739
To the mutual Diſtance BC=26-41 Miles-
1.4217286
This Solution will likewiſe give Anſwers to the ſeveral Queſtions
following.
1. There are two Iſlands B and C, a Ship from the Northermoſt fails
S. E. by S. 15 Miles, another from the Southermoſt ſails N. E. 19 Miles
and meets the former. I demand the Bearing and Distance between the
two Iſlands,
Anſwer.
The Mands bear from each other. N. by E. or S. by Wi nearly
Diſtance 26.41 Miles
2. Two Ships ſail from the ſame Road, the one fails N. W. by N. 15
Miles, the other ſails S. W. 19 Miles. I demand their Bearing and
Diſtance.
Anſwer.
Their Bearing is N. by E. and S. by W. fere. Diſtance 26.41,
Miles.
3. A
S.
202
Navigation.
:
3. A Ship from a certain Port ſails S. E. by S. 15 Miles, then S. W.
19 Miles; I demand the direct Courſe and Diſtance back again to the
Port.
Anſwer.
The direct Courſe N. by E. Diſtance 26:41 Miles.
Caſe III.
Coaſting along the Shore I ſee two Headlands, the firſt bears from
me N. N. W. the ſecond N. N. E. : Eaſterly, then ſtanding away
E. by N. * Northerly 16 Miles, the firſt bears from me W. N. W.
the ſecond N. W. by N. $ Weſterly: I demand the Bearing and Di-
ſtance of the two Headlands.
Geometrically.
1. Having drawn the Compaſs N, E,
S, W. Let A repreſent the place of the
Ship, from whence draw the N. N. E. I
E. Line AD, the N. N. W. Line AC],and
the E. by N. N. Line AB=16 Miles.
N
A
S
B
2. From B draw the Line BD parallel
to the W.N. W. where this interſects the
W
E
N. N. W. as in D, gives the firſt Head-
Land.
3. Alſo, from the B draw the Line BC,
parallel to the NW. by N. 4 W. where
this interſects the N. N. E. $ E. Line, as
in C, gives the ſecond Headland.
4. foyn the Points D and C by the Right Line DC, then will DC
be the mutual Diſtance between the Headlands, and the Angle ADC
the Bearing from the N. N. W. Line, to find which by Calculation,
1. In the Triangle ABD are given, the Angle DAB equal to 87
Points, equal 95° 37' Ź, the Diſtance between the N. N. W. and
E. N. E. E.; the Angle ABD equal to 3 Points, equal to 39
22', equal to the Diſtance between the E. S. E. and E. by N. ; N.
the Angle ADB equal to 4 Points, equal to 45° oo', equal to the
Diſtance between the N. N. W. and W. N. W. and the Side AB e-
qual
mit
Navigation
203
qual to 16 Miles; whence to find the Diſtance AD it will be, by
the 2d Caſe of Sext. the 5th Part the ad.
As the Sine of the Angle ADB =45° oo'
-9.8494850
To the Diſtance Run AB=16 Miles
1.2041 200
So is the Sine of the Angle ABD=39° 22') 9.8023585
To the Diſtance AD=14.35
1.1569935
ز
Or the Diſtance between the Ship in her firſt Station, and the
firſt Headland.
2. Again, in the Triangle ABC are given, the Angle BAC equal
to 3 $ Points, equal to 42 deg. 11 min. equal to the Diſtance be-
tween the N. N. E. $ E. and the E. N. E. · E. .the Angle ABC
equal to 6 | Points, equal to 70 deg. 19 min. equal to the Diſtance
between the E. N. E. 1 E. and the S. E. by S. 4 E, the Angle ACB
equal to 6 Points, equal to 67 deg. 30 min. equal to the Diſtance
between the N. N. E. E. and the N. W. by N. i W. and the Di-
ſtance AB equal to 16 Miles ; whence to find the Diſtance AC, be-
tween the Ship in her firſt Station and the ſecond Headland, it
will be by the 2d Caſe of Oblique-angled Plain Triangles.
As the Sine of the Angle ACB= 67° 30'-
9.9656153
To the Diſtance fail'd AB 16-
1.2041200
So is the Sine of the Angle ABC=70° 19'
-9.9738519
To the Diſtance AC equal to 16.31 Miles 1.2123566
After the ſame manner, may the Diſtances of the Ship in her fe-
cond Station, from each of the Headlands be found, if required.
3. In the Triangle ADC are given, the Angle DAC equal 4
Points, equal 53 deg. 26 min. equal to the Diſtance between the
N. N. W. and the N. N. E. E. the Side AD=14:35 Miles, and
the Side AC=16.31 Miles; whence to find the Angle ADC, after
the manner taught in the 5th Caſe of Oblique-angled Plain Triangles.
From the Logarithm of AC + the Radius
11.2123566
Take the Logarithm of AD the lefler Side
1.1569935
The Arch anſwering to the Remaining Tango-
10.0553631
is 48° 38' }
From
.::
204
Navigation.
From which taking away 45° oo', it will be,
As the Radius
10.0000000
2
To the Tangent of the Remainder 3° 38' -8.8037619
So is the Tangent of half the Sum of the Angles
ADC, ACB, =63.17
310.2981626
To the Tangent of half their Difference=2° 12'-—9.1019245
Which added to the half Sum 63° 17', gives 70° 29' for the Angle
ADC; whence the firſt Port bears from the ſecond S. 87° or' W. or
W. by S. W. nearly, and conſequently the ſecond bears from the
firſt N. 87° ol' E. or E. by N. $ E. Whence for their mutual
Diſtance DC i will be,
As the Sins of the Angle ADC 70 deg. 29 min.--- 9.9743018
To the Diſtance AC 16:31
So is the Sine of the Angle DAC=53° 26'
1.2123566
9.9048043
To the mutual Diſtance DC 13.90–
1.1428591
After the ſame manner, may the mutual Bearing and Diſtances of
three or more Headlands be found.
This and the firſt Cafe, are of great Uſe in drawing the Plot of
any Harbour, or in laying down of
any
Sea-Coaſt.
This!Solution will give an Anſwer to the following Queſtion be-
ing of the ſame Nature with the former, but expreſſed in other Words.
Three Ships ſail from one Port, the firſt ſails N. N. W. the ſecond
N. N. E. & E. the tbird E. by N. Ž N. 16 Miles, then the firſt bears
from the third W. N. W. the ſecond bears from the third N. W. by N.
1 W. I demand the firſt and ſecond Ships Diſtance Saild, and their
Bearing and Diſtance from each other.
1
Anſwer.
The firſt Ships Diſtance fail'd is 14.35 Miles, the ſecond Ships
Diſtance faild is 16.31 Miles, their Bearing is W. by S. & W. or E.
by N. & E. nearly; and their mutual Diſtance 13.9 Miles
Care
Navigation.
205
Caſe IV.
B
6
12/E
Coaſting along the Shore at s, I ſee three
Headlands, whoſe mutual Diſtances I know, viz.
AC=12 Miles, AB=9 Miles, BC=6 Miles, A
and find the firſt A, to bear from me N. NW.
the ſecond B, N. by E. and the third C, N. E. by
N. I demand the diſtance from the Ship at S,
to each of the Headlands, A, B, and C.
S
Geometrically.
1. Having form'd the Triangle ABC, with the three given Sides,
viz. AC=12, AB=9, BC=6, from the Point C draw the Line
CD, forming an Angle with the Line CA equal to 33 deg. 45 min.
equal to 3 Points, the diſtance between the N. NW. and N. by E.
alſo from the Point A draw AD forming an Angle of 22 deg: 30 min.
equal to 2 Points, the diſtance between the N. by E and N.E. by N.
2. About the Triangle ADC deſcribe a Circle (by the 19th of the
14th) and draw the Line BD, till it meet the Circumference in S,
from whence to the Points A and C, draw the Lines SA, and SC,
then will S be the place of the Ship, and conſequently the ſeveral
Lines SA, SB and SC, the ſeveral diſtances from the Ship to the ſeve-
ral Headlands A, B, and C, to find each of which by Calculation,
1. In the Triangle ABC are given the three Sides, AC=12, AB
-9, BC=6, whence to find the Angle BAC by the 6th Caſe of Sečt.
the 5th of Trigonometry.
To the Arithmetical Complement of AC=12
8.9208188
Add. the Arithmetical Complement of AB=9 9.0457575
Alſo the Log. of half the Sum of the Sides 13.5 I.1303338
And the Logarithm of the Remainder 7.5
-0.8750613
Half the Sum of theſe 4 Log. will give
19.9719714
the Co-fine of 14 deg. 29 min. which doubled
gives 28° 58' equal to the Angle BAC
-9.9859857
From BAC =28° 58', take CAD= (by the 11th) CSD=22° 30',
and there will remain BAD=6° 28'.
2. In
Еe
206
Navigation.
2. In the Triangle ADC are given, AC=12, the Angle A=(6y
the 11th) DSC=22° 30', the Angle C=(by the fame) DSA=33° 45',
and the Angle D=123°45', whence to find AD it will be by the
2d Care of Sect, the 5th of Irigonometry.
As the Sine of the Angle ADC=123°45'. 9.9198464
To the Diſtance AC=12-
1.079.1812
So is the Sine of the Angle ACD=33° 45'— 9.7447390
To the Side AD-8.018
0.9040738
3. In the Triangle ABD are given, AB=9, AD=8.018, and the
Angle BAD=6 deg. 28 min. whence to find the Angles ADB, and
ABD), it will be, by the 4th Caſe of Sext, the 5th of Irigonometry,
As the Sum of the Sides AB, AD, =-17.018
-1,2309085
To their Difference o.982
9.9921115
So is the Tangent of half the Angles D and B=86 46' 11.2480108
To the Tangent of half their Difference=45° 37' 10.0092138
Which ſubtracted from 86 deg. 46 min. leaves 41 deg. 09
min. equal to the Angle ABS, to which if the Angle DAB=
6 deg. 28 min. be added the Sum 47 deg: 37 min. will give the An-
gle ADS, equal to the Angle ACS (by the I Ith.)
4. In the Triangle ABS are given, the Angle ASD=33 deg.
45 min. the. Anglė ABS=41 deg, og min the Angle BAS=105 deg.
06 min and the Side AB=9. Whence,
1. To find AS the diſtance between the Ship and the firſt Head-
land A, it will be by Caſe the ad of Se&t. the 5th of Irigonometry:
As the Sine of the Angle ASB=33° 45'
9.7447390
To the Diſtance AB =9
So is the Sine of the Angle ABS=41°09'
0.9542425
9.8182474
To the Diſtance AS=10.66 Miles
1.0277509
2. To find SB, the diſtance between the Ship and the ſecond
Headland, it will be (by the ſame.)
As.
Navigation.
207
As the Sine of the Angle ASB=33° 45'-
9.7447390
To the Diſtance AB=9
0.9542425
So is the Sine of the Angle SAB:= 105° 06'
99847400
To the Diſtance SB=15:64 Miles
1.1942435
* 3, To find the Diſtance SC, between the Ship and the third
Headland, in the Triangle SAC are given, ASC=56 deg. 15 min.
the Angle SAC equal to 76 deg. o8 min. and the Side AC equal to
12, whence it will be (by the farne.)
As the Sine of the Angle CSA=56° 15'
9.9198464
To the Diſtance AC=12
1.0791812
So is the Sine of the Angle SAC=76° 08'
9.9871549
To the Diſtance SC=14.01 Miles
1.1464897
The ſeveral Varieties of this Caſe, ariſing from the different Po-
ſitions or Scituations of the Point S, or place of the Ship, are very
eaſily accounted for by any who is Maſter of this Solution. For if
the Point S or place of the Ship, had been in the Point D, the Con-
ftru&tion would have been in a manner the ſame, viz. by making the
Angles ACS and CAS, equal to the reſpective Angles ADS, CDS,
c. and the Arithmetical Solution the ſame ; and the ſeveral Di-
ſtances from the Ship at D to the ſeveral Headlands, would have
been as follows, AD equal to 8.02 Miles, DB equal to 1 37 Miles,
and DC equal to 5.52 Miles.
If the Point Sor place of the Ship, had been in E, the Caſe would
have been much more eaſy, and the diſtance from the Ship to the firſt
Headland A, would have been found to be 6.29 Miles to the le-
cond Headland B 4.63 Miles, to the third Headland C5.52 Miles,
In the general Conſtruction of this Caſe, the Circles might have
been carried thro' the Points D, A, and B, or thro' the Points D, B
and C, and the Calculation would have been exa&ly after the ſame
manner.
This Caſe where it can be applied, is of extraordinary great Uſe
in drawing of Maps of Countrys
, or in laying down of Sands, Rocks,
or Sholes, in Sea-Charts; inaſmuch as it is done at one Station, or
by one ſingle Obſervation.
Ee2
And
208
Navigation.
And ſince I am upon the Buſinels of meaſuring Inacceſſable Di-
ſtances, I cannot bue recommend one way, which without Diſpute
when it can be put in Practice, is the moſt Eaſy and Expeditious
way hitherto known, and that is by firing of Great Guns.
It has been deterinined mary Years ago, by the Reverend and
Learned Mr. Flamſteed, His Majeſty's Aſtronomer at Greenwich, by
ſeveral Experiments made with a large Chamber, let off from the
Top of Shoote7 s-Hill, which is diſtant from the Royal Obſervatory a-
bout three Miles within a ſmall matter; that Sound travels at the
Rate of 1142 Feet in one Second of Time, and this by repeated
Trials was found to be the ſame, whether the Sound went with or
againſt the Wind; or ac leaſt the Difference was ſo ſmall as not to
be diſcoverable.
A Statute Mile contains 5280 Foor, and conſequently a Sound
requires 4.623 Seconds, or 9.246 half Seconds, or 9 4 half Seconds,
to travel an Engliſh Mile.
A Degree of a Great Circle, by the moſt accurate Obſervations,
contains 69.12 Engliſh or Statute Miles.
And becauſe a Geographical or Nautical Mile, is zó part of a De-
gree, it follows, that an Engliſh Mile is to a Geographical or Nauti-
cal Mile, as 60 to 69.12.
Whetefore, as 60 to 69.12, ſo is 4.623 Seconds, the time that
Sound requires to travel an Engliſh Mile, to 5.326 Seconds; the
time that it requires to travel a Geographical or Nautical Mile.
Again, becauſe every League contains three Nautical Miles, there-
fore Sound requires 15.975 Seconds, or 16 Seconds nearly, to travel
a Nautical League.
Wherefore, as i to 5.326, ſo is any given diſtance in Miles, to
the time that Sound will require to move over that diſtance; or as
I to 16, ſo is the diſtance given in Leagues, to the time that the
Sound will require to arrive at the given Place. On the contrary,
As 5.326 to 1, ſo is the number of Seconds between the time
that a Sound is generated, and the time that it arrives at any given
Place, to the diſtance between thoſe two Places in Miles; or as 16
to 1, ſo is the number of Seconds between the time of its generation,
and the time of its being heard, to the diſtance between the two
Places.
Hence if the diſtance between any two Places be given, the time
that Sound will require to travel over that diſtance may be found, and
on
Navigation.
202
on the contrary, the time that a Sound is going between any two
Places being known, the diſtance between thoſe two Places is
eaſily had.
And becauſe the Motion of Light is in a manner Inſtantaneous, it
moving at thc Rate of rocoo Miles at leaſt, (by the beſt Obſerva-
tions) in a Second of time, or whilſt a Pendulum of 39.2 Inches
makes one Vibration.
Hence, if the difference between the time that an Exploſion is
firſt feen, and the time that the Sound of it reaches the Ear be given,
the diſtance between the place where the Gun was Fir'd, and the
Place of the Obſerver, is eaſily found.
For, as 5.326 to 1, ſo is the number of Seconds, to the diſtance
in Sca Miles: Or, as 16 to 1, ſo is the number of Seconds, to
the diſtance in Sca Leagues.
Wherefore, if the number of Seconds between the time that the
Flaſh of a Gun is ſeen, and the time that the Report becomes
audible, be divided by 5.326, the Qliotient will give the number of
Miles of diſtance, between the place where the Gun was Fir'd and
the Place of the Obſerver; or if the ſame number of Seconds be di-
vided by 16, the Quotient will give the number of Leagues of
Diſtance.
Example 1.
Being at Sea, I counted 50 Seconds between the time that I ſaw
the Flaſh of a Gun and heard the Report ; I demand the diſtance.
IF 50 Seconds be divided by 5-326, the Quotient 9.388, or 9.4
Miles, will be the Diſtance required : Or, if the ſame number of
Seconds, viz. so, be divided by 16, the Quotient 3.125, or 3
Leagues, will give the Leagues of diſtance; fo that the Gun when
it was Fir'd was 9.4 Miles, or 3 Leagues from the Obſerver.
1
8
Example 2
Being by the Sea-ſide, there was a Ship in the Offing, who Fir'd
ſome Guns, and Obſerving, I found there was 40 Seconds between
the time that I ſaw the Flaſh and heard the Gun; I demand the
diſtance of the Ship from the Shoar.
Anſwer, 7 Miles, or 23 Leagues,
If
210
Navigation
If the Obſerver have not a Minute Watch by him, let them take
a String or piece of Wire of 32.2 Inches long, and faſten a Plumet
at one end of it, and hang the other upon a Nail or Peg, or hold it
between his Finger, and count the number of Vibrations between
the Flaſh and the Report, which will give him the number of Se-
conds, or it he has a mind to be more nice, let him take a String or
Wire of 9.8 inches, and prepare it as before, and the number of
Vibrations in this Cafe, will give the number of half Seconds of time.
It is generally found, that the number of Pulſations of the Artery,
or contractions of the Heart, in a Man of a good habit of Body, is
Seventy-Five in a Minute ; wherefore, this being ſuppoſed it fol-
lows, that Sound travels at the Rate of one Nautical Mile in o Pul-
ſations, or League in 20, and conſequently we can never be at a
loſs for putting of this method in Practice, for if the number of Pul-
ſations between the time that a Man ſees the Exploſion of a Gun
and hears the Report, be divided by 6} (or 7, for round Number)
the Quotient will give the number of Miles of diſtance, or the ſame
number of Pulſations being divided by 20, will give the ſame di-
ftance in Leagues.
edo
seosed ads Djeddato:d.edu.bdsuddedicado
va
Die wiederier
Wiederheduidendidosos
Section V.
Of Turning to Windward.
o know how near the Wind a Ship will lie, obſerve what Courſe ſhe
goes on each Tack, then in the middle between both is the Wind;
therefore the half of the Arch, or number of Points between each Courſe
(viz. that Arch the Wind is in) is the diſtance of the Courſe from the Wind,
TO
Caſe 1
.
A Ship that makes her way good within 63 Points of the Wind,
at N. by E. is bound to a Port bearing N. N. E. E. diſtance 84
Miles ;
I demand the Courſe and Diſtance upon each Tack, to gain
the intended Port.
Geo-
Navigation.
211
B
ERNE
F
NWBIZ!
X
C
W
A
S
Geometrically.
1. Having drawn the Com-
Nd
paſs N, E, S, W. Let A re-
preſent the place the Ship is
in, and draw the N. N. E. E.
Y EBNE
Line AB, equal to 84 Miles,
then will be the Port the is
bound to.
2. Set of 6 Points on each ſide of the N. by E.or Wind Line Ad,
to x and y, and draw Ay, Ax, then will A y be the E. by in. & E.
Line or Courſe, upon the Larboard Tack; and A x the N. W. by
W. W. Line, or Courſe upon the Starboard Tack.
3. Thro' B draw BC, parallel to the N. W. by W. W. Line,
or Courſe upon the Starboard Tack, till it meet the E. by N. & E.
Line in C, then will 4C and CB be the diitances upon each Tack,
to find which by Calculation,
In the Triangle ABC are given, the Angle BAC equal to 53 deg.
26 min. equal to 48 Points, equal to the diſtance between the N. N..
E. į E. and the E. by N. ^ E. the angle ABC equal to 87 deg. 11
min. z, equal to 73 Points, the diſtance between the N. N.E.E.
and the N.IW. by W. ^ W.the Angle ACB equal to 39 deg. 22 min. ,
equal to 3 Points, equal to the diſtance between the N. W. by W.
W. and the W. by N. ^ W. and the diſtance AB equal to 84 Miles,
whence to find
I. The diſtance upon the Larboard Tack, it will be,
As the Sine of the Angle ACB=39° 22' }
9.8023585
To the diſtance AB=84
1:9242793
So is the Sine of the Angle ABC=87° 11' }
9.9994781
To the diſtance AC upon the Larboard Tack 132.25 2.1213989
2. For the diſtance upon the Starboard Tack it will be,
As the Sine of the Angle ACB=39° 22' }
-9.8023585:
To the diſtance AB=84
1.9242793
So is the Sine of the Angle BAC=53° 26'
9.9048043
To the diſtance upon the Star. Tack 106.35 Miles-- 2.0267251
So
2
212
Navigation.
So that the Ship in failing E. by N. Å E. 132:25 Miles, and then
N. W. by W. & W. 106.35 Miles, will arrive at her Port.
In the fornier Conſtruction, thc Line EF might liave been drawn
parallel to the E. by N. \ E. Line, and the Solution would have been
the ſame, and in this Caſe the Ship in ſailing iſ N. W. by W. } W.
106.35 Miles, with her Starboard Tacks on board, and then Ē. by
N. E. 132.25 Miles, with her Larboard Tacks on board, would
have arrived at her Port B in the ſame time.
This Caſe is of Uſe izr hiting a Port, or in doubling a Cape or Head-
land to Windward the Leeway being known.
Whoever underſtands the former Problem, will very eaſily re-
folve the following, the Operation being omitted for Brevity's fake.
Being at Sea I law a Ship directly to Windward, bearing N. E. a-
bout tliree Leagues diſtant, it was then about fix in the Evening,
fhe ſtood away with her Starboard Tacks on board, and ſo did we till
9, we had run at the Rate of 6 Knots, and then the Ship bore from
us E. N. we both made our way good within 6 Points of the Wind,
but we fore-reaching her, and being willing to ſpeak with her at 2
next Morning, I would know when I muſt Tack, and how far I
muſt ſail on each Tack, to be with her at that time, ſuppoſing the
Wind to continue the ſame, and ſhe to keep on the ſame Courſe.
Anſwer. I muſt Tack at 12h. 26m.or at 26 min. after 12, after I
have failed 25.73
Miles
upon
the Starboard Tack, and then in failing
11.76 Miles on the Larboard Tack, I ſhall meet the Ship.
This Queſtion is of great Uſe in hiting a Ship under fail.
Caſe II.
Being at Sea, the Wind at North, I ſaw a Headland bearing N.
by E. E. diſtant three Leagues, then I ſtood away between the
North and Eaſt eight Leagues, with my Larboard Tacks on board,
and then the fame Headland bore from me Weſt by North; I de-
mand the true Courſe.
Geo-
..
Navigation.
213
**
B
N
Geometrically.
1. Having drawn the Compaſs N,
E, S, W. Let A repreſent the Place
of the Ship at Sea, and draw the N.
by E. E. Line AB, equal to 27 Miles,
then will B repreſent the Headland.
E
2. From B draw the Line BC, paral-
lel to the E. by S. Line, till it interſects
S
AC at the diſtance of 24 Miles, from
the Point A, then will AC be the Ships Courſe; to find which in
the Triangle ABC are given, AC=24 Miles, AB=9 Miles, and
the Angle ABC equal to 95 deg. 37 min. 1, equal to 8 Points, e-
qual to the diſtance between the N. by £. { E. and the W. by N.
whence to find the Angle. BAC, by the firſt Caſe of Se&t. the 5th of
Trigonometry, it will be,
As the Diſtance fail'd AC=24
1.3802113
To the Sine of the Angle at B=95° 37' } 9.9979037
So is the Diſtance giveri AB=9
0.9542425
To the Sine of the Angle at C=21° 55'
9.5719350
Whence the Angle BAC will be 6 2deg. 27 min. į, and conſequently
the Courſe is E. by N. oo deg. 35 min. Eaſterly, or ſeven Points
from the Wind.
Hence by knowing how near the Ship can lie up to the Wind, may
the Leeway be found.
For ſuppoſe ſhe can lie up within o Points of the Wind, then in
this Cafe ſhe has made one Point Leeway.
When a Ship is near the Land as in the former Caſe, the Leeway
may be found much more exactly by the following Example.
Ff
Supporc
214
Navigation.
Suppoſe a Ship at A, lying up N. by W.
towards B, but that inſtead of keeping on
the Courſe AB, ſhe is carried on in the
Line AE, or N. by E.
B
N
A1 BIL
NBE
'Tis evident that the Point E will al-
ways bear the ſame from the Ship, becauſe
ſhe keeps on in the ſame Courſe AE, and
that the bearing of every other Point alters.
W
E
А A
Hence therefore, the difference between
S
the Courſe AE, the Ship is carried upon,
or N. by E. and the Point of the Compaſs the Ship Capes at, or the
Angle BAE, equal in this caſe to two Points, is the Leeway ſhe
makes.
This is of great Uſe when you are about to donble a Cape,
Headland, c. the Leeway. the Ship makes being known.
The Solution of the ad Caſe will give Anſwers to the following Queſtions.
1. There are two Ports bearing N. by E. } E. and S. by W.IW.
distant 9 Miles, a Ship from the Northermoſt Port fails E. by S. and
then ſailing 24 Miles farther, arrives at the Second or Southermoſt Port;
I demand the fiift Diſtance ſaild, and Courſe to the ſecond Port.
Anſwer.
The firſt Diſtance fail'd is 2:1.38 Miles, and the Courſe to the
ſecond Port W. by S. oo deg. 35 min. Wefterly.
2. Suppoſe two Ports that lie W. by N. and E. by S. a Ship from the
Westermoſt fails S by W. { W.9 Miles, another from the Southermoſt
Sails 24 Miles, and meets the former ; I demand the Courſe Steer'd by
the last Ship, and the diſtance between the two Ports.
Anſwer.
The Courſe Stcer'd by the laſt Ship is W. by S. co deg. 35 min.
Weſt, and the diſtance between the two Ports 21.38 Miles.
3. Two Ships ſail from the ſame Port, the one ſails N. by E. į E. 9
Miles, the other 24 Miles North Eaſerly, and then the firſt Ship is found
do bear. Weft by North ; I demand the ſecond Ships Courlé, and Diſtance
between the Shios,
Anſwer.
Navigation.
215
4.
Anſwer.
The ſecond Ships Courſe is E. by N. oo deg. 35 min. E. and diſtance
between the Ships 21.38 Miles.
Either of the former Queſtions might have been ſo propoſed, as
to have been Ambiguous and admitted of two Anſwers, as the
Reader will eaſily perceive, if he Conſults the 2d Example of the 3d
Caſe of S-Et. the 5th of Irigonometry. I ſhall give an Inſtance of one,
and leave the Operation as an Exerciſe for the Practitioner.
There are two Ports that lie E. by N. 35 min. Eaſterly, and W.
by S. 35 min. Westerly, a Ship from the Weſtermoſt Port fails North Eaſt-
erly 9 Miles, another from the Eastermoſt ſails W. by N. and meets the
former ; I demand the firſt Ships Courſe and ſecond Ships Diſtance.
Anſwer.
The firſt Ship’s Courſe may be N. by E. E. and then the ſecond
Ships diſtance will be 21.38 Miles; or the firſt Ships Courſe may be
N.' E. and then the ſecond Ships diſtance will be 23.15
Miles.
5. Being at Sea, the Wind at S. E. by S. we lay by under our Main
Sail, and ſaw two Iſlands, which by the Chart bore from each other E. by
S. and W. by N. diſtant 4 Leagues or 12 Miles, the firſt bore N. E. by
N. the ſecond N. W. by N. tben 4 Hours after, the firſt bore E. N. E.
the ſecond N. N E. I demand the Ships true Courſe and Drift per Hour.
The Solution of this, being after the ſame manner with Caſe the 3d of
Setion the 4th, I ſhall only give the Anſwer.
Anſwer.
Her true Courſe was W. N. W. 10 deg. 12 min. W. fo that ſhe
made her way good within 12 Points of the Wind nearly, and drove
at the Ratc of 2.58 Miles an Hour.
Caſe III.
There are two Ports that lie N. N. E. E. and S. S. W. 1 W.
diſtant 84 Miles, a Ship from the Southermoſt Port, cloſe haui'd
upon the Wind at N. by E. with her Larboard Tacks on Boara,
making one or more Trips, fails 132.25 Miles, then 106.35 Miles,
Ff
with
216
Navigation.
with her Starboard Tacks on Board, and then arrives at her North-
crinoſt Port; I demand the Courſe upon each Tack, and how near
the Wind The made her way good.
1. Having drawn the Compaſs N, E,
S, W. Let A repreſent the Southermoſt
Port, and draw the N. N. E. Ź E. Line
AR, equal to 84 Miles, then will. B re-
preleni the Northermoſt Port:
NBE
B
N
w
E
S
2. With 132.25 Miles, the diſtance
upon the. Larboard Tack, ſetting one
foot of the Compaſſes in A, deſcribe a
ſmall Arch at C, with 106.35 Miles,
the diſtance upon the Starboard Tack,
ſetting one foot in B, croſs the former Arch in C, and draw the
Line AC, the Courſe upon the Larboard Tack, and BC the Courſe
upon
the Starboard Tack ; to find each of which by Calculation,
in the Triangle ABC are given; tlie three Sides AC equal to 132,25
Miles, CB equal to 106.35 Miles, AB equal to 84 Miles; whence
to find the Angle BAC (by the 6th Caſe of Set. the 5th of Irigono-
metry.
To the Arithmetical Complement of AC=132.25 7.8786043
Add the Complement Arithmetical of AB=84
8.0757206
Arfo the Log. of Sum of the three Sides 161.3
2.2076344
And the Log. of the Remainder 54.95
-1.7399677
Half the Sum of theſe four Log. will give the Co-ſine of 19.9019270
26° 43',
43', which doubled gives 53° 26' for the <BAC
9.9509635
Hence, the Courſe upon the Larboard Tack is E. by N. 1 E. and
conſequently ſhe makes her way good upon this Tack; within 6 å
Points of the Wind.
Again, to find the Courſe upon the Starboard Tack, it will be by
Caſe the ift of Seit. the sth of Irigonometry.
AS
Navigation.
217
2.0 267375
As the Diſtance fail'd BC=106.35 Miles
To the Sine of the Angle at A=53° 26'
So is the Diſtance faild AC=132.25 Miles
To the Sine of the Angle at B=87° og'-
9.9048043
2.1213957
9.9994625
Hence the Courſe upon the Starboard Tack is N. W. by W. W.
and conſequently fee has made her way good upon this Tack within
6 of the Wind as bcfore.
By the help of this Cafe, may the Leeway a Ship makes be found,
provided it be firſt known how near ſhe can lie up to the Wind, as
the foliowing Example will fhew ; the Solution of which being of
the ſame kind with the former, ſhall be omitted.
Suppose two Port: bearing N. N. E. Ź E. and S. S. W. W. diftant
aſunder 9 Miles, a Ship lying up within 6 Points of the Wind, at N.
by E. ſails from the Southermoſt Port, 24 Miles with her Larboard Jacks
on board, and 21.3 Miles with her Starboard Tacks on board, and then
arrives at the Norther moſt Port; i demand her Leeway on each. Tacki
Anſwer.
The Courle upon the Larboard Tack is E. oo deg. 03 min. S. and
confequently ſhe has made 1 point_3 min. Leeway upon this Tack.
Her Courſe upon the Starboard Tack is W. N.W. 27 min. N. and
conſequently upon this Tack, ſhe has made to deg. 48 min. or i
Point n arly Lecway.
Whar the uſual Allowances made for Lecway are, the Reader
may ſee in Sect the 6th l'Art the 5th.
The foregoing Solution will likewiſe give an Anſwer to the follow-
ing Queſtion.
A Ship fails from a certain Port N: N: E. E. 9 Miles, then be-
trueen the South and Eaſt 21.3 Miles, and then is forced back by foul l'ea-
ther 24 Miles, to the place from whence ſhe came; I demand her C024.64
from the ſecond place to the third, as alſo her Courſe from the third place
back again to her Port.
Anſwer.
Hér Courſe from her ſecond Place to her third, is E.SE.
Southerly, and from the third Place back again to her l'ost Weſt
ou deg. 03 min. North
Suction1
27 min,
218
Navigation.
Jac
on GO
Section
VI.
Of Currents, and how to allow for them.
Def.
URRENTS are certain ſettings of the Stream, by the means
Co
of which all Bodies moving therein, are compell’d to alter
cither their Direction or Velocity, or both; and ſubmit to the
Laws impoſed upon them by it. Whence,
Cor.
1. If a Current ſet with the Courſe of the Ship it increaſes her
Morion, by as much as is the Drift of the Current.
Caſe I.
1
Thus, If a Ship fails N. N. E. 20 Miles in 4 Hours, in a Current
that ſets N. N. E. at the Rate of 2 Miles an Hour, che Ships true
Courſe in this caſe will be N. N. E. and her Diſtance failed 28 Miles
in the ſame time.
2. If a Current ſets directly againſt the Ships Courſe, it leſſens
her Velocity juſt ſo much as is the Currents Drift.
Caſe II.
So that if a Ship fails N. E. 30 Miles in 5. Hours, in a Current
that fets S. W. at the Rate of two Miles an Hour, then her-true
Courſe will be North Eaſt, and Diſtance run 20 Miles, in the ſame
time. But,
3.
If the Drift of the Current exceeds the Velocity of the Ship,
the is driven backward with a force equal to the Exceſs of the Rate of
the Current, above the Rate the Ship fail'd at.
Caſe III.
Thus; If a Ship fails N. E. 5 Hours, at the Rate of 6 Miles an
Hour, in a Current that ſets under foot S. W. at the Rate of 8 Miles
an
:
Navigation.
219
an Hour, then in this Caſe the Ship falls a Stern, and her true Courie
will be S. W. 10 Miles in the ſame time. But,
4. If the Ship thwart the Current, it not only leſſens or augments
her Velocity, but gives her a new Direction, compounded of the
Courſe ſhe ſteers and the ſetting of the Current, as is manifeſt from
the following
Lemma.
If a Body be agitated by two Morions at
the ſame time, the one with a certain Velo-
city that will carry it according to the Di-
redion of the Line AB, the Length AB, in
a given ſpace of time, the other according to
the Direction of the Line AD, with a Velocity that will carry it to
the Diſtance AD in the ſame time, I ſay that the Body will move
in the Diagonal AC, and at the end of that time, will be found in
the Point C. Wherefore,
A
B
D
Caſe IV.
If a Ship fails N. E. 110 Miles in 15 Hours, and a Current is.
found to ſet S. S. W. at the Rate of 2 Miles an Hour, and it be re-
quired to find the Ships true Courſe and Diſtance fail'd, in that time.
N
NNE
NE 110
Geometrically.
1. Having drawn the Quadrant N, A, E.
Let A repreſent the Place the Ship ſail'd from,
and draw the N. E. Line cqual to 110 Miles,
then will be the Place the Ship Caped at.
В
2. From C draw CB parallel to the S. S. W.
Line, cqual to 30 Miles, the Drift of the Cur-
rent in 15 Hours, thien will B be the Place of
the Ship, AB her true Diſtance fail'd, and NAB A
the Angle of the true Courſe; to find which,
In the Triangle ABC are given, AC 110 Miles, BC 30 Miles,
and the Angle ACB equal to 22 deg 30 min. equal to two Points,
equal to the diſtance between the N. N. E. and the N. E. whence to
find the Angle at A, it will be by the 4th Caſe of Sect. the 5th of
Trigonometry.
As:
NE9.56€
JE.
2:20
Navigation.
As the Sum of the Sides AC and CB=140-
2.1461280
To their Difcrence-80
I.go30900
So is the 1'angent of half the Sum of the Angles
A and B=78 deg. 45 min.
} 10.7013382
To the Tangent of half their Difference=70.49–10.4583002
Hence the Ang'e at A, will be found to be 7 deg. 56 min. and
consequently the Ships true Courſe will be N. 32 deg: 56 min. Eaſt-
erly; or N. E. 7 deg. 56 min Eaſt, whence to find the direct
Diſtance fail'd it will be, by the 2d Caſe of Sect. the 5th of
Trigonometry.
Ås the Sice of the Angle at.4=7° 56'
9.1399475
To the Drift of the Current BC=30
1.4771213
So is the Sine of the Angle at C=22° 30'
9.5828397
To the Diſtance run 83.18 Miles
1.9200165
j
The former Example might have been thus expreſſed :
A Ship fails N. E. 110 Miles, in a Current that ſets S. S. W.
30 Miles
in the ſame time, what is the true Courſe and Diſtance faild.
Anfuer.
Direct Courſe is N. 52 deg. 56 min. E. Diſtance 83.18 Miles.
Or it might have been thus expreſed
A Ship fails N. E. 110 Miles, in a Current that ſets S, S. W. at the
Rate of o Miles in the time the Ship Sails 22 Miles; I demand, &c.
Anſwer.
True Courſe N. 52 deg. 56 min. E. dire& Diſtance 83.18 Miles.
Caſe V.
If a Ship from the Latitude of 36 deg: 30 min South, ſails N. E.
by N. 40 Miles in 8 Hours, then N. E. by E. 36 Miles in 7 Hours,
in a Current that lets S.S. E. at the Rate of 2 Miles an Hour, and it
be required to find the dire& Courſe and Diſtance between the Ship
and her firſt Port, as alſo the Latitude the Ship is in.
1. Having
Navigation
2 2 1
N
صور)
NE B1140
955E
11
BNEDE 26
OG
1. Having drawn the Quadrant N, E.
Let A repreſent the place the Ship fail'd
from, and draw AC the N. E. by N.
Line equal to 40, alſo CB Parallel to
the S.S. E. Line cqual to 20 Miles, cqual
to the Diift of the Current in 8 Houis.
E
2. From B draw BD parallel to the N. E. by E. Line, and DG
parallel to the S. S. E. or Current Line, equal to 17 į Miles, the
Drift of the Current in 7 Hours ; and draw the Line GA, then will
G repreſent the Place of the Ship at the end of her Runings, GA tlie
direct Diſtance faild, and the Angle NAG the Angle of the truc
Courſe.
This Example being no other than a Repetition of the former, the true
Courſe and Diſtance anſwering to each Courſe Steer’d, may be found as in
the former Example, and thence the whole Difference of Latitude made,
by Caſe the 1ſt of Se&t. the 2d of Navigation, and conſequently the La-
titude the Ship is in: But the eaheſt and readieſt way, is to conſider it
as a Traverſe of four Courſes, and find the Difference of Latitude and
Departure to each courſe, after the manner taught in Traverſe Sailing,
as follows,
Dif. of Lat.
N. S.
Departure.
E.
W.
Courſes.
Diſt.
1
33.3
presente nemen
18.5
N. E. by N.-
S. S. E.
N. E. by E.
S.S. E. -
40
20
36
17.5
22.2
7.6
29.9
6.6
E
20.0
16.1
53.3
34.6
34.666.3 Eaſting
H
18.7 N.erly
Or, the two Settings of the current may be added together, and
then it may be conſidered as a Traverſe of three Courſes, as follows,
Gg
Courſe
sunt
222
Navigation.
Diff. of Lar.
N. S.
Departure.
E. W.
Courſes.
Diſt.
22.2
N. E. by N.
N. E. by E.
S.S.E
40.
36.
37.5
33.3
20.0
29.9
34.6 | 14.2
E
53:3
34.6 66.3 Eaſting
34.6
18.7 N.ing
Hence to find the Latitude the Ship is in, becauſe ſhe ſails from a
South Latitude Northerly.
From the Latitude fail'd froin --wys
36° 30' S.
Take thc Diference of Latitude made
18 7 Nly
Remains the Latitude the Ship is in
36 11 3 S.
To find the dire&t Courſe it will be, by Caſe the 4th of Se&t. the
ad of Navigation.
As the Difference of Latitude 18.7-
1.2718416
To the Departure 66.3
So is the Radius-
1 8215135
10.0000000
To the Tangent of the Courſe 74° 15'
10.5496719
Which becauſe the Difference of Latitude is Northerly, and De-
parture Eaſterly, is N. 74° 15' E. or E. N. E. 6° 43' E., Whence for
the direct Diſtance it will be,
As the Radius
10.0000000
To the Difference of Latitude 18.7
So is the Secant of the Courſe 74° 15
1.2718416
10.5663254
To the Diſtance fail'd 68.9 Miles
-1.8381670
After
Navigation.
223
After the ſame manner, may the Latitude the Ship is in be found,
and thence her direct Courſe and Diſtance fail'd, after ſhe has Run
any Number of Courſes in a Current.
Caſe VI.
L
E
B
A Ship coming out from Sea in the Night, has ſight of Scilly Light,
bearing N. E. by N. Diſtance four Leagues, it being then Floud
Tide, Tetting E. N. E. about two Miles an Hour, the Ship running
after the Rate ots Knots an Hour; I demand upon what Courſe
the muſt fail and how far, to hit the Lizard; it bearing from Scilly
E. į S. Diſtance 17 Leagues.
Geometrically.
1. Having drawn the Com-
paſs N, E, S, W. Let A re-
preſent the Place of the Ship w
at Sea, and draw the N. E, by
N. Line AS equal to 12 Miles,
then will S repreſent Scilly. -
S
2. From S draw SL equal to 51 Miles, and parallel to the E.;S:
Line, then will I be the Place of the Lizard.
3. Draw LC parallel to the W. S. W. Line equal to two Miles,
and CD equal to five Miles, and parallel to it draw AB, till
it meet LC produced in B, then wiil AB be the Diſtance re-
quired, and the Angle SAB the true Courſe ; to find which by Cal-
culation,
1. In the Triangle ASL are given, Ąs equal to 12 Miles, SL c-
qual 5 i Miles, and the Angle ASL equal 118 deg. 7 min. „, equal
to 101 Points, equal to the diſtance between the N. E. by N. and W.
N. Whence to find 1ſt the Angle SAL it will be, by the 4th Cafe
of Se&t. the 5th of Irigonometry.
As the Sum of the Sides AS and SL=63
1.7993405
To their Difference=39
1.5910646
So is the Tangent of ļ the Angles A and L=30° 56' 9.7776284
To the Tangent of half their Difference=20° 21'—9.5693525
Whence
Gg
g
2 24
Navigation.
1
Whence the Angle A will be found equal to srº 17', and conſe-
quently the direct Bearing from the Ship to the Lizard, is N. 85° 02'
E. or E. by N. 6° 17' E. and for the Distance it will be,
As the Sine of the Angle at A=51° 17'
-9.8922329
To the Diſtance from Scilly to the Lizard 5 I 1.7075702
So is the Sine of the Angle at S=118" 7'
-9.9454298
To the Diſt. between the Ship and the Lizard 57.65M. 1.7607671
Again, in the Triangle DLC are given, the Angle L equal to
17° 32', the diſtance between the E, N. E. and the N. 85 deg.
02 min. E. the Side CL equal to the Currents Drift in an Hour, e-
qual to 2 Miles; and the Side CD the Ships Diſtance run in the ſame
time, cqual to 5 Miles; whence to find the Angle at D it will be,
by the 2d Caſe of Sect. the 5th of Trigonometry.
As the Diſtance run CD=5 Miles-
0.6989700
To the Drift of the Current LC-2 Miles
So is tlic Sine of the Angle at L=17° 32'
0.3.0.10300
9.4789423
To the Sine of the Angle at D=6° 55'
-9.0810023
Hence, becauſe the Angle CDI is equal to BAL, the Courſe the
Ship muſt Steer, is S. 88 deg. oz inin. E. and for the Diſtance it
will be, by Caſe the 1ſt of Sect the 5th of Trigonometry.
As the Sine of the Angle ar C=155° 3
9.61.68944
To the Diſt. between the Ship and the Lizard 57.65 1.7607671
So is the Sine of the Angle at L=17° 32'
9.4789423
To the Diſtance ſhe muſt ſail 41.96 Miles
1.6228150
So that the Ship in failing 8h. 24m. S. 88 deg. 03 min. E. will ar-
rive at the Lizard.
This Example is of Uſe in hiting of a Port or doubling a Hendland, in
a Current or Tides-way,
Caſe
1
Navigation.
225
The uſual way to try Currents at Sea, is to let down a heavy
Caſe VIL.
A Ship that makes her way good within 6 Points of the Wind,
at N. N. E. is bound to a Port bearing N. E. by N, diſtant 16
Leagues or 48 Miles, and there is a Current under Foot, that ſets
E. N. E, at the Rate of two Miles, in the time the Ship fails five
Miles: I demand the Courſe and Diſtance upon each Tack, to
fetch the Port.
This Queſtion being a Compound of the 5th Caſe of the laſt seft.
and the 411 Caſe of this sect. I hall omit the Operation, and only
inſert the Anſwer.
The Courſe upon the Larboard Tack is N. 83 deg. 39 min. E.
Diſtance 40.64 Miles, and the Courſe upon the Starboard Tack,
is N. 22 deg. 11 min. W. Diſtance 37.98 Miles.
This Queſtion is applicable to hiting a Port, or doubling a Cape or.
Headland to Windward in a Current.
Weight of Lead, Stone, doc. out of the Head of a Boat, about
100 Fathom deep, which will ride the Boat as faſt as if ſhe were at
an Anchor; and then by the help of the Compaſs may the ſetting of
the Current be determined, and by the Log, the Drift or Rate of
Driving ; in the ſame manner as the Courſe upon which the Ship fails,
and her Diſtance run is found.
How it may be found, when the Ship is within fight of Land, the
following Examples will abundantly hew.
Cafe VIII.
A Ship from a certain Headland in the Latitude of 34 deg. oo min.
North, fails S. E. by S. 12 Miles in three Hours, in a Current that
fers between the North and Eaſt, and then the ſame Headland is
found to bear W. N.W. and the Ship to be in the Latitude of 33 deg.:
52 min, North ; I demand the ſetting and Drift of the Current
Geo
***
A
N
A
W
B
226
Navigation
Geometrically
1. Having drawn the Compaſs N, E, S,
W. Let A repreſent the place of the Ship,
and draw the S. E. by S. Line AB equal to
12 Miles, alſo the E. S. E. Line AC.
E
D
2. At the diſtance of 8 Miles the Diffe.
rence of Latitude, draw DC parallel to the
S
Eaſt and Weſt Line, where this cuts the E.
S.E. Line AC in C, gives the Place of the Ship.
3. Toyn the Points B, C, then will BC be the ſetting and Drift of
the Current, to find which by Calculation,
1. In the Triangle ADC, Right-angled at D, are given, the Dif-
ference of Latitude equal to 8 Miles, the Angle DAC equal to 679
30', equal to 6 Points, equal to the diſtance between the South and
E S. E. Whence to find AC, the Diſtance the Ship has fail'd, it
will be,
As the Radius
10.0000000
To the Difference of Latitude 8 Miles
So is the Secant of the Courſe 67 deg. 30 min
0.9030900
-10.4171603
To the Diſtance Run 20.9
-1.3202503
2. In the Triangle ABC are given, AB equal to 12 Miles, AC e-
qual to 20.9 Miles, and the Angle BAC equal to 33 deg. 45 min.
equal to three Points, equal to the diſtance between the S. E. by S.
and E. S. E. Whence to find the Angle at B it will be,
As the Sum of the Sides AC and AB=32.9
1.5171959
To their Difference=8.9
0.9493900
So is the Tang, of į the Angles at B and C 73° 07'1-10.5180008
To the Tangent of half their Diff.=41° 43'z -9.9502549
Whence the Angle ABC will be found to be 114 deg. 51 min. and
conſequently the Current ſets N. 8 Ideg. o6min. E. or Ě. by N. 2 deg.
25 min. E. whence to find BC the Currents Drift, it will be,
As
Navigation.
227
As the Sine of the Angle at B=114° 58'
99578041
To the Diſtance run AC=20.9
So is the Sine of the Angle at A=33° 45
1.3202503
9.7447390
To the Drife of the Current in 3 Hours 12.8 Miles 1.1071852
And conſequently it ſets at the Rate of 4.266 Miles an Hour.
He that underſtands what has been ſaid upon this Head, will find it no
difficult matter to give Solutions to the following Questions, and there-
fore I ſhall ornit the Operations for Brevitys ſuke; and only inſert the
Anſwers.
Cafe IX.
Being at Sea I ſaw two Headlands, one bearing S by E. tie o-
ther E. N. E. E they bore from cach other S. S. W. W. and N.
N. E. į E. Diſtance 5 Leagues, then I ſtood away E. S. 4 Hours,
and run 8 Leagues, and then the firſt bore from me W. by S.and the
fecond N. W. I demand the Setting and Drift of the Current.
Anſwer.
The Current ſets S 45 deg. 50 min. W.cr S. W. so min. W. at the
Rate of 1.7 Miles an Hour.
1
Caſe X.
A Ship takes her Departure from an Iſland in the Latitude of
36 deg. 50 min. North, bearing E. N. E. diſtant 4 Lcagues, and
fails W. by N. 15 Leagues in 9 Hours, and then ſaw another Iſland,
whoſe Bearing from the firſt (by the Chart) is N. W. by N. diſtant
14 Leagues, bearing E. by N. and then by Obſervation the Ship is
found to be in the Latitude of 37 deg. 19 min. North; I demand ihe
Ships true Courſe and Diſtance fail'd, alſo the Setting and Drilt of
the Current which is ſuppoſed to be in this place.
Anſwer.
The Ships true Courſe is N. 51 deg. 18 min. W or N. I. 6 deg.
18 min. W. her dire&t Distance Saildis 53.74 Miles, the Current fers
N. 4 deg. 54 min. E. at the Rate of 2.77 Miles an Hour.
Care
228
Navigation.
Caſe
XI.
A Ship from the Latitude of 48 deg. 30 min. North, fails 24
Hours on ſeveral Courſes, till by Dead Reckoning ſhe came into the
Latitude of 49 deg. 50 min. North, and her Departure was 60 Miles
Weſt, the next Day ſhe went on ſeveral Courles, till her Departure
was 120 Miles Eaſt, and Latitude by Dead Reckoning 49 deg.
30 min. North, bue by a good Obſervation ſhe is found to be in the
Latitude of 49 deg. oo min. and all this while there was a Current
that fet S. W. I demand the Drift of the Current, and the true
Courſe and Distance faild cach Day, alſo her Bearing and Diſtance
from her firſt Porr.
Anſwer.
The Drift of the Current is after the Rate of Mile an Hour, the
firf Diry's Courſe is N. 45 deg. 46 min. W. or N. W. 46 min, W.
diiect Diſtance 104.6 Miles. Second Days Courſe is S. 67 deg. 47 min.
E.0; E. S. E. 17 min. E. dire&t Distance 113.7 Milcs, Bearing from
the first Pnit N. E. Diſtance 42.42 Miles; Error in each Days Difference
of Latitude and Departure 15 Miles.
Caſe XII.
064
There are two Ports bearing N. E. by N. and S. W. by S. diſtant
200 Lcagues, a Ship fail'd from the Northermoft W. SW: and in
two Days fix Hours arrived at the Southermoſt, there was all the
way a Current under Foot, ſetting South by Eaſt; I demard the
Ships Diſtance failid and Rate of the Current.
Anſwer.
The Ship run at the Rate of 2.67 Miles an Hour, and the Current
Set 2.1 Miles in the ſame time,
Caſe XIII.
A Ship from the Latitude of 48 deg. oo min. North, fails for
Hours on ſeveral Courſes, till the be got into the Latitude of
47 deg. 10 min. North, her Departure by Dead Reckoning being
24
70 Miles
Navigation:
229
70 Miles Eaſt, but by a good Obſervation the ship is found to be
in the Latitude of 46 deg. 40 inin. North; the Reaſon of the Error
being ſuppoſed to proceed from a Current, ſetting between the South
and Weſt at the Rate of sį an Hour ; I demand the ſetting of the
Current, and the Ships true Courſe and Diſtance faild.
Anſwer.
The Current Sets S. 33 deg. 33 min. W. or S. W. by S. nearly; the
Ships true Courſe S. 32 deg. 04 min. E or S. S. E. 9 deg. 34 min. E.
direct Diſtance 94:39 Miles.
Caſe XIV.
Being at Sea two Leagues E. by S. from a certain Headland, I
ſtood away N. N. E. 5 Leagues in 3 Hours, and then ſaw an Iſland,
which by the Chart bore from the Headland N. N. W.43 Leagues,
bearing from the Ship W.by N. N. there was a Current under Foot,
detting between the South and Eaſt 2 Miles an Hour; I demand
the true Courſe and Diſtance fail'd, as alſo the ſetting of the Current.
Anſwer.
I be true Courſe is N. 52 deg. 30 min. E. or N E. 7 deg. 30 min. E.
Diſtance ſaild 12.84 Miles, ſetting of the Current S. 58 deg. 52 min. E.
or S. E. by E. 2 deg. 37 min. E.
Caſe XV.
Being at Sea, I ſaw two INands that bore from each other by the
Chart S. S. W. and N. N. E. diſtant aſunder 8 Miles, the North-
ermoſt bore from me W.S. W. Diſtant 12 Miles, then I ſtood away
S. E. 3 Hours, and run 15 Miles, and then the Southermoſt bore
W. by S. there was all the while a Current under Foot, that ſet W.
N. W. 1 W. I demand the Ships true Courſe and Diſtance, and the
Drift of the Current.
Anſwer.
The Ships true Courſe is S. 23 deg. 12 min. E. or S. S. E. oo deg.
42 min. E, direct Distance 9.37 Miles, during which time the Current
ſets at the Rate of 2.46 Miles an Hour.
Нh
Care
230
Navigation.
$
Caſe XVI.
Taking my Departure from the Land I ſaw a certain Cape, that
bore from me N. N. E. then I ſtood away W. by N, 3 Leagues in
3 Hours, and then the Cape bore from me E. Ž S. but there was
all the while a Current under Foot, ſetting N. E. by N. 1 E. at the
Rate of i Mile an Hour ; I demand the Ships true Courſe and Di-
ſtance ſail'd, as allo her Diſtance from the Cape.
Anſwer.
The Ships true Courſe is N. 48 deg. 46 min. W. or N.W'. 3 deg.
46 min. W. direct Diſtance ſaild 7.94 Miles, true Diſtance from the
Cape 7.86 Miles.
Caſe XVII.
Being at Sea I ſaw two INands, one bearing from me S. E. by S.
the other E. by N. then I ſtood away E. by S. 2 Hours, and Runs
Leagues, and then the firſt bore W. by S. and the ſecond N. W. by
N. during the whole time there was a Current that ſet S. W. by W.
at the Rate of zź Miles an Hour ; I demand the Bearing and Di-
ſtance of the two Iſlands.
Anſwer.
The two Iſlands bore from each other N. 10 deg. or min. E. or S.
10 deg. or min. W their mutual Diſtance is 8.2 Miles.
Caſe XVIII.
A Ship at Sea Eſpies an Inand in the Latitude of 40 deg. oo min.
N. bearing W. S. W. diftant 4 Leagues, and fails S. E. Ž E. 15 Leag.
in 9 Hours, and then ſaw another Iſland in the Latitude of 39 deg.
os min. N. bearing W. by S.and by a good Obſervation the Ship is
found to be in the Latitude of 39 deg. 22 min. North, and daring
the whole time a Current is found to ſet between the South and Weſt,
at the Rate of iš Miles an Hour; I demand the Ships true Couiſe
and Diſtance, ſetting of the Current, alſo the Bearing and Diſtance
between the two Iſlands.
Anſwer
Navigation.
231
..
Anſwer.
The Ships true Courſe S. 39 deg. 09 min. E. or S. E. by S. s deg.
min. E. Diſtance Run 49.01 Miles, the ſetting of the Current is S.
14 deg. so min. W or $. by W. 3 deg. 5 min. W. the Rearing of the
Irianus S. 38 deg. 18 min. W.or N. 38 dcg. 18 min. E. their mutual
Diſtance 70.08 Miles.
Caſe XIX.
24
A Ship takes her Departure from a certain Cape bearing N.N.W.
Diſtance by Eſtimation three Leagues, and having ſaild W. by S.
twelve Hours at the Rate of 7 Knots an Hour, arrives at an Iſland
bearing W. { N. from the former diſtant 17 Leagues, there is a Cur-
rent ſuppoſed to ſet between the North and Eaſt; required the Ships
true Courſe and Diſtance fail'd, alſo the Setting and Drift of the
Current.
Anſwer:
The Current ſets N. 42 deg. 49 min. E. or N. E. by N. I E. nearly,
at the Rate of 3.43 Miles an Hour, the Ships true Courſe is N. 75 deg.
46 min. W.or W. N.W. 8 deg. 16 min. IV. and direct Diſtance ſaid
18.7 Leagues, or 56.13 Miles.
Caſe XX.
A Ship under Sail diſcovers an Iſland dire&ly to Windward, bear-
ing E. by N. diſtant by Eſtimation two Leagues, and having ſtood
away N. W. by N. for four Hours, at the Rate of 4 * Knots an
Hour, another Iſland appears in ſight bearing from the firſt N. N.
E. E. and from the Ship E. by N. N. there is a Current all the
time ſetting under Foot S. S. W. at the Rate of 1 of an Hour ; I
demand the Ships true Courſe and Diſtance failed, as alſo the di-
ſtance between the Iſlands.
Anſwer.
The Ships true Courſe is N. 56 deg. 33 min.W.or N. W. by W.
18 min. West, the dire&t Diſtance 15.93 Miles, and the diſtance between
the Iſlands 11.48 Miles.
ne:
Hh 2
Se&.
--
232
Navigation
REZETA RENTALMENTELEZETLEREREDELE
Section VII.
Of the PLAIN-CHART
S
INCE a ſmall part of the Earths Superficies differs very little
from a Plain Surface, Ingenious Men were firſt led for (Ordinary
Ules) ro deſcribe the Sea-Coaſt between Places not very remote, on
Planes which they called Charts ; in which they made the Degrees of
Longitude equal in all Places to the Degrees of Latitude. So that,
A Plain-Chart, is a Repreſentation of ſome ſmall Part of the
Superficies of the Terraqueous Globe, in which the Meridians are
drawn parallel to each other, and conſequently the Degrees of
Longitude are equal every where to each other; and alſo equal to
the Degrees of Latitude, and is generally made to ſerve for ſome
particular Voyage, but when made, may ſerve for any Voyage, whoſe
diſtance does not exceed the Limits of the Chart. Wherefore,
If it were required to make a Chart that ſhall ſerve for a Voyage,
between the Parallels of 31° 30' and 30° 30' of North Latitude, and
to contain 3° 30' of Longitude. Or which is the ſame thing,
Suppoſe it were required to make a Chart that ſhall contain five
degrees in Latitude, and 3 deg. 30 min. in Longitude.
Having pitch'd upon a convenient Length, for the Breadth of a
Degree.
1. Draw the Meridian MN, and prick off the Number of Degrees
the Chart is to contain, from M towards N. And
2. At Right-angles to the Meridian MN, from the Point M
draw the Line MŇ, which may Repreſent the Parallel of 31 deg.
30 min. North, and fet off the Number of Degrees it muſt contain
in Breadth from M towards S.
3. Thro’the Point S draw SQ parallel to MN, as alſo the Lines
00, II, 22, c. parallel to MN, thro' the ſeveral Diviſions of
the Line MS, and theſe will Repreſent ſo many Meridians.
4. The Lines 31 31, 32 32, (c. drawn parallel to MS, thro'
the equal Diviſions of the Meridiān, will Repreſent ſo many Paral-
Jels of Latitude.
2
5. Each
.
Navigation.
233
5. Each of the Right-angles M, N, Q, and S, being divided into
eight equal Parts, the Lines drawn from the ſeveral Angular-Points
thro' the ſeveral Diviſions of the Arch, will be the Rhomb Lines
upon the Chart; and are of Uſe for the ready finding of the Bearing
of Places from each other.
6. Having divided each Degree into as many equal Parts as its
Breadth will admit of, the Chart is rendered fit for any manner of
Uſe that it can be applied to. And,
Firſt, let it be required to lay down a Place upon the Chart, from
its Latitude and Longitude.
1. Thro’the Degree in the, Meridian antwering to its Latitude,
draw a Parallel of Latitude.
2. Lay a Ruler over the Degree of Longitude in the Graduated
Parallel, and draw a Meridian, where this cuts the Parallel of Lati-
tude before drawn, gives the Place required.
By the help of this Method, may any Place be laid down by its
Difference of Latitude and Difference of Longitude from any Place
in the Chart, viz. by counting its Difference of Latitude from the
Latitude of the given Place, either Northwards or Southwards, ac-
cording as the Caſe requires ; and its Difference of Longitude from
the Meridian of the Place, either Eaſtward or Weſtward, and draw-
ing Lines thro'the Points thus found, as in the former Cafe.
N. B. According to the Principles of Plain Sailing, the Difference of
Longitude, Meridional Diſtance and Departure are all one ; being equal
among themſelves, and therefore what is ſaid of one, bolds good of the
other.
After the former manner, may a Map of any particular Country,
or of any part of the Sea-Coaft be drawn, if the Latitudes and Lon-
gitudes of the Places it muſt contain be known, or their Differences
of Latitude and Differences of Longitude from any known Place
whatſoever.
By the Inverſe Method, viz. by drawing Lines parallel to the
Meridian, and to the Parallel of Latitude paſſing thro' any Place
propoſed, may its Latitude and Longitude be known ; and confe-
quently the Difference of Latitude and Difference of Longitude, be-
tween any two or more Places, by the firſt Section of Navigation.
2, Let it be required to lay down a Place upon the Chart by its
Bearing and Diſtance from another place.
tom
After
FO
234
Navigation.
After having found out the Rumb-Line anſwering to the Bearing
given, thro' the Place given and parallel to that Rumb draw a Litie,
and ſet of upon it the given diſtance in Parts of the Graduated Me-
ridian, and the Point thus found, will Repreſent the Place required
to be laid down.
By the Inverſe Method, may the Bearing and Diſtance between
any two Places in the Chart be known, viz. by taking their Di-
stance between the Points of the Compaſſes, and applying it to the
Graduated Meridian, in order to find the Miles of Diſtance, and by
trying which of the Rumb Lines is parallel co the Line paſſing thro'
them, in order to diſcover the Bearing.
After this manner, may the Induſtrious Sailor diſcover upon what
Courſe and how far he muſt fail, to arrive at his deſired Haven ; as
alſo to lay down his ſeveral Runnings, and thence find the ſeveral
Places that he arrives at in the end ; and ſo be able to alter his
Courſe a new, in order to gain his Port in the ſhorteſt time poflible.
But,
Tho’this way be very Expeditious, and in ſome Caſes exact enough
for common Uſe, yet it is by no means proper to depend altogether
upon it, but rather to keep the Account of the ſeveral Runnings in
Numbers, finding the ſeveral Eaſtings, Weſtings, Northings or
Southings, and Tabulizing of them after the former Methods, and
then if at any time his Curioſity ſhall lead him to Examine how exact
he has been in his Workings, it will be of Uſe to him.
3. Let it be required to lay down a Place upon the Chart, by its
Bearing and Difference of Latitude from another Place.
Count the Difference of Latitude either Northward or Southward
(as the Cafe requires) on the Graduated Meridian from the given La-
titidi, and thro' the Point laſt found draw a Parallel of Latitude,
and lay a Ruler over the given Place, and parallel to the given Rumb
Line, where this cuts the Parallel before drawn, gives the Place re-
quired.
4. Let it be required to lay down a Place upon the Chart, from
its Diſtance and Difference of Latitude from another place.
Having drawn the Parallel of Latitude as before, by the help
of the Difference of Latitude, take the given diſtance between the
Points of the Compaſſes, and ſetting one foot in the Place given,
with the other croſs the Parallel before drawn, and the common In-
terſection will give the Place required.
5. Let
.
Navigation.
235
5. Let it be required to lay down a Place upon the Chart, from
its Bearing and Meridional Diſtance from another place.
Count the Meridional Diſtance in the Graduated Parallel from
the Meridian, paſſing thro' the Place given, either Eaſtward or
Weſtward (as the Caſe requires) and thro' the Point laſt found draw
a Meridian Line ; where this interſe&s the Rumb Line drawn paral-
lel to the Bearing, will give the Place required.
6. Ler it be required to lay down a Place upon the Chart, from
its Diſtance and Meridional Diſtance, from another place.
By the help of the Meridional Diſtance, draw the Meridian pro-
per to the Place you would find, then with the diſtance of the two
Places between the Points of the Compaſſes, ſetting one foot in the
Place given, turn the other about, till it interſects the Meridian laſt
drawn, which will give the Place required.
By the help of theſe Rules, may the Place of a Ship in the Chart
be found from any poſſible Data, if inſtead of the Bearing of two Pl.
ces, you make Uſe of the Courſe the Ship has Steerd.
It would be needleſs to illuſtrate theſe Rules by Examples, ſince
ne that underſtands what has gone before, will find it no matter of
Difficulty to reſolve either of theſe, or any other of the ſame Nature
with thoſe that have been propoſed in failing, by the help of the
Chart only
wwwbiedriweb www.wibiers
belius dicidio Dwiedsriddiesa
Sect. VIII. .
.
HE
Of ſailing under a Parallel.
THER TO we have conſidered the Superficies of the
Earth as a Plain Surface, and the Meridians as parallel to each
other, and conſequently the Degrees of Longitude equal to each other
in all Places, and equal to the Degrees of Latitude; which tho' it be
notoriouſly falſe in it ſelf, yet when a Voyage is performed near the
Equator, or a large one is divided into ſeveral ſmall ones, eſpecially
if the Ship return Home on or near the oppoſite Rumb, a tollerable
good Account may be kept thereby.
For
*
0
3
1
21
30
36
e е
f
35
35
g
h
S
34
3+
K
H
33
33
32
32
P
R
D
M
S
T
1
3
**************************
:
236
Navigation.
P
o
CI
a
For that the Degrees of Longitude grow leſſer and leſſer, the
nearer you approach the Pole, is manifeft from the adjacent Figure ;
in which P repreſents the Pole, PLL
a Quadrant of the Meridian, CQ the
Semidiameter of the Equator, and OL
the Semidiameter of any Parallel of
Latitude, which is to the Semidiame-
L
ter of the Equator Cl, as the Sine of
the Arch P L, to the Sine of the Arch
PQ: That is, as the Co-line of the
Latitude of the Place L, or Sine of the
Diſtance of that Parallel, from the Equa-
tor to the Radius.
Whence, (becauſe the Circumferences of all Circles are as their
Radij directly, as has been already ſhewn) it follows,
1. That the Circumference or Length of any Parallel, is to the
Circumference or Length of the Equator, as the Co-ſine of the La-
titude, or diſtance of that Parallel from the Equator, to the Radius.
And ſince fimilar Parts have the ſame Ratio with their Wholes,,
it follows,
2. That the Length of a Degree of any Parallel, is to the Length
of a Degree of the Equator, as the Co-line of the Latitude, or di-
ſtance of that Parallel from the Equator, to the Radius.
3. Hence the Number of Miles contained in a Degree of the Equa-
tor being given, the Number of Miles contained in a Degree of any
Parallel may be readily found, and conſequently
4. The Length of any Portion of the Equator being given, the
Length of the correſpondent Arch of any Parallel is likewiſe given;
and the contrary,
And ſince the Difference of Longitude is no other than the Length
of the Portion of the Equator, intercepted between the Meridians
palling thro' the given Places, Hence we are taught an eaſy way,
how from the Difference of Longitude between any two or more
Places, lying in the fame Parallel, to find the diſtance between them
or from the diſtance firſt given, to find the Difference of Longitude ;
as fall be Exemplified in the following Problems.
;
PROB.
#
Navigation.
237
Problem 1.
The Difference of Longitude between two Places, both lying
under the ſame Parallel of Latitude being given, to find the Diftance
between them.
Example:
Suppoſe a Ship from the Lizard in the Latitude so deg. oo min.
North, fail due Weſt, until her Diference of Longitude be 48 deg.
so min. equal to 2930 Nautical Miles, and it be required to find the
Diſtance fail'd in that Parallel.
Geometrically.
1. Make AQ equal to the Difference
of Longitude 2930 Miles,
2. With the Sine of go deg. oc min.
upon AQ, deſcribe an Iſoſceles Triangle
APQ, then will P repreſent the Pole,
AQ a Portion of the Equator, PQ the
Meridian of the Lizard, and PA the Me-
4
a
ridian of the Place the Ship is in.
P
*
N
L
3. With the Sine of the Complement of Latitude of the Lizard,
ſetting one Foot of the Compaſſes in P deſcribe the Parallel NL,
then will L repreſent the Place of the Lizard, and N the Place of
the Ship, and let the Liue NL be drawn, and it will be by the 17th
Prop..of Part the ist.
As PQ: PL:: LA: LN. That is,
As the Radius
10.0000000
To the Co-line of the Latitude 50° oo'
So is the Difference of Longitude 2930
9.8c80675
3.4668676
To the Diſtance in that Parallel 1883.36
3.2749351
Problem 2.
If the Diſtance fail'd in any Parallel of Latitude be given, to find
the Difference of Longitude, i. e.
If a Ship from the Lizard in the Latitude of so deg. oo min.
North, fail due Weſt 1883.36 Miles, and it be required to find how
much the has altered her Longitude.
Ii
Geome-
:.
238
Navigation.
L
Q
Geometrically.
P
1. Upon NL equal to 1883.36 Miles,
with the Sine of 40 deg. oo min. equal
to the Complement of the Latitude, de-
ſcribe the Iloſceles Triangle NPL, and
produce the Sides PL, PN, to Q and A,
till PQ and PA be each go deg. oo min.
and draw Al, which will be equal to
A
the Difference of Longitude. Wherefore,
As PL: PQ::LN: QA. That is,
As the Co-fine of the Latitude soºoo!
9.8080675
To the Radius
10.0000000
So is the Diſtance faild 1883.36-
3.2749333
To the Difference of Longitude 2930
3:4668658
Which converted into Degrees, make 48 deg. 50 miu.
So that in failing 1883.36 Miles, in the Parallel of so deg. oo min.
the Ship will alter her Longitude 48 deg, so min.
Problem 3
The Diſtance fail'd under any Parallel of Latitude, and the Diffe-
rence of Longitude being given, to find the Latitude of the Parallel
Example
If a Ship in failing 1883.36 Miles Weſt, alters her Longitude
48 deg. 50 min. or 2930 Miles, and it be required to find the La-
titude ſhe is in
Geometrically.
P
1. Upon Ag equal to the Difference of
Longitude 2930 Miles, with the Sine of
90 deg. oo min, deſcribe the Ifoiceles
Triangle APQ, and let fall the Perpen-
dicular PR.
N
L
2. With half the Diſtance fail'd, draw
DL parallel to PR, where this interſeas
A
2 the Meridian in 1, gives the Place of
the Ship, and QL will be the Latitude
of the Parallel required ; to find which by
Calculation it will be,
As
R
D
Navigation
239
As AQ: NL::PQ: LP. That is,
As the Difference of Longitude 2930
To the Diſtance fail'd 1883.36
So is the Radius
To the Co-line of the Latitude 50 00
3.4668676
3.2749333
10.0000000
-9.8080657
Problem 4.
3
S
Suppoſe two Ships in the ſame Parallel of Latitude but under
different Meridians, fail both directly North the ſamne Number of
Miles; let it be required to find their Diſtance aſunder,
Example
Suppoſe two Ships, one at the Madera the other at Bermudas, both
Places lying in the Latitude of 32 deg. 20 min. North, and diſtant
from each other 2380 Nautical Miles, fail directly North 600 Miles'
or till they are arrived in the Latitude of 42 deg. 20 min. North
and it be required to find their diſtance aſunder.
P
Geometrically.
1. Upon B M equal to 2380 Miles de-
ſcribe an Ilocſeles Triangle BPM with the
Sine of 57 deg. 40 min. the Complement
of the Latitude of the two Places.
S
2. Set oft'the Co-ſine of 42 deg. 20 min. B
M
the Latitude the Ship fail'd into from P
to S and S, then will S, S, be the reſpe&ive Places of the Ships, and
SS their diſtance aſunder ; to find which by Calculation it will be
As PM: PS:: MB: SS. That is,
As the Co-line of the Latitude they ſail'd from 32° 20' 9.9268314
To the Co-line of the Latit. they ſail'd into 42° 20' 9.8687851
So is their diſtance firſt given 2380
3.3765769
To their diſtance required 2082
3:3185306
Whence it appears, that
two Ships in failing 10° direaly N.one
from Madera and the other from Bermudas, are got nearer to each o-
ther by 298 Miles, which is į part nearly of the diſtance firſt given;
and is another Inſtance of the Imperfection of the Plain Chart, which
ſuppoſes the diſtance between theſe two Ships, to be the ſame in both
Latitudes.
Theſe are the Principal Queſtions in this kind of Sailing, tho'o-
thers may be propoſed; an Example or two of which I ſhall give
for the Learner's Practice.
1. Suppofc
li 2
-40
Navigation
2.L
1. Suppoſe two Ships in the Latitude of 32° 0' N. diftant aſunder
2380 Miles, fail due N. till their mutual Diſtance be 2082 Miles;
let it be required to find the Diſtance fail'd and Latitude they are in.
Anſwer. Diſtance fail'd 600 Miles, Latitude come into 42° 20 'N.
2. Two Ships in the Latitude of 42 deg. 20 min. North, differ-
ing in Longitude 48 deg. 50 min. fail both directly South, till ar-
riving at two INands, both (by Obſervation) are found to be in the
Latitude of 32 deg. 20 min. North ; let it be required to find each
Ships Diſtance fail'd, and how far the two Iſlands are aſunder.
Anſwer. Each Ships Diſtance fail'd is 600 Miles, and the mutual.
Diſtance between the Iſlands is 2380 Miles.
By the help of the firſt Problem is the following Table made;
flewing the Miles of Latitude anſwering to a Degree of Longitude
in any Parallel of Latitude, of great uſe for the ready reſolving all
Problems of this kind.
A TABLE ſewing ho.v many Miles Anſwers to a Degree:
of Longitude, at every Degree of Latitude.
Miles. D.L. Miles. IDL. Miles.
DLJ Miles. UD.L. Miles.
59. 99 23 55. 23 45 42. 43 67
23. 45
59. 96
24
46
68 22. 48
59.
62 25
54. 38 47 40.92 69
50
4
53. 93
48
15 70
52
59... 77 27 53. 46
49
39. 36 71 19. 54
59. 67
28
52. 97
501 38. 57
72
59. 56
29 52. 47 51 37. 76
73 17, 54
59. 42
Si. 26 52
36.94 74
59.
31 51. 43 53 36. 11 75 15. 52
32
54
76
58. 89
So. 32
34. 41 77
50
37 49. 74
56 33. 55 78 12. 48
13 58. 46 35 49. 15 57
79
45
14 58. 22
36 48. 54
58 31. 79
80
42
15 57. 95 37 47 92 59 | 30. go
81
16
57. 67 38
601.30.
82
35.
17 57. 37 39 46... 62
29. 09 83
7 32
18
40 450 95
62
84.5 28
19
63 | 27. 24 85 5 23
56. 38 42 44. 59
64 ( 26.30
86
4.
56. 01 43 43.
88
25. 36
14
55. 63 44
24. 41
88
29
55. 23 45 42 43 67 23.. 45 89
OS
Tho"
I
2
54. 81
41.. 68
21.
an AW N
59. 86
26
40.
20.
18. 55
olna
7
8
16. 53
26
9
10
WW WW NN
5.9. 08
50. 88
35. 26
14, 51
II
55
13.
12
58. 68
32. 68
II.
1.
9.
38
47. 28
00
61
57. 06
28. 17
1
56. 73
41
45. 28
W
an
20
21
18
65
87
3.
2.
22
43. 16
66
23
1.
i
२
Navigatior.
247
!
'Tho this Table be made only to whole Degrees of Latitude, yet
it will not be difficult to make it ſerve for any Latitude whatſoever;
by ſaying as i Degree is to the Difference between the next greater
and letler Tabular Latitude, ſo is the Exceſs of the Latitude propo-
fed above the next leſs Tabular Latitude, to a proportional Part;
which added to the Miles anſwering to the next leſs Tabular Lati-
tude, will give the Miles anſwering to the Latitude propoſed. Thus,
If it were required to find the Miles anſwering to a Degree of
Longitude, in the Latitude of 51° 32', it will be,
As i Degree or 60 Minutes, to 82, the Tabular Difference be-
tween the. Miles anſwering to 51° and 52°, ſo is 32' the Exceſs of
the Latitude given above 5 I", to 44.7 ; which therefore ſubftracted
from 37.76, the Miles anſwering to si Degrees, gives 37.31, the
Miles anſwering to i Degree in the Latitude of si° 32'.
By the help of this Table, it will be eaſy to Convert Degrees of
Longitude, into Miles of Diſtance, or Miles of Eaſting or Weſting,
into Degrees of Longitude, in any Latitude whatſoever. For,
1. If it were required to Convert Degrees of Longitude into
Miles of Diſtance, it will be
As 1 Degree to the Miles anſwering to 1 Degree in the Latitude
propoſed, fo is the Difference of Longitude to the Miles of Diſtance.
Wherefore,
If two Places lie Eaſt and Weſt of each other, in the Latitude of
43° 20' whoſe Difference of Longitude is 2° 27', and it be required
to find the Diſtance between them, having found out the Miles an-
fwering to i Degree, in the Latitude of 43° 20', viz. 43.64.it will be,
As 60 Minutes to 43.64, ſo is 2° 27' or 147 Minutes, to 106.92
Miles; the Diſtance between them.
Again,
2. If it were required to Convert Miles of Eaſting and Weſting
into Degrees of Longitude,
Having found out in the Table the Miles anſwering to a Degree
of Longitude; in the Parallel under which you fail it will be,
As the Tabular Miles to i Degree, ſo is the Miles you have
fail'd, to the Degrees of Longitude.
Suppoſe I have faild 104 Miles under the Parallel of 43° 30'
and it be required to know how much I have altered my Longitude,
it will be,
As
B
242
Navigation.
.
As 43.52 the Miles anſwering to i Degree, in the Latitude of
4.3 deg. 30 min. to 104 the Miles of Diſtance ; ſo is i Degree, to
2 deg. 23 min, the Difference or Alteration of Longitude, and ſo
for the reſt.
By the help of this Table, may all the Problems propoſed in this
Seâion be eaſily reſolved, but this I leave to Exerciſe the Ingenuity
of the Young Student.
gegen
3
OCouce ge
ngo
Jeg
RM
Section IX .
Of Sailing by the Middle Latitude.
By
the Method propoſed, and made Uſe of in the laſt Section,
tho a perfect Solution can be given to no Queſtion, except ſuch
wherein the Courſe is dire&tly under fome Parallel or Meridian, yet,
if the Places are not very far to the Northward or Southward of the
Equator, and even in this caſe not very far aſunder ; and their Me-
ridional Diſtance be converted into Miles of Longitude, according
to the Proportion of the Middle Latitude between them, to the Ra-
dius, the difference will be ſo very Imall, that almoſt in all Caſes it
may ſafely be rejected; as will manifeſtly appear, if the following
Solutions be compared with thoſe deduced from the true Method.
Caſe I.
Given both Latitudes, and the Difference of Longitude, to find
the direct Courſe and Diſtance.
Example
Let it be required to find the dire& Courſe and Diſtance between
the Lizard in the Latitude of so deg. oo min. North, and Barbadoes
in the Latitude of 13 deg. 30 min. North, differing in Longitude
52 deg. 58 min. Barbadoes lying ſo much to the Weſtward of the
Lizard.
Geome-
Navigation.
243
Saild
Diff. Long
bod om Diff:Lat.
Dist:
Geometrically.
1: With 90 deg. oo min. taken
EG
of the Line of Sines, ſetting one
А
Foor of the Compaſſes in A, fweep
the Arch DE, till DE be equal to
the Difference of Longitude 3178
Miles, and draw AE.
2. With the Sine of 58° 15' the
Complement of the Middle Lati-
C
tude, deſcribe the Arch FG, and Merid: Dist
B
draw FG
3. Make AB equal to the Difference of Latitude 2190, and fet
of the Meridional Diſtance FG from B to C, and draw AC, which
will be the Diſtance required ; and BAC the Angle of the Courſe.
To find which by Calculation, becauſe it is,
AF Co-ſine of Mid. Lat.: AD,R :: FG Mer. Diſt.: DE Diff. Long,
AB Dift. of Lat.
:R :: BC Mer. Diſt. : t, BAC Courie.
it will be (by the 12th of the 15th.)
As AB : AF::DE:t, BAC. That is,
As the Difference of Latitude 2190
3.3404441
To the Difference of Longitude 3178
3.5021539
So is the Co-line of the Mid. Lar. 31° 45'
9.9295989
To the Tangent of the Courſe soº so
10.0913087
For the Diſtance AC it will be,
As R: AB :: Sec. BAC : AC. That is,
As Radius
10.0000000
To the Difference of Latitude 2190
So is the Secant of the Courſe 50° 59'-
3:3404441
10.2009722
To the direct Diſtance 3478787
ट
-3:5414163
Whence it appears, that the direct Courſe is S. so deg. 59 min.
W. or S. W. 5 deg. 59 min. W, Diſtance 3478.87 Miles, differing
from the true Courſe obtained (by Mercators Sailing) o deg. 53 min.
and from the true Diſtance 64.73 Miles, (See the ad Problem of
Mercators Sailing) which conſidering the great Diſtance, is not very
Conſiderable.
Carc
i
244
Navigation.
Caſe II.
One Latitude Courſe and Diſtance fail'd being given, te find the
other Latitude and Difference of Longitude.
Example.
Suppoſe a Ship from the Lizard, in the Latitude of so deg. oo min.
North, fails 150 Miles upon the direct Courſe to Barbadoes, or S.
so deg. 06 min. W. and it be required to find the Latitude the Ship
is in, and how much fhe has altered her Longitude.
Geometrically.
With the Courſe so deg. c6 min. and
Diſtance fail'd 150 Miles, Conſtruct the
Triangle ABC according to the Directions
given in che ist Cafe of Plain Sailing, then
will AB be the Difference of Latitude, to
find which by Calculation it will be,
As R: AC :: CS, BAC: AB. That is,
As the Radius
E. G
C
la
D
10,0000000
To the Diſtance faild 150
So is the Co-ſine of the Courſe 5006
2.1 760913
9.8071626
To the Difference of Latitude 96.2
1.9832539
Hence, becauſe the Ship fails from a North Latitude Southerly,
the Latitude come into is 48 deg. 23 min. 1: North, and conſequent-
ly the Middie Latitude is 49 deg. 11 min. 16.
Conſtr. On A, as a Center, with the Sine of 40 deg. 47 min. iš
the Complement of the Middle Latitude, deſcribe the Archi FG,
and make FG equal to the Meridional Diſtance BC, and produce
AG to E, till AE=AD, be equal to the Sine of 90 deg. oo min.
and draw DE, which will be the Difference of Longitude. And
becauſe,
AC Diſtance failid: R:: BC Merid. Diſtance : S. BAС the Courſe.
AP Coline Mid. Lat : AD, R:: FG Mer. Diſt.: DE Diff. of Long.
it will be by the 12th of the 15th,
As AF: AC :: S. BAC: DE. That is,
As
Navigation.
245
As the Co-line of the Middle Latitude 49.1118
9.8152074
2.1760913
9.8848889
To the Diſtance fail'd 150
So is the Sine of the Courſe 50° 06'
To the Diference of Longitude 176.025
Or 2 deg. 56 min, in this Caſe Weſterly,
Caſe III.
2.2457728
EG
А.
Given both Latitudes and Courſe, to find the Diſtance fail'd and
Difference of Longitude.
Example
Suppoſe a Ship from the Lizard in the Latitude of so deg. oo min.
North, fails S. so deg. 06 min. Welt, until by Oblervation flie be
found to be in the Latitude of 48 deg. 23 min. 1: North, and it be
required to find the Diſtance fail'd and Difference of Longitude.
Geometrically.
1. Having found the Difference of La-
titude AB equal to 96.2 Miles, by the
Rules given in Se&t. the ift. Conſtruct
the Triangle ABC, according to the ad
Caſe of Plain Sailing.
2. Make AF equal to the Sine of 40 deg.
B
48 min.' ə the Complement of the Middle
Latitude, and FG equal to BC the Meridi-
onal Diſtance, and produce AG to E, till AE equal to AD be equal
to the Sine of go deg. oo min. and draw ED; which will be the
Difference of Longitude required. And becauſe,
AB, Diff. of Lat. :R:: BC Merid. Diſt. : t,BAC, the Courſe.
AF,Co-line of Mid. Lat.:R:: FG, Merid. Diſt.: DE, Diff. of Longit.
it will be, (by the 12th of the isth)
AF: AB::t, BAC: DE. That is,
F
D
Kk
.
246
Navigation.
As the Co-fine of the Middle Latitude 49° 11';:
9.8152074
To the Difference of Latitude 96.2
1.9831751
.So is the Tangent of the Courſe so° 06'.
10.0777263
To the Difference of Longitude 176.02
2.2456940
The Diſtance fail'd may be found by the ad Caſe of Plain Sailing.
For, R: AB :: Sec. BAC : AC. That is,
As the Radius
-10.0000000
To the Difference of Latitude 96.2
So is the Secant of the Courſe 50° 06'
1.9831751
10.1928374
To the dire& Diſtance fail'd 149.95
2.1760125
Cafe IV.
Both Latitudes and Diſtance fail'd being given, to find the dire&
Courſe and Difference of Longitude.
Example.
Suppoſe a Ship from the Lizard in the Latitude of 50 deg. oo min.
North, fails between the South and Weſt 150 Miles, and then by
Obſervation is found to be in the Latitude of 48 deg. 23 min. 15
North, and it be required to find the direct Courſe and Difference
of Longitude
Geometrically.
1. Having found the Difference of La-
A А
titude AB=96.2 Miles, with it and the
Diſtance fail'd Conſtru& the Triangle
ABC, according to Caſe the 3d of Plain
Sailing
2. With the Sine of 40 deg. 48 min.15
the Complement of the Middle Latitude,
deſcribe the Arch FG, till FG be equal
to BC the Meridional Diſtance, and
make AE and AD each equal to the
Sine of 90 deg. oo min. and draw DE the Difference of Longitude,
and it will be
As
EG
с
B
D
Navigation.
247
As AC:R:: AB:t, BAC. That is,
As the Diſtance fail'd 150
2.1760913
To the Radius
So is the Difference of Latitude 96,2-
10.0000000
1.9831751
To the Co-fine of the Courſe soº 06.1
2.8070838
Again, for the Difference of Longitude. Becauſe,
AC Diſtance fail'd
:R::BC Merid Diſt. · S, BAC the Courſe
AF, Co-line of Mid. Lat. :AD, R::FG Merid. Diſt.: DE Diff. Long.
it will be (by the 12th of the 15th.)
As AF: AC ::S BAC: DE. That is,
As the Co-line of the Middle Latitude 49° 11'tó 9.8152074
To the Diſtance fail'd 150
2.1760913
So is the Sine of the Courſe soº 06'1
9.8849417
To the Difference of Longitude 176.12–
2.2458256
Hence the direct Courſe is S. so deg. 06 min } W. Difference of
Longitude 176.12, equal to 2 deg. 56 min. it? Weſterly.
Caſe V.
Given one Latitude, Courſe and Meridional Diſtance, to find the
other Latitude, Difference of Longitude and Diitance fail’d.
Example.
Suppoſe a Ship from the Lizard in the Latitude of so deg. oo min.
N. fails S. 50 deg. 06 min. Weſt, until her Meridional Diſtance be
115.1, and it be required to find the Latitude come into, direct Di-
ſtance fail'd, and Alteration of Longitude.
By the help of the Courle so deg. 06 min.
and Meridional Diſtance 115.1 Miles; let
the Triangle ABC be Conſtru&ed, accord-
ing to the Directions given in the 6th Caſe
of Plain Sailing. Then for the Diſtance
fail'd it will be,
EG
А A
het en hoe het
Kk 2
AS
*
248
Navigation.
As S. BAC: BC::R: AC. That is,
As the Sine of the Courſe so deg. 06 min.
To the Meridional Diſtance.115.1
So is the Radius
9.8848889
-2.0610753
10,0000000
To the direct Diſtance 150.03-
2.1761864
For the Difference of Latitude it will be,
Ast, EAC: BC :: R: AB. That is,
As the Tangent of the Courſe so° 06'-
10.0777263
To the Meridional Diſtance 115.I
So is the Radius-
2.0610753
IO 0000000
To the Difference of Latitude 96.2.
1.9833490
Hence, becauſe the Ship fails from a North Latitude Southerly,
the Latitude come into will be 48 deg. 23 min. 8 North, and con-
ſequently the Middle Latitude 49 deg. II min.;..
Conftru&tion. Make AF equal to AG, equal to the Sine of 40 deg.
48 min.TŐ the Complement of the Middle Latitude, and GF equal
to BC the Meridional Diſtance, and produce AG to E, till AE e-
qual to AD, be equal to the Sine of 90 deg. oo min then will DE be
the Difference of Longitude ; to find which it will be,
As AF: FG :: AD:DE, That is,
As tlie Co-line of the Middle Latitude 49° 11' 9.8152074
10.0000000
2-06 10753
To the Radius
So is the Meridional Diſtance 115.1
To the Difference of Longitude 176.12
Or 2 deg. 56 min. it, in this Cale Weſterly.
Cafe VI.
2.2458679
>
Given both Latitudes and Meridional Diſtance, to find the Courſe,
Diſtance, and Difference of Longitude.
Example
Wir
Navigation.
249
Example
Suppoſe a Ship from the Lizard, in the Latitude of sodeg.co min. N.
fails between the South and Weſt, until her Meridonal Diſtance be
115.1, Miles and then by Obſervation is found to be in the Latitude
of 48 deg. 23 min. 1 North, and it be required to find the Courſe,
direct Diſtance and Difference of Longitude.
Geometrically.
By the help of the Difference of Latitude
96.2, and Meridional Diſtance 115.1. Let
the Triangle ABC be Conſtructed, accord-
ing to the Directions given in the 4th Caſe
4
of Plain Sailing. Then to find the Courſe
it will be,
As AB : BC ::R:t, BAC. That is,
As the Difference of Latitude 96.2
1.9831751
E G
to you
To the Meridional Diſtance 115.1.
So is the Radius
2.0610753
10.0000000
To the Tangent of the Courſe soº 07'
10.0779002
Or S. W. 5 deg. 07 min. Whence for the direct Diſtance it will be,
As the Sine of the Courfe so deg. 07 min.
9.8849945
To the Meridional Diſtance 115.1
So is the Radius
2.0610753
10.0000000 :
To the direct Diſtance 150
2.1760808
Conſtr. Make as before AF equal to AG, equal to the Co-line of
the Middle Latitude 49 deg. ir min. the ; FG equal to BC, the
Meridional Diſtance 115.1, AE equal to AD, equal to the Radius,
then will DE be the Difference of Longitude ; and it will be,
As the Co-line of the Middle Latitude 49° 11'i: 9.8152074
To the Radius
So is the Meridional Diſtance 115.1
10.0000OCO
2.0610753
To the Difference of Longitude 176.67
2,2458079
250
Navigation
Cafe VII.
Given one Latitude, Diſtance fail'd and Meridional Diſtance, to
find the other Latitude, Courſe and Difference of Longitude.
Example
Suppoſe a Ship from the Lizard, in the Latitude of so deg. co min.
North, fails between the South and Weſt 150 Miles, and is then
found to have departed from her former Meridian 115,1 Miles, and
it be required to find the Latitude ſhe is in, her direct Courie and
Difference of Longitude.
Geometrically.
With the Diſtance Run 150 Miles, and
Meridional Diſtance 115.1, Conſtruct the
Triangle ABC by Caſe the 5th of Plain
Sailing. Then is BAC the Angle of the
Courſe ; to find which by Calculation, it
will be,
As the Diſtance fail'd 150
2.1760913
EG
А A
To the Radius
So is the Meridional Diſtance 115.1--
10.0000000
-2.061073
To the Sine of the Courſe so deg. 07 min.
-9.8849840
Whence to find the Difference of Latitude it will be,
As the Tangent of the Courſe so deg. 07 min. 10.0779830
To the Meridional Diſtance 1 15.1 -
-2.0610753
So is the Radius
10.0000000
.
To the Difference of Latitude 96.2
1.9830923
Whence the Latitude come into will be 48 deg. 23 min. ' North,
and coníequently the Middle Latitude 49 deg. II min. .
2. With the Middle Latitude and Meridional Diſtance Conſtruct the
Triangles AFG, ADE, as in the three former Caſes and, it will be,
As
}
Navigation.
251
For the Difference of Longitude,
As the Co-fine of the Middle Latitude 49° 11'i:
To the Radius
So is the Meridional Diſtance 115.1
To the Difference of Longitude 176.12
9.8152074
10.0000000
-2.0610753
2.2458679
Caſe VIII.
Given one Latitude, Difference of Longitude and Meridional
Diſtance, to find the other Latitude, Courſe and Diſtance fail'd.
Example
Suppoſe a Ship from the Lizard, in the Latitude of 50 deg. oo min.
North, ſails between the South and Weſt, until her Difference of
Longitude be 176.12 Miles, and then is found to have departed
from fier former Meridian 115.1 Miles, and it be required to find
the Laticude ſhe is in, her direct Courſe and Diſtance faild.
Geometrically.
1. Make AD equal to the Sine of 90 deg.
oo min. and draw DE Perpendicular to it, equal
to the Difference of Longitude 176.12.
2. Draw G m parallel to AD, at the diſtance
of the Departure equal to 115.1, where this in-
terſeas AE, as in G, draw GF parallel to ED,
then will AF be the Co-fine of the Middle La-
titude; to find which, it will be,
As the Difference of Longitude 176.12
2.2458679
1
1
B
C
G
D
111
To the Meridional Diftance 115.1 -
So is the Radius
2.0610753
10.0000000
To the Co-line of the Middle Latitude 49° 11':-9.8152074
Hence the Latitude come into is 48 deg. 23 min. 7: North, and
conſequently the Difference of Latitude is 96.2 Miles.
Conſtr. Make AB equal to 96.2 Miles, and BC equal to FG
the Meridional Diſtance, and draw AC, which will be the Di-
ſtance faild, and the Angle A the Angle of the Courſe; to find
which, it will be
As
252
Navigation.
Ly
10.0000000
10.0000000
>
As the Difference of Latitude 96.2
1.9831751
To the Meridional Diſtance 115 1 ---- 2.0610753
So is the Radius - -
To the Tangent of the direct Courſe 50° 07'
10.0779002
For the direct Diſtance fail'd, it will be,
As the Sine of the Courſe 50 deg. 07 min. -9.88.49945
To the Meridional Diſtance 115.1
2.0610753
So is tiie Radius
To the direct Diſtance 150-
2.1760808
This Caſe according to the Rules of Middle Latitude Sailing is un-
determined, and capable of as many Anſwers as there may be aſſigned
diffcrent Latitudes.
To the end that nothing may be wanting to render this Seation as
Compleat as poffible, I fall add one Traverſe, with its actual Solu-
tion; and omit the Proceſs, judging it needleſs to any one that un-
derſtands what has been ſaid upon this Subject'
A Ship from a Port in the Latitude of so deg. oo min. North,
fails W. S. W. 26 Miles, then S. E. by E. 30 Miles, then S. W. by
W. W.36 Miles, then W. 64 Milcs, then S. W. { W. 15 Miles,
then S. E. by S. 37 Miles. I demand the Latitude the Ship is in,
and how much ſhe has altered her Longitude.
Solution.
ز
Diff. of Lat:
Diff. of Long.
Courles.
Pts. Dift.
N.
S.
E.
W.
al
6
37,30
26
30
9.95
16.67) 38.56
38.5
30
16.97
W.S. W.
S. E. by E.
S. W. by W. W
Weſt.
S. WW.
S. E. by S.
5
5
8
43
3
64
III
48.78
98.09
17.74
9.51
IS
37
30.76 31.45
Diff
. Lac.
83.861 70.01 201.91
70.01
Diff. Long.-131.90
Hence
Navigation.
253
ma
.
Hence, becauſe the Difference of Latitude is Southerly, the Ship
will be found to be in the Latitude of 48 deg, 36 min. 16. North;
having altered her Longitude 2 deg. 11 min. i Weſterly, differing
but of a Mile from the true Difference of Longitude; as the
Reader will eaſily ſee, if he compares this Solution with the Solution
of the faine Traverſe at the End of the 11th Seation.
And indeed, whoever will be at the Pains to compare the Solutions
of the ſeveral Queſtions in this section, except the firſt, with the So-
Intions of the ſame problems according to Mercator, in Sect. the rith,
(notwithſtanding the Places are ſo far to the Northward) will find
ſuch an Agreement between them, that he would readily conclude
that both Methods are equally true, if the contrary were not De-
monſtrable.
The abſolute Departure or Weſting in the preceding Traverſe,
according to Plain Sailing, is 85186 m. which reduced into Miles of
Longitude, by the Rules of Middle Latitude Sailing, is 131.67
Miles, equal to 2 deg. 11 min. 767, differing but of a Mile from
the Sum of the ſeveral Differences of Longitude, deduced from the
ſeveral Courſes and Diſtances ſeparately, and but of a Mile from
the true Difference of Longitude, deduced by the Rules of Mercators
Sailing, which alone is a fufficient Inducement to any one, who will
not make Uſe of the true Method by Meridional Parts, to make Uſe
of this at leaſt, ſince it will ſerve him without any conſiderable
Error, in Runnings, of 150 Leagues between the Equator, and the
Parallel of 30 Degrees; of 100 Leagues between that and 60 De-
grees of Latitude; and of sc Leagues, as far as we have any Oc-
caſion, as is eaſy to be proved.
.
Section X
Of Mercator's Sailing:
Hoever conceives the Superficies of the Terraqueous Globe to
be Inveſted with a Rete or Ner, compoſed of Meridians and
Parallels, cannot but eaſily apprehend when that the fame is Extended
on a Plane (and fo diſtorted, as to make the Meridians become parallei
to each other, and the Degrees of Longitude every where equal to
LI
each
W
254
Navigation.
each other, and to thoſe of Latitude) that very great and notori-
ouis Errors will unavoidably follow.
For ſince, (as we have already proved) the Parallels decreaſe in
the Proportion of the Sines of their Diſtances from the Pole, or
which is the ſame thing, that a Degree of Longitude under any Pa-
rallel, is to a Degree of Longitude under the Equator, as the Radius
is to the Secant of the diſtance of that Parallel from the Equator. The
Repreſentation of Places in this Chart remote from the Equator, will
be lo diſtorted, thar (for Inſtance) an Iſland in the Latitude of 60 De-
grees (where the Radius of the Parallel is but half ſo great as that of
the Equator) would have its Length from Eaſt to Weſt, in Compa-
riſon of its Breadth from North to South, Repreſented in a double
Proportion of what indeed it is ; (as Mr. Wright very well obſerves
in his Correction of Errors of Navigation.
To Rectify which, amongſt many other Inconveniencys he there
takes Notice of, and to contrive a Projection in Plano, that might
be liable to none of theſe, was the great Study of many of the An-
cients; and tho’a Method was hinted by Ptolomy himſelf, yet it was
never happily accompliſhed, till undertaken by our Ingenious Coun-
try-Man Mr. Edward Wright.
Which he did by letting the Meridians remain parallel as before,
and protra&ting the Degrees remote from the Equator, in like. Pro-
porton with thoſe of Longitude. That is,
By making a Degree of Latitude fo protracted, to a Degree of
Latitude (which is every where equal to a Degree of Longitude in
the Equator) as the Radius is to the Sine of the Diſtance of ſuch
Parallel from the Pole, or as the Secant of its Diſtance from the
Equator is to the Radius: The Reaſon of which is manifeft from
the following Figure.
Where the Arch PTA repreſents a Quadrant
of the Meridian, P the Pole, CA the Semidiame-
ter of the Equator, and NT the Semidiameter
P
of any Parallel of Latitude, which is to the Se-
mediameter of the Equator, as the Sine of the
Arch PT is to the Sine of the Arch PA; that is
as CS to CA, as hath been proved in Sect. the 8th.
From A draw AM parallel to CP, or perpen-
S
dicular to CA, till.it meet CT produced, in M.
And
:
M
N
T
Navigation.
255
And becauſe NT equal to CS, is ſuppoſed every where equal to
CA, and that CS is to CA, as CT is to IM; when CS becomes e-
qual to CA, CT will become every where equal to CM, that is,
the Semidiameter of the Meridian in any Parallel, is equal to the
Secant of the Diſtance of ſuch Parallel from the Equator.
Whence it follows,
1. That a Degree of the işlarged Meridian, is every where equal
to the Secant of the Arch contained between it and the Equator.
2. That its diſtance from the Equator, ſhall be equal to the Sum
of all the Secants contained between it and the Equator.
3. That the diſtance between any two Parallels on the ſame ſide
of the Equator, ſhall be equal to the Diference of the Sum of the
Secants of all the Arches, contained between each of the Parallels
and the Equator.
4. That the Diſtance between any two Parallels on contrary ſides
of the Equator, is equal to the Sum of the Sum of all the Secants,
contained between each of the Parallels and the Equator.
The Eflential Property of this incomparable Projection, (which
in moſt caſes is preferable even to the Globe it ſelf) being thus found
out and Demonſtrated, the next thing requiſite was a ready way for
finding the Sum of the Secants of any given Arch.
This Mr. Wright did by the continual Addition of the Secants of
I', 2, 3, 4, 5, 6C. and thence formed a Table for the eaſy Di-
viſion of the Meridian Line.
After him the Famous Mr. Oughtred Conſtructed new Tables, by
the continual Addition of the Intermediate Secants, viz. of }, 17,
2, 31, c. min. which Tables have been ſince Corrected, and
brought nearer the Truth by Sir Jonas Moor; and are thoſe that
I formerly made Uſe of.
And tho' either of theſe are lufficiently exad for the Mariners Uſe,
at leaſt as far as Navigation is pra&icable, yet it is abſolutely neceſſa-
ry to let the Learner know how the ſame Table may be Calculated,
indepently from the Length of the Arch of Latitude firſt given ; as
well to make him intire Maſter of this moſt uſeful Branch, as to
open a Way for him to Calculate the Meridional Parts to greater
Degrees of Exactneſs it required.
It has been already ſhewn (in Part the 2d, Seation the 2d, Page 79.)
that if a be put for the Length of any Arch, the Radius being
Unity, the Secant of the lame Arch will be, it ža? +2a+tarla
LI 2
:
6
+
256
Navigation.
3 6 2 6 8
A,
7257
وو
II
1991
399163
I
+382 +32 lata?!, &c. this therefore multiplyed by as the
Fluxion of the Arch, will give a +in’atzfata tanto
at
Cc. the Fluxion of the Sum of the Secants
its Integral or Flowing Quantity at alt=4a' tra? +
,&c. will be the Sum of all the Secants contained in,
or which can be ſet upon the Arch a.
This again Multiplyed by - where A ſtands for the Length of
A
the Arch which you deſign ſhall be the Integer or Unity, in your
Meridional Parts, (whether it be a Minute, a League, or a Degree)
will give the Meridional Parts of the Latitude propoſed.
Now becauſe in the common Tables of Meridional Parts, the
Arch of 1 Minute is taken for Unity, if the Series atlaitiat
+
at
&c. bé multiplyed by..0029088871, c.
which Number 000.290.888.20, &c. is the Length of the Arch of
I Minute, the Radius being Unity, the Product will give the Me-
ridional Parts anſwering to the propoſed Arch of Latitude a.
Let it be required to find the Meridional Parts anſwering to the
Arch of Latitude of s Degrees.
The Length of the Arch of 1 minute is
0902.9088.8208.66
This multiplyed by the minutes in 5 deg. =
300
Gives the Length of the Arch of 5 deg.= 0872:6646.2599
399 Pod
3
3
Wherefore, put a
Then will
to a
And
And
can'
0872.6646.2599
+1.1076.2019
+21.0875
+406
6 1
040
And a tým total ta' =
0873.7243.5959
This therefore divided by ooc2.9088.8208.66, will give 300.
381498, for the Meridional Parts antwering to ş deg: true to fix
Places of Decimals, or the Millionth part of a Mile.
Again, Let it be required to find the Meridional Parts anſwering
to the Arch of Latitude of so deg.
The
- **
--
Navigation.
257
The Length of the Arch of 1 min.is
This multiplyed by the min. in 10 deg. =
Gives the Length of the Arch of 10 deg.
0002 9088.8208.665
600
-1745.3292.5199
1
3
Wherefore put a as before =
Then will
And
Alſo
And
And
1745.3292.5199
+ 8.8609.6155
+674.8008
+5.9709
240
2
61
277
10402
szca
9
+573.
75 76
505.21
399168
1 1
+ 58
61
S04
in
ll
II
And Conſequently, a tiritza't
1754.2852.9702
tica't, insa
This again divided by 0002.9088.8208.06, &c. will give 603
10695795, the Meridional Parts anſwering to 10 deg of Latitude.
After the ſame manner, may the Meridional Parts anſwering to
any other Degree of Latitude be found : But if it be required to
find them to one Place of Decimals only, which is as far as the largeſt
Tables Extend, and which is fufficiently exact for all Nautical Uſes,
above three fourths of the Labour will be faved.
Again, inaſmuch as in the Stereographic Projection of the Sphere
upon the Plane of the Equator, the Rumb Lines are proportional
Spirals about the Pole, and the Differences of Longitude or Arches of
the Equator, are the Exponents or Logarithms, of the Rationes of the
ſeveral diſtances of the Pole from the Curve, or Arches of the Me-
ridian intercepted between the Pole and the Rumb Line ; which:
Arches are the Tangents of half the Complements of the ſeveral
Latitudes in the ſame Projection : It follows, that the Differences
of Longitude are the Logarithms of the Rationes of thoſe Arches one
to another,
And ſince the Nauticat Meridian Line, is no other than a Table
of the Longitudes, anſwering to each Minute of Latitude on the
Rumb-Line, making an Angle of 45 deg. with the Meridian, it is
manifelt that the Meridian Line is no other than a Scale of Logarith-
mic Tangents, of the half Complements of the Latitude.
Now, ſince in every Point of any Rumb Line, the Difference of
Latitude is to the Departure, as the Radius to the Tangent of the
Angle
:
258
Navigation.
Angle that Rumb makes with the Meridian ; and that thoſe equal
Departures are every where to the Differences of Longitude, as the
Radius to the Secant of the Latitude ; it follows, that the Differences
of Longitude on any Rumb, are Logarithms of the lame Tangents,
but of a different Specics, being proportioned to each other, as are
the Tangents of the Angles made with the Meridian.
Hence, from any Table of Logarithmic Tangents, it will be eaſy
to find the Number of equal Miles of Longitude, anſwering to the
Diſtance of any two Points in the Meridian Line of the true Chart,
and confequently to make a Table for the ready Diviſion of the ſame.
For having determined the Angle which the Rumb Line makes
with the Meridian, under which the Differences of the Logarithmetic
Tangents are the true Differences of Longitude, to every Degree
or Minute of Latitude ; it will be, as the Tangent of that Rumb is
to the Tangent of the Rumb under which the Differences of Longi-
tude anſwer to the Diviſions of the Meridian Line, ſo is the Diffe-
rence of the Logarithmetic Tangents of half the Complement of any
two Latitudes propoſed, to the Difference of Longitude on the pro-
poſed Rumb; or to the Meridional Difference of Latitude, between
the ſame two Places on the true Chart: And fo is the Logarithmic
Tangent of half the Complement of the Latitude of any Place, to
the Diſtance of the ſame Place from the Equator.
Now, If we put Unity for 1 Minute of Longitude, as in the com-
mon Meridian Line, becauſe the Length of the Arch of 1 Minute,
the Radius being Unity, is oo029088820866, 6c. and the Index
of the Brigg's Logarithms (from which Vlacq's Tables are made)
230258509299, c. it will be, as 230258509299, c. to
290888208665,6o. fo is the Radius, to 126331143874, &c. the Na-
tural Tangent of 51° 38'9"; the Angle which the Rumb Line makes
with the Meridian, under which the common Tables of Tangents
are the true Differences of Longitudes.
Wherefore, as 126331143874, c. to i Minute, ſo is the Diffe-
rence of the Logarithmetic Tangents of half the Complements of
the Latitudes of any two Places, to their Meridional Difference of
Latitude in Minutes. Hence and from the foregoing; it follows,
1. That the Difference of the Logarithmic Tangents, or
Tangent Complements of half the Complements of the Latitudes
of any two Places, on the ſame fide of the Equator, divided by
126331143874; &c. will give their Meridional Difference of Latitude.
2. That
.
Navigation.
259
A
2. That the Sum of the Differences of the Tangent of 45 deg. and
the Tangents or Co-rangents of half the Complements of the Latitude
of two Places, lying on contrary ſides of the Equator, divided by
126331143874, c. will give their Meridional Difference of Latitude
3. That the Difference of the Logarithmic Tangent or Co-tan-
gent, of half the Complement of the Latitude, and the Radius or Tan-
gent of 45 deg. divided by 126331143874, c. will give the Meridi-
onal Difference of Latitude in Minutes. Or which is the fame thing,
To half the given Latitude add 45 deg. and having found out the
Logarithmetic Tangent of the Sum, reject the Characteriſtic, and
divide the Remainder by 126331143874, c. and the Quotient will
give the Meridional Parts of the Latitude propoſed.
Hence we are taught a more ready way to find the Meridional
Parts contained between any two Places, or to form a Table for the
more eaſy Diviſion of the Meridian Line in the true Chart, by the
help of the Logarithi Tangents only, of which I ſhall give an
Example or two as follows,
Let it be required to find the Meridional Parts anſwering to
s deg. of Latitude.
From 10.00000.000 do, the Logarithmic Tangent of 45 deg, e-
qual to half the Diſtance of the Equator from the Pole. Take
9.9620524617, the Logarithmic Tangent of 42° 30';. cqual to half
the Complenient of the given Latitude. Or;
From 10.0379475383, the Logarithmic Co-tangent of 42° 30'
equal to half the Complement of the given Latitude, take 10.00000
.00000, the Logarithmic Tangent of 45 Degrees. Or,
To 2° 30' half of the given Latitude, add 45 Degrees, and from
10.0379475383, the Logarithmic. Tangent of their Sum 47° 30',
take the Index or Characteriſtic 1o, and divide the Remainder
the Quotient 300.38 1498, will be the Meridional Parts anſwering to
the Latitude propoſed.; agreeing exa&ly with the former Calculation.
Again, let it be required to find the Meridional Parts anſwering
10 10 deg. of Latitude.
* Take 9.9238135302, the Logarithmic Tangent of 40 deg, equal
to half the Complement of the given Latitude, from 10.00000.00000
the Logarithmic Radius, or Tangent of 45 deg. and the Remainder
will be 761864698. Or,
Take 10.00000.00060 the Logarithmic Radius, from 10.07618
64698, the Logarithmic Co-Cangent of 40 deg. and the Remainder
will be 761864698 as before. Or,
Haying
II
260
Navigation
vide the Remainder 382580215 the faine as in the former Caſe, by
Having found the Logarithmic Tangent of so deg. equal to the
Sum of half the given Latitude, and 45 deg. equal to 10.0761864698,
and reje&red the Index or Characteriſtic, and divided the Remainder
761864698, which is the Same in cach Cafe, by 12633.11438, the
Quotient 603.069579, will give the Meridional Parts of the Latitude
propoſed, viz. of rodeg. exactly the ſame with the former Calculation.
Again, Suppoſe it were required to find the Number of Meri-
dional Parts contained between the Latitudes of 5 and 10 deg.
From 9.9620524617, the Logarithmic Tangent of 42 deg.
30 min. equal to half the Complement of the leſler of the two given
Latitiides, take 99238135302, the Logarithmic Tangent of 40 deg.
equal to half the Complement of the greater of the two given Lati-
tides, and tlie Remainder will b: 382389315. Or,
Take 10.0379475383, the Logarithmic Co-tangent of 4odeg: equal
to half the Complement of the leſler of the two given Latitudes,
from 10.0761864698, the Logarithmic Co-tangent of 42 deg. 30 min.
half the Complement of the greater of the two given Latitudes, and di-
----
1 263311438, and the Quotient 302.688081, will be the Number
of Meridional Parts contained between the Latitudes of 10 and s deg.
or thc Meridional Difference of Latitude between them, this therefore
added to 300 381498, the Meridional Parts of s deg. will give 603.
.069579, the Meridional Parts anſwering to 19 deg of Latitude as
before found : Or taken from the Meridional Parts of 10 deg. will
give the Meridional Parts of s deg. equal to 603.069579.
After the ſame manner may the Meridional Parts anſwering to any
given Latitude be found, or the Number of Meridional Parts con-
tained between any two given Latitudes,but for the farther Application
of theſe Rules to Mercator's Sailing, I refer the Reader to Se&t. the 12th
To gratify the Curious, as well as to Obviate a fooliſh and filly 06-
je:7ioz, uſually made by Ignorant and Unthinking People, agninst the
Truth of Mercator's Sailing, (grounded upon a Notion they have
got, that becauſe the Tables of Meridional Parts now in Uſe, being made
by a continual Addition of the Tabular Secants, are not ftri&tly true; and
that therefore, the Concluſions drawn from them muſt be falſe.) I have
been at the Pains to Calculate, de novo, a New and CorreTable of Me-
ridional Parts, after the manner taught in this section, and to, much
greater degrees of Exactneſs than any that have hitherto been Publiſk'd
.
A
NEW and CORRECT
T A B L E
O E
Meridional Parts :
Toevery Degree and Minute of Latitude;
Computed after the manner taught in the prece,
ding Pages
Mm
262
A new and correct Table of Meridional Parts.
O.
I
2
Mer. Partsy Differ. Mer. Parts Differ. Mer. Parts | Differ.
Minutes! 1-nin
Minutesli
1.0006
2
63.0035 1.0002
ICO 6
1 0006
IC027
650038 | 1.0002
con alu
12.0000 1.0000
14
0.0000
1.000C
I
1.000C
1.0oor
2 2,000
1.000O
3
3'000
1.0000
4 4 0000
1.000O
5
5.0000
10000
6 6.0000
1.000C
7 7.0000
І осоо
8 8.0000
1.000:
9 90000
1.0000
IO 10,0000
1.0000
IL 11.0000
12
І орсо
13 13.0000
1.0000
14 0000
1.0000
15 15.000
1.0001
16,000
1.0000
17 17.0001
1.0000
18
1,0000
191 190001
I, 000
20
20.0OCI
1.0000
21 21 OCOL
1.0000
22 22.0001
1.0001
23 23.0002
1.0000
24 24.0002
1.0000
251
25.0002
1,0000
26 26.0002
1.0001
27! 27.0003
1.OCO
28! 28.0003
I cooo
29 29.0003
1.0001
30 30.00041
M
h0.2030
120 0241
2 1.0002
61.0032
1 2 1.02;
1.00)
62.0033
122 0256
1.0002
123.C 262
64.0037
124 0264
1.0001
[15.0275
1.0002
66.0040
1 26.0282
1,00O2
67.0042
127 0288
I-0002
68.0044
1 28.029
1.0002
69.0046
129.0302
1.0102
70.0048
130 0316
1.00:2
71.0050
131.0317
1.C002
72,005 1.2002
132.0324
730955 1.0702
133.0332
74.0557
134. 339
1,0002
75.005
13503 +7
1.0003
70.0062 | 136.2355
1.0002
77.0064
1.2003
137.0362
78.0067
138.0371
1.5003
79.0070
139.0379
1.0002
80.0072
140.0387
1.0003
81.0075
141,0395
1.0003
82.0078
142,0404
1.000.2
83.0080
143.0412
I ono3
84.0083
144.0421
1.0003
85.0084
145 0430
1,0003
86.0089
146.0439
1.0004
87.0093
147.0448
1.0003
88.0096
148.6457
1.0003
89.0099
149.0467
1.000+
90.01 03
150.0476
16
3
1000614
1.1007
1.00c6
1.0007
1,0007
1.00089
IO
1.CO 17
1,0007
(2
I 0208
1
1.0007
1.008 4
19
1,0008
6
I.coc8
1.0008
17
18
1. 208
1.000819
2
1.0008
21
1.0009
22
10008
1.0009
23
1.0009
24
25
1.0009
26
1.0009
1.0009
27.
38
1.0010
29
1.0009
30
M
18 0001
1
2
A new and corre&t Table of Meridional Parts.
263
2
Mer. Parts, Differ: Mer. Parts | Differ. Mer. Parts Differ .
Minutes 101:
30
31
3?
3?
34
31
36
32
38
39
40
41
42
43
nost in
0.0004
1.0000
31.0004
1.0000
22.0004
1.0001
33.0005
1.0000
340.05
1.0001
35,000 6
1.0000
36.00co
1.000)
37.0007
1.000
38 0008
I OCOU
390007
I 000
40'0009
1,001
4.0010
1.000I
42,0011
1.0000
43.001
1.0001
44 00:2
10001
45.0013
1.0 201
46.2014
1.000I
47.00151 1.0001
48 0016
1.0001
49.0017
1.0001
SO0018
1.OCOT
51.0019
I.CO01
52.0020
1,0001
53.0021
1.0001
54,0022
1000I
55.0023
J.0002
56.0025
1.0001
57,0026
1.0001
58,0027
1.0002
59.0029
1.000)
60.COJO
90.0103
150.0+76
3
1.0002
1.0010
91.0106
151.0486
3
1.0004
1.0010
92.0110
152.0496
1.0004
1.0010
93.0114
153.0500
3
I 000
LOCI
94.0117
154.0516
3 -
1.014
1.0010
95.0121
155 052
1.0004
1,0010
96.0125
156.0536
I 0094
1.0010
97.0129
157:0546
37
1.000
1.0011
98.0133
158.557
38
1.0004
1.0010
99.0197
159.0567
39
1.0004
1.0011
10 014:
160,0578
C
1.0 04
1.0011
101.0!45
161,589
+
1,0005
1.0011
102,015
162.06
+?
1.0004
1,001)
103.015+
163.0611
4:
1.0004
1.0011
104.0158
164.0622
I.coOS
1,0012
105.0163
165.0034
S
1.0004
1.00II
IC6.0167
166.0045
6
1.0005
1.0012
107.0172
137.065
1.0005
1.0012
108 0177
168.0669
1.0005
I.1012
109.0182
169.2681
19
1.0006
I 0012
I10.0188
170.069
5
1,0005
1.00 12
U11.0193
171.0705
1.0005
1.0013
112.0198
I.COOS
172.0748
I 2012
113 0293
3
1.0006 173.073
1.001
114.0 209
174.074
1.001
115.0214
175-075
1.0006
1.001
1 16.0220
176.0765
50
1.0006
1.0013
1 17.0225
1.0000
177.0782
1.0014
118.0232
5
178.0790
1.0006
1.0013
119.0238 1.0000
179.0806
1.0.14
55
1 20.0244
180.08 2:
60
2
M
Mn 2
+8
2
5
2
53
54
1,00
la
50
57
58
59
60
M
O
I
264
1 New and Corre&t Table of Meridional Parts.
3
4
5
Mer. Parts. Differ. Mer. Parts Differ. Mer. Parts Differ.
Minutesl 01
Minutes 01
2
243.2026/ 1.0025
1
180.0823
240.1952
181.0836
1.0013
1.0024
241.1976
1.0025
I.CO14
182.0850
2
242.2001
1.0014
31 183,0864
1.025
1,0014
4
184,0878
244.2051
1.0025
1,0015
51 185.0893
245.2076
1,0925
6186.0907 1.2014
246.2101
1.0026
7/ 187.0922 1.0015
247.2127
1.0026
8
TCOIS
188,0937
248.2153
1.0026
9 189,9952.1.0015
249.2179
1.0016
1.0027
IC 190.0968
250,2206
1.001S
1.0026
u 191.0983
251.2232
1.0010
1.0027
12 192,0999
252.2259
1.0016
1.0027
131 193.1015
253.2286
1,0016
I 0027
14 194.1031
254.2313
1.0016
1.0028
151 195.1047
255.2341
1.0016
1.0027
161 196.1063
256.2368
1.0016
1.0028
17 197.1079
257.2396
18 198,1096
1.0017
1.0028
258.2424
1.0016
1.0029
191 199.1112
259.2453
1.0017' 260,2482
1,0029
200. I 129
21 201.1146
1,0029
22 202,1163
1.007
I.CO29
262.2540
1.0018
23 203.1181
2632570
1.0330
1.0017
1,0029
264.2599
24 204.1198
1.0018
1.C030
25 205.1216
265-2629
1.0018
26 206.1234
265.2659
1.0030
271 207.1252
267.2689
1,0030
28208.1270
I'0018
1.0030
1.0018
29 209.1288
1,0030
269.2749
1.0019
1.0031
210,1207
270.2 780
MI
3
4
300.3815
301.3853
302.3891
303.3930
304-3969
305.4008
306.4048
307.4088
308 4128
309.4169
310.4210
301.4251
3124292
313.4334
314.4376
315.44.18
316 4460
317.4503
318.4545
319-4588
320.4631
321.4075
322.4719
323'4763
324.4808
325.4852
326.4897
327.4942
328.4988
329.5034
220 5080
5
1.0038
1
1.0038
1'003
1.0039
3
4
1.0039
5
1.0040
6
I 004
7
1,0040
8
1.041
9
1.004.1
IO
LU041
II
1.0041
I2
1.0042
1.0042
13
14
1.0042
1.0042
16
1.0043
17
I 0042
1.0043
1.0043
19
20
1.0044
21
1.0044
22
1.0011
23
1.0045
1.0044
25
1.0045
26
1.0045
27
1.0046
1.0046
29
1.0046
30
M
20
1.00171 261.2511
24
I 0018
268 2719
28
30
A new and correct Table of Meridional Parts.
265
3
4
S
Mer. Parts Differ. Mer. Parts | Differ. Mer. Parts | Differ.
Minutes 1 olimon
Minutes and
270,2780
1.0019 272.2841
1.0019 273.2872
1.0?
32
32
214.1333
2
3Ć
3
38
1.024
330.5080
30
1.0046
331.5126
31
1.0047
332.5173
32
1.0047
333.5220
32
1.0047
371 5267
34
1.00.14
335.7315
1.0048
230.536;
1.0048
337-5441
1.0049
338.5460
339.5508
35
1.00.19
3405557
1.0049
3+1.500
+
1,0049
342.5655
4
1.005
343.5705
ti
1.005
344.5755
fa
1.005
345-5805
OS
1OOST
346.5851
1.005)
347.5 907
+
1,0051
348'5958
1.0052
349.6010
45
I 0052
5
1.0052
351,0114
5
1.005?
352.0167
1 005)
1.0053
1.0014
2 10.1307
30
1.0019
1.0030
211.1326
31
1.0031
212,1345
3
213.1354
1,0019
1.0032
274.2904
34
1.0019
1.7032
215,1402
275.2935
35
1.0020
1.0032
216.1422
276.2968
36
1.0020
1.0032
217.1442
277.300
37
1.0020
1,0033
218.1467
278 3033
38
10020
1.0033
219.1482
279.3966
39
I 0021
1.0.24
220. I 503
280.3 100
10
1,0320
1.031
2211,3
281.3134
41
1.0021
1.003+
222.1544 1.0021
2 3 2.3168
1.0034
223.156
283.3202
1.0021
1.035
284.3237
1,0)2 i
1.0034
285-3271
45
1,022
1.0035
236.3305
1.0022
1,0035
237,3341
10022
1.0035
288.3376
1.0022
10035
289.3411
1.0323
1,0035
290.3445
1.0022
1.0030
231.1740
291.3482
5!
1.2023
1.0036
$2232,1763
292.3518
1,002
1.0036
53.233.1786
293.3554
1:0023
1.0037
54 234 1809
294.3591
1.0023
1,0037
; 5
235,1832
295.3628
I 0024296.3665
1,0037
50 236.18;6
1.0023
1.007
571 237.1879
297.3702
1.002+
1.0037
$8 238.1903
298.3739
1024
1.0038
59
239.1927
299.3777
1.0025
1.0038
50 2401952
209,2815
3
4
42
43
14) 224,1586
451_225.1607
40 226.1629
+7) 227,1651
178 228,1673
49
229.1595
230.1718
(
48
350.6062
܃
353.6220
354.6273
355.6327
356.6380
357.6434
358.6488
359.6543
360.6598
1.0053
ic
1.0054
1.0054
1.0055
$9
1.0955
M
58
M
S
7
Minutes 1
Minutes or
3
1
+
10 57
IC
1.0086 509.8590
266 A New and Correct Table of Meridional Parts
6
8
Mer. Part s. Difer. Merid. Prs. Differ. Mer. Parts Differ.
360 6598
42L0487
481.5673
361 6553
1.0055
1.007
422 056?
482.5771
1.0098
1.0056
2 362.6709
1.0076
423.0638
483.5870
1,0099
2
& 363.6765
1.0056
10076
1.0099
424.0714
1.0056
484.5969
3
I C070
364.6821
1.0100
4
425.079
485.6069
4
30568781.0057
1.0077
10100
426.6867
486.6169
61 36 6.6924
1.0056
1,0077
1.0101
427.0944
487.6270
6
7 30706991
1.0101
428.1022
1.0078
487.6371
7
8 368 7048
1.0057
1.0078
1.0101
429.1100
489.6472
8
1 0078
1.0102
93697100 1.0058
430.1178
4906574
1.0058
104 3707164
1.0079
1,0102
431.1257
491,6676
U 371.7222
1,0.58
1.0079
I 0103
492.6779
432,136
1,0059
12372.7281
1.0079
1,0103
433.1 +15
493,0882
1,0359
1.0080
1.010
131 373.7340
434.1 +95
494,6905
3
1,0059
141 3747399
1.0080
435.157)
1,0010
495.7087
1,0104
1.0080
15 375 7459
436.1675
496.7193
1.010.
1.000
1.0081
16 376.7519
497.7298
1.0105
1.0060
437'1736
1.0081
17 377-7579
498.7403
1,0105
181 378.76.40
1.0082
1.0106
1.061
439.1899
499.7509
TE
1.0082
1.010
1 379.7701
1.0082
50.7615
20 380 7762
9
1.0107
441.216;
501,7722
20
1,0062
21 381.7824
1.0083
502.7829
1,0107
7886
1.0082
1.0107
443.22:9
1.0063
:31 383.7949
503,7936
22
1.0083
J.0108
504.8044
1,0084
1.0108
2 :
24 384-800 1.0062 444.23 12
505 815
1.0084
23_385.8074_1.063 445 2396
1.0109
446 2480
505,8261
1.0003
261 386.8137
1.0184
1,010G
1.0064
447.2564
26
271 3-87.8201
1.0110
1.0063
448 2049
27
28 388.8254
1.01 10
1.0064
449.2734
2
29 389.8 328
1.01 IC
1 0064 450,2820
39200.8392
1.00861 5 10.8700
25
1.010
451.2906
511.8811
30
M
6
7
8
M
I 2
1
1.0661 438.1817
!?
1.0061 440.1981
20
2
1.0085 507 8370
1.0.85 508.8480
-
1
1 New and Correet Table of Meridional Parts. 267
6
ៗ
8
Mer. Paris. Differ. Mer. Parts | Differ. Mer. Parts Differ.
Minuteslili na
Minutes and more
1.0065
1. C065
1.0065
1,0036
1,0066
32
451.2905
452.2994
453.3079
454.3166
455.3254
455.3342
457.3439
458.3519
459.3608
400.3098
461 3788
462.3078
463.3969
38
1.6090
40
41
390.8292
391.8457
32 92 8522
331 393.8587
4 394 8653
25_395.871
3,90
297.8853
38 398.8920
91 399,
349,8988
400,9056
401.912
+1 402,9192
31 703.9262
144 4.9331
15 4 5 9401
16 4.6.9+71
+? 407.95 + 1
181 408.9612
+9 4 9.9683
411,9754
511 411 9826
1+ 412.9898
13
131 413.9970
541 415.0043
55 416.0116
16 417.0189
571 +18.0263
419.0337
591 420.0412
61
421.6487
M м
6
1.0067
IC067
1.0068
1.0068
1.0003
1,0069
1.0067
1 0069
1.0070
1.0070
I C070
1.0071
1.0071
1.0071
464.400 1.0091
465.4152
460.4244
467.4336
468.4429
469.4522
511.880
30
1.0086
1.011
512.8922
1.0057
LC112
1.0087
513.9034
10112
1.07885149146
33
1.0113
1.0088 515.9259
34
ION 2
$10.9372
35
1.0208
1.0:14
1.0088
$17.9406
36
101 14
518.COLO
1.0089
1.0114
37
519.9714
1.0090
1.0115
520.829
$21.9944
1.0115
1.099
1.0071
523.006; 1,0116
10116
524.0176
10117
42
I 0092
525.1293
43
1.0117
1.0092
526.0410
4
527.0527
1.6117
45
1.0092
528.0645
1.0118
1.909
1.0092
529 0763
I
1.0119
S3IIdol
1.0094
49
10120
522,112
1.0094
so
1.0120
1.009)
533.1241
1,0121
1.0095
534.1262
52
1.0121
1.0096 535-1483
53
1.0122
I 0096
536.1605
54
1.01 22
537.1727
1.0097
5)
1.0123
1.0-97
538.1854
I 0123
56
1.0097
539.1973
57
10123
1.0098
540.2096
58
1.0124
1.0098
541.2220
:59
1.0124
542.2.244
30
8
M
1.0118/46
530.0882 10119
47
48
470,4610 1.0094
51
LCO72
1.0072
1.CO72
1.0073
1.0073
1.0073
1 0074
1'0074
1.0075
1.0075
471.4710
472.4804
473 4899
474+994
475 5093
475.5106
477.528
478.5380
479.5477
480.5575
481.5672
7
58
168 A New and Correct Table of Meridional Parts
IO
II
Minutes
Minutes!ol
1,0188
2
1.0126
10126
1.0189
542.2241
il 543.2459
2 514.2594
3) 545 2720
8 546.2846
547.2973
548.3100
549.3228
El 550.3356
9 551.3485
IO
151557.4268) 1,0132 618.30591
9
Mer. Parts. Differ. Merid.Prs. Differ. \Mer. Parts Differ.
603.0696
664.0922
1.0125 604.0850
1.0154 665.1109
1.0187
1.0125 605:1005
1.0155 | 666.1297
10156667.1486
1.0189
606.1161
3
6071317
10155668.1675
4
1.0190
1.0127
609,474
1.0157 669.1865
S
1.0127
1.01906
0.9.162
1,0157 670.2055
1.0191
10128
1.0158
610.1789
671.2246
7
1,01928
1.0128
611.1947
10158
672,2438
1.0129 612.21 06
10159
1.0192
673,2630
9
1.0193
1.0129 613.2265
1.01591 674.2823
552 3614
1,0130
1.0160
I 0193
ul 553.3744
614.2425
675.2016
1,0130
1.0160 676,3210
1,0194112
12
615.2585
1.0131
1.O6I
616,2746
677,3404
1.0131
1,0161
617.2907
678.3599
1.0196115
1.0162
679.3795
1.0132
1,0162
1.0196
619.3231
680.3991
1.0132
1.0163 681.4188
620.3394
1.0163 682.4385
1.029711€
is 560.4665 1.0133 621.3557
1.0164 683.4583
1.0298
19
i 501.4798 1.0133 622 3721
1.0134
1.0165 684.4782
1.0299
20 562.4932
623-3886
1,0199
1,0134
1.0165685.498
624.4051
1,0166686.5181
1.0200
1.0135
22
625.4217
1.0166 687.5381
1.0200
21
1.0135
626.4383
1.0201
1.0136
1,0167 | 688.5582
24
221 566.547 72
627.4550
1.0136
1.0167 689.5784
1.0202
628.4717
1.0202
1.0137
1.0168
690.5986
26
26 568.5745
629.4885
1.0137
1.0168
630.5053
691,6189
1.0138
1,0203 28
1.0169
2570.0020
631.5222
692.6392
1•0139
1.0169
1.0204
632.5391
69396596
29
I 0139
1.0204
1.0170
633.5561
69416800
30
3
JO
M
99
1.0194113
1,019514
554-3874
13 555.4005
141 556.4136
1 558.4409
171 559.4532
I
1,0297117
20
21 563.5066
22 564.5201
: 5655336
2 567.5608
25
1,020327
2-1 569.5882
2
571,6159
$72.6298
II
M
What are
A new and correct Table of Meridional Parts.
269
g
II
Mer. Parts Differ. Mer. Parts | Differ. Mer. Parts | Diffe
Minutes 10 min
Minuies
3C
1.0205
31
1.0200
32
1.0206
3?
1.0207
1.020834
36
38
gi
141
42
42
35
1.0208
36
1.0209
1.0209
1.0210
39
1.021)
40
1.0211
1.0212
1.0213
1.0253
42
44
1.0214
1.0214
1.0215
46
1,0216
47
1.0210
4 €
45
1.0217
5C
1.0217
51
1.0218
I 0219
52
1.029
75
572.6298
633.5561
1.0139
694.6800
1.0171
573,6437
634.5732
695.7005
31
1.0140
1.0171
574.6577
635.5903
696.7211
32
1.01 40
1.0172
575 6717
636.6075
697:7417
32
1.0141
576,6858
1.0172
637.6247
6987624
34
1.0141
577.6999
638 6420
1.0173
699.7832
35
1.0142
5787141
1.0173
639.6593
1.0142
700.8040
1.0174
579.7283
37
640.6767
L0143
1.0174
701.8249
38 580 7426
641.6941
I 0143
702.8458
1.0175
581.7569
39
642.7116
10144
582.7713
703.8668
1.0176
40
643.7292
70.1.8879
5837858
1,0145
1.0176
1.0145
644.7458
1.0177
705.9090
584.8003
1.0145
645.7645
706.9302
585.8148
1.0177
43
1.0146
646.7822
586.8294
1.0178 707.9515
44
647.8000
1,0146
$ 87,8440
1.0178
708.9728
45
648.8178
709.9942
1.0147
1.0179
40 580.8587
1,0147
649.8357
1.0180
7110156
47
1.0148
650.8537. 1.0180
712.0371
48 590.8882
1.01481651.8717| 1.0181 713.0587
49
591 9030
L.0149
652.8898
1,0181
714.0803
50
592.9179
653.9079
715 1020
1,0149
1,0182
51 593.9328
1.0182
716.1237
2
594 9478
1,01 go
655.9443
1.0183
717.1455
531 595.9628
656.9626
1.0151
718:1674
L.0183
541 596.9779
657.9809
1.0152
719.1893
1,0184
55
597.9931
658.9993
720.2113
10152
1.0185
599 0082
56
660.0178
121.2334
1.0153
1.0185
600.0236
57
661,0363
1.0153
1.0186
722.2555
$8601.0389
662,0549 1.0186
723.2777
1.0133
59
602.054
663.0735
1.0154
724.2999
60 603.0696
1.0187
664.6922
725.3222
M
9
NA
1
$89.8734
1.0150054.9261
53
1.022054
59
1.0221
$0
1.0221
57
1.0222
1.0222
59
1.0223
60
M
58
2
IO
II
270
A New and Correct Table of Meridional Parts
12
13
14
Mer. Parts. Differ. Merid. Prs. , Differ. Mer. Parts i Differ.
Minutes 1
Minutes of
2
IC
10314
801.1 549
725 3222
786.7799
848.4856
1.0224
1.0264
Il 726-3446
1.0307
787.8463
849.5163
-1.0224
1.0264
1,0357
727 3670
788.8327
850.5470
1.0225
I 0265
31728.3895
1.0308
789.8592
851.5778
3
1.02 26
1 0266
4 729'4121
1.0309
790.8858
852.6087
4
1.0226
1.0266
10309
s 730'4347
791.9124
853 6396
IS
1.0227
1.0267
1.03.10
6 731.4574
854.6700
6
792.9391
1.0228
1.0267
1.0311
7 732.4832
7
855.7017
793.9658
1.02 28
1.0268
1.0312
& 733.5030
856.7320
8
794.99 26
1.0229
1.0269
1,0313
9 734'5259
857.7642
9
756.0195
1.0229
1.0269
1735 5488
1.0313
858.7955
797'0464
1,0230
1,0270 859.8269
u 736.5718
11
798.0734
1,0231
1.0271 8608584
1,0315 12
2 737-5 949
799.1005
1.0272 861,8900
1,0316
13 738.6180 1.0231
800.1277
13
1,0232
1.0272 862.9216
1,0316
14
14 739,6412
1,C233
1.0273 862.9533
10317
15) 740.6545
802.1822
15
1.0233
1.0274 864,9851
1.0318
16 741.6878
803.2096
1,0275
1,0319
17| 742 7112
17
1,0235
1.0275
1.0319
181 743.7347
80s 2646
867.0489
18
1.0276
1.0320
19744-75821.0234 806,2922
868.0809
9
1.0226
1.0321
20
20 745.7818
807.3199
1.0277
1.0322
211 746.8055
870.1452
1.0278
1.0322
22 747.8292
871.1774
22
1.0238
1.0279
J.0323
231 748.8530
810.4033
872.2097
1.0239
1,0280
1.0324
241 749.8769
811.4313 1.0280
873,2421
24
1.0325
251 750-9008
7509008
874.2746
25
1.0240
1.0281
1.0325
26 751.9248
813.4874
875.3071
26
1.0240
1.0282
1.0326
27 752.9488
814.5156
876.3397
27
1.0241
1.0282
1.0327
281 753.9729
815.5438
877.3724
1.0242
1,0283
1:03 28
291 754.9971
816.5721
878.4052
29
1.0242
1.0284
1.0329
30756.0213
817,6005
879.4381
M
. 2
12
14
M
10
1.0234 804.2371
866.0170
1.0277 869.1130
1,0237808.3476
1.0237 809.3754
21
23
1.0239 812.4593
28
30
A new and correct Table of Meridional Parts.
271
Minutes 11
vingies
31
32
33
34
35
36
37
38
39
40
1.0334139
30
1.0335139
38
1.0336139
4C
828 9174
845.3941 | 1.0304
1 2
13
14
Mer. Parts Differ. Mer. Parts Differ. Mer. Parts | Differ.
756.0213
817.6005
879.4381
30
1.0243
1.0284
1.0330
818.6289
880.4711
757.0456
31
1.0244
1.2285
881.5041
1.0330
819.6574
758.0700
32
1.0244
1,0286
882.5372
1.033
75 9944
820.6860
1.0245
1.0287
760 1189
1.0332
821.7147
833 5704
34
1.0246
1.0287
884-6037
1.0333
822.7434
761'1435
35
1.0246
1.0288
762,1681
1,0313
823.7722
885 6370
1.0247
1.0289
763.1928
824.8011
886.6704
1,0248
1.0290
7642176
1.0335
825.8301
837:7039
1.0249
1.0290
765,2425
826.8591
888.7374
766.2674
I 0249
1.0291
827.8882
889.7710
1,0250
1,0292
890.8047
1.0337
767,2924
41
41
1.0251
1.0293
768.3175
1,0338
829.9467
891.8385
42
1,0251
1.0293
769.3426
1.0339
830.9760
892,8724
43
1.0252
1.0294
832.0054
1.0340
893.9064
+4
1.0295
894.9404
1.034)
771 3931
1.5
1.0253
1.0295
1.0341
834.0644 1.0298
895.9745
1.0254
835.0940
1.0342
897.0087
47
1.0297
898.0430
1.0343
48
1.0344
837.1534 1.0298
49
1.0256
1 0344
838.1832
50
1.0256
1,0299
1.0345
839.2131
901.1463
51 777.5459
51
1.0300
1.0346
1.0257
840.2431 1,0300
902.1809
1.0347
841•2731
903.2156
531 779.5974
153
10258
10301
1.0347
842.3032
904.2503
1'0259
1,0302
843.3334
905.2851
55
I 0260
1,0303
1.0349
906.3200
1.0261
844.3637
1.0304
1.0350
571 783.7012
907.3550
57
1.0262
1.0351
$81 784 7274
1°0262
846.4245
908-3901
1.0305
1,0352
847.4550
909.4253
59
1.0263
1.0306
60 786.7799
848.4856
910.4600
13
14
M
Nn 2
42
43
44 770.3678
1,0253 833.0349
40 772.4184
47| 773.4438
148 774.4692
49 775.4947
501.776.5203
1.0254
1.0255
836.1237| 1.0297 899,0774
900.1 118
52 778,5716
1,0258
52
54 780.5232
551781.6491
50 782.6751
1.034854
156
58
59 785.7536
1.0353160
M
12
272
A New and Correct Table of Meridional Parts
10
IS
1
17
Mer. Parts. Differ. Murid. Prs. Differ. Mer. Parts Differ.
Minutes 1
Minutes! ol
974.8058
ago
1046711
I 2
910 4606
972.7260
il 911.4959
1.0353
973.7064
21 9125313
1.035+
1.0355
31 913.5668
975.8473
914-6024
1.0356
4
976.8879
915.6380
1.0356
9 77.9286
6 916.6737
1.0357
978.9694
21 917.7095
1.0358
980.0103
81 918.7454
1.0359
986.0513
9919.7814
1.0360
982.0923
1.0350
920*8174
10
983.1334
921.8535
1921.
1,0361
984.1746
12 922.8897 1,0362
13 923.9260 1.0363 985.2159
141 924.9624
1.03611 986,2573
1,0365
98762988
15 925.9989
988.3404
! 6 927.0354
1.0365
1.0366
989.3821
191 928.0720
18] 929.1087
1.0367/ 990.4238
1.0368
991.4656
19 930.1455
992.5075
20 931.182
1.0369
4
993.5495
1.0369
21 932.2193
1.0370
994.5916
22 933.2563
1.0371
995.6338
231 934'2934
996.6761
1.0372
24 935 3306
1.0273
997.7185
291936.3679
998.7610
1,0374
26 937-4053
999,8035
271 938.4428
1000.8461
28 939.4803
1001.8888
1.0376
291 940.5179
10C2.9316
1002.9745
M
5
16
1035,339
1.0404
1.0457
1036.3496
1,0404
1,0458
2
1037.3954
1.0405
1.0457
1038.4413
3
1.0406
1.0460
1039.4873
4
1.0407
1.046!
1040 5334
1.0408
1.0462 6
1041.5796
1.0409
1.0463
1042,6259
7
1.0410
1.04651 8
1943,6724
1.0410
1.0465
1044.7189
9
1.0411
1.0466
IO
1045.7655
1.0412 1046 8122
1.0413\10478590
1,0468
1.0414
1.0469
1048.9059
13
1.0415
1,047014
1049.9529
1.0416
1.0471
105 10oCo
IS
10'417
' 10471|15
105 2.0471
1.0417
1,047217
1053.0943
1.0418
1.0473118
1054.1416
1.0419
1.0474
1055.1890
9
1.0420
1.0475
1056.2305
2c
1,0421
1.0476
10572841
21
1.0422
1.0477
1058.3318
22
1,0423
J.0478
1059.3796
1.0479
24
1.0480
45
1,0425
1.0481
1062.5236
26
1.0426
1.0482
1062.5718
27
1.0427
1.0483
1064,6201
28
I 0428
1.0484
1065.6685
29
1 0429
1.0485
1066.7170
30
17
MI
23
1.0424 1060 4275
1,0425 1061,4755
1.0375
1.0375
34_941.55551 1.0377
:
al New and Correct Table of Meridional Parts.
273
16
1.7
15
Mer. Parts. Differ. Mer. Parts Differ. Mer. Parts Differ.
Minutes 21
Minuteslim
1.0486
1.0487
1-0488/32
33
1,0381
34
35
36
1.0489
1.0490
1.0191
1 0491
1.0492
1,0493
1.9494
1 0495
37
38
39
40
1049041
42
1.0388
1.0389
30 941.5556
31 9+2,5934
32 943.6313
331 944.6692
341 945.7072
351 946.7453
36 947.7835
371 948.8218
38 949.8602
391 950,8987
10951.9372
411 9529758
+21 954.9145
431 955.0533
441 956.0922
+51 957.1312
40 958.1703
47 959.2094
481 960.2486
49.961.2879
50 962,3273
I
521 964,4064
531 265.4469
541 966.4857
55967.5255
56 968.5654
571 969.6054
58
970.6455
59 971.6857
60 972.72 60
IS
1003 9745
1066.7170
1.0378
1.0430
1005 0175
1.67.7650
1.0379
1.0431
1006,0606
1008,8143
1.0379
1.0432
1007.1038
1069.8631
1,03 80
1008.1471
1 0133
1070.9120
1,0434
1009.1905
1071.9610
1.0382
1010.2339
1.0434
1373.0101
1.0383
1.0435
L011.2774
1074.0592
1'0384
1012.3210
1.0430
1075 1084
1 0385
1.0437
1013:3647
1076.1577
1.0385
1014-4085
1.0438
1977.207
1.0380
1.0139
1015-4524
1078.2566
1.0387
1.0440
1016.4964
1079.3062
1017'5405
1018.5847
-1.0390
1019.6290
1.0391
1020.6734
10391
1021:7178
1.0392
1C22.7623
108540000
1.0393
1,0446
1023.8059
1086:6564
1.0394
1.0447
1024.8516
1087.7069
1,0395
1025.8964
1.0448
1088.7575
1.0449
1089.8082
1,0396
1,0450
1027.9863
1090.8589
1.0397
1.0451
1091.9097
1029.0314
1•0398
1.0452
1030,0766
1092.9606
1.0399
1.0453
1031.1219
1094.0116
I 0400
1.0454
1032.1673
1095 0627
1'0401
1.0455
1933.2128
1096.1139
1,0402
1.0455
1034.2583
1097.1652
1.0403
1.0456
1035.3039
1098.2166
16
17
1,04411080.3559
1.0412 1081.4057
1.0443 1032 4556
1.0444,1083.5050
1.044411084.5557
1.0445
77
48
49
5
963.3668
1,0396 1026.9413
1.0457
43
1.0498
44
1.0499
45
1.osoo
46
1.0501
10503
1.0504
1.0505
1.0506
51
1.0507
52
1.0507
52
1.0508
54
1.0509
55
1.0510
1.0511
57
1.OS 12
1.0513
59
1.0514
60
56
58
M
274
A new and correct Table of Meridional Parts.
18
19
20
Minutesi
Mer. Parts, Differ. Mer. Parts Differ. Mer, Parts Differ.
Minutesi -
present
2
mo
1098.2 166
1161.4871
11C99 2681 1.0515
1162.5448
21109.3197
1,0516
1163.6026
1.0517
i101.3714
3
1164.5605
1.0518
4
1102.4232
1165 718
1.0519
1103 4751
155.7766
1.0520
C1104 5271
1167.83481
1.0521
7 IUS.5792
1168.8931
I 0522
81106.6314
0169.9515
1.0523
9 11076838
1171.010
1108.7362
1.0524
1172.0686
1.0525
1111109 7887
1173 1273
1.0526
121110.8413
11 74.1861
13
1111.8940
1.0527
1175,2450
1.0528
141[112.9468
1176.3041
1.0529
151113.9997
1177.3633
1.0530
161215.0527
1178.4226
171116.1058
1.0531
1179.4820
180117.1590
1.0532
1180.5415
19/11 18.21 24
1,0534 1181.6011
201119.2658|1.0534 1182.6608
21 120.3193
22 1121.3729
1184.7805
231 122.4266
24 11 23.48.94
1,038
251124.5343
261125.5883
27 11 25.6425
281127.6967
1191,1424
291128.7511
1.0544
1192.2031
1129.8056
M
18
19
1 2251390
1.0577
1.0578
12 26.2022
1.0579
1227.2676
1.0580 1228.3321
1.058111229.3967
1230 4614
1.0582
1,05831231.5262
1.0584
1232.5911
1.0585
1233.6561
1.0586 1234.7212
12357865
1,0587
1.0588 1236 8519
1.0589
1237.9174
1.0591
1238 9830
1,0592
I 240.0487
1241,1145
1.0593
1242.1804
1.0594
1.0595
1243.2464
1.0596
1244.3126
1245-3785
1.0597
1246.4453
1.0598
1.0599
1 247.5118
1.0600
1248.5784
1.0601
I 249.6451
1.0603
1 2 50.7120
1 251.7.790
1.0005
1.0606
1252.8461
1.0607
1253.9133
1.0608
1254 9806
1,0609
1256.6480
1257.1156
20
1.0642
1.0644
1.0645
1.0646 3
1.0647
41
1.0648
6
1.0649
7
I, 650
8
1.0651
1.0653
9
10
1,06541
1,0655
[ 2
1055
1.065
13
1.065
4
55
1,0659
6
1,0660
1.0662
17
18
1.0063
19
1.0661
ܝܐ
1.05351183.7206
1.0530 |181.780;
1.05371185.8405
1186.906
10539 1187.6609
1.0540 1189.0213
1.0541 1190.0818
1.0543
1.0665
21
1.0660
22
1 0667
1.0669
23
1.06724
25
27
1.0671
26
1.0672
1.0672
28
1.0674
29
1.0674
39
IM
1.0545|1193.2639
.
A New and Correct Table of Meridional Parcs.
275
20
PU
Minuteslim
Minutes and in mm
1.0678 31
34
36
1.0617
1.0553 1202.8560
1.0554
-1.610931
18
19
Mer. Parts, | Differ. Mer. Parts Differ. Mer. Parts Differ.
[129.8050
1193.2369
1257.1156
1.0546
1.0609
1130.8602
1.0677
II
194.3248
1258.1833
32 131.9149
110547
1.0610
1195.3858 1259.2511
1.0548
1.0611
33
1 -0679
132,9697
1196.4469
12 60.3190
1,0549
1 0612
1.0680
33
24
1134,0245
1197.5081
1261.3870
1,0550
1.0613
1.068
25/1135.0796
1198.5694
1262.4551
35
1.0551
1.061
36 1130.1347
1.0682
1199,6309
1263-5233
1.0551
137.1898
1.0616
1200.6925
1 0684
1264.5917
I'0552
37
1138.2450
38
1201 7542
1.0685
12656602
1.0618
29
1,068638
91139.3003
1266.7288
1 140.3557
1.0619
39
1203.8779
1267.7975
1.0687
45
1.0555
1.0620
11141.4112
1 0688
12049399
1208.8663
1142,4669
1.0557
1,062,
1 206.0020
1269 9352
1 0689
1.0622
4311143.5227
42
1207.0642
1.0691
1271 0043
441'144.5786
1.0569
1208,1265
1,0623
43
1.0692
1272,0735
1.0624
441
+5 1145.6340 1.0550
1 209.1889
1273.1428
10 1146.6907
1.0561
106:6
1210,2515
1.0694
1274 2122
1.0562
1,0627
+7 1147.7469
46
1211.3142
1.0695
1275.2817
48 1148.8032
1.0563
1.0628
47
1212.3770
1 0697
1275.3514
1.0564
1.0629
19 1149.8596
1213'4399
1.0598
1.0620
1277.4212
15
1.0566
501150,9162
12145029
1278.4911
J.0699
5
SIT151.9728
1,0566
1.0631
1215.5660
1.0567
52 1152.0295
1.0632
1216.6292
51
1.0701
1280.6312
1,0568
52
1314153.0863
1,0633
12176925
1281.7015
1.0703
1.0569
51955.1432
53
1218.7559
1.0634
1282.7719
1.0704
5511156.2002
1.0570
54
1.0636
1219.8195
1283.8424
1.0705
55
561157.2573
1.0571
1,0637
1220.8832
1.0706
10573
1284.9130
57 058.3146
56
1.06 38
1.0707
1221.9470
1159.3720
1285-9837
1.0639
12 23.0109
1287.0545
59 160.4295
1.0640
1.0710
1288.1255
1224.0749
SS
6011161.487 i
1.0576
1.0641
1225.1390
1.0711
1289.1966
66
M
18
19
1.0558
18
1279.5611 1.0700
10574
1.0575
1.070837
58
20
276
A new and correct Tahle of Meridional Parts,
2 I
22
23
Mer. Parts. Differ. Mer. Parts. Differ. Mer. Parts Differ.
Minutes !
Minutes
1.0786
1418.6295
10864
I
1.07'311355 8422
2
421.8892 1.0867
alur two
1.08748
1
9 1
10
II
1.07241365 5565
1.07251366.6365
1.087912
13
01289.1966
1353 6849
1.0712
11 290.2678
1354.7635
1419.7159
21291.3391
1.0787
I 0866
1.0714
1420,8025
3 [292.4105
1356.9211
I 0789
10716
3
41293 4821
1358 0001 1.0790
I'0868
1422.9760
4
1'08 o
5 1294.5538 1.0717
1359 0792) 10791
1424.0630
61295.6256 1.0718
1360.1584 10792
1 0871
1425.1501
6
1 0719
7.1296 6975 1363.2378 1.0794
1426.2373
1'0872
7
81297.7696 10721
1362 3173 1.0795
1427.3 247
9.1298 8418
1.0722
1363 3969 1.0796
1428.4122
1.0875
101299.9141
1.0723
1364 4700 1.0797
1.0876
1429.4998
11300.9865
1 0799
1.0878
1430 5876
1 0800
I 2
1/02.0590
1 0720
131303.1316
1431 6755
1.0801
1367 7166
I 0880
1432.7635
141304.2044
1.0728
1368 7969 1 0803
Io889
14338917
14
151305.2773
1.0729 1369 8773
1 0804
1'0880
1434.9400
15
161306.3503
10730
1370.9578
1.0805
16
1.0806
1.0731
171307.4234 1372 0384
1'0886
17
1437.1171
1.0733
181308.4967
1.08 8
1473.1192
1.0887
1438.2058
18
1 0734
191309.5701
1374.2001
1,0809
1.0888
1439:2947
1.0810
19
201310,6436 1.0735/1375.2811
1 089
1440 3837
211311.7172
1.0736
1.0812
1376 3623
221312-7909
1 0737
+377 4436
1442.5621 1.0893/22
23|1313,86481 10739113785250
1.0894
1443.6515
110815
241314.9388 1.0740 1379.6065
1 0896
1444.7411
24
251316.01291 1.0741 1380 6882
I 0897
1445.8308
25
2013170871
1 0742
138.7700
1.0818
26
27 13 18 16 15 1.0744
1382.8519
1.0819
I 0900
27
281 319,2360
1.0745
1.0821
1383.9340
1. 07491385.0162
1.0822
1449 1007
1.0902
I 0823
1450.1909
29
30 132 1.38531 1.07471386 0985
1.0904
1451.2813
30
M
23
1436.0285 | 1.0885
20
21
1441.4728 1.0891
1.0813
1.0814
23
1•0817
1476 9206 1.0898
1448 0106
1.090128
29 1320-3106 10746
21
22
M
A
New and Correct Table of Meridional Parts. 277
21
22
23
Mer. Parts. Differ. Mer. Parts Differ.
Differ. Mer. Parts Differ.
Minutes in
Minutes wall
34
36
1.0757
38
39
41
1.0840
1.0922
42
1.0843
2011321.3853 1386.0985
1.0748
1.0825
31 1322.4601
1387.1810
1,0750
110826
3211323.5351
1388.2636
1.07SI
1.0827
331324.6 102
1389.3463
1.0752
1.0829
3411325.6854
13904292
1,0754
1.0830
3511326.7608
1391.5 122
1,0831
361327.8363/ 1.0755
1392.5953
3711 328.9119
1.0756
1.0832
1393.6785
1.0834
381329.9876
1394.7619
1'0835
3911331.0634
10758
1395.8454
1.0760
1.0836
4011332.1394
1396.9290
411333.2155
1.0761
1'0838
1398 0128
1,0762
1•0839
4211334,2917
1399.0967
431335.3680
1.0763
1400,1807
1.0842
44 1336.4445
1.0765
1401.2649
451337.5211
1.0766
1402.3492
401338.5978
1.0767
1.0844
1403.4336
47 1339,674.6
1'0846
1.0768
1404-5182
48 1340.75 16
1.0770
1'0847
1405.6029
1:084.8
49 1341.8287
501342.9059
1•0849
1.0772
1407.7726
1.0851
511343,9832
1408.8577
5211345 0607
1409.9429
5311346.1383
1.0776
1.0853
1411,0282
54 0347 2160
1.0855
1.0777
1412.1137
1.0778
5511348.2938
1.0856
1413.1993
56
1.0780
[•0858
1349.3718 1414.2851
1.0781
571350.4499
1415-3710
$81351.528
1.0800
1416.4570
1.0783
5911352,6064
1'0862
1417.5432
6011353.6849) 1.0785
1.6863
1418 62951
MI
1451.2813
}
1.0905
1452.3718
31
1.0907
1453.4625
32
1-0908
1454 5533
33
1.0909
1455.0442
1.0911
1456.7353
35
1.0912
1457.8265
10913
1458.9178
37
1.0915
1460.0093
1,0916
1461.1009
1.0917
1462 1926
Io919
1463.2845
41
1.0920.
1464.3765
1465 4687
43
1.0923
1466.5610
44.
1.0925
1467.6535
45
1468.7461
1.0926
46
1.0927
1469.8388
47
I 0929
1470.9317
1.0930
1472.0247
49
1.0931
1473.1178
so
1.0933
14742111
51
1,0934
1475 3045
$2
1.0936
1476.3981
52
1.0937
1477.4918
94
1.0939
14785857
55
1.0940
1479.6797
1480.7738
, 1.0941
57
1.0943
1481.8521
1.0944
1482,9629
55
14840571
1.0946
6c
23
M
1.0771|1406.6872
48
1.0773
1.0775
1.0852
1.0859
56
1.0782
$8
21
22
Oo
Minures
Minutes 1
1.0947 1551.0987
1.09491552.2023
1.09501553.3061
1.09531555.5141
1.10381618.6976
1 09501557.7226
1493 9146 1.0958
101495.0100 1.0960
i 105016287203
10963|1563 24661
278 A new and corre&t Table of Meridional Parts,
24
25
26
Mer. Parts. Differ. Mer. Parts. Differ. Mer. Parts | Differ.
01484.0571 1549 9952
1616.4721
1.1035
11485.1518
1617.5848 1.5127
1.1036
1.1128
2 1486.2467
311487.3417
16198100 1•1130
3
10951
11039
4 1488 4368 1554.4100
4
1620 9238 1'1132
I 1041
11133
51489.532 1
1622.0371
5
1.09541556 6183
I 10.42
61490.6275
I 113516
1623.1506
I.1043
11136
7.1491.7231
1624.2642
7
I1045
1625.3780
1.1138
81492.8388 1095715588271
8
I'1046
1559 9317
1626 4919
1'1139
9
I'1048
1561:0365 1627.60601:1141 10
1.0961
11 1496.1067 1562 1415
1628
1.105 I
12 1497.2030
1629.8347
I 09641564.3519
L.1053
13 1498.2994
I 1146
1630.9793
13
11054
1:1148
341499.3960
1632.C641
14
1.0967
1'1149
151500.4927
1633-1790
15
1634.2941
1•1059
1'1152
1568 7745
1635.4093
17
1:1060
181503,7837) 1.0971 15698805 1636.5247
1.1155
191504.8810
I 097311570 9866 1637-6402
19
1.0974
I.10(-3
11157
201505.9784
1572 0929 1638.7559
1.0976 1573.1994
1.1065
1.1158
201507.0760
21
1 09771574.3000
231508.1737
I. I 160
1640 9877
22
1.1162
1 0979
23
2311509,2716
1.1164
15765197
1643.2203
24
1.0981
11071
25115 11:4677
1644-3368 25
1.1072
311578 7340
2015 13.5660
1645 4535
1.1074
1.09841579.8414
1,1168
2715136644
640.5703
27
1.09861580 5489
I.1075
281514.7630
1647.6873
1582.0566 1.1077 1648-8045
1.1079
301516.9606 1.0989-7583.1645
1649'9218
M
25
25
M
1.1.14311
I.1144112
1.09661565 4573
1.115116
1566.5629111056
16 1501.5896 10969
1567 6686 1.1057
171502.6866 1.0970
1.115418
1,1061
20
1:1066639.8717
1:1068
I•10691642.1039
1.1165
1.1167 26
1575.4128
241510 3696 1.0980
1577 6268
I 098
291515.8617 1098
1.117028
1.117229
1173130
24
1 New and Correct Table of Meridional Parts.
279
:
25
24
Mer. Parts,
Differ. Mer. Parts Differ. ||Mer. Parts Differ.
Minutes to
Minutes1.01.
1.1085 16543921
$
.
1,1088
I 1094
1•1095 1662.2231
39
41
42
30 1516.9606
RUISI
518.0596
321519.1988
3311520,2581
3411521.3576
3510522.4572
3611523.5577
3711524.6569
381525.7569
391526.8571
1011527.9574
111529.0579
42 1530,1585
431531.2593
4411532,3602
45 1031.4613
401534,5625
47 1535.6639
48115367654
19 1037.8671
5011538.9689
5111540,0 '09
520 541.1730
53 1542.2753
54 1543,3777
S51544-483
16
1945.5830
57 1546.6858
$8
1547.7883
591548.8919
6011549 0052
M
24
1583.1645
1.0990
1.1080
1584,2725
1.0992
1.1082
1585.3807
1 0993
1.1083
1986.4890
1.0995
1587.5975
1.0996
1.1086
1588.7001
1.0998
1589.8149
1.0999
I.1089
1590.9238
I.1000
L1091
1.1002
1592.0329
l'IC92
1593.1421
I.1003
1594.2515
1.1005
I.I 006
1595.3610
1•1097
1.1008
1596.4707
I.1059
1.1009
!597'5800
1.I 100
1.1011
1598:6906
1.II02
1599.8008
1-1012
1.1103
1600 9111
1.1014
1602.0216
1,1015
II 106
1603.1322
1.1012
Intos
16042430
1.1018
1.III
1605 3540
1.1020
1.1111
1606:4651
1.1021
1.1113
1607.5764
1,1023
1,1114
1608.6878
1.1024
1.1116
1.1020
[609.7994
LI117
1610.9511
1.1027
1:1119
1612.0230
1. 1028
1.11 20
1613.1350
I.1030
I. I 122
1.1031
1614.2472
I'1124
11033
1615.3596
1,1125
1616.4721
25
1649.9218
1.1175
1651.039;
31
1117
1652.1564
32
11178
165 3.2747
33
11180
34
1018
1655.5108
35
1.II
183
1656.6291
26
I 1185
1657.7476
37
1,1186
1658.8662
38
1,1188
1659.9850
1661.1040
1.1190
1,1191
41
1.1193
1663:3424
1.1194
1664.4618
43
1.1196
1665.5814
44
1.1198
1656.7012
I. 1199
1667.8211
40
1. 1201
47
1670.0615
I 1203
1.1204
1676.1819
1.1206
1672.302
1.1207
1673.4232
1,1209
1674.5441
1.12II
1675.6652
1.1213
1676.7865
1.2214
1677.9079
1.1216
1679 0295
$
1.2217
1680.1512
57
1.1219
1681.2731
1.1221
1632.3952
$$
1.1222
6C
1683.51741
26
M M
4
1.1105 1068.9412
48
45
inlnunla
158
002
A
280 A new and corre&t Table of Meridional Parts,
27
28
29
Minures
Mer. Paris. Differ. Mer. Parts. ID iffer. Mer. Parts Differ.
Minutes 1
I'1228
754.5602
1755 6934 1.1332
4 antro
1826 3000
1.1234
L:1457113
14
0 1683.5174
1751.1617 1819.4366
1•1224
1.1327
11684.6398 1752.2944
1820 5800
1.1434
I. 1226
21685.7624 1753.4272
1.1328
1821.7236
1.1436
3116868852
I
11330
1822.8674
1.1438
3
I 1229
41688.0081
1824.0114
1.1449
4
I'1231
1756 8268
I 1334
51689.1312
1825.1556
I'1442
61690.2544
1.1232
1.1335
I 14446
1757.9603
1759 0940
1.1337
71691.3778
11446
1827.4446
81692.5014
1,1236
1760 2279
15339
1828.5894
!'1448
8
I•1237
11341
9 1693,6251 1761.3620
1.1449
1829 7343
9
I.1239
I'1342
1011694.7490
1762 4962 18308794
1.1451
111695.8731
1. I 241
I 1344
1763 6306
1.145311
183.2 024.7
121696.9973
1.12421764.7652
I 1346
1.1455 12
1833.1702
13 1698.1217
1.12441765.9000
1.1348
1834:3159
141699.2463
1.1246
1.1350
1767 0350
1.1459
1835-4618
1.1248
1 1351
151700'3711
1768.1701
183.6.6078
I•1460
15
1769.3054
1.1353
1837-7540
1.1251
I.1355
1741702.6211
1'1464
1770 4409
1838.9004
17
1771.5765
1.1356
1:1406
181703,7464) 1:1253
18
1941704.8718. 12541772:7123
1.1468
1841.1938
19
201705.9974
I•1256
1842.3408
21 1707.1232
I.f258
1843.4880
221708 2491
1•125911776 1209
1:1364
1844.6354
1:1366
1:1475
231709.3752
1845 7829
23
1.1263
241710 5015
1778.3942
1.1367
1.1264
1846.9306
1.1369
251711.6279
1779:5311
25
I 12661780 6682
201712.7545
1.1371
1849.2266 1.1481127
1.1268
271713'8813
1781.8055
1.1373
1850.3.749
1.1483
27
1.1374
1851.5234
1,1485
281715.0083
28
J. 1271 1784.0805
1.1376
2910916.1354
1852.6721
I.1487
29
1.1378
301717.2627 | 1:13731785.2183
1853.8209
1:1488
30
M
27
28
29
M
1.1462 16
701.4960 1:1 249
1,13581840 0470
1773.8483_1.1360
1774.984511:1362
1 1470 20
1.147221
1.1474/22
1.1 261 1777:2575
I 147724
1848.0785 1.9479
1.L270 1782.9429
...
A new and correct Table of Meridional Parts.
281
2
Minutesland
Minutes and
32
2
1.14963
34
54
36
1.150413)
I
Hermann
1.1506/38)
27
28
Mer. Part , Differ. Mer. Parts Differ. Mer. Parts Differ
1717.2627
1785.2183
1853.8209
30
1,1274
1.1380
1718.3901
1,1490
1786.3563
31
1854.9699
31
1.1276
1.1382
1.1492
1719.5177
178704945
32
1856.1191
1.1278
1.1383
1 1494
1720.6455
1788,6328
1857,2685
3?
1.1280
1.1385
1721.7735
1789.7713 1858.4181
11282
1.1387
1.1498
1 722.9017
1790.9100
1859.5679
35
31
1.1283
1.1389
1.1500
301724 0300
179 2,0489
1860.7179
1.1285
1,1391
1.1502
37
1725.1585
1861.8681
1793-1880
I 1287
1.1392
3811726.2872
1794.3272
1863.0185
1.1.289
1.1394
391172714161
17954666
1864 1691
1,1290
1,1396
1.1508
40 728.5451
1796-6052
1865:3199
40
1.1 292
1.1398
1.15.10
1729.6743
41
1866.4709
1797 7460
41
1.1400
1.1294
1730.8337
1,151
1798.8860
1867.6220
42
42
1.1296
1.1513
1.1402
1800.0262
1731.9333
43
1.1403
1.1515
44 1733 0630
1863.924.8
44
1.1 405
1,1517
4511744.1929 21802.3070
1871.0765
I 1407
I.1919
401735-3229
1803 4477
1872,2284
1.1 302
1.1409
1.152)
+7 1736.4531
1804.5886
1873'3805
1.1411
1.1523
47
1211737.5835
1874.5328
1.1412
491738:7! 40
1875.6853
49
1.1527
1.1414
1739.8447
1876.8380
50
50
1.1416
1.1529
51740.9756
1877 9909
51
1,1418
1:1531
521742. 1067
1879.1440
1.1420
1.1533
5311743.2380
5.3
1.1422
1.1535
$4/1744.3694
1881.4508.
54
1.1316
1.1536
1745.5010
1813.7222
55
1,1318
1814.8647
1.1425
56 1746:6328
1.1538
1883.7582
111427
571747.7648
1884.9122
1.1429
1.1542
581748.8969
1886.0664
5
1,1431
I.1544
5911750,0292
1887.2208
159
1.1432
6011751.1617
1888.2754
M
27
28
29
M
1868.7733
46
1.12971804.1665
1.1299
1.1300
- 1
46
1,1525148
1.1304 1805.7297
1.1305 1806.8709
1,1307 1808.0123
1.1309 1809.1539
1.1311|1810.2957
1.131311811.4377
1.1314 1812.5799
52
1880.2973
1.1423 1882.6044
1.1540
50
1.320 1816.0074
1,1321|1817,1503
1.1323 1818.2934
1.132511819.4266
1.1546160
!
282
A new and corre&t 'Table of Meridional Pares.
32
30
Mer. Parts Differ. | Mer. Parts Differ. Mer. Parts Differ
Minutesl 01
Minutes 1 mm
2
I
2
ro
1.1681
1,1573
1.1821
1888.3754
1958.0124
2028.3837
1.1548
1.1668
i 1889,5302
1.1794
1959.1792
2029.5631
1.1550
1.1670
211890.6852
1960.3462
2030.7427
1.1796
1.1552
1,1672
1891.8404
3
1961.5134
1.1798
3
2031.9025
1.1554
1,1674
1.1 800
4
1892.9958
1962-6808
4
2033.1225
1,1506
1,1679
1.1802
5
1894.1514
1963.8484
2034.2827
1.1558
1.1678
1.1803
61895.4072
1965.0162
2035.4630
1.1560
916.1841
1.1679
711896.4638
2036.6435
1.1805
7
1.1561
81897.6193
1967.3522
1.1 807
8
1.1563
2037.8242
1,1683
1968 5205
9 1898.7756
1.1809
9
2099.005 1
11555
1.1685
1.1811
101899.93.21
1969.6890
2040.1852
1,1567
1.1687
1.1813
I 1901.0888
1970.8572 1.1689
2041.3675
1.1569
1.1816
121902.2457
972.0260
1,1571
12
2042.5491
1.1691
1,1818
131903.4028
1973.1957
2043.7309
13
1.1694
14.1904.5601
14
1,1575
1974.3651 1.16072044.9130
151905.7176
1.1823
2046.0958
1975.5147
15
1,1698
1.1577
161906,8753
1.1825 16
1976.7045
2047.2778
1.1 701
1.1579
17 1908.3332
1.1828
1977.8746
2048,4606
1.1582
17
18 1909.29.4
1.1584
1979.0449
1.1705
2049.6436
1,1832
19 1910.3498
2050.8 268
1980.2154
1.1586
1.1707
1.1825
2011911 5094
40
1981.3861
2052,0203
211912.6671
1,1709
1.1587
1982.5570
2053.1940
1.1711
1.1989
22 1913 8260
1983-7281
1,1591
1.1713
2054.3779
23 1914.9851
1984.8994
1.1719
2055.5620
lil593
241916.1444
1986.0709
1.1595
2056.7463
1,1716
1911917'3039
1987.2425
2057:9307
26 1918 4636
1.1719
11597
1988.4144
1.1599
2059.1154
1•1721
271919.6235
1989.5865
2.60.3003
27
1.1601
1•1723
1.1851 18
28 1921.7836
206 1.4854
1.1603
1.1725
29 1920.9439
1.1605
2062 6707
1991.9313
1.1727
1.1855
30 1923 1044
2063.8562
M
30
M
31
32
1.1703
1,183018
19
1,18371-1
1.183922
1184123
1.1843 124
44 25
1.1844
1.184726
1.1849 127
1990.7588
1.1853 29
1993.1040
30
M
..
A new and correet Table of Meridional Parts. 283
30
31
32
Mer. Part Differ. Mer. Parts Differ. Mer. Parts Differ.
Minuteslim
Minutesi
30
1,1857
1.1860 31
11862132
1.1864
3?
1.1862134
1.1869
1.1872
36
1.1874
37
1.7876
35
38
1.1878135
1.1880
1,188214
1.188414?
1923 1044
1993.104
1,1607
1924 2651
1994.2769
1,1606
32
1925.4260
1995-4500
1.1611
33.1926.5871
1996.6233
1.1613
34
1927.7484
1.1615
1997.7968
1928.9099
1998.9706
35
7.1617
30/1930.0716
1.1619
2000.1446
32 1931.2335
2001.3188
I 1621
38 1932.3956
2002.4932
1.1623
39 1933 5579
1,1624
2003 6679
+91944.7208
2004.8428
1,1626
11/1935 8829
2006.0179
1.1628
4211937.9457
2007.1932
1.1631
431 1938.2088
2008.3687
1. 1633
+4) 1939.3721
1.1635
2009.5 444
45 1940.8316
2010.7203
46 1941.6993
1.1639
47 1942.8632
1.1641
2013.0727
181944.0273
2014.2492
1,1644
4411945.1917
2015-4258
1.1640
1946.3563
2016,6026
5111947-5211
2017'7796
1.1 649
521948.6860
2018.9569
1.1651
531949.8511
2020.1344
1.1653
541951,0164
202 1.3 122
1952.1819
1.1655
2022.4902
156 1953.3476
1.1657
2023.6684
57/1954.5135
1,1659
2024.8469
1.1.661
58 1955.6796
2.026.0256
59 1958,8459
1.1663
2027.2045
6919$8.0124
1.1665
2028,3837
30
31
2063.8562
1.1729
1.1731
2065.0419
2066.2279
1.1733
2067-414
1.1735
2008.6005
1.1738
2069.7872
1.1740
1,1742
2 2070 9741
1.1744
2072.1613
1.1747
20733487
1.1749
2074.5363
207507241
1.1751
2076.9121
1.1752
2078 1°C3
1.1755
2079.2887
1.1757
2080 4774
1,1759
2081.6663
1.1761
I 1763
2082.8554
2084.0448
1'1765
2085-2344
1'1766
1086.4242
I'1768
2087.6142
1,1770
2088.8044
1,1773
2089 9948
1.1775
1.1778
2091.1855
1.1780
2092 3764
2093.5675
1.1782
1.1785
2094.7589.
1.1787
2095.9505
1.1789
2097.1423
2098.3344
1.1792
2099.5267
32
1.188714
1.1889
ta
IS
1.1891
1.1894
46
1.1637 2011.8964
1,1890 +7
1.1898 48
1.1648
1.1900
49
50
1. 1902
51
1.1904
I 1907
1.1909
1.1919
155
1.1914
52
53
54
1.1916150
1.1918/52
5
8
1.1921
59
160
1.1923
M
284 A new and correčt Table of Meridional Parts.
33
34
35
Minutes
Mer. Parts Differ. Mer. Parts Differ. Mer. Parts| Differ
Minutes on
1
1.1943 12182:3462
1.1945
2099.5267
1 2100.7192
22101,9119
3/2103.1048
4.210.4.2979
5 2105.4913
6 2106,6849
72107.8787
8.2.108.07 28
9
2110 2671
2111.4616
11 2112.6564
1 2 2113.8514
132115,0406
142116.2420
1521174377
162118.6335
172119.8297
8 2121.0260
1912122,2225
2012 12 3.4192
21 2124.6162
222125.8134
232 127.0109
242128.2086
25/2 1 29.4065
262130.6047
272131.8031
28|2133.0017
292134.2006
302135.3997
M
33
2171.4807
2244.2868
1.1925
1,2063
1.2210
2172.6870
1.2065
1.1927
2245.5078
1.2212
2173.8935
2 246.7290
2
1.1929
1,2058
1.2214
2175.1003
2247.9504
3
1.1931
1.2070
1.2217
2176.3073
2249:1721
4
1.1934
1.2073
1.2219
2177.5 146
2250.3940
1.1936
1.2075
1.2 221
2178.7221
2251.6161
6
1.1938
1.2078
1.2224
2179.9299
2252.8385
7
I.1941
1.2080
1.2226
2181.1379
8
2254.0611
1.2083
1.2228
2255.2839
9
1.2085
1.2231
2183.5547
2256.5070
1,1948
1.2087
1.2234
2184.7634
2257.7304
1.1950
1.2090
1.2236
2185.9724
2258,9540
(2
1.1952
1.2092
1.2239
2187.1816
2 260.1779
13
1.1954
1.2094
1.2241
2188.3910
226 1.4020
1,1957
1.2097
14
2189.0007
1.2244
2262.6264
15
1.1959
1,2099
1.1247
2190.8106
2263.8511
1.1961
1.2101
1.2249
2192.0207
2265.0760
7
1.1963
1.2104
1.2252
2193.2311
2266.3012
1.1965
18
1.2106
1.2254
2194.4417
2267.5266
1.1967
19
1.2109
1,2257
2195.6526
2268.7523
20
1.1970
I.2 III
1.2260
2209.9783
21
1.1972
1.2262
21980751
22
·1,1975
1.2116 2271.2045
1.2264
2199.2867
1.1977
1.2118 2272.4309
23
1,2267
2200.4985
24
2 273.6576
1,2121
1.1979
1.2269
2201.7106
2274.8845
25
1.1982
1.2123
1.2272
2202.9229
2276.1117
26
1.1984
( 2126
1.2274
1.1986
2204.1355 1.2128 2277.3391
27
2205.3483
1.1989
2278.5668
1'2130
1,2279
2206.5613
2279.7947
29
1.1991
1.2133
1.2282
2207.7746
2281.02291
30
34
35
M
16
2196.8637 102114
1.2277 128
.
SA's iet
A new and corre&t Table of Meridional Parts. 285
Minuteslimling
Mduies
?1
1.2000 2212.6302
1.2 29 113?
36
38
1.2338/51
2311.8111
33
3+
༢5
Mer. Part Differ. Mer. Parts Differ. Mer. Parts Differ
2 207.774
2281 0229
30/2135-3997
30
1.1993
1.2135
1,2285
2208.9881
2282.25 14
311?136.5990
31
1.2138
1.2 287
2 210.2016
137-7086/ 1.1996
22
2283-4801
3
1.2140
1.1998
1.2290
2138.9984
22114159
2284.7091
37
33
1.2143
2140,1984
2285.9383
34
34
1.2003
1.2145
1.2295
3512141.3987
2213-8447
2287.1678
35
1.2005
1.2147
1.2297
36/?142.5992
2288.3975
2215.0594
1.2007 2216.2744
1.2150
1.2300
3712143.7999
2289.6275
37
I 2009
1.2152
1.2302
38/2145.2008
2217.4896
2290.8577
1.2012
1.2155
1.2305
342146.2020
2218.705!
2292.0882
39
1.2014
1,2157
1.2307
40214714034
22 19.9208
2293.3189
40
1,2016
1,2100
1.2310
412148.6050
2221,1368
2294 5499
41
1.2019
1.2162
1,2313
42 2149.8059!
22223530
2295.7812
1.2021
1.2165
4312151,0090
2223-5695
2297.0127
1.2023 2224*7862
1.2167
+42152:2113
2298.2445
1.2170
4512053.4139
2299.4765
15
1.2028
1.2172
1.2323
4012154.6167
2227.2204
2300 7088
[•2174
47 2155.8198 1:2031 2228.4378
2301.9414
1.2033
1.2177
4812157.0231
2229.6555
2303'1742
1.2035
1.2 179
492158.22 66
2230.8734
2304.4073
1,2181
1,2037
1.233?
49
2159.4303
2232.0915
2305.6406
50
1,2184
1.2336
512160,6343
2306 8742
1.2186
522161.8385
2308.1080
1.2189
1.2045
5312163.0430
2235-7474
2309 3421
I.2047
1.2191
1.2344
542164.2477
2236.9665
2310.5765
1.2194
552165.4526
55
1.2052
1.2196
502166.6578
1.2349
2239.4055
2313.0460
1,2054
1.2199
57 2167.8632
1.2352
2240.6254
1.2202
1.2056
582169.0688
1.2354
2241.8456
1.2058
1.2205
5912170.2746
1.2357
2243.0061
2316.7523
602171.4807
1.206
1.2207
2244.2868
2317.9883
6
M
23
34
35
M
1.2315/42
1.231843
1.2320/44
1:202612226.0032
1,232646
1.2328/47
1.233148
1.204012233.3099
1,2042 2234.5 285
1.2341
52
153
1.2049 2228.1859
! 2346,54:
1
isc
57.
2314.2812
2315.5166
sy
1.2360
Рp
Whe
286
A new and corre&t Table of Meridional Parts.
30
37
38
Minutesi
Mer. Parts | Differ. Mer. Parts | Differ. Mer Parts | Differ
Minutes 1
1
2
2+72,0692 1.2697
1.2531
1.2534
1.2539 2477.1510
TO Ovalad
1 2385
1.2724
1.2353
2484.7826 1.2720
1.2729
14
2317.9883
2392.6305
12319.2245
1.2362
2393.8828
1,2365
2 2320'4510
2395.1354
1.2367
2321.6977
2396.3882
42322.9347
1.2370
2397.6413
52324.1720
1.2373
239888947
1.2375
6-325.4095
1.2378
2400:1483
712326 6473
1.2380
2401.4022
8,2327.8853
2402.6564
1.2383
92329.1236
2403.9109
09!23.0.362 I
2405.1656
012331.6009
1,2388
2406.4206
122332.8400
1.2391
2407.6759
13 2334.0793 2408.93.15
142335.3189
1.2396
24.10.1873
1,2399
152336.5588
2411.4434
162337.7989
1.2401
1.2404
2412.6998
1712339.0393
1.2407
2413.9565
182340.2800
2415.2135
1.2410
192341.5210
1416.4708
1.2412
20 2442.76 22
2417.7284
2112344.0037
1.2415
2418.9862
1.2417
222345.2454
2420.2443
232346,4874
1,2420
1.2423
2421.5026
2412347.7297
2422.7612
1.2425
25|2348.9722
2424.0201
26 2350.21501 2428
1.2431
2425.2793
272351.4581
2426.5388
1.2433
282352.7014
2427.7986
1.2436
292353 9450
2429.0587
1.2439
302355.1889
2430.3191
M
36
37
2468.2609
1.2523
2469.5301
1.2692
1.2526
2470.7995
1.2694
1.2528
3
1.2700
2473.3392
4
24"4.6095
1.2707
1.2536
2475.8801 1.2706
6
2+77.1510 1.2709
1.2842
7
2478.4222
1.27 1 2
8
1.2545
2479.6937
1.2715
1.2547
9
2480.9655
1.2718
IO
1.2550
2482.2376
1.2721
1.2553
2483.510C
[ 2
1.2556
1.2558
13
2486.0555
1.2561
15
1.2564
2488,6022 1.2735
1.2567
17
1.2570
1.2741
2491.1501
1.2573
18
2492.4245
1.2576
2493.6992
20
1.2578
2494.9742
21
1.2581
1.2753
2496.2495
22
1.2583
2497.5251
23
1.2586
2498.8010
1.2759
1,2589
1.2762
2500.0772
25
1.2592
2501.3536
1.2764
26
1.2595
1.2767
2502.6303
1•2598
27
1'2001
2503.9073
1.2604
29
2505.1846
2506.4622
30
38
M
2487.32871 1.2732
16
2489.8760! 1.2738
19
1,2747
1.2747
1.2750
1 2750
24
1.2770128
1,2773
1.2777
A new and correct Table of Meridional Parts.
282
27
Minutesland
36
38
Mer. Part, Differ. Mer. Parts Differ. Mer. Parts Differ
Minu'es
30
1.2786
3!
1.2782
32
1.2788
32
34
2
1.2612 2509.0185
1 2785
1.2617 2513.5758
1.2791
; ܐ
2437.08701,2623
1.2794136
37
1.2797
1.2790
1.2803
38
1.280639
40
1.2 809
1,2812/41
1.2474
1.2815142
1.281843
1.2479
1.26462524.3803
2355.1889
2430.2191
2506'4623
1,2441
1.2606
3112356.4330
2431.5797
1.2444
1.2009 2507.7403
32
212357.6774
2432.8400
1.2447
3212358.9221
2434.1018
2510.2970
1.2615
1.2449
2360.1670
34
24:5.3632
1.2452
2361.4122
39
2436.6255
251 2.8549
1.2455
1.2620
362362.3577
2514.1343
1.2458
3712363 9035
1 2460
2439.1493 1.2626 2515.4140
3812365 1495
2440.4119
2516.6940
1.2629
1.2463
58
342356.395
2441.6748
1.2466
1.2631
25179743
702307.6424
2442.9379
2519.2549
1.2468
1.2634
4112368.8892
2444.2013
1.2471
1.2637
2520 5358
4212370.1363
2445-4650
1.2640
2521 8170
432;71.3837
1.2476
2443.7290
1.2643
2523'0985
14412372.6313
24479933
4512373.8792
2449.2579
2525.6624
1.2482
1.2649
462375.1274
1.2485
2450.5228
1.2652
2526.9448
472376.3759
245 1,7880
1.2487
2528.2275
1.2654
1812377.6246
2453.0534
I 2490
2529.5105
49|2378.8736
24543191
1,2492
1.2660
2530.7937
2455.5851
2522.0773
5112381 3723
1.2663
1.2495
24'56.8514
1.2498
1.2666
<212 382,6221
2533-3612
2458.118
1.2500
1.26692534.6454
532283.8721
2459.3849
1.2671
2531.9299
542285.122+
2460.6520
1'2506
1.2674
2537.2147
55 2286.3730
2461.9194
2938.4998
502 387.6 239
1.2509
1.2677
2463.1871
2539.7852
1.2680
1.251 2
572388.8751
2464.4551
9812390.1266
1.25 IS
2541.0709
1.2683
2465.7234
2542.3569
1.2686
1.2518
592391.3784
2466.9920
602202.6303
2543.6432
1.2521
1.2689
2468.2609)
2544 9298
M
36
37
38
us
1.2830/47
1.2657
1.2821
44
1.2824
1.2827
40
48
1.2832
1.2836
49
50
1.2839
51
1.2842
I 2845
52380 1228
$2
1'7503
1.284853
1.2851
54
5
I 2354
50
1.2857
1.286015?
1.2863
SA
1.286655
6C
M IM
Рp2
288 A new and correct. 'Table of Meridional Parts.
41
Minutes!
39
40
Mer. Parts Differ. Mer. Parts | Differ. Mer. Parts | Differ
Minutes 01 -
1.3252
1.325
1.3259
1.3262
1.2265
4
1.3068
1.2387|2631.8359
vala
1.3-81
I 2
2897
21
2544.9298
26 22.6 902
1.2869
2546,2167
1.2872
262 3.9958
212547.5039
2625.3617
2548.7914
1.2875
2626.6079
3
1.2878
425500792
1627.9144
1.2881
2.551.3673
2629.2212
5
1.2884
012552.6557
2630,5 284
7
2553 9444
1.2891
812555,2335
2633.437
1.2894
9/2556°5229
2634:4518
1
102557.8126
2635.7602
1,2900
11/2559.1026
2637.0690
1.2903
122560'929
2038 3781
1.2906
132561.6835
2639.6875
1.2909
142562.9744
264009972
1,2912
1512564.2656
2642'3073
1.2915
162565.5571
2643.6177
1.2918
172566,8489
2644.9284
1.2921
18/25081410
2646.2394
1.2923
1912569.4333
264.7.5507
1.2927
20 2570 7260
2648.8624
1.2930
212572.0190
2650. 1744
22 2573-3123
1.2933
2651.4857
1,2936
2312574.6059
2652.7994
1.2940
24 2575.8999
2654 1124
1.2943
2512577.1942
2655.4257
26 2578.4888 1.2946
2656.7393
1.2949
272579.7837
2658.0532
1.2952
28 2581,0789 2659.3675
1.2955
2912582.3744
2660.682)
1.2958
30|2583.6702
2661.99 70
M
39
40
2701.5979
1.3056
2102.9231
1.3059
2701.2487
1.3062
2705.5746
1.3065
2706.90 8
2708.2273
1,3072
2709.55 +2
1.3075
2710.8814
1.3078
271 2.2089
1,3084
2713-5367
2714.8648
1.3-88
2716.1933
1.3091
2717.5222
1.3094
2718,8514
13097
2720.1809
1.3101
2721:5108
1.3104
2722.8410
1.3107
1.3110
2724 1716
2725.5025
1.3113
1.3117
2726.8338
2728.1655
1.3120
2729.4975
1.3123
2730.8298
1.3127
2732 1624
1.3130
2733.4954
1.3133
2734.8287
1.3130
2736.1623
1.3139
1.3143
2737.4963
1.3146
2738 8306
2740.1653
1.3149
2741.5003
41
alt
1.3269
1.3272
1.327
8
1.3273
1.321
1.3285
1.328
[ 2
1.3292
13
1.3255
4
1.3299
113302
1.3306
1.3309
1,331
19
1.3317
1.3320
LI
1.332
22
13326
23
1.3330
24
1.3333
25
1.3336
26
1.3340
io
7
18
܀
27
1.3343 128
1.334729
1.3350
30
M
M
i New and Corre&t Table of Meridional Parts. 289
39
40
41
Mer. Parts. | Differ. Mer. Parts Differ. Mer. Parts Differ.
Minutes in
Minutes and
1 296712065.9438
31
1-330132
1.3364/33
1.336734.
2753.53101.3381
3&
39
4
***
al
1
+2
2680 4124
1.3005 2681.7623
2012583.6702
2661.9979
2 741.500
3C
1.2961
1.3354
1.3153
21125849663
2653-3123
2742 8357
1.2964
1.3357
1.3156
3212586.2027
266.4 6279
2744 1714
1.3159
3312587.5594
2745 5075
1.2971
3412588.850;
1.3164
2667.2600
2746.8439
I,2974
1.3165 2748 1806
2668.5755
351-590.1539
35
1.2977
1.3169
1.3371
362591.4516
2669 8934
2749.5177
36
1.2980
1.317-
13374
3712572,7496
2671.2106
2750.8551
37
1.298
1.31751 2752.1929
1.3378
7812-594.0479
2672.528
1.2986
13179
291595.3465
2673.8450
1.2989
1.3182
1912596.6454
1.338;
2754.8695
2675 1642
1.2994
13183
+1 2597 9+16
1.3388
2676.4828
2756.2083
1.2993
13189
12/2592,2442
13392
2757.5+7
2577.8317
1:3'92 2758.8870
1,2999
+32600.5441
1.3395
2679.1 209
44/2501.8443
1.3002
1.3195 2760.22691.3399
4704
1.3197.2761.5671
1.3402
+5|2003.1448
4
4612604 4456
1 3008
1.3202
2683.3805
2762.9077
1.3406
4 i
+7 2607.7407
1.3011
2684.4010
13205 | 2764 2486
2764 2486 1.3409
47
1.3014
1.3208
$82607.0481
:685-7218
2765 5898 1 3412
19 2608.3499
1.3018
2687.0429
1.3211
2766-9314
1.3446
79
; 2609.6920
1.3021
2688.644
1.32152768.2733
1.3419
26109514
1.2024
1.3218
2689.6862
2769.6156 1.3423
51
12 26 12.2571
1.3027
1 3221
2601.0083
1.3427
2770.9583
52
32613,5601
1.3030
1.3225
2692.3308
:
1.303+
3+12614,8635 2653.6536
2773,6447
1.3434
1.3037
1.3232
$9
512516.1072
1.3437
2.694 9768
2774-9884
59
;612617.4712
1.3940
1.3440
2696-3003
17 2618.7755
1.3043
269706242
193239
1812620.0801
57
1.3646
1.3447
2698.9484
13242
2779.0215
13246
5912621.3850 113049
1.3451
2780:3666
2 700:2730
6012622.69621 1.3052
1.3249
27015979
M
39
40
41
48
so
53
52
1.3228
is
1.3235 2776.3324
56
2777.6768 1.3444
$8
}
2781.71201 1.3.454
IS
290
A new and corre&t Table of Meridional Parts.
42
43
44
Mer. Parts. Differ. Mer. Parts. Differ. Mer. Parts Differ.
Minutes !
Minutes 1
1
2
312
1.3686
1 369012951.3780
ali tu
I
8
1•3486 2875.4163
)
IO
I 3712
1.394311
1.394712
012781.7120
2863.0953
2945.8142
1.3458
1.3675
12783.0578
1.3904
2864.4628
2947.2046
1.3462
1.3679
22784.4040
2865.8307
1.3908
2948.5954
I'3405
2785.7505
13083
2867 1990
I'3911
1•346928685676
3
2949 9865
1°3915
412787 0974
4
I'3919
312788.4446 -13472 28699366
2952.7699
1.3476
62789.7932
1.3694
2871.3060
1.3923
2954.1622
6
- 3479 2872.6757
1.3697
1.3927
72791.141
2955.5549
7
1.3483
8/2792.4884
1.3701
2874 0458
I'3931
2956 9480
1.3705
92793.837
13935
2958.3415
9
I'3709
1'3939
102795.1860 1:3440 2876 7872
2959.7354
112796.5354
1.3494 2878 1584
2961.1 297
1.3497
1212797 8851
13716
2879.5300
2962.5244
1-35012880 9020 1.3720
1 3951
1312799,2352
2963 9195
13
142800 5850 1.3504 28*2.2743
1 3723
13955
2965.3150
14
1.35-82883.6470
1.3727
2966.7109
1'3959
IS
1.3511 2885.0201
1.3731
2968.1072
1.3963
1.3735
17|2804.6390 1.35152886 3936
2969.5038
2887.7675
1.3739
18/2805.9908 1.3518
18
13743
1 352212889.1418
192807.3430
1.3974
2972.2982
19
202808.6956 1:35202890 5165 1'3747
2973.6960
13978
1.3750
212810.0485 1:352928918915
12975 0942
I'3754
222811.4018
1 3533 2893-2669
2976.4928
2312812.7555
1.3758
23
1.3761
1.354012896.0188
24128141095
1 3994
2979-2912
24
25281564639 1-35442897.3953
1.3765
25
26 28168186 1 3547 2898.7723
1.3769
2982.0912
1.3772
27 2818 1737
1.4006
2983.4918
27
1.3776
28 2819.5291
1.401128
292820 8849
2984.8929
1.3558
1:3780
2986.2944
1.4015
29
302822 2410
1.3561
I 3784
2904.2834
2987-6963
I'4019
M
42
43
44
152801'93641_1.35-8
102803,2875 13511
16
17
13966
2970 9008 1.3970
20
21
22
1 3537 28946427
6
1.3982
1.3986
2977 8918 1.3990
2980.6910 1.3998
1.4002126
1.35512900.1494
1.355412901.5270
2902.9050
30
M
1 New and Correct Table of Meridional Parts.
291
42
43
Mer. Parts. Differ. Mer. Parts Differ.
44
Mer. Parts Differ.
Minutes 21
Minutes
2905-6622 1.3788
36
38
39
40
11
42
30 2822,2410
2904.2834
312823.5975
1.3505
32128249544
1.3569
2907.04.14
332826.3116
113572
2908.4210
34 2827.6692
1.3576
2909.800
1.3580
3512829.0272 2911.1812
361830.3855) 1.3583
2912.5619
37/2831.7442 1,3587
2:13.9430
382833.1033
1.3591
2995:32 +5
3912834,4627
1.3594
2916.7063
40 2835.8225
1.3598
2918 0885
412837.1827) 1.3602
2919.4711
+212838.5433
1.3606
2920.8541
432839,9042
1.3609
292 2.2375
44/2841.2655
1.3613
2923.6213
4512842 6274
1.3616
2925.0055
402843,9891
2926.3900
47 2845.3514
1.3623
29277749
48 2846.7141
1.3627
2929,1602
49 2848.0771
1.3630
2930.5459
;02849.4405
1.3634
2931.9320
112850.8043
1.7638
29333185
52 2852,1685
1.3642
2934 7054
132853.5330
1.3645
2936.0926
54 2854.8979
1.3649
2937.4802
552856.2632
1.3653
2938.8682
5612857.6289
1.3657
2940,2566
572858.9949
2941,6454
1.3664
2943.0346
592861.7281
1.3668
2944.4242
1.3672
2945.8142
MI
42
43
1.3620
2987.6963
2989.09 86
1.3792
2990.5013
1.3796
2991.9044
1.3799
2993.3079
1.3803
2994 7117
1.3807
2996.1159
1.3811
1.3815
2997.5 205
2998 9255
13818
3000.3309
1 3822
3001.7367
13826
1.3830
3033.1429
1.3834
3004.5496
1.3838
2005.9507
1.3842
30073042
3008.7721
1.3845
2010.1804
1 3849
1.3853
3011,5891
30129,82
1.3857
2014.4077
1.3861
3015.8176
1.3865
1.3869
3017.2279
1.3872
3018.6386
3020.0497
1.3875
1.3880
3021.4612
3022.8732
1.3884
1.3888
3024.2856
1 3892
3025.6984
I'3896
3028.5252
1.3900
3029.93921
44
30
1.4023
3
1.4027
32
1-4031
33
1.4035
34
1.4038
35
1.4042
1 4046
37
1.4050
1.4054
1.4058
1.4062
I 4067
1.4071
+2
1.4075
ti
1.4079
1.4083
40
1.4087
47
I 4091
1.4095
49
1.4199
1.4103
50
1.4107
52
1.4111
1.4115
S.
1.4120
5
1.41 24
56
1.4128
57
1.4133
1.4136
1.4140
6c
M м
48
:ر
1.3060
$82860.3613
3027.1116
58
59
60 2863.1953
292
A New and Corre&t Table of Meridional Parts
45
40
47
Mer. Parts. Differ. Merid. Prs. Differ. Mer. Parts | Differ.
Minutes - 1
Minutesiol
1.4398
30799292
3115.5456
3031.35 36
1.4144
3116.9854
3032 7084
1.4148
318.4256
1.4153
3034-1837
3
3119.8662
(3035.5994
1.4157
3121.3073
53037.0155
1.4161
3122.7488
63038.43 20
I.4165
3124.1997
73039.8489 1.4169
3125.6331
813641.2652) 1.4173
3127'0759
93042,6839
1.4177
3128.5192
133044.1026 1.4181
3129.9629
113045 5206 1,4186
3131.4070
12:046.9396 1,4190
3132.8516
133048.3590 1.419+
3134.2966
14304907788 1.4198
3135.7420
153051•1940 1,4202
3137. 1879
1630526196 14206
3138.6342
1713054 04-01 1.4210
31 40.0810
18 3055 4021 1.4.215
3741'5282
193056.88401 1.4219
3142 9759
2013058.3063 _1:4223
31444240
211305 9.7290 1.4227
3145.8725
223061 1521
1.4231
2313062.5757
3148.7709
24/3063.99971.424 .
3150.2208
23 3065.4241
3151.6711
263006.8489) 1.4248
3153.1218
273068.2741 1.4252
3154.5730
283069.6998 1:4257
1.4261
3156.0246
2913071.1259
1.4265
3157.4767
301072.55241
13158.9292
M
05
46
3203.7136
1.4665
3 207.1801
1.4402
1,4670
3205 6471
I 4406
1.4674
3207.1145
I 4411
1.4679
3 208.5824
4
1.4415
1.4683
3210 0507
1.4419
1.468817
3211.5195
1.442+ 13212.9888
1.4693
1.4428
1.469718
3214.4585
1 44333215 9287
1.4702
9 9
1.4437:217.2993
1.47061
1.4441
I 4711 1
218.8704
1.4446
1.4716
12
3220 3420
1:4450
1.4720
3221,8140
13
1.4454,3223,2865
1,472512
1.4452 3224,7595
1 47315
1.4463 13226.2329
1 47341
1,4739117
3227.7068
1.4472 3229.1811
1.474311€
1.4477
1.4748
3230 6559
9
1.4753120
3232.1312
1.4.485
1.475721
3232.0069
1.4490
1.4762122
3235.0831
1.4494 3236.5598
1.476722
1.4499
1.4771
24
3238.0369
1.4503
1.477625
3229.5145
1.4307
26
3240.9926
1.4512
3242.4711
1.4516
1.4790128
3243.9501
1.4521
1:4795129
3245.4296
1,4525
1.4800130
3246 9096
47
M
1.1408
1.4481
1.4236131473215
1.4244
1.4781
1.478527
al New and Correct Table of Meridional Parts. 293
ak
Minutes in
Minutes and
3160.3822 1.4530
32
1.4828
1.4575
45.
46
47
Mer. Parts, Differ. Mer. Parts Differ. Mer. Parts Differ.
3013072.5524
31589292
3246.9096
30
1.4269
1'4805
313073,9793
3248.3901
1.4274
31
1.4809
3212075.4067
3101.8356
1.4534
3249.8710
1 4278
3313076.8345
1.4539
316362895
1-4814
3251.3524
33
1.4819
I.4543
34 3078.2627 1.4282
3164 7438
1.4286
3252.8343
34
353079.5913
3.166.1985
1.4823
1.4547
3254 3166
35
36305 1.1203
1.4290
3367.6537
1.4552
325 5.7994
3713082.5498
36
1.4295
3169.109,3
1.4556
1.4832
3257.2826
38 3083.9797
1.4299
37
I 4837
1.4501
2170.5654
3258.7663
38
1.4841
39130854100 1.4303
3172.0219
1.4565
3260.2504
1.4570
4013086.8407) 1.4307
3173.4789
1.4840
3261.7350
41|3088.2719 1.4312
3174.9364
1.4851
326312201
423089.7035
1.4310
3176.3943
1.4579
3264.7057
433091.1355) 1.43 20
3177.8527
1.4584
3266,1918
443092.5679 1.4324
3179.3195
1.4588
3267.6784
453094 6008_1.43291180 7707
1.4592
3269.1654
463095.4341
1.4333
3182.2304
1.4597
1.4875
3270.6529
47/3096.86781 1.4337
3183.6905
I 4001
48 3098.3020
1.4342
3272.1409
3185.1911
3273•6294
49 3099.7366 1.4346
3186.6121
1.4610
1.4890
327 511184
503101.1717
1.435.1
49
1-4615
3188.0736
3.276.6079
1.4895
513102,6072
1.4355
so
3189.5355
1040191
52|3 104,0431
1.4359
I 4624
3190.9979
51
1.4904
1.4363
5313105.4794
3279.5882
$.4629
52
3192.4608
3281.0791
1,4909
5413105.9161
1.4367
3193: 9241
1.4633
53
1.4914
3282.5705
5513108.3533
1.4372
3195.3879
1.4638
54
1.4918
3284.0623
55
563 109.7909
1.4376
3196.8521
1.4642
1.4923
3285.5546)
1.4647
573111.2289 1.4380
56
3198.3168
3287.0474
1.4928
$8|3112.6674 1.4385
1.4652
571
1.4933
3199.7820
1.4384
3238.5407
5913114.1063
3201.2476
I'4656
3290.0344
59
603115.54551 1.4393
3202.7136
1.4660
3291.52861
14942
60
MI
45
47
09
39
40
1,485641
1.486142
1.486643
1.4870144
45
1.488046
1.4885147
1.4606
48
3270.0978 1.4899
1.4937
58
so
Minutes
Minuteslo
1.556c
1
2
1.52873486.9360
13388.1833
1.49801394.2927
294 A New and Correct Table of Meridional Parts
48
49
Mer. Parts. Differ. MeridiPrs. f Differ. Mer. Parts | Differ.
13291 5 286
2382,0823
3474.4720
13293.0233
1.4947
3383.0068 1.5245
1.4952
3476.0289
23294 5185 3385.1318
1.5250
1,5565
3477.5854
31:296.0142
1.4957
3386,6573
1.5255
1.5571
1.4962
3
3479.1425
3297.5104
I 5260
4
1.5576
3480.7001
1.4967
4
(3299 0071
3389,7099
1.5266
1.5582
3482 2583
S
6,300 5042
1.4971
3391.2370
1.5271
1.5587
3483,8170
713302.0018 1.4976
3392.7646
1,5276
1.5592
3485.3762
7
813303.4999
1.4981
1.5281
1.5598
8
3395.8213
3488.4903
1.5603
13306.4976 1.4991
3397,3504
1.5291
1.5019
349°0572
IC
113;07.99?2 1.4996
3398.8800
1.5.296
1.5014
3491.6186
123309.1973) 1.500
1.5302
1.5620
3400.4.102
3493.1806
I 2
1313310.9978 1.50CS
1.5307
1.5625
3401.9409
3494.7431
13
143312 4988 1.5010
1.5312
1,5631
3403.4721
3496.3062
1.5015
1513314.0003
1.5317
1.5636
3405.0038
3497 8698
19
16|3315.5023) 1.5020
3406.5300
1.5322
1564
3499.4339
17|33170048 1.5025
1,5647
3408.0687 1.5.327
3500'9986
17
183318-50781 1.5030
3409.6019
1.5332
1.5652
3502 5638
1913320.0113
1.5035
1.5338
1.5658
3411,1357
1,5040
3504,1296
9
2013 321.5152
1.5343
15663
341 2.6700
:505.6959
213323.0198) "5045
1.5348
I5639
3414.2948
3507.2628
223324.52481,5050
1.5353
1.5674
34 5.7401
3508.8302
22
23326.0303 1.5055
1.5359
1.568
3510,3982
22
24/33 27.5363
3418.8124
1.568s
1.5064
3511,9667
2513329.0427
1.5691
1,5369
3420.34.93
3513 5358
25
2613330.5496
1.5069
1.5374
1.5696
3421.8867
26
1.5074
273332 0570
1.5702
34234246
1.5079
3516.6750
27
2813333.5649
1 5385
1,0708
1.5084
3424.9631
28
3518.2464
15390
293335.0733
1.5713
3420.5021
1'5089
35198177
29
3013336.58221
3428.04.16
1,5395
1.5719
35213895
30
M
SO
IM
14
1
2
?1
1.50603417.2760
1.5364
24
13379 3515.105
48
A New and Correct" Table of Meridional Parts. 295
49
Minutes 1
48
so
Mer. Parts. Differ. Mer. Parts Differ. Mer. Parts Differ.
Minutes ago
3429 5816 1.5400
3434 2048 1.5416
3521.6826
36
3437.2890 1,5427
3438,8328 1.5432
3440.37051 1.5437
1.5774
40
1:5458
41
42
2013336.5822 3428.0416
3521.3896
30
1.5094
30:338.0916
1'5724
3522.9620
31
1.5099
32 3339.6015
1.5730
1.5405
3431,1221
3524:5350
32
I S104
3333711119
3432.6632
1.5411
3526.1085
1-5735
33
1.5109
34 3342.6228
1.5741
34
1.5114
251334 4.1342
1.5421
1,5746
3435.7469
3529.2772
35
3613345.6401
1.5119
1.5752
3530.8324
3713347 1585/ 1.5124
15757
3532.4081
37
38 3348.6714
1.5129
35339844
1.5763
3
393350.1848 1.5134
1'5443
1.5769
3441.9208
3535 ; 613
39
4013351.6987
1.5448
1.5139
34434550
3537.1387
4113353,2131
1.5144
3445.0109
1'5453
3538.7167
1.5780
423354,7280) 1.5149
34+6.5567
15786
3540.2953
43/3356.2434
1.5154
3448.1031
1.5464
3541.8744
1.5791
4
443357.7593
1.5159
3449.6500
1.5469
1.5797
3543.4541
41
4513359.2757
L.5164
3457.1974
1.5474
1.5802
35450343
4
4013300.7926
1.5169
1.5479
3452.7453
1.5898
3546.615
44
4713362,3100
1.5174
1.54&5
1.5814
3454.2938
3548.1965
47.
+83363.8279 1.5179
1.5197
I 5819
3455.8428
1.5184
3549.7784
493365.3463
1.5496
1.5825
3457.3924
1.5189
3551.3609
5013366.8652 3458.9425
1-5501
1.5830
3$ 52.9439
50
5113368.3846 1.5194
3460.493
1.5506
1.5836
3554 5275
51
5213369.9045
1.5199
3462.0443
15512
1.5842
1.5204
35561117
52
5313371.4249
3463.5900
145517
1.5847
3557.6964
1.5209
5413372.9458
5?
1.5522
3465.1482
1.5853
3559.2817
553374 4673
1.5215
3.
560.4541
55
563375.9093
1.5220
3468.2543
1.5533
1.5863
56
3562.0411
5713377.5118
1.5225
1:5538
3469.8081
1.5870
3564 6287
57
$83379.0348 1.5230
3471.3025
!:5544.
1.5876
3565.2169
59 3380.5583) 1.5235
15549
1.5882
3472.9174
3,567.8056
se
60 3282.0822/ 1.5240
1.5555
1.5887
3474 47291
3568.86761
MI
48
49,-
50
1
48
49
54
1.5859
3466.7010 1.5528
58
60
M
Qq 2
296 A Neid and Corre&t Table of Meridional Parts
Minutes
Minutesl of
1.589913668.4439
*571.9848) 1.5899
31 573.5752
3767.08481,6626
3575,1662] 1.5910
3673-3230 / 1.6275
1,6632
1.0282
159393679.8370
1.630
13,701.07)
5
52
13
Mer. Parts.). Differ. Murid, Prs. Difer. Mer. Parts | Differ.
3508 8056
:66 1.1940
3767.7602
1.5893
1.6246
1.6620
3 70.3949
3666.8186
3765.4222
1,6252
2
1:5904
3670.0098
1.6258
3768.7480
1.6632
3
13571.6900
1.6264
3770.4119
4
1.596
576.7578
1.6645
3772.0764
1.5922
1.6276
1.66526
63578.3500
1,5927
3674.9506
3773,7416
1.6658
73579.94271 3576.5788
3775.4074
1.5933
1.6288
1.666518
358153601 3678.2076
3777.07739
1.6294
1.6671
413583.1299
3778.7410
9
1.5945
)
1.6678
13881.4670
135847244
3780.4088
IO
113586.3194
1.5959
1 6684111
3683:0977
37810772
1.5950
1.669112
123587.9150
3684.7290
3783-7453
1.5962
1313589 $112
3686.3609
1.5968
37854166
0.1,6325
1413591.108C
1.5973
37870864
153592 7053
3689.6.265
1.6331
3788.7574
1.5979
1,6337
1691.2602
16 3594 3032
1.5985
3790'4291
1213595.9017
3692.8945
379211014
1.5991
183597 5008 3694.5294
1.6349
3793.7744
1.635$
1913 599 1005
3795.4480
1:6362
3697.801 1
2013600.7008
37971222
1.6009
213602.3017
1.6368
3798.7971
1.6014
1.6374
22:603.9031
800.4727
1. 6020
233605 505!
1.6380
3702.7133
1.6026
3802.1489
1.6387
2413607.1077
370443520
3803,8258
25126087109
1,6393
3705.9913
3805:5033
?S
1.6399
26136 10.3140
1.6043
3707.6312
3807:1815
16405
2713611.9189
3709.2717
1.6049
3808.8604
1.6411
28|3613.5238
3710,9128
3810,5399
1.6417
2913615.1293
1.6061
3712.5545
3812.2201
1,6424
1.6899
30/3616.7354 371 4.1969
3813'9010
M
50
52
53
M
1,6307
1.6313
1,6319
1.6697113
1,6704114
3687-99,34
1.6343
1.5997696.1649
1.6003
3699 4379
..
1.6714
15
T67171
1,67232
1,6739118
1 67319
1674220
1.674971
1.675022
1.6762
1,6769
1.6775
1,6782 26
1.6789
27
1.6795128
1.6802
29
30
22
24
1.6032
1.6037
9.6055
A new and correct Table of Meridional Parts. 297
SI
Minutesi 1
Minules
31
32
32
818.9475
36
38
1
1.6480 3827.3716
1.5486 3829.05 84
143|3637.66841
52
52
Mer. Part Differ. Mer. Parts Differ. Mer. Parts, Differ
3616.7354
3714.1969
3813.9010
3.
1.6067
1.6430
1 6815
3618.3421
3715.8399
1.6073
3!
3815 5825
1.6436
1.6822
3619.9494
3717.4835
1.6079
32
3817.2647
1.6442
1 6828
3621.5573
3719.1272
3
32
1.6085
1,6449
1.6835
3623.1658
3720.7726
1.6090
34
1.6455
3820,6310
34
1.6842
3624.7748
35
3722.4181
3822.3152
1.6096
1.6461
1.6878
36 3626.3844
3724.0642
1.6102
1964673824.0000
1.6855
37|3627.9946
3726.7109
3825.6855
1 6108
37
I-6474
1.6861
38/3629.6054
1.6114
3727.3583
1.6868
393031,2168
3729.0063
1.5120
39
1.6875
403632.8288
3730.6544
3830.7459
10
1.6126
1.6493
1,6881
473634.4414
37323012
3832.4340
16499
+
1 6888
423636.0546
3733.9541
1.6505
3834,1228
1.6138
1.6895
42
3735:6046
3837.8123
1.6511
4
1.6902
44
3639.2828
3737.2557
1.6150
383.7.5025 1.6908
1.6518
451 3640:8978
3738.9075
3.819.1933
is
1 6156
1.6924
1.6915
403642.9134
1.6161
3740.5599
1.6530
3840.8848
1.6922
473644.1295
3742.2 129
1.6167
3842-5770
1'6537
1.6928
483045.7452
1.6173
1'6543
3844.2698
493647.3635
3745.5299
1*6549
3845,9633
49
36 +8.9814
5
1.6942
3747,1758
3847.6575
50
1.618
1.6556
1.6949
51 3650:5999
1.6191
3748.8314
3849.3524
51
1.6955
$213652.2190
1.6562
3750.4876
3851.0479
52
1 6962
5313653.8387
3752 1444
1•6203
3852.7441
53
1.3969
5413655.7590
3753801
1'6210
3854.4410
1.6581
553657,0800
3856.1386
155
1.6216
1.6588
1 6982
503658.7016
:.6222
3857.8368
1 6989
57 3660.3238
3859.5357
1.6228
1.6600
5813661.9406
57
I 6996
591 3663-5700
3861,2353
1.6234
1.66:17
1.7003
5013665.19401
1.6240
1.6613
3862.9356
59
1.7010
28646366
6c
M
51
52
5?
M
1 6132
1.6144
14
16
+
3743.8666
1.6935
148
1.6179
1.6197
1.6568
16575
1.697654
1.6594
SO
3755.4600
3757,1188
3758.778;
3760.438.
3762.0984
3763.7002
5&
298 A new and corre&t Table of Meridional Parts.
54
Minutesi
Minutes 1
mojo
1413888.5220
55
S6
Mer. Parts | Differ. Mer. Parts Differ. Mer. Parts' Differ
3864.6366 3967.9661
4073-9042
1.7017
1 3856.3383
3969.7099
1.7438
1.7887
4075.6929
1.7023
23868.0406
3971.4544
1.7445
4077.4824
1.7895
1.7030
313869.7436
1.7453
3973.1997
4079.2726
1.7902
3871'4473
1.7037
1.7460
3974.9457
408110636
1.7910
4
1.7044
3873 1517
1.7467
1976.6924
S
1.7918
4082.8554
1.7051
3874.8568
1.7474
978.4398
1.7925
4084.6479
6
1.7057
:876,5025
1.7482
7
3980. 1880
7
1.7064
486.4412 1.7933
1.7489
8|2878.2689
3981.9369
4088.2353
1.7941
8
17071
913879.9760
1.7496
3983.6865
1 7949
490.6302
9
1.7078
13881.6828
3985.4369
1,7504
1.7956
4091 0258
IO
113883.3923
1,7085
39871880
1.751)
1.7964
40G3.6222
11
1.7092
1.75 18
1213885,1015
3988.9198
1.7972
4095.4194
I 2
1.7099
1.7526
1313886 8114
3890.6924
1.7980
4097.2174
1.7106
13
17533
:992.4457
4099.0162
1.7988
14
1,7113
1:7540
1513890 2333
3994.1997
1.7996
4100.8158
1.7119
16 3891 9452
1.7548
1.8004
4102.6162
3995.9545
1.7126
1.7555
1.8011
1713893.6578
3997-7100
17
1.7133
4104.4173
183895.3711
1.7562
1.8019
1.7140
4106.2192
399904662
1.7570
193897.0851
1,8027
4108,0219
4001.2232
1.7147
203898.7998
4002.9809
4109.8254
1.7154
1:7584
21 3900.5 152
4111.6297
1.7592
2213 902.2013
4006.4985
1.7599
4113-4348
233903.9481
4008.2584
4115.2407
1.7175
1.7607
1.8066
24 3905.6056 4010.0191
1.7182
4117.0473
1.7614
253907.3838
4011.7805
4118 8547
1.7189
1.7621
2613909.1027
+013.5426 1.7629
4120.6629
1.7196
271391008223
4015.3055
1.7203
4122 4719
1.7637
28 3912.5426
1.7210
4017.0692
1.8106
293914.2636
4018.8336
4126.0923
1.7217
1.7652
303915.9853
4020.5988
4.127.9037
30
M
54
55
56
M
ala
16
1.7577
19
1.8035 120
1.804321
1.7161 40047393
1,7168
1.8051
22
1 8059
23
24
1.8074
25
1.808226
1.8090
27
1.8098 28
29
1.8114
1.7644
412 4.2817
102
A new and corre&t Table of Meridional Parts. 299
Minutesl olm
Minules and
1.7238
1.8130
18138
1.8146
1.81.54
36
1.7260
1.81703
1.8178138
39
4160 6459 1.82671
54
55
50
Mer. Part, Differ. Mer. Parts Differ. Mer. Parts Differ
3915.9853
4020.5.88
4127.9037
30
30
1.7224
1.7659
1.8122
3917.7377
4022,3647
31
1.7231
1.76674129.7159
3919.4308
4024.1314
37
1.7674
4131 5289
32
323921,1546
4025.8988
1.76824133-3427
32
1.7245
3922.8791
4027.6670
34
1.7689
1.7252
4135.1573
34
35
3924.5043
4029.4359
4136 9727
35
1.7259
1.7096
1.8162
30 926.3302
4031.2055
1,7704
+138.7889
373928.0568
403249759
1.7711
1 7273
4140.6059
3813929.7841
403 7.741
414264237
1.7719
1.7281
1.8186
3913931 5122
4036.5189
1.7286
1.7726
41 4+2 123
1.8194
403933.2410
4038.2915
4146 061)
10
1.7295
1.7734
1,8202
413934.9705
4040.J649
1.774.1
4147.8819
41
1.7302
1 8210
14213936.7007
4041,8390
4149.7029
1.7749
4.7
1.7309
1.8218
4313938.4316
404396139
4151.5247
1.775
4:
1.73164045 3896
4413940:1632
1.8226
4153.347
1.7764
10
1.7323
1.8234
45/3941.8955
4047' 1660
4155.1707
1-7712
1.8242
403943.6285
10
1.7337
1.8251
+739+5 3622
4050.7212
+7
1.7344
1.8259
403917.0965
4052.4999
48
197795
1.7352
3948.8318
4054.2794
1.7359
4162.4726
1.7803.41
395935677
4055:0597
4164 3001
1.7366
1.7810
i 8283
5139523043
4057.8407
4166 1284
51
1.7373
1.7818
1.8291
39;4.0416
4059.6225
1.7825 4167.9575
52
4061.405
4169.7874
1.7833
53
17388
1.8308
53957.518+
4063 1883
1.7841
4171 6182
17395
54
53959.2579
4064.9734
4173.4498
5$
1.7402
1.7848
103960.9981
4066.757
50
573962.7390
1.7409
1.7856 4175 2822
4268.5428
4177.1154
1.7864
583964.4806
57
1.7416
1070.3292 4178.9494
1.7871
17424
5913966.22 30
4072, 1163
4180.7842
55
1.78.79
1.7431
503967.95611
4073 9942
4182.6199
6
M
5+
55
56
M
1.733014048.9+32
1.7780 4156.9949
1'7787 14158.8200
1.8273150
49
ܕܼ
1 8299
313955.779617380
1.8316
183241
1.8332
1.8340
1.834
1.8357
8
158
300 A new and correct Table of Meridional Parts.
57
58
59
Mer. Parts. Differ. Mer. Parts. Differ. Mer. Parts Differ.
Minures 1
Minutes
1.8875
1.8884
1
2
1 83904299 9631
ş
1.9468
184144307.5290
841414305 6362
18937
1.84314311.3173
91
1.84504317.0065
1.6973
2 2
1.9525112
19534
1.8481 4322.7035
04182-6199
4294.2979
1.8365
117184.4564 4296.1854
1.8373
214186.2937
4298'0738
1.8381
314188.1318
44189.9708
4301.8533
1.8398
54191 8106
4303.7443
1.8400
64193.6512
74195.4926
4307
84197 3349
I 8423
4309 4227
9,4199,1780
1.8440 )
10.4201.0220
4313.2 1 2 28
114202.8668 1.84184315:1092
4204.7124
13.42065588) 1-8464 43 18 9046
14.4.208 4061
1.8473
4320 8036
15 4210.2542
164212.1031
1 8489 4324 6043
1742 13.9529
4326 5060
1842 15.8035
1.85054328 4086
194217 6549
1 851443303121
2014219.5072
1.852314332.2 165
214221-3603 11.8:31 4334.5218
234223,2143
2342 25.0691
244226.9248 1.8557 4339843
254228.7813
1.8565 4341 7520
264230 6386 18573
4345.5735
28 4234:3558
294236.2157
1.85994349 3966
3014238 0764
1:8607
4351.3100
57
1409.1399
4411.0820
1 9421
1.9430
I 8893
1413.0250
1.9440
1.8902
4414.9690
3
19449
1 8910
4416 9139
4
1 9458
+418 8597
1.8919
1.8928
4420 8065
1.9477
4422.7542
7
1.9487
8
+424.7029
1•8946
4426 6525
1.9496
9 ୨
1.8955
4428.6031
1.9506
1 8964
1430.5546
1.9515
II
1898 14432,5071
1 8990
4434.4005
13
1.9544
4436.4149
14
1 8999
1'9553
44;8.3702
IS
1.9008
1.9563
16
4440.3265
1.9017
1'9573
4442.2838
17
1 9033
4444.2420
1.9592
4446.2012
19
199044
1 9602
4448.1614
20
1.9053
1.9611
4450 1225
21
1.9062
1.9621
4452.0846
22
1.9071
1.9630
1.9080
4454.0476
1.9610
24
4456.0116
1.9089
1.9650
4457.9766
25
1.9098
1.9659
4459.9425
26
1.9669
1.911614461.9094
27
4463 8773
1.9679
28
1.9688
4465.8401
29
1.9698
1.8498 4326
4328 4086 1.90264442.2838
1.9582
18
1 85401433
4336.0 280
1.8548 4337.9351
23
1.9107
43 43-6018
274232 4968 1.8582
1.8590 4347 4841
1.9125
191344467.8159
I
30
2
M
58
M
59
sammen
. New and Correct Table of Meridional Parts.
301
57
58
59
Mer. Parts. Differ. Mer. Parts Differ. Mer. Parts Differ.
Minutes in
Minutest
1 8633
1.8642
1.9757
20+238.0764
+351.3100
1.8616
214239.9380
1.9143
1353.2243
1.8625
3211241.8005
+355.1395
1 9152
3314243.6638
1.9162
+357.0557
34 +245 5280
1.8650 +358.9726
+358.9728 1.9171
3514.247.3930
4360.8908 1.9180
36:249.2588
1.8658
1.9189
4362.8097
1.8667
37142 51.1255
+364 7295
1.9198
:811252.9930
1.8675
+366 6502
1.9207
394254 8614
1.8684
+368,5718
1.9216
+256.7306 1.8692
4370.4944
1.9226
+4258.0007) 1.8701
+372.4179
+2426.4716
1.8709
+374.3423
131+262,3434
1.8718
1.9253
4376.2676
4414264,2161
1.8727
4378 1938
1.9262
+5 4265.0897_6.8736
4380.1210
1.9272
40
4267.9642
4382.0491
1.9281
+7 +269.8395
+383.9781
48
1.8762
4271.7157
+385.908
1.9299
494273,5928 1.8771
+38768389 1.9309
+389.7707
1.9318
514277.3796
1.8788
+391.7034
1.9327
5214279.2293
1.9336
+393.6370
1.8805
534281.1098
4395.5716
1 9346
54 +282.9912
4397-5071
551284.8735
1.8823
1399.4435
1.9364
5614286.7566
1.9374
4401.3809
57 +288.6406
1.8840
403.3192
19383
4405.2585
1.9393
5911292.4113
+407 19821 1.9402
1.8866
60 +294.2979
+409,1399
1.9412
M
57
58
19235
1.9244
30
1-9700
1.9718/31
1.9727132
1'9737133
1.974734
35
1.9766 36
1.9779137
197863
1.9796 39
1.980642
1.9816 43
1.9825 42
1.9835
44
45
4467.8159
4409.7867
4471.7585
4473.7312
4475.7049
4477 6796
4479.6553
4481.6319
4483-6095
4485.5881
+487.5677
4489.5483
4491.5299
4493.5124
4495.4959
449 7.4804
4499.4659
4501.4524
4503.4399
4505.4234
4507.4179
4509.4084
4511.3999
4513.3924
4515.3859
45173804
4519.3759
4521.3724
4523-3699
4525.3683
4527.3677
59
1.8745
1.8753
1.9290
47
48
49
+275,4708_1.8780
1.8797
1.8814
1,9845
1.9855
1.9865
1.9875
1.9885
1.9895
1.9905
1.991
51
1.9925
52
1.9935
53
54
1.9945
1 9955
$8
19965
1.9975
57
1.9984
58
1.9994
60
M
1'9355
1.8831
581+290.555
1.8849
1.8858
58
Rr
OL
Minutes
Minutesi ola
ont al
4813.51331 2.1518
302 A New and Correct Table of Meridional Parts
60
O2
Mer. Parts. Differ. Merid. i'rs. Differ. Mer. Parts Differ.
01:527.3677
4649.2 253
4774.9820
1+529.3682
2.0005
4651.2885
2.0032
2,1300
4777.1126
2.0643
2.2015
21+531.3697
2 13 i8
4653:3528
4779.2444
2
2 0025
2.0654
314533.3722
2.1330
4055.4182
+7813774
3
2'0035
2.0665
2.1341
4/4535,3757
4557-4847 4783-515
2.0676
$15.7-3802 2.0045
4659 5523
2.1353
4785.6468
2 0606
2.136517
6 4539.3858 2.0950
4661.6209
47877833
2:C066
2.0097
2.1377
7+54 1.3924
4789.9210
2.0708
2,1388
14543-4000 2.00764063,6906
4665.7614.
+792.0598
2,0714
2.14.0
945454086 2.0086
4667.8333
+794.1998
2.0730
2.1412
1° 4547-4183_2.0097
4669.9063
4796 3 10
2.0741
2 1424
11 +549.4290 2.0107
4671.9804
+798.78.34
2.0117
2.0752
12 4551.4407
2,1730
4674.0556 48c06270
1 2
2.0127
2,0763
2.1440
13 +553.4534
4676.1319
802.7718
2.0774
2,1459
1414555 4671 20131
4678.2093
4804.9177
2,0785
2.1471
15+557.4818_2.0147
4680.2878
7807 0648
2.0796
2.148
161559.49762.0158
4682,3674
4809.2131
16
2.0.68
2.0807
2,1495
1714561.5144 4687.4481
4811.36.26
17
2.0178
2.088
18+5635322
2.1507
4686,5299
18
2.0188
1514565.5510 4688.6128
2.0829
4815,6651
19
2.0199
2.0840
2014567 5709
2.1530
4690.6968
4817.8181
20
2.0 209
2.0851
2.1542
214569.5918 4692.7819
4819.9723
2.0219
2.0862
2.1554
221571.6137
4694.8681
22
4822.1277
2.0873
2.1506
23/45736367) 2 0230
4824.2843
2.0885
2.0240
2.1578
24|45756607
4699,0439
4826.4421
24
2.0251
2.0896
2.1591
25/4577:6858
4701.1335
4828.6012
2.0261
2.0907
2.1603
2614579.7119
4703.2242
26
27/1581.73.90 2.0271
201615
4705.3160
4832.92:30
27
2.0282
28]+583.7672
20929
2.362728
4707.4089
4835.0857
2.0292
2,0941
2914585.79641
2:1639
4709 5030
4837.2496
29
2.0952
2.165!
30 +587.82661 2.0302
4711.5982
4839.4147
39
MU
60
61
62
M
13
will
2
LI
2 023014696.9554
23
S
209184830.7615
- New and Corre&t Table of Meridional Parts. 303
00
62
Minutes
Minutes 1
31
35
36
2.174837
39
40
pi51
2.17601381
61
Mer. Parts. Differ. Mer. Parts Differ. Mer. Parts Differ.
3014587.8266
1711.5982
4839.4147
2.0313
2.0963
2,166
3t+s 89.8579
4713.6945
4841.5810
2.0323
2.0974
32.591.8902
2.1675
471 5.7919
4843.7485
20334
2.0986
2'1687 32
331+59319236
4717.8905
4845.9172
2.0344
2.0997
2.1699
331
24/+595.9580
4719.9902
4848.087
2.1008
34
2.1711
351+597-9938
4722.0910
4850.25 8.2
2.1019
3614600.0 300
2.1724
47241929
4852.4306
2.0376
4726 1960
2.1031
37602,0676
4854.6042
2.1736
2.0786
2:1042
384,104,1062
4728 4002
4856.7790
2.0397
391 +605.1459
4730.5055
2.1053
+858.9550
2.0408
2.1065
11008.1867
473 2.6120
4861,1323
2,1773
474610.2285
2.0418
2,1076
4734.7196
4863-3108
2.1785
4214612.2714
2.0429
4736.8283
2.1797
4314614.3153
2,0439
4738.9382
2.1099
1867.6714
2 0450
2.IIO
1414616.2003
42
2.1822
4741.0492
4869.8536
2.0460
2,1122
4514618 4063
44
4743.1614
2.1834
4872'0370
45
4614620.4534
2 0471
4745.2747
2.11334874.2216. 2.1846
+24022.5016 2.0482
4747.3892
2,1145
4876.4074
2,1156
2.1871
+8 4624.5508 2.0492
4749.9048
1914626.6011
2.0503
2.1168
4751.6216
2.1883
4880.7828
2.1179
+6286525 -2.0514
4753.7395
2.1896
4882.9724
51 +630.7049
2.0524
2.1191
4755.8586
4885.1632
2.1908
52 +632 7584
2.0535
2.1202
2.1921
47.5 7:9788
4887.3553
2,1214
534634 8136 2.0546
4760.1002
4889.5486
2.1933
2.1225
53
54 1636.8686 2.0556
4762.2227
4891.7432
2.1946
554638.9253
2.0567
4764.2464
2,1 237
54
4893.939
2 1958
56 4640-9831
2.0578
2 12.8
4766.47 12
4896.1.36
2. 1970
5714643.04.20
2.0589
2.1200
4768.5972
2.1983
184645.1020
2.0600
2.1271
4900.5338
2 1995
2.061 4770.7243
5914647.1631
2-1283
6014640.2 253
2.0622
47728526
4902.7346
59
2.1294
2.2021
4774.9820
4904.93671
MI
60
61
62
RI 2
2.1087 14865.4905
41
2.1809 42
4878.5945
2.1858/41
47
48
49
50
51
52
4898.3343
56
$7
58
2.2008
304
A New and Correct Table of Meridional Parts
03
04
OS
Minutes
Mer. Part s.] Differ. Merid.tis. Differ. Mer. Parts | Differ.
Minutesi ol
+920.3864
2.21475 062 2017
9314600
I
9358983
904 9267
1029.4216
2,2033
11907.1400
1041.7034
2,2046
6 909 3446
5043.9866
$11911.5507
2 2058
5046-2712
2.2071
1913.7575
2.20841
5048.5572
915.9659
5050.8445
+918 1755
2.2090
5053'1332
2.2109
SO55 4.233
2.2 1 22
+922 5986
5057.7147
9740120 2213413660.0075
927 0267
2.2160
929.2427 5064.5973
2.2173
2,2185
5666.8942
193306785
5069.1925
2 2198
5071 4922
2.221
13119381194
5073.7933
2.2224
IC 940 3418 5076.0958
2.2237
1942.5655
5078.3997
2.2249
I944.7904
Is +947.0166
5083.0116
2-1949.2441 5585:3196
44951 4729
5087.6290
2.2301
2 1953.7030
2.2314 5089 9398
33/4955.9244
5092.2520
5094.5656
2.2340
2514.960.4011
5096,8807
2.2353
2014962.6364
2.2366
5099.1972
? 14964.8730
5101.5151
2.2379
2 967.1109 5103,8344
22392
2
5106 1551
3: 971.5906
M
5178.808)
2,2818
151811751
2.3670
I
2.2832
5183.5435
2,3684
2
2.2846
5185.9134
2.3699
3
2.2860
5188.2848
2.3714
4
2.2873
2.3729
5190,6577
S
2.2887
2.274.46
5193,0321
2.2901
2.375817
5195.4079
2.2914
2.3773
5197.7852
2,2928
2.3788
$ 200.1640
2.2972
2.3803110
52025443
2.2956
2-3818 II
5204.9261
2.2969
2,383312
5207.3094
2.2 983
2.2848113
5209.6942
2.2997
2.3863/14
5212.0805
2,3011
2:3878
IS
1214.4083
2.9025
2389316
5216,8576
2.3039
2,3909117
5219.2485
2.3053
2.3924118
$221,6409
2.3066
2-3939119
5224.0348
2.395420
2.90945228.8271
2.396921
2.3984122
2:3122
2.3999123
2 3136
5236.0269
2.401524
2.3151
5238.4299
2.403925
2.3165
5243.8344
5.404526
2 3179
5.4060127
5 243.2404
23193
5.4070 28
5245.6480
2.3207
5.4091129
5248.0571
2.3222
5:4107130
5250.4678
65
M
I
22262 5080.7056
2.275
2.3680
5226.4302 2.3954126
2.2288
2.3108
5231.22555
5233,6254
24 4958.1677 2.2327
1969.3501
2.240515108.4773
63
64
A new and correct Table of Meridional Parts. 305
63
64
Minuteslim
Minu esimi
304971596
31
2.1444
3
$4
34
30 +985 0612
2.3292 5260.1258
ܐ
38
2 4562/5
2.4578591
65
Mer. Part , Differ. Mer. Parts Differ. Mer. Parts Differ
511-8.4773
5250•4678
2,2418
2.3236
1973.6324
2,4122
5110.800g
5252.8800
2.3250
2.2431
+976'075;
3
2.4137
5113'1259
2-3264 5255.2937
+9783199
24153
5115.4522
2.2458
2.3278
0.5657
5257.7090
3
S177801
2 4168
2.2471
4932.8128
34
2.4184
5120*1093
5262-5442
31
2 248+
2.306
2.4199
5122-439
2.3320
5267.9641
2.2497
37 1937.3109
5124.7719
2.4215
5267.3856
13:
2.3335
381-787.5519
12510
5127.1054
2.4230
5269.8086
2,3349
2.252
3-1991.8142
j 129.4493
2.4246
2.3364
$ 272.2332
39
Ic+9240792.2537
503.7767
2.4261
5274 6593
2.3378
+790.32291 2.2550
1134.1145
2.4277
5277.0870
42 +998.5792
2.2563
2.3292
5136.4537
2.429
5279.5163
43 5600 8368 | 2.2576
138.7914
2.3407
5281 9471
14:
2.3421
4.15003.0958
2.2590
5141'1365
5284.3795
2.4324
2.2603
2.3436
44
45 5005.356
5143.4801
5286.8135
2.4340
16 50076177
2,2616
2.3450
5145.825 :
5289.2490
2.4355
47 5009.8807
2.2530
5148.1716
2.3465
2.4371
5291.6861
2.2643
4&5012.1450
47
2-3479
5150.5195
2.4387
5294.1248
2.2656
48
495014.4106
2.3493
5152,8688
5296.5651
2.440?
55016.6776
2,2670
49
2.3508
5155.2196
2.4419
5299.0070
501 5018.9459
2.2683
2.3522
5157.5718
2,4434
2.2697
$215021,2156
2.3537
5301.4504
51
5159,9255
2.4450
5303.8954
535023.4865 2:2710
2.352
5162.2807
5306.3420
2.7460
53 3
54 50 25 7590
2.2724
51646373
2.3566
2.4482
555028.0327
5308.7902
2.2237
2.3581
5166.9954
5311.2400
55
50 5030.3078
2.2751
5169.3550
2.3596
57 5032.5842
2'2764
5313.6914
2.3611
5171.7161
$316 1444
2 4530
5034.8620 2.277
5174.0786
2.3625
5318.5990
595037.1411
2.2791
5176.4426
2.3540
5321.0552
6) 3029.4216 2.2805
$178.8081
2.3655
5323.5130
M
63
64
05
M
+
2,4308142
19
152
2.449854
2.451450
2.45469?
60
Minutes)
Minutes ou
2
315330896
55335.8260
alu
IO
11546.3623 2.6194/5705.3278
306 A new and corre&t Table of Meridional Parts.
60
07
68
Mer. Parts Differ. Mer. Parts Differ. Mer. Parts Differ
5323.513U
5474.0057
5630.8184
2.1594
2,5602
2.6705
15325.9724
1476.9659
$633.4884
24616
2.9619
2.6724
2 5328.4334
5636.1613
479.1278
2 4626
2-5637
2.6743
5481.6915
5638.8356
3
2.5654
2.4642
2.6762
415333.3602
34842569
5641.5118
4
2 4058
2-5672
2.6781
1486 8241
5644.1899
2.4075
2.569
2.6800
65338.2935
5489.3931
5646.8699
2.4691
2,5708
2.6820
715340 7626
5491.9639
5649.5519
7
2.4707
2.5725
2.6840
8:53+3.2333
5494 5364
5652.2359
8
2.4723
2.5743
2 6859
95345.7050
1497:1107
$654.9218
2 5761
9
2.4740
1.53491796
2.6879
54 9.6868
5657.6097
2 4750
2.5779
2.6898
U1153506552
902,2647
5660.2995
2.5797
2.6918
125353.1324
5564.8444
294788
25814
1311355.6112
5507.428
5665.6850
2.5832
4:358.0917
510.0050
5668,3807
2.4821
2.5850
55:00.5738
5512.5940
2,4838
2,5468
1615363.0576
5515.1808
2.4854
5673.7780
6
2.5886
: 2.7016
1715305 5430
5676.4796
5517 769
2..4871
2.7036
18.5368.0301
5520.3598
2.4887
5679.1832
18
25922
195370 5188
522.5 520
56818887
2.4904
2.5940
2015373.0092
5525.5410
5684.5962
2.4920
2.5958
2115375.5012
5528.1418
5687.3057
21
2.4936
2.5976
2.7114
225377.9948
5530.7394
5690.0171
22
2.4953
2.5994
1315380.4901
5533.3388
5692.7305
2.4970
2.6012
2415 82,9871
2,49865535.9400 2,6031
5695.4459
24
15385 4857
5538.5431
5698.1633
25
2 5003
2.0049
265387.9860
5541.1480
2,6067
5700.8828
2.5020
27.5 390.4880
5543-7547 2.6086
5703.6043
27
2.5037
205392.9907
2.5054
217256
295395.4971
$709 0534
29
2 5071
15 398 0042
5551.5866
57117811
M
66
68
M
24772
662 9913
12
2.6937
13
2.6957
4
24005
5671.0784) 2.6377
2.6996
117
2.5904
19
2.7055
2.7075
2.7095
lio
2.7134
23
2.7154
2.7174
2:7195
2.7215
2.7235128
26
3548.9737 2.6123
2.7276 30
67
..
Minuteslimli
inues in
27314
55620533 2.6214
633
5567.2979 2.6251
$ 69,92 31 2.6270
2.73762
2.739
2.7416
2.7473
2.7457139
2.6307
15315456.133411
A new and correct Table of Meridional Parts. 307
66
67
68
Mer. Part Differ. Mer. Parts Differ. Mer. Parts Differ
3C1398.0942
5:51.5860
571117810
31
2.5087
2.6141
2.7296
371.5 100.5127
5554.2001
3
5714.5 100
2.510
2.6:59
3:15403.0233
5556.0100
57 17 2422
21 20
2.6177
273369
3215405 $353
5559.433)
2.5137
57199758
2.6196
27356
5408:0491
+
34
5722.7114
2.5154
35 5410-5644
564.6747
5727.4490
?
2.5171
2 6232
30 5413.0815
5728.1886
130
2,5188
5415:500}
5730 9302
3
2.5205
3815418.1205
2.5222
5572-5500 26288 573 3.6739
395420 6430
1575.78
5736,4195
2.523S
2.7177
40 5423.166
3577.8095
57398673
2.,250
2.6726
2.7498
41 5425*6925
1580.742:
57119171
$'
2.5273
4215+23.0195
5583.0765
2.7518
5741.6589
42
2.5290
2,7539
435430.7488
55857120
4
2-5307
14/5433.2795
580.3510
2 7560
2.5324
5750.1788
455435.8119
2.758
5520.9910
5752.9364
2.760
4015438.3460 25341
5553.6329
76
5755 6970
2.5359
47 5440.8819
2.7622
5596.2767
5758.4592
4815443,4195
2.5376
5598.9224
5761,2235
48
2 539
4915 4+5.9588
50015699
5763.9899
545448.4999
2,5411
5604 2193
5766 7584
2.7685
5
55451.0427
2.5428
5606 8706
2.6513
5769.5289
$215 453.5872
2.1445
2.6532
5609.5238
5772.3015
2'5461
5612,1789
2'5480
545458.6814
5775.0762
$614,8359
5777.8530
2.7768
5515461.2311
54
2'5497
2.6589
5617.4948
5680.6319
So 5463.7825
25514
5620'1557
5783.4129
2.7810
57 5466.3357
2'5532
2.6628
5322:8185
5786.1959
57
5815468.8906
2.5549
56254832
5788.9810
5?
5915471.4473
2:5567
2.6666.
5628.1498
605474,0057
2.5584
5791.7082
2.6686
5630.81841 5794.5575
M
66
67
68
M
2.6344
2.5363
2,6382 5747.4228
2.5400
2-6419
2.5438
2.6457
2.6.475
2.6494
47
2.7643
2.7664
49
ܐܝ
2 7705151
2.7726
2.7747
52
2.6551
2.6570
53
27789155
2.6609
SC
2.7830
2.7851
2.6647
2.7872159
2.789316
308 A new and correct Table of Meridional Parts.
09
71
70
Mer. Parts.
Differ. Mer. Parts. Differ. Mer. Parts | Differ.
Minutes !
Minutes ºn
3.0729
1
2
2.797915977.6319
mtr
2 806415 989 3834
1
2.81496001.1730
3 101712
3.1043 13
14
15
5965.9179
6145.7012
015794:5575
2.7914
2.9 250
5968:8429
6148.7741
15797-3489
2.7936
2.9273
3.0755
16151.8496
215 800.1425
5971-7702
2.7957
2.9297
3.0781
315802.9382
5974 6999
6154.9277
3
2.932016158.0084
3'0807
4
4,5805-7361
2.8000
3 9343
3.0833
6161.0917
5'5808 5361
5980 5662
2.8021
2936716164.1776
3.0859
65811.3382 5983-5029
2 8043
2.9391
2.0885
6167.2665
5986.4420
75814.1425
7
2.9414
3 09118
85816.9489
61703572
2.8085
2.9438
3'0937
9 5819.7574
9
5992 3272
6173.4509
2.8109
2:9462
6
3'0964
5995 2734
10
176 5473
105822.5681
2.8128
2 9486
3.0990
5998 2220
15825.3809
6179.6463
2 95 TO
125828.1958
6182.7480
2.81716004.1263
2.9533
6185 8523
1315831.0129
2 81936007.0820
2.9557
61889593
3.1070
3 45833.8322
282146010 0401
2.9581
6192.0699
3:1097
1558366536
2.8236 6013 0006
2.9605
3.1123116
165839.4772
6195 1813
2.825860159635
3.1150
17
1715842 3030
2 82806018.9288
209653
1815845.13.10
3.1179118
6201.4140
2.83026027.8965
3.1204
19
195847 9612
2.9702
3.1231
2058507936_2.83246024-8667
6207.6575
2.9726
3.1258
6210 7833
2515853.62812 83456027.8393
63
2.9750
3.1285
2215836.4648 2 8367160308143
62139118
2.83896033 7917
2.9774
6217.0430
3.1312
2315859,3037
23
2.9798
245862:1448 2.84116036.7715
6220.1769
2.843316039-7538
2 9823
3.1365
2515864,9881
6223:3134
25
2.84556042.7385
2.9847
3.139226
265867.8336
6226-4526
2-8477|6045.7256
29871
3.141927
2715 870.6813
2 9896
6229 5945
2.84996048.7152
3.1447128
28 5873.5312
5232.7392
2852 116051.7073
2.9921
3.147429
295876,3833
6235.8866
3015879.2377
2.9945
2.85446054-018
M
M 69
70
71.
2 96296198.2963
2:967766204.5344
20
21
22
31339124
6239 0368/ 3.150 lid
l New and Correet Table of Meridional Parts. 309
70
71
Minutes or
69
Mer. Parts Differ. Mer. Parts Differ. Mer. Parts Differ
Minuteslim!
60;7.0988) 2.9970
3:1529
31557
31
3.1584/32
3 0068
2.86556069,7114
2.8677
3.0092
30118
2.874510081.7639
3.0168
38
39
40
3:02:9
3.1612
33
3.1639
341.
35
3.1667
3 1695
36
3.1722
37
31750
3.1778
3.1 E06
3 1834/41
3
1890
43
3.1918
45
3.1946
44
3.1974
3.2003
3.2031
2
2.8881
3011879.2377 6054-7018 6239.0355
2.8565
2015882.094
6242.1897
2.8588
6065.6982
2.9995
321;884.9531
6245,2454
2.8610
2315887.8141
3:0019 6248 503)
5053.7002
2 3532
3.00446251.6652
24 ; 890.6773
6066 7946
35 5893.5428
62 54.8289
3615896.4108
1072.7207
6257.9956
2.8700
27899.2805
075.7325
62611105!
2.8722
3.0143
181192.1527
1078.7468
62
264.3373
6267.512 1
3915905.027?
2.8769
3.019+
4105907.9039
6084.-3.79
6270.6901
2.8794
+5910.7829
6087.8049
6273.8707
2.8812
3..24+ 6277.054
+215913.6641 6090 8293
2.8835
3'0269
135915 5476
6280*2403
6093 8562
2-8858
3 0294
145919.4334
6096.8856
5283.4293
3.0319 0286.62 11
+559?2,4215
6009 9171
2 8901
3.0344
165925.2118
102.9519
5289.8157
2.8926
3 qgts
+715928.1944
6105.9888
5293.0131
2.8949
3,019-6296.2134
+85930 9993
6109.0282
3.04!!
1915933.865
6299.4165
2:89956115.1106
30145
io 1936.7960
6302.6225
2.9018
3.0471
51939.6978
6081617
6305.8313
2.9011
3.0496
52 5942.6019
6 1 21.2113
6309.043"
2.9064
3.2522
i's 5945,5083
1124.2635
6312,2575
2.9087
30548
1545948.4170
6 127,3183
6:15.4749
2.9110
3.05741 6318.6951
5511951.3280
61203757
3.05.99
;65954.2413
63219182
2.9156
3 0525
5711957.156
0136.4981
5325.144
5815960.0749
2918c
30051
61395032
5328.3729
5911962.992
2.9203
3.0677
6142 6309
6 11965.0 1791
29227
3.0702
5145.7012
M
69
7
Ss
44
2.89725112.0701
48
49
3.2060
50
3.2038
50
32117
52
3.2143
53
3.2174
54
3.2 202
3 2231
3.2259
3.228
55
2.9133.6133.435€
56
57
1
58
6331.6040 3.2317
6324.81923.2346
59.
60
M
70
Minutes
Minutes
3 42196749 3731
3. 24° +0571 2703
6753 0060 3.6335
2
3 246 26548 1304
3.249265515654
3.25216555.0037
-3 43176760 2847
3:43505763.9293
-)
67675776 36183
326096565.3385
3.26326568.7901
3428416756 6438
86360.8209/ 3.25806561 8902
3.484916819:0490
3 505416841.3377
3:512416848.7980
3:5159 16852.5340
310 A new and correct Table of Meridional Parts,
72
73
74
Mer. Parts Differ Mer. Parts. Differ. Mer. Parts | Differ.
06334 8392
6534.4233
6745.7433
16338,0767
3.2375
6537 8452
3 6298
216341 3171
3.4251
3
46347.8066
3:6409
4
5 625 1 0558
3 6440
66354.3979
3-43-83
3
76357.56 29 3:255065584453 3.4416
6771.2296
3.6520
3 4449
7
6774 8854
3.65581
8
9,6364,0818
3 4483
6778
3.6595
9
10,63673457
6782.2082_3.6633
116370.6125
36670
12,6373-8823 6575 7032
6789 5468 3.6708
13,6377.1550
3.6745
14,6380-4307 3 27576582 6296 3.4649
6796.898836783
14
1562837094
6800.5809
266386.9910
3.2816
589.5693
3 4715
17,6390 2756
8,6393 5632
3
3.4782
3.5936
18
19,6396.8538
3.2906
3.4816
6400.1474
37013
21,6403.4440
3 1883
6822.7542
3.7052
226406 7436 3 2996,06:04689
3 2996,06:04689 3 49"716826.4632 3
23,6410.0463
23
246413.35201 3.3057,6617-4626 3.4986
6833 8927 3.7167
24
25,6416.6607
3.5030
25
265419.9725
276423.2873
3.5089
5845.06.59
3 7282
27
286426.6052 | 3'31796631.4913
296429.9265 3 32096635.0072
3 7360
29
300433:2501| 3*324046638.5266
5856.27391 3-7399
30
7
74
74
M
3 2668
3:451685778 5449
3.269890572 2450
32 3:4582
3.45496785.8752
3:46151679.3 2205
3:2727,6579.1647
13
3 27876586.0978
3.4682
3.682 1
6804:2668 36859
3.689811
16
18
3 2846 6593-0412
6600.0049
3.2936, 6603 4889
6606.9772
3.474916807.9566
6811.6502
6815.3.477
3.6975119
20
20
3 2966
21
22
3.3027,6613
9640
3:495116830 1760
632 3 7090
3.7128
6837.6133
372 16
3:30876620 9646
3.31186624 4700
3.314866279789
3724420
35194|5856.2732
A New and Correct Table of Meridional Parts. 311
74
Minutes and
Minutes non
3.5228
3.5262
3*7478/31
3.751832
3.5356
3.330116645.57561
16939.5628) 382841
72
2
Mer. Parts. Differ. Mer. Parts Differ. Mer. Parts Differ .
3015433.2501
6638.5266
6856.2739
3.3270
3016436.5771
1672.049+
68600178
37439
3216439.9072
6863.7656
3.3332
6679.1053
333443.2404
3.5297
5867.5174
3.533110871.2731
3.3363
33
37557
3416446.5767
66 52 6384
34
3.3394
353449.9161
6656.1750 6875.0328
37597
35
3.3425
3.5400
3615453.2580
3.7637
6659.7150
6878.7965
3.5435 6882.5642
36
3.3456
37677
:76 +56.6042 6663.2585
3.3487
3.5470
3.816459.9529
37
3 7717
6666.8055
6886.3359
8
3.351
3916463.3047
3.5505
3º7757
6670-3500
5890.1116
1016466.6596
33549
3:5541
6673.9101
6893.8914
1715470.0175
3.3580
6677.4677
3-5576
6897.6752
1216473.3788
3.3612
6681.0288
3.5611
6901'4630
+3164767431 3.36436684.5935 35647
+46480,1166 3.36756688.1617
6909.0506
1516483.4812
3.3706
6691.7335
6912.8504
45
3.5754
106486.8550 3.3738
6695.3089
6916.6542
47/6490.23 20
3.3770
3.5790
6698*8879
6920.4621
3.8079
3:5826
+816493.6 121/ 3.3801
47
3.8119
6702.4705
6924.2740
+96496.9954
3.3833
3.5862
6706.0567
6928.0900
500 500.3819
3.3865
35898
6709.6465
6931.9101
50
5116503.7710
3.3897
3.5934
6713 2399
6935.7344
3.8243
525507.1645
3.3929
3.5970
51
6716.8369
3.6006
533510956063.3961
52
6720.4375
6943.3953
3.8325
541651389599
3.3993
3 6342
6724,0417
6947.2320
55155173624
3.4025
6727.6495
3.6078
69510728
55
565520.7681
3.4057
3.8449
6954.9177
5716524.1770
3.4089
3.6151
6734.8761
6958.7668
58
5527.5892
3-4122
6738 4948
57
3.6187
3.8532
6962.6200
5916531.0046
3.4154
6742,1172
3.6224
3.8574
3 4187
6745.7433
6970.3390
60
M
72
73
74
SS 2
38
3.7798/39
3.7838
40
37878 41
0905.2548 3.7918 42
3.7958/4?
37998/44
3.8038
46
3 5682
3.5718
3.816648
3.8201 49
3 8367 53
3.840854
6731,2610/ 3.6115
3.8491 56
58
60165 34.4.2331
3.62615966.4774
3.861659
ki
312
A new and correct Table of Meridional Parts,
75
tinutes
70
77
Mer. Parts Differ. Mer. Parts. Differ. Mer. Parts Differ.
Minutes 1
I
16974 2048 3.8658
17218 3454
2
278417226 6413
7485 0309| 4.4650
7493978; | 44765
4.4822
76997-4882 3.8912
4.180
3 7017.0082 3:9126
7255 8325 4.18:0
12
3:91697264.2174
4.5166
13
4.5224 141
45282
970-3390
7210.0688
7467.2048
4 1359
17214 2047
7471 6529
4 4481
269482748 3 8700
4.1407
7476 10661 4.4537
3.8742
316981.94 o
7222 4909
4 4 1455
7489 5659
4 4593
3
4,6985 8274
3
4 1504
4
4
6989.7101
3.8827
7230 7957
4.1554
7489.5016-44707
66993,5970
38869
72349570
4.1603
7239.1 222
4.1652
7498.4003
87001.3830 3.89547243 2923
4:1701
7502 9483
4 4880
97005,2833/ 389977247.4674 4.175
7507 4420
4 4937
1070091873_3.90407251.6475 7511.9414
4.1994
10
117013.0956 3 90837255 8325
4.5052
7260 0225
4
7520 9575
45109 12
13 70209151
4:19+9
7525 4741
147024.84631 322127208 4173
4.1999
7529 9965
42048
1312028.7718_3.9255
7272-6221
7534 5247
IŞ
167032.7017 3.92997276.8318142097
7539 0587 45:40 16
1770366359 3 2342,7281 0465 4:21
4
7543.5985
4.5398
18
7548.1441
4.545818
97044 5174
3.94297289.497 4.2246
7552.6956
4.5515
19
19 7048
46 47
4 2297
7557.2529 45573 20
217052.4164
4 2347
7561.8161 4.5632 27
2217056 3724
7566-3852
237060.3328, 3696047306-4398 4.2449
7570.9603
2417064 2977
3.9649731068991.42501
4581
17575 5414
24
2517068.2671
4.2553
7580 12851_45871
25
9797.2409/ 3.9738
7319 2057
4.2003
7584 7217
277076.2192) 3:9783
7323.4714
7589.3209
4'5992
27
87080.2020 3. 828
7327 7423
4 2704
7593.9261
605
297084.1893 3 9873
7332 0 1 84
4.2761
5985374
46112
30/70881811/ 3.9918
3991813 36.2996 4 2812
7603.1547!
75
76
17
3 9386
7285 2601 4.2190
3.9473729367204
395177297.255
3 95607302 1949
7302 1949 42398
4 5691 22
4.5751 23
3.96947314 9452
4593226
4.265
329
.617:30
77
A new. and correct Table of Meridional Parts.
3.13
75
76
77
Mer. Part Differ. Mer. Parts Differ. Mer. Parts Differ
Minutesi 1
M. CS
4.29167007.7780
4.6233
3
4.629.
4635
3
401! 5
6.
5
3
54
4 0476
4.0600
*
3
36
31
666:
4,"72434
f
C
4
437140.4852 4.0508
188 1811
7396.2990
760301547
3 9963
4.2864
7092.1774
31
7340.586
4.000€
7086.1782
3
7344.8774
7612.4073
7100.1835
4.0053
4.2967
3.
23-79,1745
7617 6427
4 0090
4.2019
7104.1933
7353476
7621.6842
108.2070
4.0143
1,3072
3:
7357.703
7626.3318
4.0iod
4.3124
30711 2.2264
7302:09,8
7630.985€
27116.2497
4.023
7266 413
4.3176
76156456
187120.2775
4.0272
4.3229
7370.7363
7640.3113
03241
31.124.2094
4.3281
73750644
7544.9842
128.340%
4,0369
4.3334
379.1974
7549 6625
4.1413
4.3387
+132.3883
7383.7365
7654.34:8
427136.4344
4.0461
7388 0806
4.3441
70590289
4 3494
7392.420
7663.7362
7144 5407
40555
4'3548
7396.784
7768.439,8
4'3603
7401145
7673:1497
4012152.9658
.
4 0644
4.3657
7405510
767.005
4.c656
471?156.7354
4.712
144-9.8820
7682.588..
j7160.8-97
40743
7414258
7687.3172
4.0790
4.3821
441164.8887
7418-6407
7692.6524
Ś 17168 9724
4.837
4.3876
7423.C 283
7646.7940
5 7173.0608
40884
4.3930
7427.4213
7:01 5424
27177 1538
4.093
7431.8198
4.3985
7706.2965
5:
4.0978
7436.2227
4.4039
77110574
5:11857542
4 1025
1.4094
7440.6331
7715.8247
55°
4.14.7"
74450179
44148
7720.:98
507193.5733
4.1117
4.4299
7449*.602
7725.3788
4.1167
$77:97.69.00
7453.8941
4 4258
7730.1656
5:201.8115
4.1215
45*.3254
205.9378
4 1 26
7462.7622
4:4369
7739.7588
5
7210. 6881 4.1210
4.4425
7457.204
7744 5652
M
75
: 70
77
4.6;8
4.08.9
4.5911
4.64731
4.736
4.099
4.7162
1.7225
4.72881
4€
4:7352
+52148 nCOo_46.602
4.3760
47410149
5
4 7480
51
+7515
52
4 70095
7181;2517
189.4514
Si
4.7073
5
4.7738
S
4.7803
47060
4.7933
15
4.42147734.959
4.70995
4 06+
M
Minutes)
Minutes)
8380.9666 5.7630
1
2
4.8263 8061 4629
4.83293066.7314
8072, 077
5.811517
4,852918082.5843
4.859618087.8846
4 87313098.5095
nagi-
.
314 A new and corre&t 'Table of Meridional Parts.
78
80
Mer. Parts
Differ. Mer. Parts Differ. Mer. Parts Differ
017744.5652
1945.7049
8375.1970
4.8130
5 2149
3050.9498
17749.3782
4.8196
૪
217754.197
5.2526
3056.2024
5.7731
8386.7337
5.2005
317759.0241
5.7827
8392.5164
3
52685
4177638570
5.7922
8398.3086
4
4,8396
5.2763
5.812
5
7768.6966
8704.1103
Ş
617773.5428
4 8462
8077.2920
5.28.43
8709.9218
5.2923
717778,3957
8415.7430
58212
7
$:3003
8!7783.2553
8421.5739
5.8309
8
4.8663
3093.1930
5.3084
97788.1216
5.8407
8427.4146
5.3165
107-92.9947
8433.2651
5.8505
4 8798
107797 8745
5.3246
8103.8341
5.8603
8439.1254
4.8857
5.3327
127802-7012
5.8701
8199. 1608
8444.9953
12
5.3497
5 880
8450.8755
4.9304
1478125550 8119.8565
5.3490
8.456.7055
5.8900
4
4.9072
8125.2136
5.3571
8452-6656
5.900
5
5.3653
8130.5789
8468.5755
5.9099
5.3726
1717827.2970
5.9199
8135.9525
8174.4954
7
4.9279
5,3818
187832.2249
5.9301
8141.3343
8480.4255
4.9347
5.3901
5.9402
19/7837.1566
8146.7244
8486.3657
19
4'9418
5.3986
5.9503
207842,1014
8152,1239
8492.3160
4.9487
5.4069
217847.0501
5.9605
8498.2765
8157.5299
5.4152
5.9707
2217852.0058
8504.2472
4.9626
8163.3687
5.4230
137856.9684
5.9809
8510.2281
23
4.9697
5.4319
5.9912
24 786 1.9381 8173.8000 8516.2193
4 9767
5.4404
1517866 9148
8179.2410 8522,2207
4.9839
5.4489
267871.8987
8184.6899
8528.2326
26
4.9909
5.4574
8190.1473
27/7876.8896
8534.2549
27
4,9980
5.4659
287881.8876
8195.6132 8540.2876
28
5.0050
297886.8926
5.4745
8201.0877
8546.3306
29
S0122
5.4832
3017891.9048
8206.5709
8552.3841
3.0
M
8
79
80
M
48934181145075
13
37807.6546 48934
157817.4622
16|782 2.3762 49140
4 9208
6
18
21
4.955718162.9451
22
2.4
25
6.0014
6.0119
6.0223
6.0327
6'0430
6.0535
A new and correct Table of Meridional Parts.
315
78
79
Mer. Part, Differ. Mer. Parts Difer.
80
Mer. Parts Differ
Minuteslim
Minutes and
.60640
8558 448 60745
6.0853
8570 6079 6.09611
6.166834
38
35
3617922,1298
337927.1925
38:932 2630 5.0702
3:937.3406
10
4 1²947 5178
4717952.6175
BS
6.182614
6.1935.4:
5 11478284.251
6.2015
457467.9614
407973.0910
7891 9018
8205.5709
8552.384
30
5.019418212.062
5.4913
7896.9274
;
31
5.0265
S.9004
7901'9507
8217.5631
8564.522
32
3
5.933918223,0721
5.5090
3
7906.9346
3
50040 8228.5898
5.5177
791 2.0257
34
8576.704
5.5264
5.0485182
8234.1152
8582 8108
'917.9742
35
6,1173
5.5352
5.05; 8
8239.6514
8588 928
36
5063918245.1957
5.5439
6.1279
8595.056
13?
5.070218250.747
5.5527
61389
8601.1944
5.5516
614989
5.07768256*309€
8607:344
5.08498261.8802
6.1608)
5.5.706
79424.255
8613 505
45
5.092318257.7596
6.1715
5.5794
8619 6771
5.099718273.0479
5:5883
8615.8597
55773
4
#311957.72401 5.1078273.615
8632,253
5.6062
7952.8393
8638.2577) 6.215
ti
5.12218239.856€
5.6152
8644.473
45
5.12961827 7 4910
5.624+
6 2267
8650.699.1
46
5.6334
6.2379
4711978 2280 13708301.1244
8656.937
1
514478306.7658
55424
6.2491
4*17783.3727
8663.1863
S152
6,2502
5.6516
3312.4184
8569 477
41988.5219
45
62715
5.1578
gor 67
511993.68 +7
8318.0791
8575.7186
6.2828
51
5
5.1757
5.6791
3329.428:
8688.29551
S100.4.0272
52
3
6.3056
533009 20981 5.182.183351164
8694 601
53
5.6977
6.3171
54 3014 4002
8700.9182
9.7971
8346.5211
20175032
8707,2468
55
.3399
5.2058
$352,2374
500024 8040
8713.5767
Se
5.7257
6.3515
5213 3:57.9531
5713030.0174
87:99382
5
5.7352
.63632
5813035.238.5| 5,2 213 8363.698;
8726 3014
5.7446
6.375
52291
593040.4578
3369.4+29
8732 6764
6013045 7049
5.237118.71.1970
5.7541
8739.0632
79
IM
go
*
48
156
5167+1332367490
5.6699 86820013
6 2972
1
5.1914 8347.8!4
51930
•6.328054
5 7162
5
63860166
M
78.
316 A new and corre&t Table of Meridional Parts.
81
Minutes)
Minutes 01
8.2152
1
6.410211159.8593
a
2
8758.29371 6,4338
48764-7275| 6.4+50186-5725
6 4577 10183.84 6
78784.1005 6.4817
marlo
7 2081
8254
8.- 742
8 294
7.2529196 +7.076.0
831396
83339
8.3540
8.3742
7.31399680.4660
8.294+
7.327219188.8104
7.28249653.7378
8?
Mer. Parts Differ. Mer. Parts Differ. Mer. Parts Differ
0739.C 632
1145.4588
4588
9605.8175
6.3984
7 1928
8 745 1016
1
7152.6516
9614.0327
7.277
8.2347
218751.8718
9522 2674
6.4219
7.2227
2167.032
630.5214
3
7174'3190
7 2376
.638.799
S
8771,1731
618777 6308
76;5:08
6,4597
1196.124
7
7.2987
818790.5822
8
9293.4227
25 +2,0918
6.4935
918797.0761
6506c
2210.7360
1088c3 $821
92180659
65183
7.3444
84149
2225107
9097.2753
7.36.4
9232 7711
9705-7108
7.3762
8.458C
138823,1736
0240.1473
9714.1668
13
7.3920
8.4760
14 8829.7286
2247.5393
2722 6434
4
6.5673
7.4079
158836 2959
9254.9472
9731'1408
0.5797
7.4235
8.518 :
108842.8756
6
9262.3707
9139659
6.5922
7.4395
8.5394
78849.4678
9748.1985
9269.8102
117
6.6048
7.4554
8.; 6.,6
18,8856.0726
9756.759
18
66174
7,4715
9284.7371
9765.3408
7.1876
08869.3202
9292.2247
7773.9440
0.6431
7.5038
8,6242
218875 9633
9299 7285
9782.5682
6.6552
7.52:12
8.6466
228882.6185
22
9307.2487
791.21 42
6,568
7.5365
8.6678
23
38889.2865
6.680
93 14.7852
7.5527
8.6896
+8895.2674
9:22.3379
9808.5?16
6 6938
7.5603
8402.6612
8.7116
9229.90,77
981? 2832
25
6.7068
7.5879
268909 3080
9.37.4921
9320 0/65
67198
7 6025
78916.0878
.
9334.772
9345 0956
67327
7.619
810922 8205
9252.714
9843.5497
6,743
7.6360
9360.250
9852.3496
6.758
7.6529
08930 3250
30
9.68.927
851 170
82
.82
M
II
18810.10041 6.5?05
ial: 816.6'09165427
6.5550
8.4355112
8.4974
9277 2656
98862.690916 6302
8.581719
8.6032120
21
9799 8820
24
C
8.73?3126
87555127
877771:8
8.799929
8 8:22:
498929.5662
كم
81
***
A new and corre&t Table of Meridional Parts. 317
Minateslim
Minules
67719
8.8449
88675
6.8119
6.8254
8.9135
8.9364
6.8657
.68793
38
2011.5535 09204
9878,8844 8.8904
81
82
82
Mer. Part Differ. Mer. Parts Differ. Mer. Parts Differ
8936.3250
9368.0037
9861.1720
30
7.6696
8943 0969
9375 6733
6.7852
31
9870 0109
8949.8821
7.6867
32
9383'3600
32
6.7986
7.7039
8956.6807
B
9391.0639
9887.7748
7,7211
34
8963.4926 9398.7850
7.7383
9896,6883
34
35
8970.3180
9406.5233
990.6247
is
6.8387
7.7557
8.9596
3018977.1567
9414 2790
6.8522
30
9914.5843
7.7730
8984:0089
8.9827
32
9422.0520
37
9923.5674
7.7903
9.0062
388990.8746
9429.8423
7.80799932.5732
9.0297
393997.7539
6.8930
943 7.6502
139
7.8256
9941.602
9.0532
409004.6469
9445.4758
9950.656
40
6.9066
7.8434
9.0769
41
9453.3192
7.8610
99567131
H!
9.1010
1421 9018.4739
9461.1882
9968.834
798789
6.9341
42
9.1249
439025.4080
6.9478
9469.0591
7.8970
2977.958
45
9.1491
44 90323558
9476.9561
9987.1080
+4
7.9149
9'1734
4519039.3183
9484.8710
9996.2814
is
6.9764
40 9046.2947
7.9329
9.1977
9492.8039
10005.4791
76
7.9512
47 9053.2850
9.2224
9500.7551
7.004:
10014.7015
47
489060.2891
9.2470
7.9696
9508.7247 10023.9485
7,0180
49|9067.3071
7.9881
9516.7128
9.2717
7.0321
10033.2202
19
9.2966
5C|9074.3392
9524.7192
10042.5168
50
51 9081.3856
7.0464
8.0248
9.3217
9532.7440
7.0610
S1
10051.8385
8.0435
9.3469
52|9088.4466
9540.7875
10061.1854
8.0623
7.8756
52
539095.5222
9.3723
7.0901
9548,8498
8,0812
10070.5577
549102.6123
9.3978
7.1046
9556.9310
8.1001
10079.9555
54
55 9109.7169
9.4233
9505.0311
55
I56 9116,8359
7.1190
9.4491
10098.8279
5719123.9695
8.1381
9.4750
10108.3029
9581.2883
7,1483
58191311178
57
9.501C
9589.4455
8.1764
10117.8039
9,5273
9597.62:9
IOI 27.3312
59
69 2145.4588) 7.1780
9605.8175
10136.8847
9.5535
60
M
81
82
M
TO
(
C
6.9625
6.9903
48
8.0064
53
8.1191
10089.3788|_9.4233
7.133649573.1502
56
8.1572
SE
59/9138,2808 7.1638
8.1956
83
318 A New and Corret Table of Meridional Parts
84
Mer. Parts. | Differ.
05
Merid. Pes.
Differ.
Minutes:0
Minutes! oi
1
95801
9.6068
I
2
і
on tupo nso a
1
I 2
10136.88 7
10146.4648
210156*0716
10165.7049
3
9.6333
9.6604
4 10175.3653
10185.0;27
9.6874
610194 7073
9.7146
7 10204 5093
9.7420
810214.2789
9.7696
910224.0762
9.797.3
10233.9014
9.8252
1110243.7546
9.8532
9.8813
12 102 3.6 359
13 10263.5456
9.9097
14 | 10273.4838
99382
1510283 4506
9.9668
16 | 10293 4463
9.9957
17
10303.4710 10,0247
1810313.5246
10.0536
19 10323.6079
10.1128
20 10333-7207
10.1425
10343 8632
2210354.0350
23 | 10364.2379
24 10374.4703
10.2324
10.2628
25 | 10384.7331
26 10395.0265 10.2934
27 10405.3507
28 104'5.7061 10.3554
29 | 10726.0924
10.3863
30 10426.5102 | 10.4778
M
84
10:64.6210
:
1776.1139
51.4929
10787 0151
11.5312
10799 3150 11.5699
10810 823
8
I 1.6088
1082 2 4.718
11.6480
11'6875
10834.1593
11,7272
10845.8865
11.7672
10857.6537
I,8074
10869.4611
10881 2091
11.8480
11.8889
10393 1982
11.9299
10905 I 279
11.97.13
10917.0992
12.0131
10929.11 23
1 2.0550
10941.1673
12.0973
10953 2646
1965.4044
I 2.1398
I 2.1828
10977.5872
10989.813
12.2259
11002.c825
1 2.2694
I 2.3132
I1014 3957
11026.7530
12.3573
12.4018
[1039.1548
12.4465
11051,6013
12.49,16
I1064.0929
12.5370
11076.6299
U1089.2126
12.5827
12.6288
(1101.8414
12.6753
I1114 5167
12,7220
11127.2387
13
14
15
16
17
18
19
20
10.0833
21
21
10.1724
10.2023
22
23
24
25
$0.3242
26
27
28
29
30
M
85
a New and Correct Table of Meridional Parts. 319
l
84
Mer. Parts.
85
Mer. Parts | Differ.
Differ.
Minutes la
Minutes! '
www. w ****ia. :)
49
3010436.5102
31 10440.9595
32 10457.4404
33 10467.9534
34 10478.4984
3510489 0754
36 | 10499.6851
37 105 10.3275
3810521.0027
39 10531.7113
40 10542.4531
41 10553.2 284
42 10564.0374
43 10574.8804
44 10585,7575
4510596.6690
46 10607.6151
47 10618.5960
48 10629.6121
49 10640.6632
5016651.7499
5110062.8723
52 10674.0305
53 10685,2251
54 10696.4563
551070707241
56
10719.0290
57
10730.3708
58
10741.7499
59
10753.1666
60
10764.5210
M м
84
-10 4493
10.4809
13.5130
10'5450
10.5770
10.6097
10.6424
10,6752
10.7086
10.7418
10.7753
10.8090
10,8430
10.8771
10.9115
10.9461
10.9809
I 1.0161
I1.0511
11.0867
II,1224
11.1582
11.1946
11.2312
11.2678
11.3049
IJ.3418
11.3791
11.4167
11.4544
11127.2387
30
11140.0078 12.7691
I1152.8243
12.8165 31
10165.6887
12.8644 32
11178.0013
12.91 26 33
III91.5624
12.96134
35
11201.5724
13.0100
36
13.0593
IT 217.6317
11230.7406
13.1089 37
11243.8995
13.1589 38
13.209+
39
11257.1082
11270.3691
13.2602
41
11283.6865 13:3114
11297.0435
13.3630 42
42
13:4150
11310.4585
44
45
11337-4460
13.5202
40
13.5735
11351.0195
47
13.6271
11364.6466
11378.3278
13.6812
49
11392 0635 | 13.7357
11405.8542
13.7907
51
13.8460
11419.70C2
52
11433.6021
13.9019
53
13.9583
11447-5604
11461.5753
14.0149
14.0722
55
11475.6475
11489.7774
14.1299
11903 9655
I 4.2467
11518.21 22
59
60
85
11323.9258 13.4673
********""""""...."):
48
m'''.'............
$o
54
56
14:1881 57
1832.51801 14.3058
TI 2
320 A new and correčt 'Table of Meridional Parts.
86
8
Mer. Parts
Differ.
Mer. Parts
Differ.
Minutesi º 11
Minutes 1
I
1
alut
marjo
7
8
9
IO
II
12
13
14
I12.5 180
11546.8835
14.3055
14 4256
11561.3.191
2
11575.7954
14.4863
3
14.474
U590.3428
11604.9119
14.6091
611619.6232
14 6712
14,7341
11634-3573
7
811649,1546
14.7973
11664 0157
14.8611
9
11678,9412
149255
IO
IL
1 1693 9316 14 9904
1211708.9876 15.0560
15.1220
13
11724.1996
14 11739.2983 15.1887
1511754.5543
15.2560
16 11769.8781 | 15.3238
15.3923
17
11785.2704
18 1800-7318
15.4614
15:5316
19
11816.2628
20 11831.8642 156014
! 5.6724
21 11847.5366
11863.2807
IS.7441
22
23
15.8892
2411894.9862
IS 9629
25 11910.9491
16.0373
26 119 20.9864
16,1122
27 11943.0986
16,188
28 11959,2866
29 11975.5510
3Ci 11991 8926
M
86
I 2522,15!2
19.1605
12541.2678
19.2570
12560.5350
19 3763
12579.9119
19.4859
12599:3978
19.5969
12618.9947
19.7090
1 2638.7037
19.8226
126535262
199374
1 2678.7637
20.0535
12698.5172
20.6711
12718.6883
20.290
12738.9783
20.4103
I 27593886
20.5320
12779.9206
12800,5759
20.6553
12821.3558
20.7799
20,9062
1 2842.2620
21.0340
1286 3.2960
21. 1633
12884593
21.2942
129057536
21.4268
1 2927,1804
21 5611
12948.7415
21,6970
1297.).4385
21.8347
12992.2732
21.9741
13014.2473
22.1153
13036.3626
22.2583
13058.6209
22.403
13081 0242
22.5501
13103.5743
22-6988
13126.2731
228496
131491 27
87
17
18
19
21
22
11879.097615.8163
23
24
25
26
27
28
29
30
M
16.2644
16 3419
he
1 New and Correet Table of Meridional Parts.
321
86
87
Mer. Parts Differ.
Mer. Parts,
Differ.
Minutes olim
Minutes! non
16 4194
31
32
33
34
35
36
37
38
39
40
2011991.892)
12008.3120
321 2024.8102
33 I 2041 3877
34 12058.0455
35 | 12074.7842
36 | 12091.6046
3712108.5076
38 12125.4970
3912142.5646
4012159.7202
41 | 12 176.9617
42 12194.2900
43 | 12211'7060
44 12229,2104
45 12246,8043
4612264.4886
47 | 12282.2641
48 12300.1319
4912318.0930
50 12336.1482
51 12354.2985
5212372-5451
5312390,8889
54 12409.3309
55 12427.8723
5612446 3141
57.12465.2573
12484.1032
5912503.5528
60 | 12$22.1073
86
10.4982
16,5775
16.6578
16.7:87
16,8204
16.9030
16.9864
17.0706
17.1550
17.2415
17.3283
17.4160
17.5044
17.5939
17.6843
17.7755
17.8678
17.9611
18.0552
41
13149.1227
+3172.1250 23.0023
13195.221 | 23.1571
13218.5961 23-3140
13242.0691 23.4730
13265.7034
23.6342
13289.5011 23.7977
13313.4546 23.9635
13337.5962 24.1316
13361.8982
24.3020
13386.3739
24.4748
134110232
24.6502
13435.8513
24.8281
13460.8599
25 0086
13486.0516 25.1917
13511.1291
25.3775
13530,9952
25.5661
13562.752725 7575
13588.7045
25.9518
13614.8536
13641 2030
13667.7556
26.5525
13694.5149
26.9690
13721.4839
13754.6661
27.1822
27.3986
26.1491
26.3494
42
42
44
45
40
47
48
49
Şo
50
52
53
54
18:1503
26.7593
18,2465
18.3438
18.4420
18.5414
55
13776.0647) 27.3985
27.6186
18 6418
18.7432
18.8459
18.9496
19.0545
13803.6833
13831.5255
13859.5948
27.8422
28.0693
58 12484.
13887.8951 28.5350
/
56
57
58
59
60
M
13916.4301
87
322 A new and correct Table of Meridional Parts
88
Mer. Parts
19
Mer. Parts
Differ.
Differ.
Minutes -
Minutes
28.7737
2
3
4
3 hloooovalutu
7
8
9
IO
IL
11
12916.439)
1945.2.38
29 016.
2 1 39 74.2202
14402.4835
29.263)
3
29.5144
4
14032 9979
22,7698
5
14062.7677
6 1 4092.7975
300298
214123.0917
30,2942
814153.6551
30.5634
14184 4925
30.8374
14215,6788 31.1163
14247 0093
31.4006
12 14278.6972 31.6898
319845
13
14310.0837
14 |
14342 9684 32.2847
15
14375.5590 32.5936
16 | 144.08.4614
32.9024
17 14441.6817 | 33.220;
18 | 14475 2259
3:5442
19 14509.1005
33.8746
34.2116
20 14543-3121
21 14577.8674
34.5553
22 1461 4.7735
34.906
23 14648.0375
35.2640
2414683.6668
35.6192
36 0023
25 14719.6691
26 147560523
36.3832
27 14792.8245
36.7722
28 148.29.9942
37,1687
29 | 14867.5700
3705750
30
37 9909
14905 5609
M
88
16299.5562
16357.3379
1 6416.1071
16475.8982
16536.7477
16598.6934
166617757
16726.0372
16791.5226
16858.2795
16926.3584
16995.8126
17006 6988
17139.07774
17213.0126
17288.5727
1736;.8508
17444 8651
17525.7590
17508.6023
17693.4914
17780.5294
17869.8284
17961.5089
8055.7014
18152.5474
18252,2009
18354.3295
184.60.6 162
18569.7617
18682.486
57.7816
58 7692
59.7911
60.8495
62.9457
62.0323
6.4.2015
65.4854
66.7569
68 0789
19:4542
70.8862
72,3786
73.9352
75.560
77.2581
79,0343
80.8939
82.8433
84.8891
87.0380
89.2990
91.6805
94.1925
96.8460
99.6535
102.6286
105.7867
109.1455
112.7246
I 2
13
14
15
16
17
18
19
2
21
22
23
24
25
26
27
28
29
30
M
89
A new and corre&t Table of Meridional Parts, 323
89
Minutesla!
Minutes to
38:8493
36
38
160 16299.5563 / 56.8264
88
Mer. Parts. Differ.
Mer. Parts | Difference
30 14905.5609
18682.4863
30
31 | 14943 9762
384153
116.5464
8799.0327
31
32 14982.8255
120.6365
18919 0692
32
39 2932
33 | 15022,1187
12,0244
190+4 1932
33
129 7430
34 15665.8661 | 39.7474
19174 4362
34
134.8321
35 15 102.0783 | 40 2722
19309.2683
35
36 15142 7663
40 6880
19449 0052
140 3369
37 15183.9414
41.1751
19595.915)
146.3103
37
3815225.615643.6742
19 748 7305
152.8150
39 15267 8010 42 1854
19908 6536
159.9251
39
4015310,5 101| 42.7091
20076.3848
1677292
40
4115353-7569 43 2468
202527191
41
4315397-5543
43.7974
20438.5894
42
4315441.9167 44:3624
20635.0864
43
208
44154868589 44.9422
20843.4993
4129
44
22 1.8679
4515532.3963 45.5374
21065.3672
45
46155.78.5448 | 46.1485
21302.5479
237 1807
47 15625.3212 46 7764
21557.3130
254.7651
4815672.742047.4214
275.1672
21832.4802
48.0847
4915720.8273
22131.6038
299.1236
50 15769.5943 48.7670
22459.2565
3276527
50
5115819.0621 49.4673
22821.4598
362:2033
51
52 15869.2524
SO 1903
23226.3685
404.9087
52
53 15920.1862/ 50.9338
23685 4159
459.0474
53
54'5971.8858 516996
24215.3473
529.9314
54
59 160 24.3747
52.4889
24842.1229
55
566077.6773 53.3026
25609.2341
767 III2
57 16131.8192) 54-1419
26598.2 124
988 9783
58 16186.8274 550082
27992.0989
1393.8865
59 16242.7299 55.9025
2382 8645
30374.9634
Infinite.
M
88
89
M
176.3343
185.8703
196 4970
46
47
48
49
626.7750
56
57
58
59
60
3
!
.
ot
Navigation
285
Sect.
XI.
B.
Y the help of the Tables of Meridional Parts, made according
to the Rules laid down and demonſtrated in the former Section,
it will be eaſy to give Solutions to all the Problems propoſed in Mer--
cator's Sailing, and thence to find the true Bearing and Diſtance,
Diference of Latitude and Difference of Longitude, between any
two Places propoſed.
Problem i.
The Latitudes of any two Places being given, 'tis required to
find the Meridional Difference of Latitude between them.
i
Caſe I.
If one place be under the Equator, and the other in North or
South Latitude,
Then the Meridional Parts anſwering to the Latitude propoſed,
will be the Meridional Difference of Latitude.
Example.
Suppoſe two Places, one under the Equator, the other in the La-
titude of so deg. oo min. North, then the Meridional Difference of
Latitude will be found to be
3474.5.
:
Caſe II.
If the Places propoſed are on the ſame ſide of the Equator,
that is, both in North, or both in South Latitude, then the Diffe-
rence of the Meridional Parts anſwering to each Latitude, will be
the Meridional Difference of Latitude.
Example
If one place be in the Latitude of so deg. oo min. North, and
the other in the Latitude of 13 deg. 30 min. North, the Meridional
Difference of Latitude will be found to be 2657.9. For,
From
PP
st
286
Navigation.
From the Meridional Parts anſwering to 509 00'
Take the Meridional Parts anſwering to 13° 30'
3474.5
817.6
Remains the Meridior al Difference of Latitude
2656.9.1
Caſe III.
A
If the places propoſed are on contrary ſides of the Equator, that is,,
if ons be in North Latitude and the ot er in South Latitude. Then,
The Sum of the Meridional Parts anſwering to each Latitude, will
be the Meridional Difference of Latitude,
Example..
If one place be in the Northern Latitude of 13 deg. 30 min. the
other in the Southern Latitude of 16 deg. 03 min. then the Meridi-
onal Difference of Latitude will be found to be 1793.5. For,
To the Meridional Parts anſwering to 13° 30
817.6
Add the Meridional Parts anſwering to 16°03'
975.9
The Sum will be the Meridional Difference of Latitude 1793.5
4
Problem 2.
Given the Latitudes and Longitudes of two Places, or both Lati-
tudes and Difference of Longitude, to find the direct Courſe and..
Diſtance between them.
Example.
Let it be required to find the direct Courſe and Diſtance, betweeu
the Lizard in the Latitude of so deg. oo min. North, and Barbadoès-
in the Latitude of 13 deg. 30 min. North, differing in Longitude
52 deg. 58 min. Barbadves lying so much to the Weſtward of the
Lizard,
Confer
Navigation..
287
>
A
Conſtru£tion.
. Having found the Meridional Dif-
ference of La jude 2056.9,by che ad Cafe
of the 1ſt Problem, ſet it oti from A to D,
and erićt the Perpendicular DE, equal
to the Difference of Longitude 3178
Miles, and draw AE.
Merid.diff.of Lat.
Dit.Saild
C
Diff of Long
2. Make AB equal to 2190 Miles, E
equal to the proper Difcrence of Lati-
tude, obtained (by Section the iſt., and
draw.BC parallel to ED, then will AC be the direct Diſtance, and
the Angle BAC the Courſe.
To find which by Calculation it will be,
As AD:DE::R:t, Angle A. That is,
As the Meridional Difference of Lat.=2656.9-2.4243752
To the Difference of Longitude 3178
2.5021539
So is the Radius
10.0000000
To the Tangent of the direct Courſe soº 06' 10.0777787
Which becauſe Barbadoes is to the Southward of the Lizard, and
the Difference of Longitude is Weſterly, is S. so deg. o6 min. W
or S. W.s deg. 06 min. Weſterly.
Whence to find the direct Diſtance it will be,
As R : AB :: Sec, Angle A : AC. That is,
As the Radius
-10.0000000
:
To the proper Difference of Latitude 2190
So is the Secant of the Courſe so deg. 06 min.
2 3404441
TO 1928374
To the dire& Diſtance 3414.14 Miles
25332815
Problem 3
One Latitude, Courſe and Diſtance fail'd being given, to find the
other Latitude and Difference of Longitude.
P.p2
288
Navigation.
LA
Example
Suppoſe a Ship from the Lizard, in the Latitude of so deg. oo min
North, fails 150 Miles upon the direct Courſe to Barbadoes, or S.
so deg. c6 min. W. and it be required to find the Latitude ſhe is in,
and how much ſhe has altered her Longitude.
Geometrically.
1. By the help of the Courſe and Di-
ſtance form the Triangle ABC, accord-
ing to the Directions given in the ift Cafe
3 of Plain Sailing, then will AB be the
proper Difference of Latitude ; to find
which it will be,
As the Radius
10.0000000
To the Diſtance fail'd 150-
2.1760913
So is the Co-line of the Courſe so° 06'
9.8071626
To the Difference of Latitude 36.2
1.9832539
D
Hence the Ship will be found to be in the Latitude of 48° 23'TEN.
and conſequently the Meridional Difference of Latitude by Caſe the
2d of the iſt Prob. 147.2.
Conſtruction. Make AD equal to 147.2, and draw DE parallel to
BC, which will be the Difference of Longitude. Then,
,
As R: AD::t, Angle A: DĒ. That is,
As the Radius
10.000cooo
To the Meridional Difcrence of Latitude 147.2-2.1679078
So is the Tangent of the Courſe 50° 06'
10.0777263
To the Difference of Longitude 176.05-
2.2456341
Or 2 deg. 56 min. In this Caſe Weſterly.
Nour if the Ship were bound to Barbadoes, and it were required to
find upon what Courſe ſhe muſt fail and how far, to arrive at the Port.
Becauſe the Differences of Longitude are both the ſame way, take
the Difference of Longitude made, fom the Difcrence of Longitude
between the Lizard ard Baibadoes, and with the Remainder and the
two Latitudes, find the Courſe and Diftarce by the former Problem.
Problem
:
Navigation.
289
Problem 4.
A
Given both Latitudes and Courſe, to find the Diſtance laild and
Difference of Longitude.
Example.
Suppoſe a Ship from the Lizard, in the Latitude of 30 deg oo min.
North, fails S. so deg. 06 min. W. until by Obſervation ſhe be found
to be in the Latitude of 48 deg. 23 min.7North, and it be re-
quired to find the Diſtance fail'd and Difference of Longitude.
Geometrically.
Having Conſtructed the Triangle ABC,
by the help of the Courſe and proper Dif-
férence of Latitude, according to the Di-
B
rections given in the 2d Caſe of Plain
Sailing.
E
Find the Meridional Difference of Latitude 147.2, by the 2d Cafe
of the 1ſt Problem, and ſet it oft' from A to D, and parallel to BC
draw DE, the Difference of Longitude, and it will be,
As R : AD::t, Angle A: DE. That is,
As the Radius
10.0000000
D
To the Meridional Difference of Latitude 147.2---2.1679078
So is the Tangent of the Courſe 50' 06
I0 0777263
To the Difference of Longitude 176.05
2.2456341
The Diſtance fail'd may be found by the ad Caſe of Plain Sailing.
For,
As the Radius
10.0000000
To the proper Difference of Latitude 96.2
So is the Secant of the Courſe 50° 06'
1.9331751
10.1928374
To the direct Diſtance fail'd 149.98
2.1760125
Problem 5.
Both Latitudes and Diſtance fail'd being given, to find the direct
Courſe and Difference of Longitude,
Example
290
Navigation
Example
Suppoſe a Ship from the Lizard, in the Latitude of so deg. oo min.
North, fails between the South and Weſt 150 Miles, and then by
Obſervation is found to be in the Latitude of 48 deg. 23 min.; N.
and it be required to find the direct Courle and Ditterence of Lon-
gitude.
A
Geometrically,
E
With the Difference of Latitude 96.2,
and Diftar.ce fail'd 150, Conſtruct the
B I riangle ABC, according to the Di-
rcctions given in the 3d Caſe of Plain
Sailing
2. Let AD be made equal to 147.2, the Meridional Difference
of Latitude (obtained by Coife the żd of Prob the ift.) and draw DE
perpendicular to AD, till it meet ÁC produced in E, then is DE the
Difference of Longitude, and the Angle BAC the Angle of the Courſe.
To find which by Calculation it will be,
As AC :R :: AB : cs, Angle A. That is,
As the Diſtance fail'd 150
2.1760913
To the Radius
10.0000000
So is the proper Difference of Latitude 96.2 1.9831751
To the Co-line of the Courſe 50° 06'į
-9.8070838
Or S. W. s deg. 06 min.} Weſterly.
Whence to find the Difference of Longitude it will be,
As R : AD::t, Angle A : DE. That is,
As the Radius
-10.0000000
To the Meridional Difference of Latitude 147.2--21679078
So is the Tangent of the Courſe soº 06'}
10.0778546
To the Difference of Longitude 176.10
2.2457624
Probleme
*
Navigation:
291
->
F
A
Problern 6.
One Latitude, Courſe, and Difference of Longitude being given
to find the other Latitude and Diſtance fail'd.
Example
Suppoſe a Ship from the Lizard, in the Latitude of so deg, oo min.
North, fails S. so deg. 06 min. W. until her Difference of Longitude
be 176 Miles, and it be required to find the Latitude the is in,
and her direct Diſtance ſail'd.
Geometrically.
1. Having form’d the Angle BAC
equal to 50 deg. 06 min. Draw Ef pa-
rallel to AD, at the diſtance of 176
Miles, equal to the Difference of Lon-
gitude, and from the point of Interſecti-
D
on E, let fall the perpendicular ED,
then will AD be the Meridional Ditte I
rince of Latitude ; and it will be,
Ast, Angle EAD:R::ED: AD. That is,
As the Tangene of the Courſe soº 06
10 0777203
To the Radius
10.000.000
So is the Difference of Longitude 176--
2.2455127
V
ni
To the Meridional Difference of Latitude 147.16--2.1677864
B
:
Becauſe the Ship ſailed from a North Latitude Southerly,
From the Merid. Parts of the Lat. fail'd from 5oºoo'-3474.5
Take the Meridional Difference of Latitude
147.16
Remains the Merid. Parts of the Lat. come into 48° 23'78 3327:34
Whence to find the proper Difference of Latitude,
From the Latitude fail'd from
Take the Latitude faiļd into
50° oo' N.
48 23 18 N.
I'362
Remains the Difference of Latitude
Conſtr. Set off 96.2 from A to B, and draw BC parallel to DE,
then will AC be the direct Diſtance fail d, to find which by Calcı-
lation it willbe,
As
292
Navigation
As R : AB :: Sec. Angle BAC : AC. That is,
As the Radius-
10.0000000
To the proper Difference of Latitude 96.2
So is the Secant of the Coarſe soº 06'
-1.9831751
10.1928374
To the direct Diſtance 149.97-
2.1760125
Problem 7.
Onc Latitude, Courſe, and Departure being given, to find the
other Latitide, Diſtance fail d, and Difference of Longitude.
Example
If a Ship from the Lizard, is the Latitude of so deg. oo min. N.
fails S. 50 deg. 06 min. W. until her Departure be 115.1 Miles, and
it be required to find the Latitude ſlie is in, her direct Diſtance fail'd,
and how much the has altered her Longitude.
A
Geometrically.
With the Courſe and Departure,
let the Triangle ABC be Conſtrućt-
ed, according to the Directions gi-
ven in Cafe the 6th of Plain Sailing
B
then will AB be the proper Difference
D
of Latitude, and AC the direa Diſtance.
Wherefore,
As S, Angle BAC : BC :: R: AC. That is,
As the Sine of the Courſe 500 00'
9.8848889
E
To the Departure 115.1
So is the Radius
2.0610753
10.0000000
To the dire&t Diſtance 150.03
2.1761864
For the Difference of Latitude it will be,
As t, Angle A: BC::R: AB. That is,
As the Tangent of the Courſe so° 06'
-10,0777263
To the Departure 115.1
2.06 10753
So is the Radius
10.0000000
To the proper Difference of Latitude 96.2 -----
1.9833490
Hence
Navigation.
293
Hence the Ship will be found to be in the Latitude of 48 deg.
23 min. fo, N. and conſequently the Meridional Difference of Lati-
tude is 147.2.
Conſtr. Make AD equal to the Meridional Difference of Latitude,
and draw DE parallel to BC; and it will be
As R : AD: :t, Augle A: DE. That is,
As the Radius
10.0000000
To the Meridional Difference of Latitude 147.2-2.1679078
So is the Tangent of the Courſe so deg. 06 min, 10.0777263
To the Difference of Longitude 176.05
2.2456341
Or, becauſe the Triangles ABC, ADE, are ſimilar, it will be,
for the Difference of Longitude,
As AB : AD :: BC:DE. That is,
As the proper Difference of Latitude 96.2
1.9833490
To the Meridional Difference of Latitude 147.2-2.1679078
So is the Departure 115.1
2.0610/53
To the Difference of Longitude 176.05–
2.2456341
Problem the 8th.
Both Latitudes and Departure being given, to find the Courſe, Di-
ſtance, and Difference of Longitude.
Example
Suppoſe a Ship from the Lizard, in the Latitude of so deg. oo min.
North, fails between the South and Weſt, until her Departure be
195.1 Miles, and then by Obſervation is found to be in the Latitude
of 48 deg 23 min. I North, and it be required to find her Courſe,
Diitance fail'd, and Difference of Longitude.
A
Geometrically.
1. By the help of the proper Diffe-
rence of Latitude 94.2, and Departure
115.1. Let the Triangle ABC be Con-
B
ſtructed according to the Diređions gi-
ven in the 4th Caſe of Plain Sailing.
Qg
2. Make
.
E
D
294
Navigation.
2. Make AD equal to the Meridional Difference of Latitude 147.2,
(obtained by the 2d Caſe of Problem the ift) and draw DE parallel
to BC, and it will be,
AS AB: BC :: AD: DE. That is,
As the proper Difference of Latitude 96.2
1:9833490
To the Meridional Difference of Latitude 147.22:1679078
So is the Departure 115.1.
2.0610753
To the Difference of Longitude 176.05
2.2456341
Or having found the Courſe and Diſtance by Caſe the 4th of Plain
Sailing, the Difference of Longitude may be found by the Second
Problein.
Problem the 9th.
One Latitude, Diſtance fail’d, and Departure being given, to find
the other Latitude, Courſe, and Difference of Longitude.
Example.
Suppoſe a Ship from the Lizard in the Latitude of so deg. oo min.
North, fails between the South and Weſt 150 Miles, and then is
found to have departed from her firſt Meridian 115.1 Miles, and it
be required to find the Latitude: ſhe is in, her direct Courſe and
Difference of Longitude.
les
1. With the Diſtance fail'd and
Departure form the Triangle ABC
by the 5th Caſe of Plain Sailing; then
for the Courſe it will be,
B
As AC: CB :: R:S. <A. That is,
E
D
As the Diſtance fail'd 150
2.1760913
To the Departure 115.1-
So is the Radius
2.0610753
10.0000000
To the Sine of the Courfe so deg. 07 min.
-9.8849840
Which is S. S0° 07' W. or S.W.5ºc7' Weſtward.
Whence
Navigation
295
Whence to find the Difference of Latitude it will be,
As R : AC :: cs, Angle A : AB. That is,
As the Radius
10.0000000
To the Diſtance fail'd 150-
So is the Co-line of the Courſe so deg. 07 min.-
2.1760913
9 8070114
To the proper Difference of Latitude 96.18
1,9831027
Hence the Ship will be found to be in the Latitude of 48 deg.
23 min. 1: North ; and conſequently the Meridional Difference of
Latitude will be 147.2 Miles.
Conſtr. Set off 147.2, from A to D, and draw DE parallel to BC;
whence to find the Difference of Longitude it will bc,
As R: AD: :t, Angle A:DE. That is,
As the Radius
10.0000000
To the Meridional Difference of Latitude 147.2---2.1679078
So is the Tangent of the Courſe so deg. 07 min. 10.0779830
To the Difference of Longitude 176.15
2.2458908
Or, the proper Difference of Latitude might have been found by
the 7th Caſe of Plain Iriangles, without finding th: Courſe; and
thence the Difference of Longitude as in the preceding Problem.
A Traverſe.
Suppoſe a Ship at the Lizard in the Latitude of so deg oo min
North, is bound to the Madera, in the Latitude of 32 deg. 20 min:
North, the Difference of Longitude between them being 11 deg.
40 min. the Weſt end of the Madera lying ſo much to the Weſtward
of the Lizard, and therefore the direct Courſe (by Prob. the ift.) is
S. 26 deg. 15 min. W. Diſtance 1181.9 Miles; but by reaſon of con-
trary Winds, ſhe is forced to fail away (allowance being made for
Leeway and Variation) W.S.W.26 Miles, then S.E. by E. 30 Miles,
then S. W. by W. W. 36 Miles, then W.64 Miles, then S. W. W.
15 Miles, then S. E. by S. 37 Miles: let it be required to find
the Latitude the Ship is in, her Bearing and Diſtance from the
Lizard, alſo her true Courſe and dire& Diſtance to the Madera.
Qq2
In-
.
.
296
Navigation.
Inaſmuch as the Geometrical Cooſtruction of this problem is no
o:her than the Application of the firſt Problem, to the laying down
of the two Ports and the Repetition of the ſecond Problem, as often
as there arc Courſes I ſhall onic ir ; and haſten to what is more
uſeful, viz. the Solution of the fame Problem by Calculation. And,
For the if Courſe, viz. W. S. W. 26 Miles, it will be,
As the Radius
IC.000.000
To the Diſtance fail'd 26
So is the Co-line cíthe Courſe 67 deg. 30 min.
14149733
-9.5828397
To the Difference of Latitude 9.95–
0.9978130
Hence, becauſe the Courſe is Southerly, the Latitude come into
is 49 deg. so min. róa North. Wherefore,
From the Meridional Parts of 50° vo'
3474.50
Take the Meridional Parts anſwering to 49° 50' Táo =3459.07
Remains the Meridional Difference of Laritude
15.43
Whence to find the Difference of Longitude it will be,
As the Radius
10.00OOOCO.
To the Meridional Difference of Latitude 15-43— 1.1883659
So is the Tangent of the Courſe 67° 30'
10.3827757
To the Difference of Longitude 37.25
1.5711416
For the 2d. Courſe, S. E. by E. Diſtance 30 Miles. .
As the Radius
-10.0000000
.
To the Diſtance fail'd
So is the Co-line of the Courſe so deg. 15 min.
1.4771213
9.7447390,
To the Difference of Latitude 16.67
1.2218603
Hence again, becauſe the Courſe is Southerly, the Latitude comie
into is 49° 33' North. Wherefore,
From
Navigation
297
From the Meridional Parts anſwering to 49° 50'; }
Take the Meridional Parts anſwering to 49° 33'76.
3459.07
3433.27
Remains the Meridional Difference of Latitude
25.80
Whence to find the Difference of Longitude, it will be,
As the Radius
10.0000000
To the Meridional Difference of Latitude 25.80.-14116197
So is the Tange.it of the Courſe 56° 15'
10.175 107-+
To the Difference of Longitude 38.61–
1.5867.271
And proceeding after the ſame manner, the Difference of Lati-
tude and Difference of Longitude inay be found to cvery Couiſe and
Diſtance given; as in the following Table.
The Traverſe Table.
Diff, of Lat. Dift. of Long..
S
Couries.
Pts. Dit
N:
S.
E. |.W
9.95
16.67 38.61
37,25
W.S. W.
S. E. by E.
S. W. by W. įW,
Wert.
S. W: W.
S: E. by S:
16.97
6
5
5
8"
42
4
3.
26
30
36
64
15
37
48.75
98.09
17.75
9.51
30.76 21.22
Diff. Lat,
83.86 69.83 201.84
6983
Diff. Long.-13201
Hence, becauſe the Difference of Latitude is Southerly, the Ship
will be found to be in the Latitude of 48 deg: 36 min. 164, N. and
conſequently the Meridional Difference of Latitude will be 128.56
Whence to find the direct Courſe from the Lizard to the Ship, it
will be
1
As.
.
298
Navigation.
As the Meridional Difference of Latitude 128.56--2.1081059
10.0000000
2.1 206068
To the Radius-
So is the Difference of Longitude 132.01
To the Tangent of the Courſe 45° 49'11
10.01 25009
Which becauſe the Difference of Latitude is Southerly, and Dif-
ference of Longitude Weſterly, is S. 45 deg. 49 min. W. or S. W.
49 min. Weſt. Again, for the direct Diſtance from the Ship to the
Lizard it will be
As the Radius
10.0000000
.
To the proper Difference of Latitude 83.86-
So is the Secant of the Courſe 45° 49'
1.9235549
10.1568593
To the direct Diſtance 120.34
2.0804142
Again,
From the Meridional Parts anſwering to the Latitude
}=3345.91
the Ship is in 48 deg. 36' 16
Take the Meridional Parts anſwering to the Latitude
}= 2052.00
of the Madera 32920
I 1
Remains the Meridional Difference of Latitude be-
tween the Ship and the Madera
Again,
From the Difference of Longitude between the }=11° 40'W.
Lizard and the Madera
Take the Difference of Longitude between the
Lizard and the Ship
the } =2° 12' W.
Remains the Difference of Longitude between
? } =° 28' W.
the Ship and the Madera
Whence to find the dired Courſe between the Ship and Madera
it will be
As
Str.
Navigation.
299
1
As the Meridional Difference of Latitude 1293.91 3.1119040
To the Radius
10.0000000
So is the Difference of Longitude 9° 28'=568 M. --- 2.7543483
To the Tangent of the Courſe 23° 42'
9.6424443
Which becauſe the Ship is to the Northward of the Madera, and
the Difference of Longitude is Weſterly, is S. 23 deg. 42 min. W.
or S. S. W. i deg. 12 min. W.
Again, for the direct Diſtance between the Ship and the Madera
it will be,
As the Radius
10.0000000
To the proper Difference of Latitude 976.142.9895 1 2 I
So is the Secant of the Courſe 23° 42'-
10.0382645
To the direct Diſtance 1066.05
3.0277766
The Differences of Latitude and Longitude in the former Traverſe
Table
may be found without Calculation, by the help of the common
Tables of Difference of Latitude and Departure in Se&t. the 3d, after
the manner following,
Entering the Table with the Courſe in the Head, and Diſtance
fail'd in the Side, find the common Difference of Latitude as in Plain
Sailing ; this applied to the Latitude ſaild from according as the
Cafe
requires, will give the Latitude the Ship is in.
2. With the two Latitudes entering the Table of Meridional Parts,
find the Meridional Parts antwering to each Latitude, and thence
the Meridional Difference of Latitude.
3. Entering again the Traverſe Table, with the Courſe and Meria
dional Difference of Latitude, find the Courſe in the Head as before ;
and the Meridional Difference of Latitude under the Column of La-
xitude, and right againſt it under the Column of Departure, you will
have the Difference of Longitude required.
Examples
Let it be required to find the Difference of Longitude and Diffea
rence of Latitude, correſponding to the firſt Courſe in the former
Traverſe or W. S. W. Diſtance 26 Miles.
In
1
300
Navigation.
In the Column of ſix Points, and right againſt 26 in the Column
of Diſtances, I find 9.95 in the Column of Latitude for the Difference
of Latitude; this becauſe the Latitude faii'd from is Northerly, and
the Courſe Southerly, lubtracted from the Latitude !ail'd from so dcg.
oo mia. North, leaves the Latitude come into 49 deg. 50'S. N.
The Meridional Parts anſwering to so deg. oo min. is 3474.5,
and to 49 deg. 50 min. Tá is 3459.07, whence the Meridional Dif-
firencz of Latitude will be found by Sabftraction to be 15.43.
Again, in the Column anſwering to tne lixth Rumb, right againſt
15.31 in the Column of Latitudes, the next lefs to the Meridional
Difference of Latitude, I find in the Column of Departure 36.95 ;
alſo
againſt 15.69 in the Column of Latitude, the next greater to the Me-
ridional Difference of Latitude, I find in the ſame Column of De-
parture 15.69; then I ſay, as 38, the difference between the Tabular
Differences of Latitude, is to .92, the difference between the Tabular
Departures, ſo is . 12, the Excits of the Meridional Difference of
Latitude, above che next leſs Tabular Difference of Latitude, to 29;
which therefore idded to 36,95, the Departure anſwering to 15.31,
tlie next leſs Tabular Difference of Latitude will give 37.24, the
Difference of Longitude required ; and ſo for any other.
Whoſoever thoroughly conſiders the Nature of Departure and
Difererce of Longitude, will readily grant, that the Method made
Uſe of by our moſt Expert Navigators, and generally taught in
Books, for finding the Difference of Longitude by increafing the Ab-
ſolute Departure made in one Day's Runniggs, in Proportion as the
Meridional Difference of Latitude, is to the proper or common
Difference of Latitude, cannot be ſtriąly true.
For, tho to the ſame Courſe and Diſtance, there will always be
the ſame Departure and proper Difference of Latitude, and to the
ſame Latitude, proper Difference of Latitude and Departure, there
will always be the ſame Difference of Longitude : Yet, when ſere-
ral Departures come to be Compounded in one, as is the Caſe of
almoſt every Day's Work, the thing is quite altered, as an eaſy In-
ſtance will ſhew.
Suppoſe, for Example. A Ship in the Latitude of 60 deg. North,
fails 100 Leagues dne Eaſt, and after that N. E. till ſhe be got into
the Latitude of 70 deg. in this caſe ſhe will have made but 300
Leagues of Departure, and 2038.6 Miles, or 33.58 T of Difference
of Longitude.
Let
1
I
Navigation.
.
301
Let us now fuppoſe the fame Ship to ſail 100 Leagues due Weſt,
in the Parallel of 70 Degrees, then S. W. till the be got into the
Latitude of 60 Degrees, 'cis plain that in this Caſe flie has made but
300 L-agues of Departure as before ; but 2315.8 Miles, or 38 deg.
35 min. of Difference of Longitude greater in Proportion, than
the former Difference of Longitude; as the Co-fine of 60 Degrees is
to the Co-ſine of 70 Degrees, notwithſtanding the Meridional and
proper Differences of Latitude in both Caſes are the lame.
I have choſen this eaſy Inſtance, becauſe every one that is but
moderately Skild may underſtand it, but the ſame Abſurdity will
follow in Compounding of any other Courſes together, as the Sa-
gacious Reader will readily apprehend.
And for wait of knowing this, ſome People have found fault with
Mercator's Sailing, and condemned the 'Tables of Meridional Parts,
when in Reality the Miſtake has proceeded from their own Igno-
rance.
In this Example here given, the Places have been taken pretty
far from the Equator, and the Diſtances Run conſiderably great,
that fo the Error might be the more Obvious, but in ſmall Runnings
it is very diſcoverable, as the Solution of the former Traverſe, by
that Method will plainly flew.
The Ablolute Departure in the laſt Traverſe is 85.86 Miles, the
Latitude fail'd from so deg. oo min. North, the Latitude fail'd
into 48 deg. 36 min. 76North, and conſequently the Meridional
Difference of Latitude 128.56 Miles. Wherefore,
As the proper Difference of Latitude 83.86
1.9235549
To the Meridional Difference of Latitude 128.56 2.1081059
So is the Abſolute Departure 85.86-
1.9337909
To the Difference of Longitude 131.32
2.1183419
Differing from the true Difference of Longitude 76 of a Mile;
which tho but a ſmall Error, yet in ſeveral Days Runnings, eſpeci-
ally if the places are far to the Northward, or Southward, it will
amount to a conſiderable Error.
However, as a true Anſwer ought to be prefer'd before a falſe one,
and an exact Solution before an Approximation (tho' the Difference be
never ſo ſmall) it were to be wiſhed, that all our Ingenious Mariners
Rr
would
A
302
Navigation.
would lay a ſide their Tentative Methods, and make Uſe of the
only true Way taught in the former Traverſe, viz. by finding the
true Difference of Longitude to every particular Courſe and Diſtance,
ſince he will be always ſure of having his Deductions trus: And what
is a farther Encouragement, the difference of Time requiſite for
caſting up an Ordinary Days Work, after the manner taught in the
300th Page, by the help of the Tables of Difference of Latitude
and Departure, and the Common Way does not amount to a quar-
ter of an Hour.
Thus have I Endeavoured to render the true Sailing, in a manner,
as eaſy as the Plain Sniling; and ſhall therefore conclude this section
with propoſing of one Problem, (which may be Worked as a Traverſe
of four Courſes, after the manner of the Traverſe Citrrent, in Section
the 6th) and leave the actual Solution of it, as an Exerciſe for the
young Beginner.
A Ship from the Madera, in the Latitude of 32 deg. 20 min. North,
Longitude from the lizard 11 deg. 40 min. Weſt, fails W. S, W.
Hours, at the Rate of 5 Miles an Hour, then S. E. by S. 7
Hours, at the Rate of 6 Miles an Hour, then S. W. W. 8 Hours,
at the Rate of 0 Miles an Hour, and a Current is found to ſet
during the whole time N. W. by W. at the Rate of 1 Mile an
Hour ; I demand the Latitude the Ship is in, and Longitude from
the Lizard, as allo her true Bearing and Diſtance from the Madera.
Anſwer.
The Latitude the Ship is in, is
31 : 1410 N.
Her Longitude from the Lizard is
13 : 541: W.
Her Bearing from the Madera is-
S. 60 : 14 W.
And her direct Diſtance from the Madera is—132.74 Miles.
N. B. The Meridional Parts made Uſe of in theſe Calculations, are
taken out of the Old Tables, the New Tables being not finibed when theſe
Sheets went to Preſs.
Sect.
'
Navigation.
303
:
of the Latitudes of any two Places on the ſame Side of the Equator,
Section XII.
Containing the Solution of the ſeveral Problems in Mercator's
Sailing, by the help of the Logaritlimic Tangents only.
Prob. 1.
L
E T it be required to find the Meridional Diffcrence of La-
tituce between any two li.cus.
It has been Demonſtrated in Se7, the roth that the Difference of
the Logaritiimic Tangents or Co-tangents, of half the Complements
dividid by 12633, 6c, will give the Meridional Difierince of La-
titude between thein. Wherefore, in
Caſe 1.
If one place be under the Equator, and the other in thc Latitude
of so deg. oo min. North, and it be required to find the Meridional
Difference of Latitude between them.
From the Logarithmic Tangent of 45 deg. oo min. To.0000000
Take the Logarithmic Tangent of 20 deg. oo min. 9.5610659
w
.4389341
Or,
From the Logarithmic Co-tangent of 20 deg. oo min.—10.4389341
Take the Logarithmic Co-tangent of 45 deg. oo min. -10.000000
And divide the Remainder by 12633, &c.
-4389341
And the Quotient 3474.48, or 3474.5, will give the Meridional
Difference of Latitude.
Caſe II.
If the Places propoſed are on the ſame side of the Equator, that
is, if one be in the Latitude of 50°00' North, and the other in the
Latitude of 13° 30' N. Then,
From
Rr 2
304
Navigation.
From the Logarithmic Tangent of 38° 15'
Take the Logarithinic Tangent of 200 oo'
9.8967116
-9.5610659
Or,
:3356457
ht
From the Logarithmic Co-tangent of 20° oc' 10.4389341
Take the Logarithmic Co-tangent of 38° 15'
10.1032884
And divide the Remainder by 1263, &c.
-3356457
And the Quotient 2656.9, is the Meridional Difference of La-
titude.
Again, becauſe the Sum of the Differences of the Tangent of 45°
oo', and the Tangents or Co-tangents of half the Complements of
the Latitudes of two Places, on contrary Sides of the Equator,
divided by 12633, 66. will give their Meridional Difference of La-
titude. Therefore, in
Caſe III.
If one place be in the Latitude of 13 deg. 30 min. North, and
the other in the Latitude of 16 dcg. 03 min. Souch.
To the Logarithmic Tangent of 5 i deg. 45 min. 10.1032884
Add the Logarithmic Tangent of 53 deg. or min. į 10.1232799
From the Sum
Take twice the Radius
20.2265683
20,0000000
And there remains
.2265683
Or,
To thc Logarithmic Tangent of 38 deg. 15 min.
Add the Logarithmic Tangent of 30 deg. 58 min. į
9.8967116
9.8767200
The Sum
19.7734316
Taken froin twice the Radius
Leaves as before
20.0000000
-0.2265684
Which
Navigation.
305
Which Remainder divided by 1263, c. will give 1793.5, the
Meridional Difference of Latitude, between the two Places, the
ſame as was deduced in the former Section from the Meridional Parts.
Again, in
Prob. II.
Where the Latitude the Ship was in was 50 deg. oo min. Nortlı,
the Latitude ſhe was bound to 13 deg. 30 min. North, and the
Difference of Longitude 52 deg. 58 min. equal to 3178 Miles; to
find the Courſe and Diſtance.
From the Logarithmic Tangent of 38 deg. 15 min. 9.8967116
Take the Logarithmic Tangent of 20 deg. oo min. -2.5610652
:3356457
Or,
From the Logarithmic Co-tangent of 20 deg. 00 min.-10.438934 1
Take the Logarithmic Co-tangent of 38 deg. 15 min.--10.103288+
And it will be
:3356457
As the Remainder 3356.457
3.5258810
To the Tangent of the Log. Rumb 5 1° 38' 9" -10.1015093
So is the Difference of Longitude 3178
3.5021539
To the Tangent of the Courſe 50° 06' ---
10.0777822
Or, S. W. 05 deg. o6 min. Weſterly as before: Whence it will be
eaſy to find the Diſtance.
Again, in
Prob. III:
Where the Latitude departed from was so deg. oo min, North,
the direct Courſe S. 50 deg. 06 min. W. and the Diſtance 150 Miles
;
to find the other Latitude and Difference of Longitude
After having, found by the former Methods, the Latitude the
Ship is in, viz. 48° 23' North.
From
309
Navigation.
From the Logarithmic Tangent of 20° 48'55
Take the Logarithmic Tangent of 20° 00'
9.5796667
9.5610659
186008
Or,
From the Logarithmic Co-tangent of 20° oo'
Take the Logarithmic Co-tangent of 209 48'7
I
10.4289341
10.4203333
186008
An
10.1015093
And it will be, for the Difierence of Longitude,
As the Tangent of the Log. Rumb si 38' 0''
To tire Tangent of the Courſe 50° 06'-- 10.0777263
So is the Log. of the Remainder 196.008
2.2695316
To the Difference of Longitude 176-09-
2.2457486
Again, in
Prob. IV.
Where the Latitude ſaild from was so deg. oo min. North, the
Latitude fail'd into 48 deg. 23 min. North, and the dire& Courſe
S. so deg. 06 min. Weſt; to find the Alteration of Longitude,
From the Logarithmic Tangent of 20 deg. 48 min.; '; -9.5796667
Take the Logarithmic Tangent of 20 deg. co min. 9.5610659
186008
Or,
From the Log. Co-tangent of 20 deg. oo min.
Take the Log. Co-tangent of 20 deg. 48 min. io
10.4389341
10.4203333
186008
And
Navigation.
307
And it will be, for the Difference of Longitude,
As the Tangent of the Log. Rumb 51° 38'09" 10.IO15093
To the Tangent of the Courſe 50° 06'
10.0777263
So is thc Remainder 186.008
2.2695316
To the Difference of Longitude 176.09–
2.2457486
Or 2 deg. 56 min. jót, the fame with the former Solution.
Likewiſe in
Prob. V.
Where the Latitude departed from was 50 deg. oo min. North,
the Diſtance Run 150 Miles between the South and the Weſt, and
the Latitude fail'd into 43 deg. 23 min. 16, to find the Alteration
of Longitude.
Having found the Courle, by the help of the Diſtance faild and
proper Difference of Latitude, after the manner taught in the third
Caſe of Plain Sailing.
Then,
From the Logarithmic Tangent of 20°
9.5796667
Take the Logarithmic Tangent of 20° oo'
9.5610659
48
T
To
186008
Or,
From the Logarithmic Tangent of 70 ° '
Take the Logarithmic Tangent of 69° 11',
10.4389341
10.4203333
186008
And it will be for the Difference of Longitude,
As the Tangent of the Log. Rumb 51° 38' 91 10.1015093
To the Tangent of the Courſe soº co
So is the Remainder 186.008
10.0777263
2.2695316
To the Difference of Longitude 176.09
2.2457486
Or, 2 deg. 56 min. zás nearly, agreeing with that found by Me-
ridional Parts
Again,
man
308
Navigation.
Again, in
Prob. VI.
Where the Dificrence of Longitude was given 2 deg. 56 min. or
176 Miles, the Latitude departed from so deg. oo min. North, and
the Courſe ſtcer'd S. so deg. 06 min. Weſterly, to find the Latitude
the Ship is in, and her Diſtance fail'd,
It will be,
As the Tangent of the Courſe 50°06'
10.0777263
To the Tangent of the Log. Rumb 51° 38' 09"-10.1015093
So is the Difference of Longitude 176 Min.
2.2455127
To the Number 185.907-
-- 2.2692957
Which Number -
0.0185907
Added to the Logarithmic Tangent of 20° oo
9.5610659
Gives a Tangent
9.5796566
Or,
Correſponding to 20 deg. 48 min. 187.
From the Log. Tangent of 7oº oo
Take the Number before found-
10.4389341
0.0185907
8
TO
Rem: the Co-tang. of the ſame Number 20° 48' 10.4203434
Which therefore doubled and ſubſtracted from 90 deg. oo min.
leaves the Latitude the Ship is in 48 deg. 23 min.154 as before found;
which being obtained, the Diſtance is eaſily had.
Again, in
Prob. VII.
Where the Latitude fail'd from was so deg. oo min. North, the
Courſe S. so deg. 06 min. W. and Departure 115.1 Miles, to find
the Diſtance faild, Latitude the Ship is in, and Difference of Lon-
gitude.
Having by the help of the Courſe and Departure, found the La-
titude the Ship is in, according to the Rules in the 6th Caſe of Plain
Sailing.
From
**
Navigation
309
From the Log. of the Tangent of half the Compl.
of the Latitude the Ship fail'd into 20° 48' is
Take the Log. of the Tangent of half the Compl.
of the Latitude the Ship faild from 2000
}
9.5796667
9.5610659
.0186008
Or,
From the Logarithmic Tangent of 70° oo
Take the Logarithmic Tangent of 20° 48'.
I
10.4389341
-10.4203333
.0186008
And ſay,
10.1015093
As the Tangent of s rº 38' 09"
To the Tangent of the Courſe 50° 06'
So is the Log. of the Remainder 186.008-
10.0777263
2.2695 316
To the Difference of Longitude 176.09
----- 2-2457486
Whence the Diſtance fail'd is eaſily had.
Again, in
Prob. VIII.
Where the Latitude fail'd from was so deg. oo min. North, the
Departure 115.1 Miles, the Latitude ſail'd into 48 deg. 23 min. 79
North, to find the Difference of Longitude.
1. Find the Courſe according to the Directions given in the 4th Cafe
of Plain Sailing. Then,
From the Logarithmic Tangent of 20 deg. 48' 15
9.5796667
Take the Logarithmic Tangent of 20° oo'
9.56 10659
.0186008
t.
į
Ss
Or,
310
Navigation.
Or,
From the Logarithmic Tangent of 70° 00'
Take the Logarithmic Tangent of 69° 11';?
10.4389341
10.4203333
.0186008
And ſay,
10.1015093
As the Tangent of so° 38' 09"-
To the Tangent of the Courſe 50° 06'
So is the Log. of the Remainder 186.008-
10.0777263
---2.2695316
To the Difference of Longitude 176.09
2.2457486
Laſtly, in
Prob. IX.
Where the Latitude faild from was so deg. co min. North, the
Diſtance Run 150 Miles, between the South and Weſt, and the
Departure 115.1, to find the Difference of Longitude.
1. Having found the Courſe so deg. 07 min. and Latitude the Ship
fail'd into 48 deg. 23 min. 78, according to the 5th Caſe of Plain
Sailing.
From the Logarithmic Tangent of 20° 48' 75 -9.5796667
Take the Logarithmic Tangent of 20° 0
9.5610659
.0186008
0
Or,
From the Logarithmic Tangent of 70°
Take the Logarithmic Tangent of 69° 11' ;
10.4389341
10.4203333
9
TO
.0186008
And, it will be
As the Tangent of 51° 38' 09
10.1015093
10,0779830
2.2695316
To the Tangent of so° 07'-
So is the Log. of the Remainder 186.008
To the Difference of Longitude 176.2-
2.2460053
Το
Navigation.
311
To give the Learner all the helps neceſſary, to a right Under-
ſtanding of this moſt uſeful Part of Sailing, I shall endeavour (before
I conclude this Part) to ſet his Notions right, concerning Diffe-
rence of Longitude, Meridian Diſtance, and Departure ; and let
him fee, that tho theſe are Synonymous Terms in Plain Sailing,
conſtantly ſignifying the ſame thing, and in every Queſtion are re-
preſented by the ſame Right Line, yet, in the True Sailing they are
eflentially different one from another; and in the ſame Problem,
are, as they really ſhould be, repreſented, or exprefled by different
Lines, and are of different. Values; a thing that now and then is
apt to puzzle and confound young Beginners, and Perſons who are
not to well Grounded in the firſt Principles of this Science, as they
ought to be.
By Difference of Longitude, is meant the Arch of the Equator
contained, or intercepted, between the Meridians paſſing thro'
any
two Places; thus the Length of the Arch of the Equator con-
tained, or intercepted between the Meridians paſſing thro' the
Lizard and Barbadoes, being found by Obſervation, to contain 3178
Nautical Miles, or 52 deg. 58 min. the Difference of Longitude be-
tween the Lizard and Barbadoes is ſaid to be 52 deg. 58 min. and all
Places that lie under the Meridian of the Lizard, are ſaid to differ
in Longitude from Barbadoes, or any other Place under the Meridian
of Barbadoes 52 deg. 58 min.
By Meridian Diſtance, (in the Nautical Senſe) is meant the Di-
ſtance of one place from the Meridian of another, meaſured in the
Arch of the Parallel of Latitude paſſing thro' them, and intercepted
between them, ſo that, fuppoſing the Difference of Longitude, or
Arch of the Equator contained, or intercepted, between the Meridian
of the Lizard and the Meridian of Barbadoes, to be 52 deg. 58 min.
or 3178 Nautical Miles, and that the Lizard lies in the soth De-
gree of Latitude ; the Meridian Diſtance, or Length of the Arch
of the Parallel of Latitude, paſſing thro'the Lizard, and intercepted
between it and the Meridian of Barbadoes, will be found by the ift
Caſe of Parallel Sailing, to be 2042.8 Nautical Miles; and a Ship in
Sailing from Barbadoes, or any other Point under the Meridian of
Barbadoes, till ſhe arrive at the Lizard, is faid to have made 2042.8
Miles of Meridian Diſtance.
Again, Suppofing Barbadoes to lie in the 13th Degree of Lati-
tude from the Equator, the Length of the Arch of the Parallel of
SS 2
Lari-
:
312
Navigation
Latitude paffiing thro' Barbadoes, and intercepted between Barbadoes
and the Meridian of the Lizard, will be found by the former Me-
thod of Calculation, to be 3090.1 Miles; ſo that a Ship in failing
from the Lizard, or from any other place under the Meridian of the
Lizard, till ſhe arrive at Barbadoes, is ſaid to have made 3090.1
Miles of Meridian Diſtance, each of which Meridian Diſtances
is leſs than the true Difference of Longitude, or Arch of the Equa-
tor, intercepted between the Meridians paſſing thro’the Lizard and
Barbadies, and are to each other, as the Sines of the Diſtances of
the Places from the Pole, or Co-ſine of thcir Latitudes, or Diſtances
from the Equator, (the Difference of Longitude being equal to the
Sine of 90 Degrees) and may therefore be all exprefled by different
Right Lines, and are of diferent Values, as the Reader will
readily perceive, if he. Conſtructs theſe two Cales in the ſame Fi-
gure, according to the Directions given in the firſt Caſe of Parallel
Sailing:
By the Departure, is meant the diſtance between the Meridians,
paffing thro' two Places upon the Plain Chart, which ſuppoſes
the Earth's Superficies to be a Plain Superficies, the Meridians
Parallel to each other; and the Degrees of Latitude every where
equal to each other, and equal to the Degrees of Longitude, and is
therefore the ſame with the Difference of Longitude; ſo that, ftria-
ly Speaking, the Departure is the Difference of Longitude between
two Places, upon the Principles of Plain Sailing : And between the
Lizard and Barbadoes will be found to be 2119.2 Miles, lels than
the true Meridian Diſtance between Barbadoes and the Lizard, by
470.9 Miles, and greater than the true Meridian Diſtance between
the Lizard and Barbadoes by 576.4 Miles; and ſhort of the true Dif-
terence of Longitude between the Lizard and Barbadoes by 578.8
Miles.
And hence ariſes the Error of Plain Sailing, and to remedy this
very Thing was the Mercators Sailing-Invented, which it does to
the utmoſt Degree of Exactneſs, as every one muſt allow, that
rightly underſtands what has been ſaid upon this Head.
And indeed, properly Speaking, Departure has nothing to do with
Mercators Sailing, nor has it any ſhare in it, and 'tis for this Reaſon
that I have taken no Notice of it in the firſt Six Problems of Mer-
cators Sailing, which together with the Problems of Parallel Sailing,
are the only proper Problems of the true Sailing ; but as it happens
to
Navigation.
3.1.3
to be one conſtituent part of a Triangle, whoſe other Parts, viz.
Difference of Latitude, Courſe, and Diſtance, bear an Analogy with
the Lines they repreſent upon the Globe it ſelf; it may be made Uſe
of, as one Ingredient to make up the Data in a Problem, (tho
it antwer to neither Meridian Diſtance, nor Difference of Longi-
tude) and whatſoever Concluſions are drawn from it, as to Difference
of Latitude, Difference of Longitude, Courſe or Diſtance, are e-
qually true, and concluſive with any of the other Problems :
For it has been ſhewn in Page 144, that to the fame Courſe and
Diſtance, there will ever be the ſame proper Difference of Latitude
and Departure, and conſequently, to the fame Courſe and Departure
there will always be the ſame Diſtance and proper Difference of Lati-
tude. But by Section the roth, to the ſame Latitude, and
proper
Dif-
ference of Latitude, there will always be the ſame Meridional Diffe-
rence of Latitude, and by Prob. the 6th of Se&t. the Ith, to the
fame Courſe and Meridional Difference of Latitude, there will al-
ways be the ſame Difference of Longitude ; whence it follows, that
to the ſame Latitude, Courſe, and Departure, there will always be the
ſame Diſtance, Difference of Longitude, and Meridian Diſtance ;
and the ſame Method of Reaſoning will hold good, if the Departure
be Compoundedwith either Diſtance fail'd, or Difference of Latitude,
and 'tis upon this Account, that I have ventured according to
Cuſtom, to add the 7th, 8th, and gih Problems in Mercators Sailing,
not as they are of any real Uſe in themſelves, but as they tend to
Exerciſe the Learner, by ſhewing him all the Varieties that may
be, and rendering the Whole more compleat.
Section XIII.
XIII..
H
Aving fufficiently ſhewn the Defects of the Plain Chart, and
in the former Part of the roth Section, exhibited the Eſſential
Properties of a Proje&ion, in which all Places whatſoever (except
ſuch as lye under the very. Pole) may be laid down true, according
to their Latitude, Longitude and Bearing, I Mall now proceed to
hew how this Projection (commonly call'd the Mercators Chart) may
be Conſtructed, for any part of the Surface of the Terraqueous Globe.
And firſt, Let it be required to make a particular Chart for å
Voyage
A
3 14
Navigation.
Voyage between the Parallels of 35 and 51 Degrees of Latitude, and
to contain 22 Degrees of Longitude, which will ſerve very well for
any Voyage between the Lizard and the Streights Mouth.
Having provided a Line of Equal Parts of a convenient Length,
1. Draw the Meridian AB, (Tee Plate the 3d.) and at Right-
angles, to it from the Point A, the Right-line AD, which may re-
reſent the Parallel of 35 Degrees of Latitude. Then,
2. From the Meridional Parts anſwering to so deg.
3568.8
Take the Meridional Parts antwering to 35 deg.
2244:3
1324.5
And ſet off the Exceſs or Difference 1324.5, froin A to B, and
draw BC parallel to AD, which will reprefent the Parallel of sı
Degrees of Latitude in this Chart.
3. Set of the equal Miles of Longitude, 60 of which always
make a Degree, upon the Lines AD and BC, from A to D, and
from B to C, and draw the Meridian DC, and the Chart is bound-
cd, then for the Diviſion of the Lines AB and DC, proceed thus ;
Take 2244.3, the Merid. Parts anſwering to 35 Degrees, the leaſt
Degree of Latitude from 2318.0, the Meridional Parts anſwering to
30 Degrees, the next greater Degree of Latitude, and the Diffe-
rence or Exceſs 73.3, ſet oil upon the Meridian Line AB, or DC,
from 35 to 36, will give the 36th Degree of Latitude.
Again, Take 23 18.0, the Meridional Parts anſwering to 36 De-
grees, from 2392.7, the Meridional Parts anſwering to 37 Degrees,
and the Exceſs or Remainder 74.7, ſet off from 36 to 37, on the ſame
Meridian Line AB or DC, will give the 37th Degree of Latitude.
Or,
Take 2 244.3, the Meridional Parts anſweringto 35 Degrees the
leaſt Degree of Latitude from 2392.7 the Meridional Parts anſwer-
ing to 37 Degrees of Latitude, and the Difference or Exceſs 148.0,
ſet oft upon the Meridian from 35 to 37, will give the 37th Degree
of Latitude as before.
After the ſame manner may the whole Meridian Line AB or DC
be divided either into Degrees or Parts of a Degree. Or,
Having ſet off 2 44.3, the Exceſs of the Meridional Parts, anfwer-
ing to 35 Degrees above 2000 downwards, from 35 Deg. in the Me-
ridian Line it will give a Point; froin whence if 318.o the Exceſs of
the
Navigation
315
the Meridional Parts anſwering to 36 Degrees, above 2000 be ſet up-
wards, it will give the 36th Degree of Latitude ; and again, if 392.7,
the Exceſs of the Meridional Parts of 37 Degrees, above 2000
be ſet off from the ſame Point upwards to 37, it will give the 37th
Degree of Latitude; and proceeding in like manner, 'till you come
to the 45th Degree of Latitude, when the Meridional Parts exceed
3000, fer off after that the ſeveral Exceſſes of the Meridional Parts
anſwering to the ſeveral Degrees of Latitude above 3000, from the
Point d (whoſe Diſtance from the 35th Degree is 755.7, the Diffc-
rence between the Meridional Parts of 35 Degrees and 3000.0) viz..
30 Parts from d to 45. 115.6, from d to 46. 202.8, from d to
47. &c. by which means you will have the whole Meridian Line
divided without any previous Subſtraction, and with as much Eaſe
and Expedition as it it were a Line of Equal Parts.
After the ſame manner may each ſingle Degree be ſubdivided into
as many Parts as its Breadth will admit of, by feting off the Ex-
celles of the Meridional Parts an!wering to the ſeveral Parts of
the Degrees, as you did for the whole Degrees.
N. B. If the Chart commence from the Equator, the Meridional
Parts anſwering to the ſeveral degrees and minutes of Latitude, ſet off
in the Meridian from the Equator, will divide the Meridian accordingly
.
But before the Chart can be fit for Uſe, it will be neceſſary to draw
the Rumb Lines in ſome part of it; I have choſe for conveniency fake,
to place them in each Corner, as you may ſee in the Chart, and for
the more ready pricking down of Places, and deſcribing of the ſeveral
Runnings of a Ship, it will be convenient to draw as many Meri-
dians and Parallels of Latitude as the Chart will admit ot without
making Confuſion, and to diſtinguiſh every 5th and roth Line from
the reſt, for the more ready counting of the Latitudes, Longitudes,
and Diſtances of Places.
After this manner, may a Chart be made to ſerve for any Voyage
whatſoever, provided the Latitudes and Longitudesof the Places you
are to fail to and from be known.
Upon this Chart thus made, all Places may be laid down true ac-
cording to their Latitudes and Longitudes, and whar is of the
greateſt Benefit, and gives it the preference before all other Contri-
vances whatſoever, for Nautical Uſes, is, that the Rumb Line, or
the Path that a ship deſcribes in failing according to the Dire&ion
of
316
Navigation.
of the Compaſs, (the only Guide that Mariners have at preſent) is
repreſented or laid down by a perfect ſtreight Line, whence the true
Bearing of any two Places, or the true Courſe that a Ship muſt lail
from one Port to another, may be truly and exa&ly determined, and
that with the greateſt Eaſe poſſible.
This Rectilinearity of the Rumb Line, is the chief and primary
Property of the Mercators Projection, to which all the other Parts are
made Subſervient; and to accompliſh this very thing, after the Invention
of the Compaſs was the great Endeavour of our Fore-fathers, for many
Years, and never perfected, till undertaken by Mr. Wright ; which
happy Invention of his, will make his Name famous to all Poſterity.
And hence it is, that all Nautical Problems are reſolved by this
Projection, to the greateſt Degree of Exadneſs that Proje&tions are
capable of, and infinitely more eaſy, than by the Globe it felf; where,
becauſe the Meridians meet in a Point, the Rumb Lines (as has been
already ſhewn) become ſpiral Lines, which render the laying down
of Runnings (the Principal Data that Mariners have, to find the
Place of the Ship at all times) and determining the Bearings and Di-
tances of Places, (the next thing wanting to direct the Courſe anew)
eſpecially if the Places are pretty far from the Equator, in a manner
Iinpracticable.
As it is impoſſible by any Contrivance whatſoever, to repreſent
any Part or Portion of a Spherical Surface upon a Plane, but that
it will be diſtorted, and appear under a different Form from what
it does upon the Surface of the Sphere, and conſequently different
Projeđed Portions, when compared together, cannot have the ſame
Ratio one to another as to Magnitude, that the Parts they repreſent
have to each other upon the Globe it lelt; yet, that Contrivance that
repreſents them to, that the Dimenſions of the projected Parts, (tho’
diſtorted) when meaſured by proper Scales) retain between themſelves
the ſame Ratio that the Dimenſions of the Parts they repreſent upon
the Globe have to each other, truly and eaaly, will undoubtedly
give true Solutions to all Problems that can ariſe from it; as is the
Cafe of all kinds of Projections of the Sphere whatſoever.
For Example, In this Projection, tho' an Iſland in the Latitude
of 60 Degrees when Projected, will have its Breadth from North
to South, repreſented in near a double Ratio of what it would be,
if the fame and were placed under the Equator, and conſequently
theſe two Projeđed Iands, when compared together, will not have
the
1
Navigation.
317
the fame Ratio between themſelves as to Magnitude, that the Places
they repreſent upon the Surface of the Sphere, really have.
Yet, if the Breadth from North to South, of the diſtorted Iſland,
in the Latitude of 60 Degrees, when compared with the Length from
Eaſt to Weſt, each being Meaſured by proper Scales, be found to
have the ſame Ratio, that the Breadth and Length of the Iſland it
repreſents upon the Surface of the Sphere have to each other; it ne-
ceilarily follows, that all Concluſions drawn from this Repreſenta-
tion are equally true and concluſive with thoſe that are deduced
from the Globe it felf, that it repreſents; or which is the ſame thing,
and comes fiearer to our preſent purpoſe, tho’a Degree of Longitude
in the Latitude of 60 Degrees, be repreſented upon this Projection,
as large again as it really is upon the Surface of the Globe; yet, inaſ-
much as a Degree of Latitude at that diſtance from the Equator is
increaſed in the fame Ratio, and this Law obſerved at every Point
of diſtance from the Equator, it follows, that the true Courſe and
Diſtance between any two Places howſoever ſcituated, will be found
to be the ſame upon this Projection, as they are upon the Surface
of ehe Sphere; and conſequently all the Solutions of the ſeveral Caſes
ariſing from changing the Things given and required, will be the
fame upon this Projection, as they are upon the Surface of the
Globe it ſelf, which renders this Chart the moſt apt and proper In-
ſtrument, and the beſt adapted to the Mariners Uſe, as any that
can be thought of, till Mankind ſhall find out other Contrivances
than the Compaſs, for guiding or directing of a Ship thro' the Wide
and Pathleſs Ocean,
And 'tis upon this Account, that I have been very Particular in its
Conſtruction, as I ſhall be in its ſeveral Uſes, not doubting but the
Candid Reader will excuſe this little Digreſſion, ſince it is done with
an intent to make him a thorough Maſter of the Nature, and Deſign
of this uſeful Proje&ion, and to explain ſome few Difficulties, that
young Beginners are apt to meet with at firſt, as I have often found
by Experience.
1. Let it be required to lay down a Place upon the Chart, its
Latitude, and the Difference of Longitude between it, and ſome
known place upon the Chart being given.
Let the known place be the Lizard, and let the Place to be laid
down be St. Maries, one of the Western Ines, ſcituated in the 37th
Degree of North Latitude, and differing in Longitude from the Li-
Tt
Zard
man
318
Navigation.
Zard 18 deg. 56 min. or 19 deg nearly, St. Maries lying ſo much to
the Weſtward of the lizard.
Thro' the Lizard draw the Meridian ab, which I make the firſt
Meridian of the Chart, and throʻthe 19th deg. of Longitude counted
in the graduated Parallel D A towards the Left-hand, becauſe the
Difference of Longitude is Weſterly, draw the Meridian 19, 19,
where this cuts the Parallel of Latitude of 37 deg. gives the Place of
St. Muries upon the Chart.
After the ſame manner may any other Place be laid down, and
conſequently all that part of the Sea-Coaſt which is contained within
the Limits of the Chart.
On the contrary, by drawing a Parallel of Latitude thro'any
Place till it cut the graduated Meridian, or by taking the neareſt
diſtance of that place from the next Parallel of Latitude, and apply-
ing of it to the graduated Meridian, may the Latitude of that Place
be found.
Hence, and by the Rules delivered in the firſt Sextion of Naviga-
tion, Part the Third, may the Difference of Latitude between any
two Places before found.
And becauſe the neareſt diſtance between two Places lying under
the ſame Meridian, is equal to their Difference of Latitude, hence
we are taught how to find the diſtance between two Places lying
under the fame Meridian in the Mercator's Projection, viz. by finding
the Latitude of each, and if they are both North, or both South,
by ſubſtracting the leffer from the greater ; but if one be North and
the other South, by adding the lefler to the greater, and convert-
ing the Sum or Difference, into Miles or Leagues.
Again, by drawing a Meridian thro’any Place, till it cut the
graduated Parallel, or by taking the neareſt diſtance of that Place
from the next Meridian, and applying it to the graduated Parallel,
may the Longitude of that place be known; or the Difference of
Longitude between it and any other.
Hence, and by the 1ſt Caſe of Section the 8th, we are taught, how
to find the diſtance between any two Places, lying under the ſame
Parallel of Latitude in the Mercator's Projection, by erecting upon
the Segment of the Parallel connecting the two Places together as a
Baſe, an Eqnicrural Triangle with the Sine of 90 deg. and ſetting
off the Sine of the Complement of the Parallel, from the Vertex
downwards upon each of the Legs, for then the diſtance of theſe
IWO
Navigation.
319
two Interfe&ions applyed to the graduated Parallel, will give the
true diſtance betwcen them.
Suppoſe ir were required to find the Miles of Diſtance between
Cape St. Vincent and the Iſland of St. Maries, both lying in the Lati-
tude of 37 deg. oo min. North
On the Baſe of the Segment of the Parallel, deſcribe an Equicrural
Triangle QRT, and make TQ or TR the Radius of a Line of Sines,
fet ott thu Sine of the Complement of the Latitude 37 deg. oo min.
from T to i and W, then the diſtance XW, applied to the gra-
duared Meridian, will give so deg. 59 min. equal to 659 Miles, the
diſtance between them.
Or, make Q R the apparent Diſtance between the two Places, e-
qual to the Sine of 90 deg. oo min. upon the sector, then the diſtance
between the Co-lines of the Latitude on the fame Line of Sines, ap-
plied to the graduated Parallel on the Chart, will give the true Di-
ſtance between the Places,
Hence we are taught, how to form a Scale for the more ready mea-
ſuring of Diſtances, or laying down of Runnings under any Parallel.
Example.
Let it be required to form a Scale of 2 { Degrees, or so Leagues.
Conſtrution.
Becauſe the Breadth of a Degres of Longitude in any Parallel, is
to the Breadth of a Degree of Longitude in the Equator, as the Co-
fine of the Latitude of the Parallel, to the Radius; or as the Radius
to the Secant of the fame Latitude, if the Breadth of 2 Degrees of
Longitude in the Equator, be made the Radius of a Circle,the Secants
of the ſeveral Degrees of that Circle, will be the reſpective Breadths
of 2 { Degrees in the correſpondent Parallel of Latitude in this Pro-
jection; wherefore, make the Breadth of 2 į Degrees in the graduated
Parallel, equal to 00 Degrees upon the Sector ; then will the di-
ſtance between the Secants 35 and 35 on the ſame Lines, be the
Breadth of 2 { in the Parallel of 35 Degrees, the leaſt Latitude upon
the Chart equal to ab, and the diſtance between the Secants of s I
and si Degrees, on the ſame Lines, the Breadth of 2 | Degrees in
the Parallel of 5 1 Degrees in the ſame Chart, equal to cd.
Upon ab, the Breadth of 2 | Degrees, or so Leagues, in the
Parallel of 35 Degrees, erect the Perpendicular ac, of what Length
you think convenient, and draw cd parallel to ab, and equal to the
Tt 2
Breadth
320
Navigation
50
Breadth of 2 } Degrees, in the Latitude of s1 Degrees before found,
and joyn the Points d and b.
And becauſe the ſeveral Breadths
d 4
4 2
of a Degree of Longitude, in the
ſeveral Parallels decreaſe in the ſame
50
50
Ratio as the Secants of the Latitudes
of the ſeveral Parallels increaſe, make
ac equal to the difference of the Se-
cants of 35 and 51 Degrees, the ex-
tream Parallels, then will the difference
45
45
between the Secants of 35 and 40 fet
off from a to 40, give the Point 40 ;
thro' which if the Line 40, 40, be
drawn parallel to ab, it will be the
40
40 Breadth of 21 Degrees, or 5o Leag.
in the Parallel of 40 Degrees.
Again, if the difference between
35
the Secants of 40 and 45 Degrees, be
35
b4 2
ſet off from 40 on the ſame Line ac, to
45, or the difference of the Secants of
35 and 45 Degrees, be ſet off from 35 to 45, and from the Point 45
the Line 45, 45, be drawn, parallel to the former or to ab, it will be
the Breadth of 2 į Degrees or so Leagues, in the Parallelof 45
De-
grees of Latitude; and ſo for any other.
If inſtead of making Uſe of the Line of Secants upon the Se&tor,
(which I take to be the moſt natural and eaſy Way) you make the
Breadth of 2; Degrees of Longitude in the Equator, or in the graduated
Parallel equal to 1000, and upon the Line of Lines, or Equal Parts on
the Sector then will be 122, &c. of the fame Parts the natural Secant
of 35 Degrees, be the Breadth of 2 { Degrees in the Parallel of 35
Degrees; alſo, 1589, c. the natural Secant of 5ı Degrees, the
Breadth of 2 Degrees in the Parallel of 5 1 Degrees of Latitude, c.
Again, if the Length a c of the Scale, be made equal to 368, &C.
the difference of the natural Secants of 35 and 51 Deg. the extream
Parallels of the Chart, then will 847, &c. the difference between
the natural Secants of 35 and 40 Degrees, be equal to the Diſtance
35, 40, on the Line a c of the Scale; alſo, 93.5, 6c. the difference
between the natural Secants of 35 and 45, will be equal to the di-
ſtance between 35 and 45, on the fame Line ab, and ſo for any
other.
Afrer
Navigation.
321
After this manner may a Scale be made to contain any Num-
ber of Degrees, Leagues, or Miles in Length,
and to fit any Chart.
This Scale is not only uſeful in meaſuring of Diſtances, and laying
down of Runnings under the fame Parallel of Latitude, but is appli-
cable in many caſes, without ſenſible Error, to the meaſuring of the
Distances of Places that differ both in Latitude and Longitude; and
I have often admired, conſidering the ſimplicity of its Conſtruction,
and its general Uſe, that one of this kind is not placed in ſome con-
venient Part of every Sea-Chart that is Publiſh'd,
Hence if two Places lye under the ſame Parallel of Latitude, and
the Longitude of one be known, and its diſtance from the other, it
will be eaſy from the Place of one to find the Place of the other.
2. Let ic be required to lay down a Place upon the Chart, its La-
titude and Bearing from ſome known Place in the Chart being given,
or which is the ſame thing, having the Courſe and Difference of
Latitude that a Ship hath made, to lay down the Running of the
Ship, and find her Place upon the Chart.
Example.
A Ship from the Lizard, in the Latitude of so deg. oo min. Nortli,
fails S. W. by S. until ſhe be in the Latitude of 45 deg. co min. North,
I demand her Place upon the Chart.
Thro' the Lizard draw the Line L c parallel to the S. W.by S. Line,
till it cut the Parallel of 45 Degrees in c, then will c be the Place of
the Ship, and Lo the Line upon which ſhe has fail’d.
On the contrary, by laying a Ruler over any two Places, and
drawing a Line parallel to it thro' the Center of any Compaſs, may
the Bearing of any two Places be found; or the true Courſe
upon
which a Ship muſt fail from one Port to arrive at the other.
5. Let it be required to lay down a Place upon the Chart, its
Bearing and Diſtance from ſome known Place in the Chart being gi-
ven; or which is the ſame thing, having one Latitude, Courſe and.
Diſtance fail'd, to lay down the Running of the Ship, and find her
Place upon the Chart.
Example
Suppoſe a Ship at s, in the Latitude of 45 deg. oo min. North,
fails S. E. by S. į E. 18. Leagues, or 543 Miles, and it be required
to find the Place of the Ship upon the Chart.
Having
322
Navigation.
Having drawn the Meridian Line S m thiro's, the Place the Ship
deparced from, draw the Rumb Line Sp parallel to the S.E.by S. E.
Line, and ſet off upon it 181 Leagues from Stof, and draw the Pa-
rallelfe, then apply the proper Difference of Latitude Se, to the gra-
duated Parallel, and obſerve the Number
of Degrees it contains, count
the ſame Number of Degrees in the graduated Meridian from n to
0, and draw the Parallel o mp, where this cuts the Rumb Line be-
fore drawn in p, gives the Place of the Ship.
If the Diſtance be not very great, cſpecially if the places are near
the Equator, :ake the diſtance from the Scale anſwering to the Mid-
dle Latitude, and apply it on the Rumb Line from Stop, and it
will at once give the Place of the Ship.
Hence we are taught, how to find the diſtance between any two
Places upon the Chart, that differ both in Latitude and Longitude,
viz. by taking the Difference of Latitude between the two Places
from the graduated Parallel, and at that diſtance drawing the Line
ef parallel to nsc, till it interſect the Line Sp, connecting the cwo
Places together in f; for then the diſtance sf applied to the gradua-
ted Parallel, will give the diſtance between them: Or by laying a
Ruler over the two Places, and with the Difference of Latitude be-
tween the Points of the Compalles, carrying one Foot along the
Edge of the Ruler, till the other turned about juſt touches the Paral-
leinsc, for then the diſtance between the Pointf where one Foot of
the Compafles reſts, to the Place S, applied to the graduated Paral-
lel, will give the diſtance between them.
Or if the places are not far aſunder, eſpecially if they are near the
Equator, by taking the Diſtance Sp between the Points of the Com-
pailes and applying it to the Scale, upon that Part which anſwers to
the Middle Latitude between the two Places, for then the Diſtance
meaſured upon this Line, will give the diſtance between the two
Places.
4.
The Latitude of any Place being g ten, and its diſtance from
fome known place upon the Chart, to lay down the Place upon the
Chart; or the Diſtance and Difference of Latitude being given, to
lay down the Running of a Ship, and find her Place upon the Chart,
Example
Suppoſe a Ship at S, in the Latitude of 45 deg. oo min. North,
fails between the S. and Eaſt 543 Miles, or 181 Leagues, and then
is
1
Navigation.
323
is found to be in the Latitude of 38 deg. oo min. North, and it be
required to find the Place of the Ship upon the Chart.
Having drawn the Meridian Line Sm, take off the Difference of
Latitude from the graduated Parallel, and ſet it off downwards in
the Meridian from S to e, and draw the Parallel ef, then with the di-
ftance between the Points of the Compafles taken from the graduated
Parallel, ſetting one Foot in S, with the other curned about, croſs
the Parallel laſt drawn in f, and draw the Line Sf, till it meet the
given Parallel of Latitude in p, then will p be the place of the Ship
in the Chart : Or take the diſtance from the Scale, upon the Line
anſwering to the Middle Laticude, and ſetting one foot in Sthic
place given, with the other croſs the given Parallel of Latitude in P,
the place of the Ship.
5. To lay down a Place upon the Chart, its Bearing and Difference
of Longitude from ſome known place upon the Chart being given ;
or which is the fame thing, one Latitude, Courſe, and Difference
of Longitude being given, to lay down the Running of the Ship,
and find her Place upon the Chart.
Example.
Suppoſe a Ship at s, in the Latitude of 45 deg. oo min. North,
fails S. E. by S. E. until her Difference of Longitude be 154 Leagues,
or 7 deg. 42 min. and it be required to find her Place upon the Chart,
Having counted 7 deg. 42 min. the Difference of Longitude in the
graduated Parallel from z to y, draw the Meridian vp, where this
interſects the S.E.by S. E. Line drawn from Sas in p, gives the Place
of the Ship: Or having drawn yp parallel to Sm, at the di-
ſtance of 154 Leagucs, where this cuts the Rumb Line before drawn,
as in p, gives the Place of the Ship.
6. Let it be required to lay down a Place, its Bearing and
Departure from ſome known place upon the Chart being given,
or which is the ſame thing, one Latitude, Courſe and Departure
being given, tülay down the Russning of the ship, and find her Place
upon the Chart.
Example.
Supppoſe a Ship at S, in the Latitude of 45 deg. co min. North,
fails S. E. by S. E until her Departure be 344 Miles, and it be
required to find her Place upon the Chart.
Having
324
Navigation.
Having drawn the Meridian Line S m thro’the Place of the Ship;
and the Rumb Line Sp parallel to the S. E. by S. į E. Lint, with
the Meridian Diſtance 344 Miles, draw a Meridian parallel to jim,
till it interſect the Rumb Line in f, and draw the Parallel fe.
Laſtly, apply the diſtance S c to the graduated Parallel, and count
the Number of Degrees it contains in the graduated Meridian trom
n to o, and draw the Parallel om p, where this interfeets the Rumb-
Line Sf produced, as in p, gives the Place of the Ship.
7. The diſtance of any place from ſome know.. Płace upon the
Chart, and its Departure being given, to lay down the Place
upon the Chart, or which is the ſame thing, one Latitude, Diſtance
ſail'd, and Departure being given, to lay down the Ships Running,
and find her Place upon the Chart.
Example.
Suppoſe a Ship at s, in the Latitude of 45 deg. oo min. North,
Sails 543 Mies between the South and Eaſt, and then is found to
have departed from her former Meridian 344 Mies, and it be re-
quired to find the Place of the Ship upon the Chart.
Having drawn the Meridian S z, thro's the Place the Ship de-
parted from, ſet off the Departure in the graduated Parallel from
z tou, and draw the Meridianu f, then with the Diſtance
fail'd, ſetting one foot of the Compafles in S, with the other croſs
the former Meridian in f, and draw the Parallel fe.
Laſtly, apply the diſtance fe to the graduated Parallel, and count
the Degrees it contains in the Meridian from n to.0, and draw the Pa-
rallel omp, till it meet the Line Sf produced in p, then will p be
the Place of the Ship.
8. The Latitude of a Place being given, and its Departure
from fome known place upon the Chart, to lay down the Place
upon the Chart, or one Latitude, Departure, and Difference of La-
titude being given, to lay down the Running of the Ship, and find
her Place upon the Chart.
Examples
Suppoſe a Ship at S, in the Latitude of 45 deg. oo min. North,
fails between the South and Eaſt, until ſhe be in the Latitude of
38 deg. oo min. North, and then is found to have departed from her
furmer Meridian 344 Miles, and it be required to lay down her Run-
ning upon the Chart, and find the Place of the Ship.
From
***
Navigation
325
Thus have I ſhewn from Principles firſt laid down, and then de-
monſtrated, not only how a General or Particular Chart may be made,
but how any Place, and conſequently any Country, or any part of
the Sea-Coaſt may be laid down, from any poſible Data; as alſo
how the careful Mariner may Trace out his ſeveral Runnings, and
thence know the Place he is in at all times, and find upon what Courſe
he muſt ſteer, and how far, to arrive at any Fort; and fince I have
all along endeavoured to render the moſt uſeful Problems as eaſy to be
perform’d by this Chart as by the Plain Chart, I hope it will be a ſuffi-
cient Inducement to him to make Uſe of no other Chart for the future.
Section XIV.
Containing the Application of Splerical Trigonometry, to the
Solution of ſuch Problems as ariſe from Sailing by or upon,
1h2 Arch of a Great Circle, and uſually called Great Circle
Sailing.
A
Ltho' it has been judged proper to place this section here, as
being more immediately a part of the Nautical Science, yet
it is abſolutely neceſſary, that the Reader be well acquainted with
the Do&trine of Spherics, before he attempts to meddle with it.
And indeed of all kinds of Sailing, this is moſt certainly the trueſt;
and ought in all Caſes to be followed, if it could eaſily be put in
Practice : For, as the neareſt diſtance from one point to another
upon a Plain Surface, is meaſured by a Right-line conneæing the
Points together; ſo the neareſt diſtance from one point to another
upon a Spherical Surface, ſuch as is that of our Terraqucous Globe,
is meaſured by an Arch of a Great Circle paſſing thro' the two .
Places.
Hence it is, that the nearer any Ship keeps to the Arch of a Great
Circle, the ſhorter will her Paſlage be from one Place to another.
And therefore, How from the Latitudes and Longitudes of any
two Places, to find the Arch of the neareſt diſtance between them,
and thence the ſeveral Courſes the Ship muſt Sail upon, according to
the
Uu
AV
S.
.lt Atwo
20 19 18 17
116
15
14 113
13 19 11
110
8
Q 211
51
B
51
a
C
LL50
0
49
49
48
48
42
47
46
T
4.5
п.
45
44
43
42
42
41
41
40
el
40
***
39
39
W
38 to
X
nte
38
1
1
37
Ꭱ ;
37
***
3
36
61 D
35
y
36
TE
TE
21 20 19 18 12 16 15 14 13 12 11 10 0
326
Navigation.
the Direction of the Compaſs, is the proper Buſineſs of Great Circle
Sailing;
and ſhall be treated of as fully as the Nature of the Subject
and ihe Intent of this Treatiſe will allow of.
As the Angle which any Rumb forms with the Meridian is called
the Courſe or Bearing, ſo the Angle which any Great Circle forms
with the Meridian is called the Angle of Poſition. Whence, in
Prob. 1.
If the Places propoſed lie under the fame Meridian, and conſe-
quently differ only in Latitude, then the Angle of Poſition vaniſhes :
and the Arch of Difference of Latitude becomes the Arch of neareſt
Diſtance, and this admits of three Varietys,
Caſe I.
If one Place lie under the Equator, and the other in tlie Northern
or Southern Hemiſphere, then, in this Caſe, the Latitude it ſelfbe-
comes the Difference of Latitude, and therefore reduced into Miles
gives the Length of the neareſt Diſtance.
Thus, becauſe Cape Verd lies in the Northern Latitude of 14 deg.
-43 min. its diſtatice from the Equator in the Arch of a Gitat-Circle,
is 14 deg. 43 min. cqual to 883 Nautical Miles. Again,
Caſe II.
If the Places propoſed lie on the ſame ſide of the Equator, that
is both in the Northern), or both in the Southern Hemiſphere then
the Exceſs of the Latitude of one Piace above that of the other, will
give the Length of the Arch of neareſt Diſtance,
Thus the Arch of the neareſt Diſtance between Cape Britain on
the Conft of Newfound-land, in the Latitude of 46 deg. 32 miu North,
and Barbadoes in the Latitude of 13 deg. 30 min. North, will be
found to be 33 deg. 02 min. or 1982 Nautical Miles.
Caſe III.
If the Places propoſed lie on contrary Sides of the Equator, that
is, one in the Northern, and the other in the Southern Hemiſphere,
then in this Caſe the Sum of the Latitudes will give the Arch of
neareſt Diſtance.
Tlus
Navigation.
327
Thus the Arch of the neareſt Diſtance between Cape Palmas on the
Coaſt of Guinea, in the Latitude of 4 deg. 5 min. North, and St.
Helena, in the Latitude of 10 deg. oo min. South, is 20 deg. 5 min.
equal to 1205 Nautical Miles. And in each of theſe laft Caſes, the
dire& Courſe by the Compaſs from one place to another, is direct
North or direct South, according as the Places are ſcituared.
Prob. II.
If the Places propoſed differ only in Longitude, and are both ſcitu-
ated under the Equator, then the Angle of Poſition is a Right Angle,
the Courſe by the Compaſs is due Eaſt or due Weſt, and the Diffe-
rence of Longitude becomes the Length of the Arch of neareſt Di-
ſtance.
Thus the Length of the Arch of neareſt Diſtance between Comau,
at the Entrance of the great River Amazons, and the Iſle of St.
Thomas, near the coaſt of Africa both fcituated under the Equator,
will be found to be in the Latitude of 59 deg. 30 min. equal to 3570
Nautical Miles, equal to the Difference of Longitude between the
two Places.
If the Places propoſed lie under any Parallel of Latitude, as ſup-
poſe the Iſland of Pengrin upon the Coaſt of Newfound-Land, and
the Lizard at the Mouth of the Britiſh Channel, each being ſcituated
in the Northern Latitude of so deg. oo min. North, and differing in
Longitude 48 deg 50 min. the Iſland of Pengwin lying ſo much to the
Weſtward of the Lizard and it be required to find the Ang'e of Po-
fition, Arch of ncareſt Diſtance, greateſt Latitude of the Arch, c.
between them.
In the adjacent Figures, Let P. Q, S, A, repreſent the Me-
ridian, paſſing thro' Pengwin ifle, P the North Pole, s the South
Pole, and AQ the Equator.
4
U12
1. Ser
;
328
Navigation.
1
N
F
a
m
А A
E
D
5
PL the Complement of the given Latitude, the Angle NPL equal
1. Set off so deg. oo min. from A to
N, then will N repreſent the Coaſt of
Newfound-land.
R
2. With the Secant of 48 degrees
so minutes the Difference of Longi-
tude betwixt the Points of the Compaf-
ſes, ſetting one Foot in P, with the o-
ther croſs the Equator (if need be) in
C, about which Point as a Center,
with the ſame Diſtance, deſcribe the
Circle PLS; this will repreſent the Meridian of the Lizard.
3. About P as a Pole, with the diſtance of PN, the Complement
of Latitude, deſcribe a Parallel or ſmall Circle, to cut the Meridian
Pons before drawn, in L, the Place of the Lizard.
4. Thro’the Points N, L, D, draw the Great Circle NLD, then
in the Equicrural Spherical Triangle LP N are given, P N equal to
to the Arch Am, the Difference of Lor:gitude, to find LN the
Arch of the neareſt Diitance ; and the Angle PLN the Angle of
Poſition between the Lizard and Newfound-land, in this caſe equal
to the Angle PNL, the Angle of Poſition between Newfound-land
and the Lizard.
5. From the Point P, draw the Arch PR Perpendicular to NL,
this will be equal to the Complement of the greateſt Latitude, thro
which the Arch of neareſt Diſtance does paſs.
6. Wherefore, becauſe in the Right-angled Spherical Triangles
LRP, NRP, Right-angled at R, the Side or Arch PR is common,
and the Hypothenuſes or Arches PN and PL are equal, the Angles
LPR, NPR, are each equal to half the Difference of Longitude,
or Angle LPN, and LR equal to RN, equal to half the Arch of
neareſt Diſtance. Whence,
1. To find the greateſt Latitude the Arch does paſs thro' it will
be, As R:cs į Angle LPN::t, LP:t, LR. That is
As the Radius
10.0000000
To the Co-fine of half the Diff. of Long. 24° 25'-_-2.9593102 .
So is the Co-tangent of the given Latitude 50.00
9.9238135
To the Co-tangent of the greateſt Latitude 52° 37 9.8831237
2. To
Navigation.
329
2. To find LR or RN, half the neareſt Diſtance, it will be
AsR:S, LPN::S, LP=PN:S, LR=RN. That is,
As the Radius
10.0000000
To the Sine of the halt Dift. of Longitude 24° 25' 9.6163382
So is the Co-line of the given Latitude 5oº oo' 9.8080675
To the Sine of the half Diſtance 55° 24' 33 94244057
Which therefore doubled, gives 30 deg. 49 min. 06 ſec. equal to
1849.1 Nautical Miles, equal to the Length of the Arch of neareſt
Diſtance between the Lizard and Newfound-land.
The Length of the Correſpondent Arch of the Parallel may be
readily found by the ad Caſe of Parallel Sailing; thus,
As the Radius
10.0000000
To the Co-line of the given Latitude soº oo 28080675
So is the Difference of Longitude in Miles 2930 3.4668676
To the Length of the Arch of the Parallel 1883.4–3.2749351
Differing from the Arch of a great Circle before found 34.3
Miles..
By comparing the two latter Proportions together, it appears (by
the 15th of the 8th ) that the Sine of the half Difference of Longitude,
is to che Sine of half the neareſt Diſtance, as the Length of the Arch
of the Equator anſwering to the Difference of Longitude, to the
Length of the Arch of the Parallel correſponding to, or agreeing.
with the Arch of neareſt Diſtance.
3. To find the Angle of Poſition, or Angle PLN, it will be,
As R:t, } <LPN:: cs, PL:ct, PLN. That is,
As the Radius
10.0000000
To the Tangent of the half Diff
. of Longit. 24° 25' 9.6570280
So is the Sine of the given Latitude 5oºoo
9.8842540
To the Co-tang. of the Angle of Poſition 70° 49'4–9.5412820
Which according to the Direâion of the Compaſs, is North
70 deg. 49 min. Weft, or Weſt North Weſt z deg. 19 min. Weft.
It is abundantly manifeſt, that when the Ship arrives at R, the
Angle of Poſition, which at the Point L was only N.70 deg. 49 min.
Weſt
, becoines go deg. oo min. and therefore the Bearing is.
2
due
---
330
Navigation.
!
.
cba
due Weſt; and conſequently the Ship in failing from L to R has paf-
fed thro' all the Varieties of Angles of Poſition, between 70 deg.
49 min. į, and 90 deg. oo min. and continuing ſailing from thence
to N, returns thro' all the ſame Varieties of Angles till arriving at
N, the Angle of Poſition becomes 70 deg. 49 min. as at firſt; and
hence ariſes
the great Difficulty or rather Impoffibility, of failing by,
or upon the Arch of a Great Circle, according to the Direction of
the Compaſs: And this put Men upon contriving Methods of Approxi-
mateing as near as poſſible to the Truth, by dividing the whole
Arch into ſeveral ſmall ones, and finding the Courſe and Diſtance
upon each feparately, upon this Suppoſition, that a ſmall Arch differs
inſenſibly from a Right Line, as has been formerly ſhewn.
Hence it is, that the ſmaller the Diſtance is, the nearer does the
Line upon which the Ship fails, correſpond or agree with the Arch
of the Great Circle ; and the leſs the Difference between the Angle
of Poſition and the Courſe Steer'd by the Compaſs.
The Arch of five Degrees differs but the ta zo from the Length of
the correſpondent Tangent, and conſequently the Length of the Arch
and correſpondent Portion of the Rumb-Line will very nearly agree;
as will ſtill be more evident, if we Calculate the Length of each ſe-
parately, and compare them together.
In the adjacent Figure, Let P reprelent the Pole, PN the Meridi-
an of Newfound-land, and PL the Meridian of the Lizard, then will
NL be the Arch of neareſt Diſtance between them, and PR the
Complement of the greateſt Latitude of the Arch.
P
Let the ſeveral Meridians
Pa, Pb, Pc, &c. be drawn,
forming Angles with each
other ots Degrees, then will
the Arches La, ab, bc, &c.
be the ſeveralPortionsofthe
Arch of neareſt Diſtance,
anſwering to the equal Dif-
ference of Longitude, or
Angles LP a, aPb, bPc, &c.
N
Let it be required therefore, to find the Length of the Segment
or Portion La, of the Arch of neareſt Diſtance correſponding to
the Angle LPA, or Arch of Difference of Longitude between the
Places I and a.
Then
1 hg ferd
I
1
Navigation
331
Then in the two Spherical Triangles LRP, ORP, Right-angled
at R, are given the Angle LPR equal to 24 deg. 25 min. the Angle
APR equal to 19 deg. 25 min. and the Perpendicular PR common,
whence to find the Arch Ra, it will be by the Rules of Spheric Tri-
angles.
As the t,<LPR: t,<aPR::t, RL: t, RA. That is,
As the Tangent of the <LPR=24° 25'
9.6570286
To the Tangent of the <aPR=1925'
9.5471377
So is the Tangent of the Arch RL=15° 24' 33" -9.4403078
To the Tangent of the Arch Ra=12° 04' 45" -9.3304175
o
Which taken therefore from RL 15 deg. 24 min. 33" leaves 3 deg:
19 min. 48", equal to 199 ië Nautical Miles, for the Length of
the Arch La.
Let us now proceed to Calculate the Length of thic Portion of the
Rumb Line intercepted between the Points L and a. And,
1. In the Right-angled Spherical Triangle aRP Right-angled at
R, are given P R equal to 52 deg. 37 min. 06" and the Angle R P a
equal to 19 deg. 25 min. whence to find the Latitude of the Point a,
it will be,
As cs, <RPA:P ::t, PR:t, Pa. Tlaat is,
As the Co-Sine of the Angle RPa=19° 25
9.9745 697
To the Radius
10.0000000
So is the Tangent of PR=37° 22' 54''
9.8831237
To the Co-rang. of the Lat. of the Point a =50°59' 17'' 9.9085540
The Latitude of the Points L and a being thus obtained, and the
Arch of Difference of Longitude: the Courſe according to the Di-
rection of the Compaſs from 1 ton, and thence the Diſtance or
Length of the Portion of the Rumb Line contained between L and a
may be found, by the firſt Caſe of Mercator's Sailing ; thus,
From the Meridional Parts of 50° 59' 17''
3567.7
Take the Meridional Parts of soºoo' 00'
-3474.5
Remains the Meridional Difference of Latitude-
93.2
Whence
int
332
Navigation..
Whence to find the Courſe it will be,
As the Meridional Diff. of Latitude 93.2
1.9694159
To the Radius
So is the Dift. of Longitude sº oo'=300 Miles
10.0000000
-2.4771213
To the Tangent of the Courſe 72° 44' 30'
10.5077053
Whence to find the Diſtance or Length of the Rumb Line it
will be,
As the Radius_
10.0000000
To the Secant of the Courſe 72° 44' 30"
10.5 27711I
So is the proper Difference of Latitude 59.283 1.7729302
To the Length of the Rumb Line required 199.82-- 2.3006413
Differing but să o or só part of a Mile, from the true Length of
the Arch of neareſt Diſtance ; which alone is a ſufficient Inſtance of
the Truth of Mercator's Sailing, and ſhews the near Agreement there
is between the Length of the Rumb Line and Arch of neareſt Di-
ſtance, in ſmall Runnings.
To ſhew the great Agreement between the two different Methods
of Inveſtigation, it was abſolutely neceſſary, to determine the Lerigth
of the ſeveral Arches to Seconds; but for all Nautical uſes, if they
are found to Minutes it is more than ſufficient.
Again, If the Courſe and Diſtance from a to b be required, then
in the Right-angled Spherical Triangle RPb, Right-angled at R,
are given, RP equal to 37 deg. 23 min. and the Angle RPb equal to
24°25'-10° 80'=14° 25', whence to find the Latitude of the Point
b, it will be as before,
As the Co-line of the Angle RPb=14° 25'
9.9861045
To the Radius
10.0000000
So is the Tangent of RP=37°22'54"
-9.8831237
To the Co-tang. of the Lat.of the Point b 51° 43' 49'9,8970192
Whence
Navigation.
333
Whence to find the Meridional Dif:rence of Latitude,
From the Meridional Parts of 5 1° 43' 49".
3639.0
Take the Meridional Parts of 50° 59' 57''
3567.7
Remains the Meridional Difference of Latitude
-71.3
Whence to find the Courſe it will be, by the ift Caſe of Mercator's
Sailing,
As the Meridional Difference of Latitude 71° 3' 1.8530895
To the Radius
10'0000000
So is the Difference of Latitude 300
2 4771213
To the Tangent of the Courſe 76° 37' 51"
10.5240318
Whence to find the Diſtance it will be,
As the Radius
10.0000000
To the Secant of the Courſe 76° 37' 51"
10.6359665
So is the proper Difference of Latitude 44.53 1.6486527
To the direct Diſtance 192.58
2.2840192
After the ſame manner may the ſeveral Courſes and Diltances from
b to c, from c to d, c. be found.
For, in the ſeveral Spherical Triangles R Pc. RPb, &c. Right-
angled at R, the Side PR or Perpendicular is given, and the vertical
Angles RPc, RPd, &c. are found by a continual Subſtraction of 5.00
degrees on one ſide of the Perpendicular, and the Addition of 5.00
degrees on the other ; according as the Ship departs from L to Sail
to N, or departs from N:o Sail to L.
Thus the Angle RPc is equal to RPL-5.00, equal to 14° 25' ---5
=9° 25', the Angle RPd=4° 25'; and conſequentiy RPe=5° 00'
4-25'=00° 35', for the Ship in failing from d to e, pafles by the
vertical, or in this Caſe the moſt Northermoft Point of the Arch
LRN, and arrives on the contrary ſide of the Perpendicular, and cor-
ſequently the Angle RPE=00° 35'+5° 00'=5° 35' the Angle
RPg=10°35, the Angle RPh=15° 35', the Angle RPI=20° 35',
and the Angle RPN=24° 25' as at firſt, whence the Difference of
Longitude or Angle iPN, in the Oblique Splerical Triangle iPN,
is equal to 30 so' only.
Xx
Hence
334
Navigation
1
a to b
broc
Hence the Latitude of the Point c will be found to be 52° 14' 33
of the Point d 52° 33' 69', of the Point e 52° 37' 05", of the Point
$ 52° 29' 14", of the Point 8 52° 08' 31", of the Point h 51° 34' 41",
of the Point i 50° 46' 49", and of the Point N 50° oo' oo".
Whence the Meridional Difference of Latitude between b and c
will be 49.90, between c and d 28.82, between d and e 7.95, be-
tween e and f 12.71, between f and g 33.94, between g and h 54.76,
between h and i 76.34, and between i and N 73.45; and conſe-
quently,
The Courſe from Ito a is N 72 44 30 W Diſtance 199.82 1
N 76 37 51 W Diſtance 192,583
N 80 33 23 W Diſtance 187.353
c to d N 84 30 44 W Diſtance 184.045
N 88 28 55 W Diſtance 183.679
e to f S 87 31 03 W Diſtance 179.685
ftog
S 83 32 43 W Diſtance 184.276
S 79 39 19 W Diſtance 188.416
k to i S 75 43 23 W Diſtance 194.097
i to N S 72 17 21 W Diſtance 153.892
And the whole Diſtance from L to N=1847.847.
If the direct Courſe and Diſtance from the Lizard to the Vertical
or moſt Northermoſt Point of the Arch of the great Circle be required:
,
From the Meridional Parts of the Latitude of the Arch}=3725.87
of neareſt Diſtance 52° 37' 06"
Take the Meridional Parts of the Latitude of the Li-
zard soº oo oo".
}=3474.50
=
Remains the Meridional Difference of Latitude
251.37
d to e
& to k
.
Li-}=
Whence to find the direct Courſe, it will be,
As the Meridional Difference of Latitude 251.37 ---2.4003134
To the Radius
10,0000000
So is the Difference of Longitude 24°25=14652.1638376
To the Tangent of the Courſe 80° 15' 50". 10.7655242
Which becauſe the Difference of Latitude is Northerly, and the
Difference of Longitude Weſterly, is North 80° 15' 30" W. or
Weſt by North 1° 31' Weſterly.
Whence,
Navigation
335
Whence to find the direct Diſtance, it will be,
As the Radius
me 10.0000000
To the proper Difference of Latitude 157.1
So is the Secant of the Courſe 80° 15'50"
2.1961762
10.7718293
To the direct Diſtance 928.987-
2.9680055
Whence, becauſe the Triangles LPR and NPR are equal, and the
Ship in failing from R to N ſails Weſterly, and from a greater North
Latitude to a lefler, the true Courſe will be South 80° 15' 50" Welt,
or Weſt by South 1° 31' Weſterly; and direct Diſtance 928.987.
Hence the whole Diſtance will be 1857.974 Miles, which exceeds
the Arch of neareſt Diſtance 8.874 Miles, but is leſs than the
Length of the Parallel 25.426 Miles.
Prob. III.
Two Places differing both in Latitude and Longitude, their La-
citudes and Difference of Longitude are given, to find the Arch of
neareſt Diſtance between them, Angles of Poſition, &c.
This Problem admits of three Varieties.
1, When one of the Places is ſcituated under the Equator, and the
other in the Northern or Southern Hemiſphere.
2. When both the Places are ſcituated in the ſame Hemiſphere.
3. When the Places propoſed are ſcituated in different Hemiſpheres,
that is, one in the Northern Hemiſphere and the other in the
Southern.
Of each of theſe in their Order, And
Caſe 1.
1. Let it be required to find the Arch of neareſt Diſtance,
Angles of Poſition, Latitudes of thoſe Places thro' which the
Arch palles at every s Degrees of Difference of Longitude ; allo
the Courſe and Diſtance from each of thoſe Places to the other: be-
tween the Lizard ſcituated in so deg. 00 min. of Northern Latitude,
and the Illand of Comau ſcituated under the Equator, differing from
each other in Longitude 45 deg. oo mip. Comau lying ſo much to the
Weſtward of the Lizard.
Xx2
In
336
Navigation.
P
In the adjacent Figure, Let P repreſent
the North Pole, S the South Pole, AL
the Equator, PASQ the Meridian paſſing
thro' Comau, A the Place of Comau it
ſelf.
d
L
А
N
C
P
S
C
1. With the Secant of the Difference of
Longitude between the Lizard and Comau,
equal to 45 deg. oo min. ſetting one foot
of the Compailes in P, with the other
croſs the Equator in l, about which
Point as a Center, with the ſame Diſtance
deſcribe the Circle PLS, this ſhall repreſent the Meridian of the
Lizard.
2. Set off the half Tangent of the Difference of Longitude from
to p, this will give the Pole of the Meridian PLS.
3. Set off the Latitude of the Lizard from A to d, a Ruler laid
over d and the Pole p, will cut the Meridian PLS in the Point L, the
Place of the Lizard, thro' which and the Points A and Q, if the
Great Gircle ALL, be drawn, the thing is done.
Then in the Oblique-angled Spherical Triangle APL, are given,
AQ the diſtance of Comau from the Pole, in this Caſe equal to godeg.
oo min. or a Quadrant, PL the Complement of Latitude of the Li-
zard, and the Angle APL, equal to their Difference of Longitude,
to find the Angle PAL, the Angle of Poſition between Comau and
the Lizard, the Angle ALN the Angle of Poſition between the Li-
zard and Comau, and the Side AL the Arch of neareſt Diſtance, or be-
cauſe the Triangle APL is a Quadrantal Triangle in the Supplemen-
tal Right-angled Spherical Triangle ANL, are given, LN the Lati-
tude of the Lizard, AN the Difference of Longitude between the
Lizard and Comau, and the contained Angle ANC Right, to find
the Angle ALN the Angie of Poſition between the Lizard and Comau,
the Angle LAN the Complement of the Angle LAP, tlie Angle of
Poſition between Comau and the Lizard, and the Hypothenuſe AL
the Arch of neareſt Diſtance. Whence to find
1. The Angle ALN the Angle of Poſition between the Lizard and
Comau it will be,
As
:
resep
Navigation.
337
As the S, LN:R::t, AN:t, KALN. That is,
As the Sine of the Latitude of the Lizard soºoo'-_-9.8842540
To the Radius
10.0000000
So is the Tangent of the Dift. of Longitude of 45°00' 10.0000000
To the Tangent of the Angle ALN 52° 32' -10.1157460
Or Angle of Poſition at the Lizard.
Which, becauſe Comau lics under the Equator and to the Weſt-
ward of the Lizard, is S. 52 deg. 33 mio. W.
2. To find the Angle of Poſition at Comau or Angle LAP, it will
3
be,
As S, AN:R::t, LN:ct, <LAP. That is,
As the Sine of the Difference of Longitude =45ºoo!--9.8494850
To the Radius
10.0000000
So-is the Tang, of the Latitude of the Lizard soºoo' 10.0761865
To the Co-tangent of the Angle LAP 30° 41' 4-10.2267015
Or Angle of Poſition at Conau, which becauſe the Lizard lies to the
Northward and Eaſtward of Comau, is N. 30 deg. 41 min. & E.
3. To find the Hypothenuſe AL, or Arch of neareſt Diſtance, it
will be,
As R:cs, LN:: CS,AN : cs, AL. That is,
As the Radius
10.0000000
To the Co-fine of Latitude of the Lizard 50° 0' -9.8080675
So is the Co-fine of the Difference of Longitude 45.00 9.8494850
To the Co-line of the Arch of neareſt Diſt. 62.58--9.6575525
Or
3778 Nautical Miles.
Now to find the Latitudes of the ſeveral Points a, b, c, d, &c.
thro' which the Arch paſſes at every 5 Degrees of Difference of
Longitude ; in the adjacent Figure,
Lec
338
Navigation.
Wile/d/cba
P Let P repreſent the Pole, A the
Iſland of Comau, L the Lizard,
AL the Arch of neareſt Diſtance,
and AN the Arch of the Equator
anſwering to the Angle APL, the
Difference of Longitude between
the two Places, in this caſe 45.00
L
degrees, and let the feveral Me-
ridians Pi, Pk, Pl, &c. be drawn,
forming Angles with each other
equal to 5.00 degrees, where
N theſe interfect the Arch of neareſt
9 ponin I ki
Diſtance, as in the Poirits a, b, c,
&c. give the ſeveral Places in that Arch anſwering to every 5th De-
gree of Difference of Longitude, and the ſeveral Portions ia, kb, 1c,
&c. 5 of the Meridians Pi, Pk, P.1, &c. the Latitudes of thoſe
ſeveral Points.
And becauſe the Angles at the Pole P, are meaſured by the corre-
ſpondent Arches of the Equator in the ſeveral Spherical Triangles
Aia, Akb, Alc, &c. Right-angled at i, k, l, &c. are given, the An-
gie LAN, the Complement of the Angle of Poſition at Comau, and
the ſeveral Baſes or Arches Ai=40.00, Ak=35:00, Al=30.00,
&c. each differing 5 Degrees from the other; to find the ſeveral
Perpendiculars i a, kb, 1c, &c. the Latitudes of the ſeveral Points
a, b, c, &c. anſwering to the ſeveral Differences of Longitude Ni, ik,
kl, &c. wherefore to find, firſt the Latitude of the Point A,
ic
will be,
As R:S, Ai:: t,<i Aa:t, ia. That is,
As the Radius
10.0000000
To the Sine of 40° oo'
9:8080675
So is the Co-tangent of the Angle of Poſition 59° 19' 10.2267015
To the Tang. of the Lat. of the Point a=47.174-10.0347690
Again, to find the Latitude of the Point b, it will be,
R: S, Ak:: t,<b Ak (=LAN): bk. That is
AS
.
:
Navigation
339
hoo.4 S30 484 W Diſtance 583.65
As the Radius
-- 10.0000000
To the Sine of Ak=35.00
-9.7585913
So is the Tangent of <LAN=59.19
10.2267015
To the Tangent of the Lat of the point b=44.01$ 9.9852928
After the fame manner the Latitude of the Point c will be found to
be 40 deg. 07 min. à, of the Point d 35 deg. 27 min. i, of the Point
e 29 deg. 57 min. ', of the Point f 23 deg. 34 min. of the Point g
16 deg. 18mir. , of the Point b deg. 21 min. 4, and of the Poitit
A o deg. oo min.
Hence the Meridional Difference of Lar. between L and a, will be
found to be 246.2, between a. and 6.279.98, between b and c 316.16,
between c and d 35404, between d and e 392.5, between e and f
429.92, between f and g 463.5, between g and h 489.1, between ha
and A 503,1; and conſequently,
The Courſe from L to ais S so 37 W Diſtance 256,62
a to b S47 00 W Diſtance 286:80
W toc S 43 30 W Diſtance 323.40
c to d S40 161W Diſtance 366.48
d toe $37 231 W Diſtance 415.35
e to f S 34 544 W Diſtance 46773
3.2 55 W Diſtance 5 18.54.
& toh S31 311 W Diſtance 560.05
O
ftog
S?
And the whole Diſtance from L to A=3778.64
Differing but 168 of a Mile from the Length of the Arch of neareſt
Diſtance.
If the dire& Courſe by the Compaſs be required, it will be by the
firſt Caſe of Mercator's Sailing,
As the Meridional Difference of Latitude 34745. 3.5408923
To the Difference of Longitude 2700
3.4313637
So is the Radius
10.00000co
To the Tangent of the Courſe 37 deg. s min.
9.8904714
Which
this
340
Navigation.
Which becauſe Comau lies to the Southward of the Lizard, and
the Diference of Longitude is Weſterly, is South 37 deg. si min.
Weſt, or South Weſt by South 4 deg. o6 min. Weſterly.
And to find the Length of the Rumb Line between the Lizard
and Comail, it will be,
As the Radius
10.000000
To the proper Difference of Latitude 3000.
So is the Secant of the Courſe 37 deg. 51 min. .
3.4771213
10.1025819
To the Length of the Rumb Line 3799.3
3.5797032
Difering from the Arch of neareſt Diſtance 21.3 Nautical Miles
and from the Sum of the Diſtances by the former Calculation 20.66
Miles.
Caſe II.
F
If the Places propoſed are ſcituated in the ſame Hemiſphere, as
ſuppoſe the Lizard in the Latitude of so deg. oo min. North, and
Barbadoes in the Latitude of 13 deg. 30 min. North, the Difference
of Longitude betwixt them being 52 deg. 58 min. Barbadoes lying
ſo much to the Weſtward of the Lizard, and it be required to find
the Angle of Poſition, Arch of neareſt Diſtance, &c.
In the adjacent Figure, Let PASQ,
repreſent the Meridian of Barbadoes, P
the North Pole, Sthe South Pole, and
Al the Equator.
1
1. Set off the Latitude of Barbadoes
1? deg. 30 min. from A to B, then will
B repreſent Barbadoes.
2. With the Secant of 52 deg. 58 min.
the Difference of Longitude, ſetting one
Foot of the Compaſſes in P, with the
other croſs the Equator produced (if
need be) in C, on which Point as a Center, with the ſame diſtance
deſcribe the Circle PMS, this will repreſent the Meridian of the Li-
zard, and if the Half Tangent of the Difference of Longitude 52 deg.
58 mir. be ſet off from o to p, the Point P will be the Pole.
IR
Z
a
A
D
3 Set off
Navigation.
341
10.0000000
3. Set of the Latitude of the Lizard so deg. oo min. from A to
x, a Ruler laid over x and the Pole p, will cut the Meridian PMS
in the Point L the Place of the Lizard.
4.
Thro' the Points B, L, and D, let a Grear Circle be deſcribed
and the thing is done ; then will BL be the Arch of neareſt Diſtance,
the Angle B che Angle of Poſition at Barbadoes, and the Angle 1
the Angle of Poſition at the Lizard.
From B let fall the Perpendicular BZ, then in the Right Angled
Spherical Triangle PBZ Right-angled ar Z, are given, BP equal to
the Complement of the Latitude of Barbadoes 76 deg. 30 min. the
Angle BĖZ the Difference of Longitude, equal to 52 deg. 58 min.
Wlience to find PZ a 4th Arch it will be,
As R:cs <P::t, BP:t, PZ. That is,
As the Radius
To the Co-fine of the Diff. of Longitude 52° 58' – 9.7797981
So is the Co-tang.of the Latitude of Barbadoes 13° 30' 10.6196463
To the Tangent of PZ, or a 4th Arch 68° 16'= 10.3994444
From the 4th Arch PZ = 68 deg. 16 min. take PZ the Comple-
ment of the Latitude of the Lizard 40 deg. 00 min, and there will
remain the 5th Arch LZ equal to 28 deg. 16 min.
Whence to find, 1st, BL the Arch of neareſt Diſtance, it will be by
comparing the two Right-angled Spherical Triangles BPZ, BLZ,
together.
As cs, PZ :cs,LZ::cs,BP : cs BL. That is,
As the Cc-line of the 4th Arch PZ 68.16
9.5685387
To the Co-fine of the 5th Arch LZ 28.16 - 9.9448541
So is the Sine of the Latitude of Barbadoes 13:30 9.3681853
To the Co-line of the Arch of neareſt Diſtance 56.16
9.7445007
Equal to 3376. Nautical Miles.
2. To find the Angle of Poſition at the Lizard it will be,
As S, LZ: S, PZ:: t,<P:t,<L. That is,
As the Sine of the 4th Arch LZ 28.16
9.67538 96
To the Sine of the 5th Arch PZ 68.16 -
9.9679771
So is the Tangent of the Diff of Longitude 52.58 10.1223600
To the Tang. of <of Poſition at the Lizard 68.57
10.4149475
Yy
3. To
342
Navigation.
Rihgfedcba LR
3. To find the Angle of Poſition at Barbadoes it will be,
As S, BL:S,<P :: S, PL:S,<B. That is,
As the Sine of the Arch of neareſt Diſtance 56.16-9.9199308
To the Sine of the Dift. of Longitude 52.58=9.902 1581
So is the Co-fine of the Latitude of the Lizard 50.00 9.8080675
To the Sine of the of Poſition at Barbadoes 38.05=9.7902948
P
Now in Order to find the
Courſe and Diſtance to every fitth
Degree of Difference of Longitude
in the Arch of the Great Circie,
in the adjacent Figure, Let Pre-
preſent the North Pole, B Barba-
does, I the Lizard, and BL the
Arch of the great Circle intercep-
ted between them.
B
From P let fall the Perpendicu-
lar PR upon BL produced, this
will repreſent the Complement of
the greateſt Latitude, thro' which the creat Circle parles; to find
which Trigonometrically, in the Right-angled Spherical Triangle
PLR are given, PL the Comple:nent of the Latitude of the Lizard,
and the Angle PLR the Angle of Poſition, whence to find PR, it
will be
As R : S,LP::S,<L: S, PR. That is,
As the Radius
10.0000000
To the Co-ſine of the Latitude of the Lizard 50.00 9.8080675
So is the Sine of the Angle of Poſition 68.57
9.9700385
To the Co-fine of the greateſt Latitude 53.08
9.7781066
2 To find the Polar Angle, LPR, or Difference of Longitude
between R and L it will be,
As R: CS,LP :: t,<L:ct, <LPR. That is,
As
Navigation.
343
10.0000000
As the Radius
To the Sine of Latitude of the Lizard 50.00 9.8842540
So is the Tangent of the Angle of Polition 68.57 } 104149475
To the Co-tangent of the Angle LPR=26.393- 10.2992015
Thro'p draw the ſeveral Meridians Pa, Pó, Pc, &c. forming
Angles with the Meridian PI of s deg. 10 deg. 15 deg. &c. where
thele interfect the Arch of the great Circle, as in the Points a, b, c,
&c. they give the ſeveral places in that Arch anſwering to every fifth
Degree of Difference of Longitude from the Lizard, and the ſeveral
Portions of the Meridian Pa, Pb, Pc, &c. the Complements of
the ſeveral Latitudes of thoſe Places.
To find each of which Trigonometrically, in the Spherical
Triangle aPR, Right-angled at Ř, are given, PR equal to 30 deg.
52 min. the Complement of the greateſt Latitude, and the Angle
APR equal to the Angle LPR plus s deg. equal to 31 deg. 39 min.
Whence to find Pa the Complement of the Latitude of the Point a,
it will be
As R:cs, <aPR::ct, PR:ct, Pa. That is,
As the Radius-
10.0000000
To the Co-fine of the Angle aPR=31.29 9.9300670
So is the Tangent of the greateſt Latitude 53.08. -10 1249898
To the Tang. of the Latit. of the Point a=48.37 10.0550568
Again, If to the Angie aPR, be added 5 deg. oo min. we ſhall
s .
have the Angle bPR equal to 36 deg. 39 min. Whence to find the
Latitude of the Point b it will be,
As R:cs,<bPR::t, PR :t, bR. Thatis,
As the Radius
10.000000
To the Co-ſine of the Angle bPR=36.39. 9.9043351
So is the Tangent of the greatelt Latitude 53.08 10.1249898
To the Tangent of the Latit. of the Point b=46.56 10.0293 249
After
Y y a
3.44
Navigation.
After the ſame manner by increaſing the ſeveral Polar Angles
5 deg. oo min. the Latitude of the Point c, will be found to be
44 deg. 50 min. of the Point d 42 deg. 28 min. of the Point e 39 deg.
36 min. of the Point f 36 deg. 15 min. of the Point g 32 deg. 21 min.
of the Point h 27 deg. 51 min. of the Point i 22 deg. 46 min. of the
Point k 17 deg. 07 min. and of the Point B 13 deg. 30 min.
Hence the Meridional Difference of Latitude between L and a
will be found to be 127-3, between a and b 150.3, between b and
C 175.4, between c and d 202.0, between d and e 228.0, between
e and f 254.9, between f and g 283.4, between g and h 31 2.2, be-
tween h and i 337.6, between i and k 360.8, between k and B 225.0,
whence by the firſt Caſe of Mercator's Sailing.
b toc
The Courſe from L to n is S67 01 W Diſtance 212.57
n to b S63 23 W Diſtance 225.44
S59 41 W Diſtance 241.69
e tod S 5603 W Diſtance 261.43
d to e S52 46 W Diſtance 284.26
e to f S49 39 W
S49 39 W Diſtance 310.45
S 46 38 W Diſtance 340.78
43 514 W Diſtance 374.45
bto i S 41 38 W Diſtance 408:07
i to k S39 44+ W
S39 44+ W Diſtance 440.86
k to B. S 38 21 W Diſtance 276.71
Whence the whole Diſtance from L to B=3376.71
frog
ftoh
Differing but a parts of a Mile from the Arch of neareſt Di-
ſtance.
If the dire& Courſe by the Compaſs from the Lizard to Barbadoes,
and the Length of the Rumb Line intercepted between them be re-
quired. Then,
From the Meridional parts of soº oo
*3474.5
Take the Meridional parts of 13° 30'
-817.6
Remains the Meridional Difference of Latitude - 2656.9
Whence to find the Courſe it will be, by the firſt Caſe of Mer-
cator's Sailing
As
---
Navigation
345
***
As the Meridional Difference of Latitude 2656.9 - 2.4243752
To the Difference of Longitude 52° 58'=3178 -2.5021539
So is the Radius
10'0000000
To the Tangent of the direa Courſe 50°06' 10.0777787
Which becauſe Barbadoes lies to the South Weſtward of the Li-
zard, is S. so deg. 06 min. W. or South Weſt 5 deg. 06 min. Weit-
erly. Whence to find the Length of the Rumb Line it will be,
As the Radius
10.0000000
1
To the Secant of the Courſe 50.00
10.1928374
So is the proper Differ. of Latitude 36° 30'=2190 -2.3404441
To the Length of the Rumb Line 3414.14
2.533283.5
Differing but 37.43 Miles from the Arch of neareſt Diſtance.
Caſe III.
If the places propoſed are ſcituated in different Hemiſpheres, as
ſuppoſe Barbadoes in the Latitude of 13 deg. 30 min. North, and
St. Helena in the Latitude of 16 deg. 03 min. South, the Difference
of Longitude between them being 51 deg. 44 min. St. Helena lying
ſo much to the Eaſtward of Barbadces, and it be required to find
the Angle of Poſition, Arch of neareſt Diſtance, Courſe and Di-
ſtance from Barbadoes to St. Helena, to every fifth Degree of Diffe-
rence of Longitude:
In the adjacent Figure, Ler P repre-
ſent the North Pole, Sthe South Pole,
AQ the Equator, PASQ the Meridian
of Barbadves.
B
1. Set off 13 deg. 30 min. the Lati- A
tude of Barbadoes from A to B ; then
will B be the place of Barbadoes
P F
Q
I
112
H.
D.
X
R
2. With the Secant of sı deg 44 min.
the Difference of Longitude, ſetting
E S
:
selon
? 46
Navigation.
one foot of the Compaſſes in P, with the other croſs the Equator
(produced if need be) in C, oh which as a Center, with the ſame
Diſtance deſcribe the Circle PMS, this will repreſent the Meridian
of St. Helena, and if the half Tangent of 5 i deg. 44 min. be ſet off
from o to p, p will repreſent its Pole.
3. Set oft" 16 deg. 03 min. the Latitude of St. Helena from A to
X, a Ruler laid over x and p will cut the Meridian PMS in H the
place of St. Helena.
4. Thro' B and H let a Great Circle as BHD be drawn, and thie
thing is done ; for BH will be the Arch of neareſt Diſtance, the
Angle B the Angle of Poſicion at Barbadoes, and the Angle H the
· Angle of Poſition at St. Helena.
5. From H ler fall the perpendicular HR, then in the Spherical
Triangle PHR Right-angled at R, are given, PH equal to 106 deg.
o3 min. the Angie P equal to si° 44' whence to find PR it will be,
As R:cs<P :: T,PH: T, PR. That is,
As the Radius
10.0000000
To the Co-line of the Diff. of Longitude 51°.44' 9.7919168
So is the Co-tangent of the Latit. of St. Helena 16,03 10.5410752
To the Tang. of a 4th Arch PR
10.3329920
Which becauſe PH is greater than a Quadrant, is equal to 114°55'
,
Now if from PH equal to 114 deg. 55 min. be taken PB the Con-
plement of the Latitude of Barbadoes, equal to 76 deg. 30 min.
There will remain BR a fifth Arch, equal to 38 deg. 25 min. whence
to find the Angle of Poſition at Barbadoes, it will be, by comparing
the Right-angled Spherical Triangles PHR and BAR together.
ASS, BR:S, PR::t, <P:t, <B. That is,
As the Sine of the 5th Arch BR 38:25 -
9.7933543
To the Sine of the 4th Arch PR 114.25
9.9575697
So is the Tangent of the Difference of Longitude 51.44 10.1030286
To the Tang. of the Angle of Poſition at B 61° 36' 10.2672440
Hence to find the Angle of Poſition at St. Helena it will be,
As S, PH:S, KB::S, PB : S, H. That is,
As
1
1
Navigation.
347
As the Co fine of the Latitude of St. Helena 16.03 -9.98273 28
To the Co-fiae of the Latitude of Barbadoes 13.30 9.9878315
So is the Sinc of the of Pofirion at Barbadoes 61.36 9.9443092
Tosic of the <of Poſitioa at St. Helena 77°04' 39.9494079
To find BH the Arch of neareſt Diſtance it will be,
ASS, B: S, PH::S,KP:S, BH. That is,
As the Sine of the Angle of Poſition at B 61.36 9.9443092
To the Co-line of the Latitude of St. Helenlis 16.03 9.9827328
So is the Sine of the Difference of Longicude 51.47 -9.8949453
To the Sine of the Arch of neareſt Diſtance 59.04 9.9333689
B
а.
Equal to 3544 Nautical Miles.
To find the Latitude of the ſeveral Points in the great Circle BH,
anſwering to every s Degrees of Difference of Longitude, inſtead of
finding the Complement of the greateſt Latitude thro'which the Arch
pafles I ſhall make Uſe of the Supplemental Triangle B.AI, as being
in this Caſe much more natural and caſy.
In the adjacent Figure therefore,
P
let P repreſent the Pole, B Barba-
does, H St. Helena, and AIM the
portion of the Equator intercepted
between the Meridians PBA, PMH,
equal to the Difference of Longitude
between the two places; and let the
6
feveral Meridians Pk, P), Pm, &c.
be drawn, forming Angles with each
d
other equal to 5 degrees, where
theſe interfect the Arch BIH in the
parst M
Points a, b, c, &c. they give the ſe-
veral places in that Arch anſwering
to every fifth degree of Difference
of Longitude, and the ſeveral por-
H
tions ak, bl, am, &c; the leve-
ral reſpektive Latitudes; then in the Spherical Triangle ABI, Right:
angled at A, are given, AB the Latitude of Barbadoes, the Angle
ARI the Angle of Poſition at Barbadoos ; whence to find the Angle
AIB it will be,
C
AL
ki milOI
f
an
348
Navigation .
As R :cs, BA :: S,<ABI: cs, AIB. That is,
As the Radius
10.0000000
To the Co-line of the Latitude of Barbadoes 13.30 9.9878315
So is the Sine of the Angle of Poſition 61.36 9.9443092
To the Sine of the Angle BIA=31.12-
9.9321407
7
For the Arch Bl it will be,
As R:S,BA::t,<B:t, Al. That is,
As the Radius
10.0000000
To the Sine of the Latitude of Barbadoes 13.30 -9.3618853
So is the Tangent of the Angle of Poſition 61.36-10.2670453
To the Tangent of the Arch 1A 23.21-
9.6289306
Now becauſe the ſeveral Arches of the Equator Ak, kl, lm, &c.
are equal to the reſpective Differences of Longitudes between the
points B, n, d, and e, equal to s degrees, if from .41 equal 23.21,
be taken ſucceſſively 5 deg. 10 deg. 15 deg. &c. the ſeveral Re-
maiuders will give the Lengths of the ſeveral Arches Ik, Il, Im, &c.
ſo that in the ſeveral Spherical Triangles Ika, Ilb, lmc, &c. Right-
angied at k, l, m, are given, I k equal to 18.21, Il equal to 13.21,
I mequal to 8.21, c. and the common Angle RIA equal to 31.12,
whence to find firſt the Latitude of the Point a it will be,
AsR:S, Ik :: t,<AIB:t, ka. That is,
As the Radius
-10.0000000
1
To the Sine of Ik=18:21
9.4980635
So is the Tangent of the BIA=31.12
9.7822013
To the Tang. of the Latit. of the Point a=10.48—9.2802648
After the ſame manner the Latitude of the Point b will be found
to be 7 deg. 58 min. North, of the Point c 5 deg. or min. North, of
the Point d 2 deg. or min. North ; but becauſe the Ship in paſſing
tro'the Point I croſſes the Equator, the Latitude of the Pointe
will be i deg. oo min. South, of the Pointf 4 deg. Oj min. South, of
the Pointg 6 deg. 58 min. South, of the Point h 9 deg. si min. of the
Pointi 12 deg, 36 min. and of the Point H 16 deg. oz min. South.
Hence
Great Circle Sailing :
349
Hence, and from the 4th Caſe of Mercators Sailing.
The Courſe from B to a is S. 61.04 E. Diſtance 334.85.
a to 6 is S. 60.09 E. Diſtance 341.55
b to c is S. 59.22 E. Diſtance 346.39
c to d is S. 58.59 E. Diſtance 349.32
d toe is S. 58.50 E. Diſtance 350.71
e to f is S. 58.52 E. Diſtance 350.48
f tog is S. 59.21 E. Diſtance 347.30
ę to his S. 59.46 E. Diſtance 343.58
h to į is S. 60.421 E. Diſtance 337.07
i to His S. 62.07 E. Diſtance 442.77
Suin of the whole Diſtance=3544.02
If the direct Courſe from B to H, and the Length of the Rumb
Line intercepted between them be required, it will be by the ift Cafe
As the Meridional Difference of Latitude 1793.5
3.25 37014
To the Radius
So is the Difference of Longitude 3104
10.0000000
3.4919217
To the Tangent of the Courſe 59.59-
10.2382203
Differing but 11 Minutes from the Mean Courſe.
For the Length of the Rumb Line it will be,
As the Radius
10.0000000
To the Secant of the Courſe 59.59
So is the proper Difference of Latitude 1773
10.3008113
3:2487087
To the Length of the Rumb Line 3544.2 I
3.5495 200
Differing but 1. of a Mile from the Arch of neareſt Diſtance,
and but from the Sum of the ſeveral Diſtances before found.
Inſtead of making Uſe of the Axiom, As the Sine of half the Sum
of the Sides, &c. for finding the Angle of Poſition, I have rather
choſe to do it by letting fall the perpendicular; which Method I
take to be more inſtru&ive, and leſs burthenſome to the Memory.
Z z
As
350
Nautical Problems,
As the cliief End and Deſign of the Nautical Art, is to conduct
a Ship after the ealieſt and moſt commodious manner that may
be
from one Place to another, ſo the principal Problems upon which the
whole Buſineſs Centers, are firſt, having the Scituation of two Places
given in reſpect to Latitude and Longitude, to find upon whatCourſe,
and how far the Ship muſt fail to get from one place to the other, and
ſecondly, ſuppoſing the Ship to have failed upon the ſame or ſome
other Courle, a given and determinate diſtance, to find the Latitude
fhe is in, how much ſhe has altered her Longitude, and upon what
Courſe ſhe muſt ſail of a new, and how far, to arrive at her inten-
ded Port.
To thew how each of theſe may be reſolved by the Simple Rules
of Multiplication and Divilion, without the Aſſiſtance of thie Tables
of Logarithms, Sines, Tangents, Secants, Meridional Parts, or a.
ny Artificial Numbers, will, I preſume, be not unacceptable to the
legenious Reader
Leti thcrefore be required, firſt to find the Courſe and Diſtance
between the Lizard in the Latitude of so deg. oo min. North, and
Barbadies lying in the Latitude of 13 deg. 30 min. North, Barba-
does being ſcituated 52 deg. 58 min. to the Weſtward of the Lizard.
In the 10th Section it has been ſhewn, that if a be puc for the La-
titude of any Place, or its neareſt diſtance from the Equator in ſuch
Parts as the intire Circumference of the Earth is 3.14159, 6c. or the
Diameter 1.0000, c. that a intrattend, c. will be the
diſtance of that place from the Equator, in ſuch Parts as a Degree
of the Meridian is 60.000, &c. upon the true Chart, where the
Rumb Lines are repreſented by ſtrait Lines ; put z therefore equal
to .8726646, c. the Latitude of the Lizard, in ſuch Parts as the
whole Circumference of the Earth is 3.14159, c. equal to 3000,
the m nutes in 50 deg oo min. multiplyed by 0002.9088.2, 6c, the
Length of the Arch of one minute, and x equal to .2356194, 6c.
the Latitude of Burbadoes expreſſed in the ſame Parts; then will the
Difference between ziiz'1, 2 stjótaz?, cand x +"*+2***
-6x?, &c. equal to the Difference between .8726646 Oct
8726646, Oc. + 2.87266461 +, öc. and 2355194,' &c.+
2356194, cc. to 2356194, dc. equal to 6370452, &c. I
108:819 210573 +46642 · 1120172829+7381 198 +50, ==
7728511 divided by .0002908882, give 2656.9— for the Diſtance
between the Parallels paſting thro' the Lizard and Barbadoes, or the
Meridional Difference of Latitude between them; this therefore
+ ܐ
1
divi-
Nautical Problems,
351
divided by 3178, the Difference of Longitude between the Lizard
and Barbadoes, will give 836+for the Co-tangent of the Courſe in
ſuch Parts as the Radius is 50000, 6C
Put t equal to 836+, then, according to Section che ad of Part
the iſt, in Page the 8oth, will the Complement of the Courſe bec-
qual to t-r' titin-t?, &c. equal to 836} 8363+ } 8365---
1836' equal to 696 +, in ſuch Parts as the whole Circumfcicnce is
6.283 +- and being therefore reduced into degrees and minutes, will
give 39 deg. 54 inin. which therefore taken from go dog. oo min.
will leave so deg, o6 min. for the true Courſe ; which becauſe the
Ship is bound from a greater North Latitude to a lefler, and the
Difference of Longitude is Weſterly, is according to the Nautical
Phraſe South 59 deg. 06 min. Weſt, or South Weſt 5 deg. 06 min.
Wefterly
The Courſe 59 deg. 06 min. thus determined, being reduced in-
to minutes gives 3006, this therefore multiplied by 000 2.9088 the
Length of the Arch of 1 minute, will give .874393 for the Length
of the Arch of the Circumference, equal to the Angle of the Courſe
in the fame Parts; put a therefore equal to .874393 , then accord-
ing to Section the 2d of Part the 2d, in Page 79, will 1+12+212
+za+uza', c. equal to 1000000 + 382286-4121782 +
+-378647-11737+, &c. equal to 155898 be the Secant of the
fame Arch, this therefore multiplied by 2 190, the Difference of
Latitude between the Lizard and Barbadoes, will give 3414.1+,
for
the direct Diſtance between the ſame Places; all which is ſo very
obvious that it needs no farther Explanation.
Let us now ſuppoſe fecondly, that the Ship has failed from the
Lizard 500 Leagues or 1500 Miles, (for the larger the Diſtance the
better) upon the direct Courſe, and let it be required to find the La-
titude ſhe is in, and how much ſhe has altered her Longitude.
The dire&t Courſe is so deg. 06 min. equal to 3006 minutes, this
therefore multiplied by 0002.9088 the Length of the Arch of 1 mi-
nute, will give -874393 +- for the Length of the Arch of the Circum-
ference, equal to the Angle of the Courſe in ſuch Parts as the Dia-
meter is 1000000, &c. put a therefore equal to .874393 t, then
according to Sečtion the 2d of Part the 2d, in Page 79 will ima
tina-koa', &c. equal to 1-- 874393 +:+ 38743934-
874393 +6,&c. equal to 100000—382281+2435-62, cqual
to 6414, be the Co-line of the ſame Arch, this therefore multiplied
by 1500 the dire& Diſtance, will give 962.1 for the common Diffe-
Z z 2
T
ܐ
rence
352
Nautical Problems.
rence of Latitude, which being reduced into degrees and minutes
and ſubſtracted from the Latitude the Ship failed from so deg.
oo min. North, becauſe ſhe failed from a North Lajitude Southerly,
will give 33 deg. 58 min. for the Latitude the Ship is in Northerly.
Again, put z equal to .8726646, &c. the Latitude the Ship
failed from cqual to 3000, multiplied into 0002.9088.2, c. the
Arch of i minute, and y equal to :5928301, &c. the Latitude the
Ship ſailed inco in the ſame Parts, then will the Difference between
z+izitziz'+;6*2", yly + 4y -+ sotzy', &c. equal to
the Difference between 8726646' + + 8726646-4 21 8726646%
Gr. and 5928301? +5928301375928301', &c. equal to 2798345
*-760371 +180365 +43533 + 1085672789+734+198+59, e-
qual to 3797250 divided by 0002.9088.82, gives 1305.4, for the
Meridian Difference of Latitude.
The Meridian Difference of Latitude being thus obtained, the
Difference of Longitude will be readily had, for if a be put equal
874393, the Angle of the Courſe in ſuch Parts as the Radius or Se-
midiaineter is .50000, then by Section the 2d of Part the 2d will
atatia tita'+7$352, 6c equal to 87439+22284+
6815+2109 +683 + 2057 63+19+ 6, equal to 11950 t, mulci-
plied by 1305-4-t gives 1561.2 for the Difference of Longitude, e-
qual to 26 deg. or min. š.
Again, ſuppoſing a equal to 834393 as before, if — tikons
Hon?, Or, equal to 874393-111421 +4259–77, equal to
7671, be multiplied by 1305.4 the Meridional Difference of Latitude
before found, and that Product be divided by 1-ža’+2421-
a', &c. equal to 100000--38228+-2435--62, equal to 6414, the
Quotient 1561.2 equal to 26 deg. oi min. ,ô, will be the Difie-
rence of Longitude the ſame as before found. Or,
If a most tans 000?, &c. equal to 874393—111421-1-
4259–77, equal to 7671, be multiplied by 2035.4, the inlarged
Diſtance, the Produ&t 1561.3 equal to 26° oi', will be the Dife-
rence of Longitude, agreeing with the two former Deductions.
The Difference of Longitude 26 deg, or min. thus obtained, ta-
ken from 5a deg. 58 min. the Longitude between the two Ports
will leave 26 deg. 57 min. for the Difference of Longitude the Ship
muſt make, whence and from the Latitude the Ship is in 33 deg. 53
min. North, and the Latitude of the place ſhe is bouud to 13 deg.
30 min. North, may the Courſe upon which ſhe muſt Sail, and the
Diſtance upon that Courſe be obtained, after the manner taught in
the former Problem,
Any
1
I
720
Nautical Problems.
373
Any one who is Maſter of the ad Sestion of Part the 2d, and of
the roth Seation of Part the 3d i will readily ſee that the ſame Re-
quiſites may be found ſeveral different ways.
How operoſe and difficult foever theſe kinds of Solutions may ar-
pear to Perſons not will skill'd in the Doitrine of Series, nor very
ready in the Method of Computing, yet to Perſons that are but to-
Icrably knowing in each, they are very Eaſy and Expeditious.
The intent of mentioning this, is not to recommend this Method
to Practice, but to encourage the curious and inquiſitive Reader in
bis purſuit after true Knowledge, tor whoſe Entertainment this was
principally intended; and who in one Solution of this kind well un-
derſtood, will undoubtedly find a great deal of Satisfaction and himr
ruf well rewarded for his Pains for as every Vautical Problem may
be reſolved after this manner, fo in Solutions of this kind only, con-
fifts a true Knowledge of the Theory of the Art of Navigation.
mas
เอก
Sect. XIV.
Containing Methods for finding the Latitudes of Places,
or Heights of the Pole by the Meridional Altitude of the
Sun or Stars, and their Declinations.
T
HE Latitude of any Place, (as has been already ſhewn) is.
equal to the height of the viſible Pole, or Arch of the Meri-
dian contained between the Pole and the Horizon, and which is ever
equal to the Arch of the faine Micridian, intercepted between the
Equator and the Zenith.
For in the adjacent Figure, whicre ZONH repreſents the Hori-
zon, P the North Pole, S the South Pole, and A2 the Equator ;
alſo Z the Zenith, N the Nadir, and HO the Horizon, if from Zo:
the diſtance of the Zenith from the Horizon equal to a Quadrant,
equal to PE, the diſtance of the Pole from the Equator, be taken
away ZP the diſtance of the Pole from the Zenith, there will remain
ZE the diſtance of the Equator from the Zenith, equal to PO the
Height of the Pole above the Horizon, equal to the Lati;ude of the
Place.
354
Methods for determining
N
As the Declination of the Sun or
Star is the neareſt Diſtance of the
P
Sun or Star from the Equator, and
E
is equal to the Arch of the Meridian
intercepted between the Sun or Star
and the Equator, ſo the Meridional
Zenith-diſtance is the Arch of near-
H
0
eſt approach of the Sun or Star to
the Zenith ; and is equal to the
Arch of the Meridian contained be-
Q
tween the Sun or Star and the Ze-
S
nith of any Place, at the time of their
N
tranfit over the Meridian.
Hence we are taught, how from the Declination of the Sun or a-
ny Star, and their Meridional Zenith-diſtance, to find the Latitude
of the Place of Obſervation.
1. If the Sun or Star have no Declination, the Meridional Zenith-
diſtance is equal to the Latitude of the Place, which if the Sun come
to the Meridian South of the Zenith, the Latitude is North, but if
the Sun come to the Meridian on the North ſide of the Zenith, the
Latitude is South.
Thus, in the former Figure, where EQ repreſents the Path of
the Sun or Star's diurnal Courſe, when they are in the Equator,
ZPON the Northern Semicircle, and NSE Z the Southern, it is
manifeſt, that Z E the diſtance of the Sun to the Southward of the
Zenith, when he is upon the Meridian, is equal to P O the Lati-
tude of the Place of Obſervation, and which muſt be Northerly,
becauſe the North Pole P is elevated above the Horizon.
Again,
In the adjacent Figure, where the
Sun in the Equator at E, appears up:
E
on the Meridian to the Northward of
S
the Zenith, it is manifeſt, that ZE the
Meridional Zenith-diſtance North, is
H Η
equal to PO the Depreſſion of the
O
North Pole, equal to SH the Eleva-
tion of the South Pole above the Ho-
rizon; and conſequently the Latitude
is Southerly.
N
P.
R
N
For
من اپنے بدنه جاری
the Latitudes of Places.
355
For as the Latitude of the place is the ſhorteſt diſtance between
that Place and the Equator, it is manifeſt that when the Obſerver
is on the Northern Side of the Equator, that then the Northern
Pole becomes viſible; and on the contrary, when he is on the South-
ern Side of the Equator, that then the Southern Pole will become e-
levated above the Horizon, and become viſible, and the Northern
Pole be as much depreſſed, and conſequently become inviſible.
Hence it is, that at London where the Latitude is s i deg. 32 min.
North, that the Sun when he is in the Equator, and which happens
on the oth of March and the 12th of September, appears upon the
Meridian at the diſtance of s I deg. 32 min. on the South Side of
the Zenith, or at the Altitude of 38 deg 28 min. and which is a
proper Time in any Place, to determine the Latitude of that Place,
there being no Occafion to conſider the Declination.
If the Sun or Star appear in the Zenith, then the Latitude of
the Place becomes equal to the Declination, and is the ſame way
with it ; that is, if the Sun or Star lave North Declination, the
Latilude is North ; but if the Sun or Star have South Declination,
then the Latitude is South,
Z
E
P
H
For in the adjacent Figure it is
manifeſt, that when the Sun appears
in the Zenith at Z, that ZE his di-
ftance from the Equator equal to his
Declination, and in this Cafe North-
crly, is equal to PO the height of the
North Pole above the Horizon, e-
qual to the Latitude North, for ZO
and PE are Quadrants.
D
S
NZ
E
S
Eut if as in the adjacent Figure the
Sun appears in the Zenith, to the
Southward of the Equator, it is e-
vident, that Z E the diſtance of the
Sun from the Equator, or his Decli- H
nation in this Cafe Southerly, is e-
D
qual to SH the height of the South-
ern Pole above the Horizon, equal
to the Latitude of the Place South-
erly, for Z Hands E are Quadrants.
P
N.
3. If
356
Methods for determining
N
3. If the Sun or Star appear neither in the Zenith, nor in the E-
quator, and conſequently hare both Zenith-diſtance and Declination,
if theſe are both North or both Soutlı, and the Zenith-diſtance great-
er than the Declination, then the Exceſs of the Zenith-diſtance above
the Declination, is equal to the Latitude of the Place; and which
is South if the Zenith-diſtance and Declination are North, but if
the Zenith-diſtance and Declination are boch South, the Latitude is
North.
For in the adjacent Figure, let OD
repreſent a Parallel of Southern Decli-
nation, whence it is evident that the
Sun upon the Meridian at appears
to the Southward of the Zenith and
Equator, and that ZE equal to the
H
Exceſs of Z O, the Zenith-diſtance a-
bove EO the Declination, is equal
to PO the Latitude of the Place, and
Q
which is Northerly, becaule the
North Polc is elevated above the
D
N
Horizon ; wherefore, if upon the 19th
of February, 1720, when the Sun had
7 deg. 24 min. South Declination, his Meridional Zenith-diſtance
was found to be 58 deg. 56 min. Southerly, and it be required to
find the Latitude of the Place of Obſervation, and which way it is.
From the Meridional Zenith-diſtance
58 56 South
Take the Declination
-7 24 South
e
S
There remains the Lat. of the Place of Obſerv. 51
32 North
Z
E
S
O
But, if as in the adjacent Figure
where the Parallel of Declination : D
is on the North Side of the Equator,
and where the Sun appears at 0 on
the Northern Side of the Zenith, it is
manifeſt, that ZE the Exceſs of the
Zenith diſtance z o above the De-
clination E 0, is equal to SH the
Altitude of the South Pole above the
Horizon, and conſequently in this
Caſe the Latitude will be Southerly ;
wherefore if on the 8th of July 1720,
when
Р
Q
DN
the Latitudes of Places
397
when the Sun's Declination was 20 deg. so min. North, he was
found by Obſervation to be 72 deg 28 min. to the Northward of
the Zenith, and it be required to find the Latitude of the Place of
Obſervatio), and which way it is.
From the Meridional Zenith-diſtance-72
72 28 North
Take the Declination
56 North
20
There remains the Latitude of the Place
SI
32 South
But when the Zenith-diſtance and Declination are equal, the La-
titude vaniſhes; and the Place is ſciquated under the Equätor.
For when ſo the Zenith dift-
ance, and E @ the Declination are
equal, and both the ſame way, then
the points E and 2 coincide with
the points 2 and N, the Equator
becomes the Prime Vertical, and the
Poles P and S Iye in the Horizon, WS
PO
and conſequently the Latitude vani-
thes; and this will happen at all
'Times and in all Places under the
Equator, ſo that if upon the 8th of
July 1720, when the Sun's Declina-
D
tion was 20 deg. 56 min. North his
N
Meridional Zenith-diſtance was found to be 20 deg. 56 min. N. and
it were required to find the Latitude according to the former Law.
From the Meridional Zenith-diſtance
56 North
Take the Declination
56 North
D
20
20
There remains the Latitude of the Place
00
00
4. If in comparing the Meridional Zenith-diſtance and Declina-
tion together, they are found to be the ſame way, and the Decli-
nation be greater than the Meridional Zenith-diſtance, then the
Exceſs of the Declination above the Zenith-diſtance, is equal to the
Latitude, and which is the ſame way with the Declination and Ze-
nith-diſtance; that is, if the Declination and Zenith-diſtance be
North the Latitude is North, but if the Declination is South the
Latitude is South.
Aaa
For
358
Methods for determining
ZO
E
For in the adjacent Figure, where
OD the Parallel of Northern Decli-
nation cuts the Meridian at C, on the
P Northern Side of the Zenith, it is ma-
nifeſt, that E Z the Exceſs of O E,
the Declination, above Zo the Me-
ridional Zenith-diſtance, is equal to
PO the height of the Northern Pole
above the Horizon; and conſequent-
ly the Latitude is North. Wherefore,
H H
N
If upon the roth of June 1721.,
when the Sun's Declination is 23 deg
29 min. North, his Meridional Zenith-diſtance is found to be rodeg.
24
min. on the Northern Side of the Zenith, and it be required to
find the Latitude of the Place and which way it is.
From the Declination
Take the Meridional Zenith-diſtance
-23 29 North
24 North
IO
There remains the Latitude
'13
os North
un
Again, in the adjacent Figure,
OZ
where OD the Parallel of South-
E trn Declination, curs the Meridian
at 0 on the Southern Side of the E-
quator, it is evident that E2 the
Exceſs of the Declination OE, above
H
O
the Zenith-diſtance Z ©, is equal to
SH, the height of the Southern
Pole above the Horizon ; and con-
P ſequently the Latitude is South,
Wherefore,
If upon the roth of December 1720,
when the Sun has 23 deg. 29 min.
South Declination, his Meridional Zenith-diſtance was found by
Obſervation to be 6 deg. 02 min. South, and it be required to find
the Latitude.
NY
Fron
the Latitudes of Places.
359
From the Sun's Declination
Take the Meridional Zenith-diſtance
29 South
02 South
23
-6
There remains the Latitude
17
27 South
Z
5. If in comparing the Meridional Zenith-diſtances of the Sun or
Stars with their Declinations they are found to be contrary, that is,
if one be North and the other South, tlien their Suim will give the
Latitude of the Place ; and which is always the fame way with
the Declination, that is if the Declination be North the Latitude
is North ; but if the Declination be South the Latitude is South.
Thus in the adjacent Figure
where the Sun upon the Meridian at
O
appears to the Northward of the
Equator E Q, and to the Southward
P
E
of the Zenith Z, it is evident that
ZE equal to the Sum of zo the
Zenith-diſtance, and EO the Declina-
0
tion, is equal to PO the height of the
Northern Pole above the Horizon ;
and conſequently the Latitude is
North, ſo that if upon the roth of S
June 1720, when the Sun has 23 deg.
N
29 min. of Northern Declination, he
bé found by Obſervation to appear upon the Meridian at the di-
ſtance of 28 deg. 03 min. on the Southern Side of the Zenith
and it be required to find the Latitude of the Place of Obſervation.
To the Meridional Zenith-diſtance
28 03 South
Add the Declination
The Sum will be the Latitude of the Place
H Η
D
23 29 North
51 32 North
But if, as in the following Figure, the Sun at appears upon the
Meridian on the Northern Side, when his Declination EO is South-
erly, it is plain that Z E equal to the Sum of Zo the Meridional
Zenith-diſtance, and OE the Declination, is equal to S H the height
Аааг
of
:
!
360
Methods for determining
N
of the Southern Pole above the
O
Horizon ; and conſequently the
Latitude is South : So that if
up
on the 19th of February 1720,
when the Sun has 7 degrees 24
minutes South Declination, his
Meridional Zenith diſtance be
found to be 26 degrees si mi-
P nutes North, and it be required
to find the Latitude of the Place
of Obſervation.
H
ol
D
N
26 51
To the Meridional Zenith-diſtance
Add the Declination
51 North
24 South
7
The Sum will be the Latitude
34 15 South
It has been ſhewn at large in the Fifth Part, that when the La-
titude of the Place is greater than the Complement of the Declina-
tion, and both the ſame way, that the Sun or Star will never fet, but
tranſit the Meridian above and below the Pole ; and conſequently
in this caſe, if the Sun or Star tranſit the Meridian on contrary
Sides of the Zenith, from the greateſt Meridional Zenith-diſtance
take the leaſt, and half the Remainder will be the Complement of
the Latitude of the Place, which is North, if the greater of the two
given Zenith-diſtances or Declinations be North ; but South if the
greater of the two given Zenith-diſtances be South
Thus in the adjacent Figure,
Z
where D the Parallel of Northern
Declination, curs the Meridian in
the Points O and D, each of which
E
are above the Horizon ; it is evi-
D dent that ZP the half of zd the
H Н
Difference between Z D the greater
Zenith-diſtance, and D d equal to
Zo the leaſt, is equal to the Com-
plement of P O, the height of the N.
Pole above the Horizon in this caſe,
N
and conſequently the Latitude is
North ; ſo that iſ upon the 10th of
June
9
A
0
S
the Latitudes of Places.
361
June 1720, the Sun's Meridional Zenith-diſtance was found to be
46 deg. 31 min. South of the Zenith, and at Twelve Hours after
on the ſame Day, his Meridional Zenith-diſtance was found to be
86 deg, 31 min. to the Northward of the Zenith, and it be required
to find the Latitude of the Place of Obſervation.
From the greateſt Zenith-diſtance
86
31 North
Take the leaſt Zenith diſtance
46 31 South
Half the Remainder
40
00
equal to 20 deg: oo min. taken from so deg: oo min. will leave
70 deg. oo min. for the Latitude North,
Or,
If about the point P, at the diſtance of PO the Parallel a 0
be drawn, it is evident that if to Do the leaſt Meridian Altitude,
be added D q equal to o O the Supplement of the greateſt Meri-
dian Altitude o H half the Sum of thoſe two Arches U D and D 9,
equal to PO, will be the height of the North Pole above the Hori-
zon ; that is in the preſent Caſe,
To the leaſt Meridian Altitude DO-
3 29
Add the Supple, of the greateſt to 180=9D 136. 31
Half that Sum
140
Will give the Latitude of the Place
70 00 North,
But if the Declination of the Sun be ſuppoſed to be known or
given, then by the 5th Precept:
To the leaſt Meridional Zenith-diſtance - 46 31 South
Add the Declination
23 29 North
The Sum will be the Latitude
70 00 North
For Z Oto E is equal to Z E, equal to PO
Or,
29
To the leaſt Meridional Altitude
Add the Complement of the Declination
3
66
31 North
The Sum will be the Latitude of the Place
70
oo North
For PD+DO, is cuial to PO.
Or,
362
Methods for determining
Or,
If from the greateſi Meridional Zenith-diſtance 86 31 North
Ee taken the Complement of the Declination -66
31 North
The Remainder will be thcCompl. of the Latit. 20
oo North
S
Which therefore taken from 90 deg. oo min, will leave 70 deg, oo
min. for tlie Latitude North,
But if as in the adjacent Figure,
Z
the great:ſt Zenith-diſtance. Z D be
Southerly, then if from ZD the great-
eſt Zenith diſtance, ſuppoſe 89 deg.
P
30 min. be taken the leaſt Zenith-
o diſtance zo, 43 deg. 32 min. half
H
the Remainder 45 deg. 58 min. equal
to 22 deg. 59 min. taken from go
deg .00 min. will leave 67 deg. or
min. for the Latitude South
Bhd
v
Or,
To the leaſt Meridional Altitude
30
Add the Supplement of the greateſt to 180-133
133 32
00
Half that Sum
134
20
Will be the Latitude of the Place of Obſer.
-67 01 South
Or
To Z O the leaſt Zenith-diſtance
Add the Declination, ſuppoſe
43
23
32 North
29 South
The Sum will be the Latitude of the Place
67
01 South
But if it happen that the Complement of the Declination, or
diſtance of the Sun or Star from the Pole, be leſs than the Com-
plement of the Latitude, or diſtance of the Pole from the Zenith,
then the Sun or Star will tranſit the Meridian above and below the
Pole, on the fame ſide of the Zenith : wherefore in this caſe, if to
the greateſt Meridional Zenith-diktance be added the leaſt, lialf that
Sum will give the Complement of Latitude of the Place, and which
will
the Latitudes of Places.
363
.
N
P
A
will be the ſame way with the Zenith-diſtance or Declination ; that
is, if the Zenith-diſtance and Declination are North the Latitude will
bé North, but if the Zenith-diſtance or Declination be South, the
Latitude will be South.
For in the adjacent Figure, where
OD repreſents a Parallel of North-
crn Declination, cutting the Meri-
a
dian in the Points O and D, on the
ſame fide with the Zenith; it is ma-
nifeſt, that çif to Z D the greateſt
Zenith-diſtance, be added Dd equal H
to Z O the leaſt, half the Sum of
theſe two Arches equal to ZP, will be
the Complement of PO, the height
of the North Pole above the Hori-
zon, equal to the Latitude of the
N
Place : wherefore, if the Bright Star
in the Northern Crowns were found by Obſervation to have 56 deg.
o8 min. 23 ſec. Meridional Zenith-diſtance to the Northward, and
at the diſtance of Twelve Hours, the faine Star was found to be di-
ftant from the Zenith but 20 deg. 47 min 37 ſec. and it were re-
quired to find the Latitude of the Place of Obſervacion.
To the greateſt Meridional Zenith-diſtance 66 08
08 23 North
Add the leaſt Meridional Zenith-diſtance-10
47 37 North
ORG
Half that Sum
76 56 00
00
Will give the Complement of the Latitude 38 28
Which therefore taken from
90
00
00
Will leave the Latit. of the Place of Obſer. SI
32 00 North
Or,
If about the Point P the Parallel qo be drawn, it is evident that
if to on the greateſt Merdian Altitude, be added Og equal
to DO the leaſt Meridian Altitude, the Sum 90 equal to twice
PO is the Altitude of the North Pole above the Horizon ; where-
fore, in the preſent Cafe if
TO
364
Methods for determining the
To the greateſt Meridian Altitude of the Star 79
Be added the leaſt Meridian Altitude
23
12 23
51 37
..........
Half the Sum
103 04
04 00 N
Will be the Latit. of the Place of Oblervation si
32
oo N.
Or,
62
If from the Stars Declination, ſuppoſe
Be taken the leaſt Meridian Zenith-diſtance
19 37 N.
47 37 N.
10
There will remain the Latitude of the Place --SI
32 00N.
For EO-ZOEZ-PO.
Or,
If to the Stars diſtance from the Pole
Be added the leaſt Meridian Altitude
27 40 23
23 51 37
The Sum will be the Latitude-
-51
32
oo N.
z
S
D
For PD-DO=PO.
But if as in the adjacent Figure,
where the Parallel of the Stars De-
clination cuts the Meridian in the
points O and D, on the South ſide of
the Zenith, it is evident that if to
Zo the leaſt Zenith-diſtance, be
i
0
added ZD the greateſt Zenith-di-
ſtance, that half that Sum will give
P ZS the Complement of SH, the
height of the South Pole above the
Horizon, equal to the Latitude.
For ZDZO=2Z S; and again,
If to HO the greateſt Meridian Altitude, be added HD the feaſt
Meridional Altitude, half that Sum will give HS, the Altitude of
the Southern Pole. For O H+D H=2 Š H.
Or,
If from the Stars Declination be taken the leaſt Zenith-diſtance,
the Remainder will be the Latitude of the Place. For EOZO
ZEZ =SH.
Q
N
Org
the Latitudes of Places.
365
Or,
If from the greateſt Zenith-diſtance be taken the Complement of
the Stars Declination, or its diſtance fion the Pole, the Remainder
will be the Complement of the Latitude. For Z D-DS=Z$=
to the Complement of S H.
Or,
If to the leaſt Meridional Altitude, be added the Compicment of
the Stars Declination, or its diſtance from the Pole, the Sum will
be the Latitude. For HD+D S=SH the height of the South
Pole, or Latitude of the Place Southerly.
By comparing the ſeveral preceding Rules together, it appears,
that it in comparing the Zenith-diſtance and Declination together
they are found to be the ſame, that is, both North or both South,
that then their Sum will be the Latitude of the Place; but if thcy
are of a contrary Denomination, that is, if one be North and the
other South, then their Difference or the Exceſs of the greater above
the lefler will give the Laritude, and to find whether the Latitude
be North or South, take this Gencral Rule, the Reaſon of which
is evident from the preceding Examples.
Having drawn a Circle to repreſent the Meridian, the Horizon
HO,and the Prime Vertical Z Nas in the ſeveral preceeding Figures.
1. Set off the Meridional Zenith-diſtance from the Zenith Z on
the Right-hand, if it be Northerly, but on the Left-hand if it be
Southerly to O, this will give the Place of the Sun or Star; and
2. From this point ſet off the Declination the contrary way to
its Title, that is, towards the Left-hand if it be North, but towards
the Right-hand if it be South; and it will give the Eaſtern point of
the Equator E
3. From the point E ſet off go deg. oo min. or a Quadrant, to-
wards the Right-hand, and it gives the Place of the North Pole P.
Or,
Having found the Place of the Sun or Star O.
1. Set off the diſtance of the Sun or Star from the North Pole,
that is, the Complement of the Declination of the Sun or Star, if
the Declination be North, but the Declination increaſed by 90 deg.
oo min. if it be South, frm the point towards the Right-hand,
and it gives at once the Place of the North Pole P.
Or,
Having drawn the Meridian ZONH, and taken the point © af
pleaſure.
Bbb *
1. From
366
Methods for determining
1. From the point O ſet off the Complement of the Sun's Meri-
dional Zenith-diſtance, or Meridional Altitude, downwards towards
the Right-hand, if the Zenith-diſtance be Northerly, to 0, and it
will give the North point V of the Horizon Ho, but if the Zen.
ith diſtance be Southerly, ſet off its Complement or the Micridional
Altitude downwards towards the Left-hand, and it will give the S.
Point of the Horizon H.
2. From the ſame point o, ſer off' the diſtance of the Sun or Star
from the North Pole towards the Right-hand, and it will give the
Place of the North Pole P.
Now it in either of theſe Cafes the North Pole be clevated above
the Horizon, the Latitude is North, but if it be deprelled below
the Horizon, and conſequently the South Pole be elerated on the
contrary Side, the Latitude is South.
And the ſame Conſequences will follow, if the diſtance of the
Sun or Star be counted from the South Pole ; and if inſtead of Jay-
ing off the Polar Diſtance towards the Right-hand it be laid of
towards the Left-hand.
From a GeneralView of the ſeveral Figures, thc intelligent Reader
will eaſily perceive that there may be other Methods deduced for
finding whether the Latitude be North or South, but theſe are the
readieſt foi Practice ; and indeed this way of pronouncing the Deno-
mination of Latitude is ſo very natural and ealy, that after the Prac-
titioner has been a little accuſtomed to it, lie will readily form a
juſt Idea of it in his Mind, and determine which way the Latitude
is, without being at the Expence of drawing a Figure.
The laſt Method of determining the Latitudes of Places, by the
Meridional Heights of the Sun or Stars, obſerved above and below
the Pole, but more particularly by the Stars, is vaſtly preferable to
any
Method that I know of; inaſmuch as it requires no Theory of
the Sun to be known or given, nor no Declination of the Stars dedu-
ced from Calculation, grounded upon Obſervation firſt, but only
the ſimple Altitudes taken by Obſervation with a good Quadrant
and proper Allowances to be made for the Refraction, and how that
is to be accounted for will be thewn in the Sequel of this Section
And it the Latitude of the Place be conſiderable, even this alone
will in a manner vaniſh, as will appear hereafter; and it has this
farther Advantage, that if the Stars obſerved lye near the Pole, the
Altitudes increaſe and decreaſe ſo ſlowly, that if they are taken at
ſome ſmall diſtance from the Meridian, the Error will be incon.
fiderable.
IS
17
Methods for determining
367
It is found by Experiment, and confirmed by daily Experience)
that a Ray of Light ifluing out from a luminous Body, and falling
upon any Medium or Diaphanous Body, ſuch as Glaſs, Water, &c.
of a diferent denſity from that from whence it firſt proceeded, that
it does not go directly in the ſame ſtrait Line, but is broken or bend-
ed, and propagated, as if it had proceeded from another Point than
it really did; and if the Medium upon which the Ray falls, be denſer
than that in which it moved in before, it will be bent in wards to-
wards a Line perpendicular to the Surface upon which it fell, at the
point of Incidence ; but if the Medium upon which the Ray falls be
Rarer, in its paflage thro' it, it will recede from the ſame perpendicu-
lar : And in order to deterinine the Law that it obferves in its Par-
fage thro’ the two different Mediums, it is neceſſary to obſerve, that as
Nature takes the ſhorteſt way to accompliſh its Ends, ſo a Ray of
Light in paſſing from one point to another thro' different Medi-
ums, performs it in the ſhorteſt time poſſible ; this is an undoubted
Truth, and this being premiſed,
Let it be required to find the Point Pin the Right-line MD, lying
in the ſame Plane with the points Rand A, that a Ray of Light in
going from R to P with a given velocity v, and from P to A with a
given velocity c, fall travel from R, by Pio A, in the ſhorteſt time.
From the points Rand
A upon the Line MD, let
fall the Perpendiculars RS
R
C
and AE, and let RSa,
AE=b, SE=d, and SP
=x, then will PE=d-*,
RPSVaat xx and AE-
Р Е
Vbb dd - 2dx + xx.
D
M
S
C
But by the known Laws
of Motion, if a Body
move from R to P with
the velocity v, and from
L
a
P to A with the velocity
BA
s, if thefc velocities are
equal, then the times will be as the Spaces deſcribed PR and PA;
and if the ſpaces R P and P A are equal and the velocitys unequal,
the times will be reciprocally, as the velocitys: wherefore, if the
paces deſcribed, and the velocitys are both unequal, then the times
Bbb 2
will
X
368
Methods for determining the
will be in a Ratio compounded of the direct Ratio of the Spaces
nd the Reciprocal Ratio of the velocitys; and conſequently the
çime that a Ray of Light will take to move from R to P with the
Vaa+xx, and the time that the
velocity v, will be repreſented by
fame Ray of Light will take in moving from P to A with the veloci-
ty c, will be repreſented by
Vaa + xx + 1bb!dd2dx + xx, or cxVaat xx-tux
V
vbb dd2dx +*x; and conſequently
с
their Sum Vaa txx
с
C
2CXX
PROY
KT
du
V*
wbl +dd-2dx+-xx, will repreſent the ſhorteſt time that a Ray of
Light will require to move from R by P to A, with the reſpect-
ire velocitys vand
c;
--2dvx+ 2 uri
Wherefore,
+
2 Vaa+xx' 2Vbbidd-2dx+xx=0, and conſequently
cºt-|
=o, by dividing by 2, expunging x, and ſubſtiru-
PR PA
ting PR and PA in the room of Vaa+xx, and Vbb + dd2dx-+ **
their Equals. Wherefore, PR PA
Let us ſuppoſe P R equal to PA, for the Refraction will be the
ſame in the point P, whether the Ray P A be longer or ſhorrer;
then will cx=do-vx; and conſequently, vic::x:d-x; that
is, as SP to P E.
Now if P R equal to P A be conſidered as thc Radius of a Circle,
SP will be the Sine of the Angle SRP, equal to the Angle RPC,
the Angle of Incidence, and P & equal to BA the Sine of the Re-
fracted Angle BPA; whence it follows, that the Sines of the An-
gles of Incidence and Refraction, are directly as the velocitys v and
c of the Ray, in its paſſage thro' the different Mediums.
And becauſe the velocitys of Bodys moving thro' different Medi-
ums arc reciprocally Proportional to the denſitys of the reſpective
Medilims, it follows, that the Sines of the Angles of Incidence and
Retraction of a Ray of Light, paſſing thro' different Mediums, are
reciprocally Proportional to the Denſitys of thoſe Mediums; and this
is the Fundamental Law upon which all the Science of Dioptrics
depends.
In
the Latitudes of Places.
369
a
In the adjacent Figure,
let VR E repreſerit a Qua-
drant of a Vertical Circle
in the Heavens, OB a
Quadrant of the Earth's
Surface, and ZAH à
2
Quadrant of the Surface
of the Atmoſphere encom-
R
Z
ed
paſſing rhe Earth, all de-
Icribed about C their com-
D
mon Center, and the Cen-
ter of the Earth, and let
R repreſent any luminous
Body, from which pro-
ceeds a Ray of Light RA,
falling upoi the Surface of E
the Atmoſphere in the
H B
С
point A; now ſince this Ray in its progreſs from R to A, travis
tiro'unEtherial Matter, infinitely finer than our dir, or rather thro'
a Vacuum, when it falls upon our Acmoſphere in the point A which
is denſe in comparilon of the former; it will be refracted towards
the Line CA, perpendicular to the point of Incidence A, and in-
ficad ot going on to d, which it would have done had it ſtill
gon on
in its former Medium, it will move thro' the Path 40, and to the
Obſerver placed in the point 0, will appear as if it came from the
point r, (fince all the Objets are ſeen and judged to proceed in that
Line, according to whoſe Directions the Ray chat enters the Eye
ſtrikes upon the Senſorium) nearer the Zenith than really it is, or
higher by the Quantity of the Arch Rr
And inaſmuch as our Atmoſphere is not only infinitely denſer than
that fine Etherial Matter, it any there be, that is contained between
his and the fixed Stars, but being an Elaſtic Fluid, its lower parts be-
ing preſſed upon by the upper they become denſer, and this happen-
ing in every part of the Atmoſphere, from the uppermoſt Surface
to the lowermoſt, or that which is contiguous to the Surface of the
Earth, it follows, that every infinitely ſmall Surface of the Atmo-
ſpere is denſer than that that circumſcribes it, but rarer than that
that it circumſcribes; whence it comes to paſs that a Ray of Light
in its paſſage thro’the Atmoſphere, meets with a different Reſiſtance
in every point of it, and conſequently inſtead of procecding in the
ſtrait
376
Methods for determining
ſtrait Line A0, it is bent into the Curve Line a 0, and will appear
to the Spectatator placed in 0, as if it had proceeded all along in
the Liner A0 or Tangent to the Curve in the point of Licidence
0; and hence it is that the Sun and Moon when in dire& Oppofi-
tion, and when one is above and the other really below the Horizon,
may be ſeen born at once above the Horizon ; for the Rays of Light
by falling upon our Atmoſphere are retračied and bent inwards, and
conſequently the Bodys made to appear in different places than
where they really are; by having the Directions of their Ray's chan-
ged and their Light propogated in different Lines, than it would
have otherwiſe gon in.
And this is the Reaſon why when one End of a Stick is plunged
in the Water and the other part is in the Air, that ic appears broken;
inalmuch as that part which is in the Water appears higher than
really it is ; and that if SLaD (See che Figure in Page the 367)
be a Vettel full of Water, that a piece of Money, or any Object
placed in the point A, ſhall appear to the Eye of the Spectator fci-
tuated at the point R, as it it were placed in the point ; and that
for the ſame reaſon the whole Body of the Sun docs juſt appear abou
our Horizon, when his upper Limb is really below, and he ought
totally to diſappear.
The only certain and practical Way to find the true Quanti-
cy of the Refraaion, is by obſerving the Altitude of a fixed Star
whoſe place in the Heavens is known ; for having the Latitude of
the Place of Obſervation, the true Declination of a Star, or its
diſtance from the Pole, and the true time of the Day, which may
be obtained ſeveral ways, we may find after the manner taught in
the sth Section of the 2d Volume, the true Altitude of the Star at
that time, and this taken from the Altitude obtained by Obſervati-
on, will leave the true Quantity of Refraction at that height.
When the Sun or Star is in the Zenich the Refraction ceaſes, for
the Ray that proceeds from the luminous Body, to the Center of
the Earth, coincides with the Perpendicular to the point of Incidence,
bur the farther off the Sun or Star is from the Zenith, the more o-
bliquely does the Ray fall upon the Surface of the Air, and ſo
much the greater is the Refraction ; and conſequently at the Hori-
zon it is the greateſt of all; and at London it is found to be 33 min. oo
Sec. and at all intermediate diſtances from the Zenith it is found to
be as in the following Table deduced from the Flamſieedian Obſer-
vations, made at the Royal Obſervatory at Greenwich.
And
The Latitudes of Places.
391
A Table of Refractions, fewing how much
minſt be added to the Zenith-diſtance obler.
ved by the Quadrant, in order to find the
true Zenith-diſtance.
Zenith-diſtance.
Refraction.
Refraction.
Zenich-diſtance
Refraction.
Refra&ion:
Zenith-diſtance. Il
sce le lo
Zenith-diſtance. Leil
D
M. S.
DIM.
S. D
lon
M. S. DM. S.
90
33
00
0
4 33 65
I 4515
58
I
89
I
22
I
O
25 3879
89 23
78
88:20 1777
88
17
26 76
87 15
1575
có 164
4563
29
13
00
60
ظ دما برا نہ بن
56
54
52
50
I
4049
3648
31.47
2740
2345
O
62
61
I
O O
I
48
li
1
87
2
I
2
13
86 11
86
10
85
8
I
4852
38158
19157
2374
5373
39172
3371
48 701 2
0 46
0
45
O
44
O 42
2
I
20
44
1743
1442
I141
0949
2
21
I
56
9 33
2
85
1455
I
O
40
|
69
2
I
O
84
83
82
0735
05
2
I
20
0
26
25 1 68
3767
66
7
6
5 37
5
02
4 33 165
34
2.9
19
09
0754
ΟΙ 53
55
52
5o
SI
45 150
I
I
02
20
I
I
O
81
80
oo
In
5800
I
0
0
00
So
372
Methods for determining
tion 3
So that by this Table it appears, that if the apparent Zenith
diſtance be 75 deg. oo min. the true Zenith-diſtance is 75 deg. 03
min. equal to 7; deg. oo min.-.-03 min. oo!fcc. the Refraction ; and
on the contrary, if the apparent Altitude be 15 deg, oo min. the true
Altitude is 14 deg. 57 min. equal to the true, leſſened by the Refrac-
min. oo lec. in like inanner, if the viſible diſtance of the Sun
from the Zenith be 61 deg. 30 min. 10 ſec. the real diſtance is 61 deg.
31 min. 39 ſec. equal to 61 deg. 30 min. 10 ſec. For min. 29 ſec.
and on the contrary, the viſible Altitude being 28 deg, 29 inir. 50
fec. the real Altitude will be but 28 deg. 28 min. 21 ſec. equal to 28
deg. 29 min. so ſec.-- 1 min. 29 ſec.
And altho’the Refractions are greater in Winter than in Summer,
and in the Morning than che Evening, and at greater diſtances from
the Equator either to the Northward or Southward, than they are
at nearer diſtances; occafioned by the different denſitys of the Air,
ariſing from the different Degrees of Heat and Cold; yet this Dif-
ference is ſo ſmall in the ſame Place, as that it may be ſafely re-
jected; and the Allowancnces here made, will ſerve for all Places
within 60 deg, of Latitude, but at greater diſtances the Cold is ſo
intenſe, that the Horizontal Refractions are very conſiderably aug-
mented, as has been taken Notice of elſewhere.
'Tis upon the Account of the Refraction, that the Sun and Moon
when near the Horizon appear of an Eliptic Form, inaſmuch as their
upper Limbs are leſs retracted than their lower Limbs, and conſe.
quently appear nearer to each other than they would otherwiſe do ;
while the two Extremitys of the Horizontal Diameters being equally
refracted, keep at the ſame diſtance from each other, and the ap-
parent Diameter remains the ſame ; and 'tis upon the ſame Account
that we can look upon the Sun's Body without danger of hurting
our Sight when he is near the Horizon, ſince great part of his Rays
by palling thro'a much greater Quantity of Air, than when he is
conſiderably elevated are reflected and ſent another way, and the
Force of the reſt very much weakened, by the Reſiſtance that they
meet wit'i from ſo great a Body of Air, as they are obliged to paſs
thro', and the Number of Heterogeneous Particles that float in it.
The Longitudes of Places.
373
Vaaduka
obala S3
Section XVI.
Containing an Account of the ſeveral Methods hitherto
known, for finding the Difference of Longitude.
ND here I ſhall take no Notice of the ſeveral whimſical
Methods propoſed, ſince the paſſing of the Bill for giving
a Reward for the Diſcovery of the Longitude, which I look upon
as ſo many Effects of diſtempered Brains; but of ſuch Methods
only, as have been approved of by the beſt Mathematicians of all
Times, which are demonſtrably true ; and which have often been
put in practice with good Succeſs.
It has been fliewn in the 3d Caſe of the Second Se&tion, how from
the Latitude of the Place the Ship departed from, the Courſe upon
which ſhe failed, and the Diſtance upon that Courſe, to find at all
times the Latitude of the Place the Ship is in, and how much ſhe
has altered her Longitude : this it the Courſe and Diſtance could
be depended upon, would infallibly give the true Place of the Ship
at all times; but the many Accidents that contribute to viciate
both the Courſe and Diſtance render the deductions from it ſo un-
certain, that no poſitive or certain Concluſions can be drawn from
it. And,
The only ſure Help that our Seamen have at preſent, and by
which they Correct and Mend their Journals, being the Latitude of
the Place they are in, deduced from Obſervations made of the Al-
titude of the Sun and Fixed Stars, when they can be ſeen, after the
manner taught in the former Seation and elſewhere; and which is of
vaſt Afliſtance to them. A Way of finding out the Alteration of
Longitude made to the ſame Degree of Exactneſs as the Latitude
can be determined, is the only Thing wanting, to render the Sci-
ence of Navigation compleat ; and what Affiſtance the Heavens af-
ford them for the obtaining of this, to what degree of Exactneſs it
may be done, and what are the chief Impediments that render it not
quite ſo practicable as the Methods for finding the Latitude are,
ſhall be the principal Buſineſs of this section.
Сс с*
Every
.
374
Methods for determining
Every one who is but moderately skill'd in theſe Things, knows
that the Difference of Longitude between any two Places, is equal
to the Arch of the Equator, intercepted between the two Meridians
paſſing thro' the two Places; and that this is analogous to the quan-
tity of Time, that the Sun requires to move from the Merid, of one
Place to that of arother; or in the Language of the Copernicans, that is
elapſed between the Application of the Meridian of one of the Places to
the Sun, and the Meridian of the other; for ſince the Sun finiſhes his
Diurnal Revolution in the Space of Twenty-four Hours, or which
is the ſame Thing, ſince the Revolution of the Earth about her own
Axis is performed in the ſame time; it follows, that in every Hour
there pafles over the Meridian one Twenty-fourth part of 360 De-
grecs, or of the whole Circumference of the Equator equal to is De-
grees, in 2 Hours one Twelfth part or 30 Degrees, and in any
greater or lefler part of Time a proportional greater or lefſer part of
the Equator; whence it comes to paſs, that if the Difference of
Longitude, or Arch of the Equator intercepted between the Meri-
dian's pafling thro' any two Places be known, the Difference of the
Times of the Day in thoſe two Places is known alſo ; and conſe-
quently the Hour in one place being known, the Hour in the other
Place is known alſo; and vice verſa, if the Difference between the
Times at any two Places be known, the Difference of Longitude be-
tween thoſe two Places is known alſo, by reducing the Difference of
the Times into Degrees and Minutes, allowing 15 Degrees to an
Hour, c.
Hence it is, that if two or more Places lye under the ſame Me-
ridian, that the Hour in one will be the ſame with the Hour in the
other; and on the contrary, if in two or more Places the Hour be
the ſame, thoſe Places lye under the ſame Meridian.
And becauſe the Sun in all Places conſtantly riſes in the Eaſt, he
muſt neceſſarily apply himſelf to the Meridian of the Eaſtermoft Place
firſt, and conſequently in that place which lies to the Eaſtermoft
the Noon happens fooneſt ; and the Hour of the Day or diſtance of
the Sun from the Meridian at any other Time mult be greateſt :
Thus, for Exampler becauſe London lyes 58 deg. 15 min. equal in Time
to 3 h. 53 m. to the Eaſtward of Barbadoes, when the Sun is upon
the Meridian of London he is 3 h. 53 m. ſhort of the Meridian of
Barbadoes, that is when it is 12 of the Clock at London it is bus
after Eight in the Forenoon at Barbadoes ; and on the contrary, be-
caufe Barbadoes lyes 58 deg. 15 min. or 3 h 53 m. to the Weſt-
ward
7 m.
the Longitudes of Places.
373
ward of London when the Sun is upon the Meridian of Barbadoes,
he is 3 h 53 m. paſt the Meridian of London ; that is, when it
is 12 of the Clock at Barbadoes, it is 3 of the Clock and 53 min. in
the Afternoon at London.
Whence it appears, that if by any Contrivance whatſoever, the
Hour of the Day at the ſame point of abſolute Time in two diffe-
rent Places can be obtained, the Difference of Longitude between
thoſe Places is known alſo; and by comparing the Times together,
it is eaſy to pronounce which Place of the two lyes to the Eaſtward
or Weſtward of the other.
Wherefore, if two or more Perſons can view the ſame Appear-'
ance at two or more Places, and pronounce the Time at each Place
when ſuch Appearance was viſible, or if the time when any notable
Appearance ſhall happen at any Place be predicted, and the Time
when that Appearance was viſible at any other Place was determi-
ned, theſe Times being compared together, will give the Difference
of Meridians, or Difference of Longitude between the two Places.
Now ſince an Eclipſe of the Moon proceeds from nothing elſe but
an interpoſition of the Earth, between her and the Sun, by which
Means ſhe is prevented from refle&ing the Light ſhe receives from
the Sun, the Moment that any part of her Body
begins to be depri-
ved of the Solar Rays, it is viſible to all thoſe People who can ſee
her at the ſame time ; whence if two or more different People, at
two or more different Places, obſerve the times when it firſt began
or ended, or note the time when any Number of Digits was eclip-
led, or when the Shadow begins to cover or quit any remarkable
Spot, the Difference of thoſe Times (if there be any) when compa-
red together, will give the Difference of Longirude between the Pla-
ces of Obſervation: And of theſe we have various Inſtances, on the
21st of December 1675, at 16 h: 07m. 15 f. Mr. Flamſteed at the
Royal Obſervatory at Greenwich, obſerved the End of a Lunar Eclipſe;
and Mr. Calini obſerved that the End of the fame Eclipſe happened
at the Royal Obſervatory at Paris at 16 h. 15 m. 25 ſ. whence the
Difference of Times between the two Places of Obſervation is 8 m.
10 f. equal to oz deg. 02 min. 30 ſec. Whence it follows, that fince
the Sun was removed farther off from the Meridian of Paris than it
was from the Meridian of Greenwich ; or which is the ſame thing,
ſince it happened later ar Paris than it did at Greenwich, it follows,
that Paris Iyes to the Eaſtward of Greenwich 2 deg. 2 min. 30 ſec.
and ſo much is the Difference of Longitude between thoſe two Pla-
Сҫс 2 4
ces. Again,
12
376
Methods for determining the
On the 11th of February 1681, Mr. Flamſteed obſerved at Green-
wich the Beginning of a Lunar Eclipſe to happen at g h. 12 m. the
Beginning of the ſame Eclipſe was obſerved at Lisbon in Fortugal to
happen at 8 h. 31 m. P. M. whence the Difference of dicicians
or Ditierence of Longitude between Greenwich and Lisbon is 4 min .
of Time, anſwering to so dog. 22 min. į, Lisbon lying ſo much to
the Weſtward of Greenwich. Now by comparing the Diffcrerce of
Longitude between Paris and Greenwich, and between Greenwich and
Lisbon together, may the Difference of Longitude between Paris and
Lisbon be determined.
Greenwich Iyes to the Weſtward of Paris 2 deg. 2 min. 30 ſec .
and lisbon lyes to the Weſtward of Greenwich so deg. 22 min. 30
fec. wherefore Paris lyes to the Eaitward of Lisbon 12 deg. 25 win.
oo fec. And I my ſeit on the 11th of December 1704, at the Royal
Exchange in London, (it being a little cloudy) obſerved as near as
poflibe, that the Moon began to be eclipſed at about 41 h. 31 m.
in the Morning, and th: Beginning of the ſame Eclipſe was obſerved
at Deften in New England by Mr. Brattle, to happen at 49 min. after
Eleven, whence the Difference of Longitude between London and
Boſion by this Obſervation is 4 h. 2 min. 1, cqual to 70 deg. 37min,
Again, as near as I could judge, it being for the moſt part cloudy,
the End of the ſame Eclipſe happened at London about 5 h: 37 in.
and this was likewiſe obſerved to happen at Bufton exa&ily at 54 min
after Tw«Ive, whence by comparing theſe twoObſervations together,
it appears that the Difference of iongitude amounts to 4 h. 43 m.,
or 72 deg 52 min. and by taking the Means between then, it will
be found that the Difference of Longitude between London and Boj-
ton is 4 h. 43 m. or 70 deg. 45 min. Boſton lying ſo much to the
Weſtward of London.
One Reaſon of the Difference between the Beginning and End was
occaſioned by the badneſs of the Weather at London, it being ſo
cloudy for the moſt part, that no Body beſides my ſelf ſaw it in or
near London, however as the ſame Difference of Longitude has ſince
been found by comparing other Obſervations together, there is no
room left to doubt of the truth of it
The Longitudes of Places may alſo be determined from the Ob-
ſervations of Solar Eclipſes, but theſe being incumbered with the
Conſiderations of Parallaxes, are not near ſo apt nor commodious
as thoſe of Lunar Ecliples are; and each of thefe at the beſt
happening but rarely, another excellent Expedient has been thought
of, and that is the Eclipſes of Jupiters Satellites.
Jupi-
The Longitudes of Places.
377
Jupiter has been found to have four Satellites or Moons conſtantly
attending hin, always obſerving the fame Lams in moving round a-
bout hiin, the firſt or neareſt to his Body finiſhes his Revolution a-
bout him, in the Space of 1 Day 19 hours 28 Minutes, at the dif-
tance of 5.58 Scmi-diameters of Jupiter from his Center; the
fecond in 3 Days 13 Hours 17 Minutes, at the diſtance of 8.87
Seini diameters; the third in 7 Days 3 Hours 59 Minutes, at the
diſtance of 14. 16 Semi diameters; and the fourth in 16 Days 18
Hours 5 Minutes, at the diſtance of 24.90 Semi-diameters. -
Now it having been found, that neither Jupiter nor any of his
Attendants have any native Light of their own, but thine with a
borrowed Light from the Sun, it happens that each of theſe in every
Revolution about Jupiter, fuffer two Eclipſes, one at their Entrance
into the Shadow, the other at the Entrance of their Paflage behind
his Body, whence it comes to paſs that in each Revolution of the
Satellite, there are four remarkable Appearances by the Oblervati-
on of any one of which the Buſineís may be done, viz. one at the
Entrance into the Shadow, and one at the Emerſion out of it, one
at the Entrance buhind the body, and another at the coming out,
but the latter of thelt, viz. the Ingreſs and Egreſs of the Satellite,
into and from under the Body, is not ſo much regarded by Aſtrono-
mers as the Immerſion and Emerſion into and out of the Shadow,
becauſe in the førir.er, the difficulty of pronouncing the exact
Time is very great, it requiring in each Obſerver, Eyes equally good
and ſtrong, and Teleſcopes equally large, but the Obſervation of the
former of theſe viz. the Iminerſion into, and Emerſion out of
the Shadow, is eaſy and practicable, becauſe the ſwift Motion of the
Satellites plunge them ſo quick into the Shadow of Jupiter, that it is
110 difficult matter to pronounce with any Telescope, by which they
may be feen, the exact time of their Immerſion and Emerfion, as a-
ny one may ſoon be ſatisfied if he will but try the Experiment.
Now inaſmuch as cach of thele lappen at the ſame Moment of
abſolute Timc, if two or more Perſons in different Places, note the
Times of Oblervation, there when compared together, will give the
Difference of Longitude between the two Places of Obſervation.
Amongſt the inany Obſervations that have been made, by which
the Longitudes of many Places have been determined, I have extracted
the following out of Philos. Tranſact. No. 177. Anno 1680 Oitobi
the 23d Old Stile. Signior Joſeph Pont hea and Marco Antonio Celijo
obſerved the Total Immerſion of the firſt Satellite into the Shadow of
Jup-
378
Methods for determining
+
Jipi:er at Rome, at 10 h. 7 m. 53 f. P. M. which Immerſion was
allo obſerved by Mr. Flamſteed, at the Royal Obſervatory at Green-
wich, to happen at 9 h. 15 m. 41f. whence the Difference of Me-
ridians betwixt Rome and Greenwich is 52 min. 12 ſec. in Time, an-
ſwering to 13 deg. 03 min. Rome lying ſo much to the Eaſtward of
Greenwich.
In this Inſtance we have ſuppoſed the ſame Obſervation to be
made by two different Perſons at the ſame Time, but the Motions
of the Satellites being truly ſettled, there is need only of a Catalogue
of the Eclipſes to be publiſhed for the Meridian of any one Place,
and the Obſervations made in different Places, compared with the
Times ſet down in the Catalogue, will give the Difference of Lon-
gitude between the Place of Obſervation, and the Place for which
che Catalogue is Calculated.
And as an Inſtance of the exactneſs of the Tables for the firſt Sa-
tellite, I need only mention an Obſervation of Signior Francis Blan-
chini made at Rome in the Year 1685, when on the 28th of January,
he obſerved the total Immerſion of the firſt Satellite at 11 h. 19 m.,
which by the Tables happened at Greenwich at 10 h, 27 m. Ả,
whence the Difference of Meridians between Rome and Greenwich is
52 min. 1, anlwering to 13 deg. 07 min. į agreeing very well with
the former Deductions
When we confider the great Number of theſe Eclipſes that happen
every Year, there being more viſible in one Year than there are Days
in it, and conlequently but few Nights when Jupiter may be ſeen,
and which is near Eleven Months of the Year, but that an Eclipſe
of one or other happens, and ſometimes two or three in a
Night ; the eaſineſs with which they may be made, there requiring
only a Teleſcope of eight or ten Foot in Length, which may be al-
moſt managed with the Hand ; and the little likelihood there is of
milling the times of Ingreſs or Egreſs, they being in a manner Mo-
mentaneous: And laſtly, the great exa&nefs to which they would
give the Difference of Longitude, it being certainly as exact as the
Latitude at preſent can be taken ; it is much to be wondered at, that
the more skilful Part of our Seamer have ſo long neglected them,
and elpecially in the ſeveral Ports into which they Sail.
Beſides theſe, there is another Method equally Uſeful, Expedi-
tious, and Certain, and that is the Appulſes of the Moon to certain
Fixed Stars, and theirOccultations by the interpoſition of her Body,
for the Moon finiſhing her Revolution in the ſpace of 27 Days 7
Hours 43 Min.
The Longitudes of Plares.
379
43 Minutes there are but few clear Nights, when the Moon does not
paſs over or ſo near to ſome Fixed Star, that their diſtance from it, or
the time of her vifible Conjun&tion with it may be eaſily obſerved by
the Teleſcope and Micrometer only, and theſe when compared together,
or with the viſible Time computed to the Meridian of ſome Place
when a good Theory of the Moon ſhall be obtained, will ſhew the
Difference of Longitude of thoſe Places; an Inſtance of the Uſe of
this Viethod we have in the determination of the Difference of Lon-
gitude between London and Ballaſore in India, as follows.
On the 28th of O&tober 1680, at 8 h. 6 m. an Immerſion of the
Bulls Eye was obſerved at London, when the true Place of the Moon
corrected by Parallax was Gemini 4 deg. 32 min. 24 ſec. but at
Balaſore Road (in the Latitude of 21 deg. 20 min. North, and
about 20 Miles Eaſt South Eaſt from the Town) the true Place of
the Moon was Gemini s deg. 54 min. that is i deg. 21 min. 36 ſec.
more than at 8 h. 6 m, at London ; now according to the Moons ve-
locity at that Time, ſhe paſſeth an Arch of 1 deg. 2imin. 30 ſec.
in z h. 8 m- 40 . of Time, ſo then at 10 h. 14 m 40 ſec. at Lane
don, the Moon was in the ſame Place as at 16 h. com. at Ballafore-
Road, whence the Difference of Longitude will be 5 h.45 in. 20 ſ.
anſwering to 86 deg. 20 min. Ballafore-Road being ſo much to the
Eaſtward of London.
Again, on the 22d of December 1680, theImmerſion of the Bulls
Eye was found by Calculation to be at 14 h. 49m. at Pallafore, the
Moon's true Place was Gemini 6 deg. 30 min. 30 ſec. and at 711.
46 m. 12 f. the correct Time of the Immerſion at Dantzick the true
Place of the Moon was Gemini 4 deg. 55 in. if that is r deg. 35
min. 20 ſec. ſhort of the Place deduced from the Obſervation inade
ac Ballafore-Road, which anſwers in Time to 2 deg. 32 min. 40 ſec.
whence it follows, that 10 h. 18 m. 52 ſ. ac Dantzick, makes
49 m. at Ballafore-Road, and conſequently the Difference of Loll-
gitude between Dantzick and Ballaſore-Road will be 4 h. 30 m 8 f.
equal to 67 deg. 32 min. Balafore-Road lying ſo much to the Eaſt-
ward of Dantzick.
Dantzicklyes i h 15 m. 30 , more Eaſterly than London, where-
fore Ballafore-Road by this Obſervation, lyes o5 h. 45 m. 38 f.ee
qual to 86 deg. 24 min. to the Eaſtward of London, agreeing within
4 min. of the former Deduction, which in that Diſtance is nothing.
For a farther Confirmation, hereof, Mr. Benjamin Harris being a..
fhoar at Balafore-Town, he obſerved with very great Care and Ex-
actneſs,
14
.
380
Methods for determining
acneſs, on November the 18th, 1680, that at 9h. 13 m. the Star
which Tycho Brahe calls in Cotyla dextïa Aquarij duarum pracedens,
(and which was then in Aquarius 28 deg. 52 min. having ou deg.
46 min. North Latitude) was in a Right-line with the Cuſp of
the Moon, llie being then near the firſt Quarter, the Moon's Place
at that Time by the Horroccian I heory, viz. at 2h.5?m. at London, is
found Aquarius 29 deg. 22 min. 10 ſec. but at Ballafore her Place
was in Aquari11 29 deg. 41 min. 17 fec. that is 19 min. 07 ſec.
more than at London; which in Time gives 36 min. ſo that gh. 29m.
at London was 9h.13m. at Ballafore. Town, whence the Difference of
Longitude is 5 h. 44 m. or 86 deg. oo min. agreeing exactly with
the firſt Deduction,
The ſame may be done from the Obſervations of the Immerſions
or Emerſions of any of the Planets from behind the Body of the Sun
or Moon, as Montieur Collini pronounced the Difference of Longi-
tude between Canton in China and Paris to be 7 h. 32 m equal to
110 deg. 45 min. froin an Emerſion of Mercury from the Disk of
the Suit
, obfcrved at Canton and Noremberg; and an Eclipſe of the
Moon obſerved at Noremberg and Paris, and tho' theſe larrer fort
happen but rarely, yet when they do they contribute their Afliſtance.
In like manner, from an Obſervation of the Occultation of Mars
by the Moon, the 21ſt of August 1676, obſcrved at Dantzick and
Greenwich, the Difference of Meridians between Greenwich and Dant-
zich was found to be 1 h. 14 m 49 f. in Time, anſwering to 18
deg. 41 min. 15 ſec.
In finding the Difference of Meridians between any two Places
whoſe Latitudes are known, from an obſerved Occultation or Emer-
fion of a Star from the Moon's Limb,or its Diſtance and poſition from
her at both Places; it is not neceflary that the obſerved Times of the
Same Appearance at both Places should be known, for if the time of
the Stars Occultation or Emerſion at one place were actually ob-
ſerved, and at the other the Diſtance and Poſition of the Star to
the Moon, the Difference of Meridians may thence be determined,
as eaſily as if the Occultation and Emerfion were obſerved at both
Places, and the ſame may be done from the Diſtance and Poſition
of a Star to any of the Planets.
The Lunar Theory is already brought to very great Perfection,
compared with what it was a few Years ago, and there is no room
left to doubt, but from the great Stock of good Obſervations that
Mr.Flainfeed has left behind, of its being carried on to all the Exact-
nets
the Longitudes of Places.
381
neſs neceſſary, and then from one lingle Obfervation of an Appulre
of the Moon to any fixed Star, or of an Occultation or any Emer-
fion of a Star from the Moon's Limb, might the Difference of Lon-
gitude between the Prace of the Obſervation, and the Place to which
the laid Numbers ſhould be fitted, be readily determined.
Mr. Flamſteed has given us the Places of near 1000 fixed Stars,
confirmed by ſeveral Obſervations thar lye within the Zodiac, each
ot' which will be covered by the Moon and the reſt of the Planets, in
onc Revolution of their Node ; ſo that ſcarce one Night can bappen
but ſome or other of them will be eclipſed, or approached fo near
unto, as to come within the compaſs of a Teleſcope, in one place of
the Earth or other, add to theſe the Eclipſes of Jupiters Satellites,
and its ſcarce poſſible, that any clear Night can happen, but the
Heavens afford us ſome agreeable Phænomenon or other, by which the
Longitude of any place may be duly aſcertained.
In the Philoſophical Tranſactions Number 1, for the Month of
March, in the Year 166$, after the Invention of Pendulum Watches
by Monſieur Hugens of Zulichem ; we have an Account of an Ex-
periment made with two of them by Major Holmes, in a Voyage
from the Coaſt of Guinea homcwards, at the requeſt of ſome of the
Eminent Virtuoſo's and grand promoters of Navigation at that time,
in the manner following.
The ſaid Major having left that Coaſt, and being come to the
Iſle of St. Thomas under the Line, accompanied with four Veffels,
having there adjuſted his Watches he put to Sea, and ſailed Weſtward
Seven or Eigit Hundred Leagues without changing his Courſe; af-
ter which finding the Wind favourable, he ſteered towards the Coaſt
of Africk North North Eaſt, but having failed upon that Line a
matter of Two or Three Hundred Leagues, the Maſters of the o-
ther Ships under his Condu&, apprehending that they ſhould want
Water before they could reach that Coaſt, did propoſe to him to
ſteer their Courſe to the Barbadoes, to ſupply themſelves with Water
there ; whereupon the ſaid Major having called the Maſters and
Pilots together, and cauſed them to produce their Journals and Cal-
culations, it was found that thoſe Pilots did differ in their Reckon-
ings from that of the Major, one of them Eighty Leagues, another
about an Hundred, and the third more ; but the Major judging by
his Pendulum Watches, that they were only ſome Thirty
. Leagues
diftant from the Iſle of Frego, which is one of the Iſles of Cape Verd,
and that they might reach it next Day, and having a great Confi-
Dad *
dence
382
Methods for determining
dence in the ſaid Watches, relolved to ſteer their Courſe thither, and
having given Order ſo to do, they got the very next Day abour
Noon, a light of the ſaid Iſe of Fuego, finding themſelves to fail
directly upon it, and ſo arrived at it that Afternoon.
This and ſome other Succelles, incouraged Monſieur Hugens ro
far, that after he had improved the Structure of theſe Watches,
he publiſhed an Account at large in the Belgic Tongue, which was
afterwards Tranſlated into Engliſh, and publiſhed in the Philoſophicai
Tranſactions N° 47, for the Month of May 1669; fhewing low,
and in what manner theſe IVatches are to be uſed in finding the Lon-
gitude at Sea, with Dire&ions for adjuſting of them, and keeping a
Journal by thein, which Account the curious Reader may fee at large
in the above-mentioned Ti anſactions N° 47, in the reading of which,
if he is ignorant of theſe Matters, he will meec with forne Things
worthy of his Notice.
The chief Objection againſt the Uſe of Pendulum Clocks and."
Watches, is the Effects that Heat andCold have, (for there have been
ſeveral Contrivances to hang them, ſo as to anſwer to all the Motions
of a Ship, except in very uncommon Accidents) upon the Spring and
Pendulum, which makes the Spring in Watches draw ſtronger at ſome
times than at other times, and cauſes the Pendulum to lengthen and
ſhorten, according as the Weather is hotter or colder; but there Ef-
fees are ſo regular, that without doubt they may be accounted for.
But the principal Thing at preſent that ſeems wanting, to render
this and the former Methods pra&icable, is a true Knowledge of the
Hour of the Day and Night, this may readily be obtained from
the true height of the Sun in the Day, or of a Star in the Night, for
their Declinations at all times are given; and as a ſmall Error in the
Latitude will make no conſiderable Error in the Time, it being one
of the containing Sides of the Angle required, if the Altitude of the
Sun or Star could be taken to 1 or 2 Minutes, the Hour of the Day
or Night might be determined to all ſufficient Degrees of Exa&nels
this may be done on the Land with a common Quadrant of 18 In-
ches Radius, rightly adjuſted, and adapted with proper Sights and a
good Pedeſtal: And I cannot help thinking, that it Men would ſet
about it in good Earneſt, they would not fail to meet with Succels at
Sea ; ſince I have been informed by a skilful Commander, who took
one along with him in an Eaſt India Voyage, that by accuſtoming
himſelf to it often, he could at laſt at ſome certain times, make Uſe
of it with good Succeſs; and I am ſure that he who can contrive a-
ny
the Longitudes of Places. 383
ny Way to take the height of the Sun or any fixed Star at Sea to
the Exa&neſs before-mentioned, may fairly be intituled to a Share
of the diſcovery of the Longitude, and ought to have a proportional
Reward.
To determine the exa& Time when any remarkable Appearance
happens, is a Poſtulatum that almoſt all our Modern Pretenders to
the Diſcovery of the Longitude, and eſpecially the moſt rational a-
mong them, have taken for granted, and is one of the chief Principles
upon which ſeveral of their Methods depend, which plainly thews how
ignorant thele Gentlemen were of what they went about, and how
little Mankind can exped from them.
People that go upon new Diſcoveries ought firſt to make them-
ſelves Maſters of what is already known, and what are the principal
Things wanting to render the Means more effe&ual; this would pre-
vent them from ſpending a great deal of Time to no purpoſe, and
enable them the better to ſucceed in their Attempts.
In Page 283 and 284, I have ſhewn a Way how to find the true
time of the viſible Riſing and Setting of cither of the Limbs of the
Sun or of his Center, by making an Allowance for his horizontal Pa-
rallax, which is ſo great in our Latitude, as to cauſe the Sun at
the time of the Year to which the Calculation is fitted, to Riſe Appa-
rently 4 min. 10 lec.}, ſooner than is given by the common Method
of Computation, and to ſet ſo much later.
And as this Method of Computation will never fail to give the
true time of the Apparent Riſing and Setting of the Sun, at ali Times
and it all Places, to a very great Degree of Exactneſs, and as the
ſame Method of Inveſtigacion is applicable to the Fixed Stars, I
know of no Way at prelent that is ſo apt, and that will anſwer the
Buſineſs in Hand ſo well as this will do.
For as there requires no Inſtruments to obſerve when the Sun or
Stars viſible Rile, nor no great skill in the Obſerver, but only
a diligent and careful looking out and watching when the Sun or Star
begins to appear or diſappear, and which every one is capable of
doing, let the Ship rowl never ſo much ; and conſidering tarther,
that there are frequent Opportunitys when ever the Heavens are
clear, and which is the only time that we have occaſion for it, of
making Obſervation, there is ſcarce a Night can happen, but we may
have an opportunity of comparing our Time-keeper with the Hea-
vens, and of ſeeing how much it is too faſt or too ſlow, and be able
thence to pronounce the true time, when any remarkable Appearance
becomes viſible.
Ddid *
And
38+
:
Methods for determining
25
And as the Beginning and End of a Lunar Eclipſe, the Occultati-
on of the Stars by the interpofition of the Moon may be diſcovered
by the naked Eye, (by Perſons that are but a little converſant with
Obſervations) to a Minute of Time, and as the Eclipſes of Jupiters
Satellites may be ſeen with a Glaſs of Eight Feet ; and inaſmuch as
theſe ſhine with the greateſt Splendor a little before they are plung-
ed into it, and ſoon after they are got out of the Shadow of Jupiter,
and that unlefs the Ship be very much tofled, they may have now
and then an Opportunity of having a view of Jupiter, and ſeeing
whether the Eclipſes are over or not ; I am well perſwaded that it
People would but uſe their Endeavours of putting ſome of theſe Me-
thods in Pra&ice, they would no doubt, very often meet with Suc-
ceſs and find more practicable Ways.
It is well known, that if never fo eaſy Methods for finding the
Longitude at Sea were propoſed, they could not be put in Pra&ice
with any deſirable Succeſs, till the Longitude of the Sea-Coafts were
better determined; for moſt certain it is, that the furer any Man is
of the Longitude of ihe Place he is in at Sea, the furer he is to miſs
the Place he is deſigned for, if the Longitude of that place be not
truly determined ; ſo that if at the worſt, the Methods hitherto propo-
fed be not pra&icable at Sea, yet it cannot be denied but that they
are pra&icable at Land; and therefore the firſt thing neceſſary to
be done, is to have all our Sea-Coaſts better ſettled, and new Sea-
Charts formed: Let them attempt this firſt, and I doubt not but
the Succeſs will encourage them ſo much that they will readily
find Means to put it in pra&ice at Sea; for things that we are unac-
quainted with, generally féem more difficult than really they are,
and Uſe very often renders thofe things caſy, which at firſt light we
thought impoſſible.
'Tis to the French Miſſionarys chiefly, that we owe the Knowledge
we have of
moſt of the Sea-Coaſts, but more eſpecially the Eaſt-Indies,
and the Coaft of China, they have put theſe Methods in Pradice
with very good Succeſs, and even at Sea if we may credir a Book
written by Francis Noel-the Hefuit, 1 Entitled Obfervationes Mathemat:
et Phys, in India et China, fatta ab Anno 1684 Ad Annum 1708.
And is it not a ſevere Reflection upon us, who want no Means
and who trade almoſt to every part of the babicable World, that in
Three Seven Years; -fcarce Three Obfervations have been made, by
which the çrue Longitudele any of the Placeszthey have been in can
be truly determined
Let
#
the Longitudes of Places.
385
Let us not call out for a Diſcovery of the Longitude as if there
were no ſuch thing in being, but let us fer our felves in earneſt
to put the Methods hitherto known in practice, in their utmoſt
Extent, and no doubt but other and more effectual Means will be
found out.
I have often heard the Government blamed, for.no: ſending Two
or Three Ships abroad, to put theſe and other Methods that have
been propoſed in Practice, and thereby bring us home ſuch a Treaſure
of Obſervations, as might enable us to draw new Maps of the Sea-
Coaſts, but there is no need for ſuch an extraordinary Expence, let
every one of His Majeſty's Ships of War be provided with a good
Teleſcope, a ſmall Quadrant, and a good Tine: keeper, and let the
Teacher of Mathematicks appointed for that Ship, be obliged in eve-
ry Port he comes into, to make all the Obſervations that happen
during the time of his ſtay there; and let him be obliged at his re-
turn home, to bring them to the Royal Society, or to any perſon or
Set of Men whom the Government ſhall think fit to appoint for this
Purpoſe, who ſhould be obliged from time to time, to make ſuch
Corrections in the Charts as thoſe Helps ſhould afford them; and
there is no room to doubt, but tliat in a little time the World in
General would receive the Benefit of it, and this Nation in particul-
lar, as well as the Honour that would redound to them for lo noble
and generous an Undertaking.
And if our Commanders of Merchant-men who want,neither Lei-
Jure nor Opportunity, and who will certainly receive the greateſt Ad-
vantages from it, would lend their Affiſtance to ſo uſeful a Work,
we might foon expect to ſee a new Deſcription of țhis our Terraqueous
Globe ; but this having been ſo often requeſted and with lo little Suc-
ceſs, that I very much doubt if ever any thing will be done towards
it, ſo that thoſe Perſons who have and may ſuffer upon ſuch Account,
as too many have done, ought to lay the Fault upon themſelves,
and are not to be pitied, ſince it is impoſſible for any one to rectify
any Part of the Sea-Coaſts, without having been firſt upon them,
and making proper Obſervations.
It is neceſſary to acquaint the Reader, that if he ſhould meet with
the Subſtance of this section, in A Treatiſe of the Sphere, publiſhed
in the Year 1714, that the Author dying ſuddenly of a Fever, be-
fore he had near finiſhed the Work, this as well as a great part of
it, was wrote by my ſelf.
. Sect,
386
Methods for finding
Dildo
Duo
Udside
:Dedidos:
widowo:dadididadidada
the Sun or Star, and the Difference between theſe (if there be any)
Section XVII.
Concerning the Variation, and the Methods of at-
taining it.
S nothing is of greater Advantage to the induſtrious Seaman,
than a perfect Knowledge of the true Courſe upon which
the Ship has or does fail, ſo nothing contributes more to it than a
frequent and diligent Obſervation of the Amplitudes and Azimuths
of the Sun and Scars; ſince by the Help of theſe, he will be enabled
not only to diſcover the Errors or Faults of his Compals, if any chere
be, but by comparing the obſerved Amplitudes and Azimuths with
the truc Amplitudes and Azimuths, find how much the Needle
points to the Eaſtward or Weſtward of the true Meridian, or the
Angle that it forms with it.
For as the true Amplitude ſhews how much the Sun or Star Riſes
or Sets, to the Northward or Southward of the true Eaſt or Welt
points of the Horizon, numbred in Degrees and Minutes of the Ho-
rizon and the Azimuth, the Angle that the Vertical paſſing thro'
the Sun or Star at any time makes with the Meridian, which is
meaſured by the Arch of the ſame Horizon, contained between the
Meridian and the Vertical Circle ; ſo the Amplitude obſerved by
the Compa's, ſhews how far the Sun or Star Riſes or Sets to the
Northward or Southward of the Eaſt or Weſt, pointed out by the
Compaſs and the Azimuth, the Arch of the Horizon contained be-
tween the Magnetic Meridian and the Vertical Circle paſſing thro?
is what is called the Variation.
So that the Variation is an Arch of the Horizon, contained be-
tween the Interſection of the magnetic and true Meridian, or the
truc North and South points, and the North or South points point-
ed out by the Needle ;. and which is ever equal to the Sum or Dif-
ference of the true Amplitudes or Azimuths deduced from Calculati-
on, and the Amplitude or Azimuth obtained by Obſervation with
the Needle, as hall be exemplifyed in all its Varities.
How
the Variation of the Compaſs
387
Rih
e
How the true Amplitude and Azimuth of the Sun or any Star
at any time may be obtaired, from proper Data by Calculation,
is ſhewn at large in the 3d and 4th Setion of the sth Part, and how
the magnetic Amplitude, or Azimuth, or Amplitude and Azimuth
obtained by the Compaſs may be found, is ſo very eaſy in it ſelf,
and ſo generally known to the skilful Srilor, that it would be need-
lefs to explain it here; and therefore I ſhall proceed to fhew how
when both are known, to diſcover the true Variation, or how much
the Needle points to the Eaſtward or Weſtward of the true North
or South points of the Horizon.
In the adjacent Figure, let N, E, S,
W, repreſent the Horizon, Nche
N
true North point, n the magneti-
cal, then will E and W repreſent the
true Eaſt and Weſt points, and e and
w the magnetical, and becauſe NE,
W?
NW, are Quadrants, and confe-
E
quently equal to the Arches: ne and
az w; it follows, that the Arch of the
Horizon Ee or w W, contained be-
tween the magnetical and true Eaſt
S
and Weſt points of the Horizon, is e-
qual to the Arch Nn or Ss, contained between the magnetical and
true North or South points, equal to the Variation, and which in
this Cafe is Eaſt, becauſe the magnetical North point n lyes to the
Eaſtward of the true North point N; whence it follows.
s. That if in comparing the true Amplitude of Azimuth with the
magnetic Ampli:ude or Azimuth, they are found to be the ſame, then
the Needle points exactly true, and there is no Variation.
2. If in comparing the magnetic and true Amplitude together,
they are found to differ, but are the ſame way, that is both North
or both South, then their Difference is the Variation, which if it.
be at Sun Riſing, and the Amplitudes are both North, and the
magnetical exceeds the true, the Variation is Eaſt, but if they are
both South the Variation is Welt
For in the preceding Figure, if o repreſent the Place of the Sun,
then will E O be the true Amplitude, and eo the magnetical and
both North
and conſequently the Arch Ee equal to the Excels
of EO, the magnetical Amplitude, above EO the true Amplitude,
is equal to Nnthe Variation; and in this Cafe North, becauſe the
Arch
j
1
388
Methods for finding
Arch n E is leſs than a Quadrant, and conſequently the magnetic
North lyes to the Eaſtward of the true North." Wherefore,
If che magnetic Amplituce be 30 deg. 44 min North, and the
true Amplitude 22 deg. 15 min. North, and it be required to find
the Variation.
From the magnetic Ainplitude---
E 30 44 North
Take the true Amplitude
E 22
15 North
There will remain the true Variation
8 29 Eaſt
71
But if as in the adjacent Figure the
N
magnetic Amplitude ea, and the
true Amplitude EO, at Sun Riſing
are both South, it is manifeſt, that
the Variation Nn equal to the Arch
W
e E, the Exceſs of the magnetic Am-
E
plitude above the true is Wefterly, for
the Arch n E is greater than a Qua-
drant, and conſequently the magnetic
North point lyes to the Weſtward of
S
the true North point, wherefore in
this caſe, if the magnetic Amplitude
at Sun Riſe be Eaſt 29 deg. so min. South, and the truc Ampi tude
be 19 deg. 55 min. South, and it be required to find the Variation.
From the magnetic Amplitude
E 29 so South
Take the true Amplitude
E 19 55 South
There will remain the true Variation
E9 55 Weſt
3. If in comparing the magnetic and true Amplitude together
at Riſing, they are both found to be Southerly, and the true Am-
plitude exceeds the magnetical, then the Variation is Eaſt, but if
they are both North the Variation is Weft.
For in the adjacent Figure, if o repreſents the place of the Sun,
CE will be the true Amplitude, Oe the magnetical, and it is manifeſt
that N n the Arch of the Variation, equal to the Arch Ee the
Exceſs of the true Amplitude o E above the magnetical oe is
Eaſterly, for the Arch n E is leſs than a Quadrant, and conſequently
the magnetic North, lyes to the Eaſtward of the true North, where-
fore it by Obſervation the Sun is found to Rife sodeg. oo min to the
South-
*****
i
the Variation of the Compaſs
389
.
Fig. 3.
N
N n
Southward of the Eaſt point by
the Compaſs, when by Calculation
he ſhould Riſe 24 deg. 15 min. to
the Southward, and it be required
to find the Variation.
20
W
E
le
S کی
From the true Amplitude.
Take the magnetical Amplitude
E 24 15 South
oo South
E JO
There will remain the true Variation
14 15 Eaſt
Fig. 4.
N
N.
But if as in the adjacent Figure,
the magnetical and true Amplitudes are
N
both North, it is manifelt that Nn
the Arch of Variation, in this Cafe e-
qual to ee, equal to the Excels of
E O above e o, is Weſterly; for the
Arch n E is greater than a Quadrant,
W
E
and conſequently the North point lyes
и
to the weſtward of the true North :
Wherefore if the Sun by the Compaſs
is found to Riſe 12 deg. 10 min. to
S
the Northward of the Eaſt, when
his true Amplitude is 17 deg. 30 min. North, and it be required to
find the Variation
From the true Amplitude
E 17 30 North
Take the magnetic Amplitude
E iż 10 North
There will remain the true Variation
--5. 20 Weſt
4. If in comparing the Amplitudes together at Setting, the mag-
netical be found to exceed the true, if they are both South the Va-
riation is Eaſt, but if they are both North the Variation is Weft.
In the adjacent Figure, if o repreſent the Place of the Sun,
then will ow be the magnetical Amplitude, and O W the true, and
Eee *
con-
390
Methods for finding
#
W!
Fig. 5.
conſequently Nn the Arch of Varia-
rion equal to Ww, equal to the Ex-
N
ceſs of the magnetic Amplitude o w
n
above the true Amplitude @ W is
Eaſterly ; for the Arch no is greater
than No, and conſequently the
magnetical North n lyes to the Right
F or to the Eaſt of the trueNorth point
N; wherefore if the Sun by Obſerva-
tion is found to Set 34deg. 15 min. to
the Southward of the Weſt, when by
Calculation he ſhould Set but 17 deg.
S
55 min. and it be required to find the
Variation, and which way it is.
From the magnetic Amplitude
W 34 15 South
Take the true Amplitude
W 17 15 South
There will remain the true Variation
· Fig. 6.
But if as in the adjacent Figure, the
N
71
magnetic and true Amplitudes are both
North, it is manifeſt, that N n the
Variation, equal to Ww the Exceſs
of the magnetical Amplitude w O a-
и
bove the true Amplitude WO is Weſt,
E for the Arch n o is leſs than N, and
z
conſequently the magnetical North
lyes to the Weſtward of the true
North : Wherefore,
If the Sun be found by Obſerva-
S
tion to Set 22 deg. 18 mi), to the
Northward of the Weſt, when his
true Amplitude is only 18 deg. 10 min. North, and it be required
to find the true Variation, and which way it is.
From the magnetical Amplitude
18 North
Take the true Amplitude
10 North
17 00 Eaſt
u
W 22
W 18
There remains the true Variation
4 08 Weſt
S, IF
the Variation of the Compaſs.
391
N n
5. If in comparing the Amplitudes together at Sun Setting, the
true Amplitude exceed the magnetical, if they are both North the
Variation is Eaſt, but if they are both South theVariation is Weft.
For in the adjacent Figure, if o
Fig. 7.
repreſent the Sun, then will öW be
N
the true Ariplitude, and Ow the
magnetical, and conſequently the
Arch of Variation Nnequal to Ww
ti
the Exceſs of the true Amplitude
oW above the magnetical w O, is
W
E
Eaſterly ; for the Arch n O is greater
than No, wherefore the magnetic
e
North lyes to the Right-hand of the
true North. Wherefore,
S
If the Sun be found to Set 14 deg. 30 min. to the Northward,
when his true Amplitude is 22 deg. 10 min. North, and it be re-
quired to find the Variation, and which
way
it is.
From the true Amplitude-
W 22
Take the magnetical Amplitude
W 14 30 North
10 North
7 40 Eaſt
7
There remains the true Variation
Fig. 8.
But if as in the adjacent Figure,
N
the magnetical and true Amplitudes
are both North, and the true exceed
the magnetical, it is manifeſt that
the Arch of Variation Nn equal to
Ww, equal to the Exceſs of Wo W
the true Amplitude, above the mag-
E
netical w O is Weſterly ; becauſe the w
Arch N O is greater than no, and
conſequently the magnetic North
lyes to the Weſtward of the true
S
North.
6. If in comparing the magnetic and true'Amplitudes together they
are found to be contrary, that is if one be North and the other
South, then their Sum is the Variation; which if it be at Riang
and the magnetical Amplitude be North, the Variation is Eaſt;
but if the magnetical Amplitude be South the Variation is Weft.
Eee* 2
For
392
Methods for finding
Fig. 9:
N n
HE
For in the adjacent Figure, if o
N
be the true Place of the Sun, then
will Eo de the Sun's true Ampli-
tude, in this Caſe South, ana e the
magnetical Amplitude, in this Cafe
w
North ; whence it is manifeſt, that
W
Nn the Variation, equal to Ee is
equal to the Sum of E O the true
le
Amplitude, and eo the magnetical,
and Eaſt, becauſe the Arch NO is
greater than n , and conſequently
S
the magnetic North lyes to the Eaſt-
ward of the true North ; wherefore,
if the Sun's magnetic Amplitude be found to be 10 deg. 25 min.
North, when by Calculation he ſhould Rile 12 deg. 24 min. to the
Southward of the Eaſt, and it be required to find the Variation and
which way it is.
To the magnetic Amplitude
E 10 25 North
Add the true Amplitude
E 12 24 South
The Surn will be the true Variation
22
49 Eart
Fig. 10.
But it as in the adjacent Figure,
N
N
the magnetic Amplitude be South,
it is manifeft, that Nn the Variation,
equal to E e the Sum of the true Am-
plitude EO and the magnetical Ampli-
tude e O is Weſt, for becauſe the Arch
W
n O is greater than No, the mag.
netic North will lye to the Weſtward
of the true North : Wherefore,
If the Sun is found to Riſe 4 deg.
S
25 min. to the Southward of the Eaſt
when by Calculation he ſhould Riſe
10 deg. 30 min. to the North, and it be required to find the Vari-
ation, and which way it is.
То
the Variation of the Compaſs.
393
To the magnetic Amplitude
Add the true Amplitude
E10
E4 25 South
30 North
The Sum will be the true Variation
14 55 Weſt
7. If in comparing the Amplitudes at Setting together, they be
found to be contrary, and the magnetical be South, the Variation
is Eaſt; but if the magnetical be North the Variation is Weſt
For in the adjacent Figure, if o be the Place of the Sun, then
will Ww be the true Amplitude Northerly, and w the magnetical
Amplitude Southerly, whence it is manifeſt that Nn the Variation,
equal to w W, the Sum of the magnetic Amplitude o w and the
true Amplitude Wis Eaſterly, for
Fig. IL
the Arch No is leſs than n , and
N
72
therefore the magnetic North muſt
point to the Eaſtward of the true
North ; wherefore when the true Ain-
plitude by Calculation is found to be
6 deg. 30 min. North, and the Sun is. W ve
E
found to Set 4 deg. 45 min. to the
Southward of the Weſt, and it be re-
quired to find the Variation, and which
of
S.
To the magnetical Amplitude
W 4 45 South
Add the true Amplitude-
W 6 30 North
e
way it is.
II
15 Eaſt
N
N
е e
The Sum will be the Variation
Fig. 12.
But it as in the adjacent Figure,
the magnetical Amplitude be North,
and the true Amplitude South, it is
plain that N n the Variation, equal
to the Sum of Wo the true Ampli-
tude, and wo the magnetical Ampli-
tude is Weſt, becauſe the Arch N O
is greater than n and conſequently
the magnetical North muſt lye to the
Left-hand of the true North; where-
fore when the Sun thould Set jo deg
Tv
E
W
S
20 min
.: :
... Y
394
Methods for finding
20 min. to the Southward of the true Weſt point, he is found to Set
4 deg. 30 min. to the Northward of the magnetic Weſt, and it be
required to find the Variation, and how much it is.
To the true Amplitude
20 South
Add the magneric Amplitude
W4 30 North
Wro
The Sum will be the true Variation
14 So Weſt
By comparing the ſeveral Solutions together, it is manifeſt, that if
the Amplitudes either at Riſing or Setting are the ſame way, that
is both North or both South, their Difference or Exceſs is the Va-
riation ; but if they are contrary, that is, if one be North and the
other South, their Sum will give the Variation; and to find whe-
ther it be Eaſt or Weſt, ſet off the true Amplitude if it be in the
Morning from the true Eaſt point, but if it be in the Evening from
the true Welt point, either Northerly or Southerly as the Caſe di-
rects, and it gives the Place of the Sun ; from which point let off
the magnetic Amplitude che contrary way to its Title, that is, if
the magnetic Amplitude be North ſet it off Southerly, but if it be
South fet ir oft Northerly, and it gives the magnetic Eaſt or Weſt
points, from whence ſet off 90 Degrees or a Quadrant and it gives
the magnetic North : Now it the magnetic North lyes to the Right-
hand of the true North the Variation is Eaſt, but if the magnetic
North lyes to the Weſt of the true North, the Variation is Weſter-
Jy.
If in comparing the Azimuths together taken at any time, viz. the
magnetical and true, they are found to diſagree, their Difference is
the true Variation ; which if it happens to be in the Morning, and the
true Azimuth numbred from the North, exceeds the magnetical
counted from the ſame point, the Variation is Eaſt; otherwiſe Weſt.
For in the iſt, 3d and 10th Fig. if o be the true Place of the Sun,
then will No be the true Azimuth, and n the magnetical, and
conſequently Nn equal to the Exceſs of the true Azimuth NO above
the magnetical Azimuth no, the Variation, is Eaſterly; for becauſe
the Arch No is greater than no, the magnetic North muſt lye
to the Eaſtward of the true North ; wherefore if the Sun by Com-
putation is found to be 89 deg. 40 min. to the Eaſtward of the true
North, and by Obſervation at the ſame time he is found to be but
78. deg 30 min to the Eaſtward of the true North, and it be re-
quired to find the Variation, and which way it is.
From
the Variation of the Compaſs 395
10 Eaſt
?
From the true Azimuth
N 84 40 Eaſt
Take the magnetical Azimuth
N 78 30 Eaſt
There remains the true Variation
6
But if as in the 2d, the 4th, and the 10th Figures, the magneti-
cal Azimuth exceeds the true, it is plain that Nn the true Varia-
tion, equal to the Exceſs of no the magnetical Azimuth above NO,.
the true Azimuth is Wefterly ;, for the Arch NO is leſs than no
and conſequently the magnetic North lyes to the Weſtward of the
true North. Wherefore,
If the magnetic Azimuth by the Compaſs is found to be 89 deg.
40 min Eaſt, when the true Azimuth by Calculation is 74 deg. 30
min. Eaſt, and it be required to find the Variation.
From the magnetic Azimuth-
N 89 40 Eaſt
Take the true Azimuth
N 74 30 Eaſt
The Remainder will be the true Variation
15
10 Weſt
9. If in comparing the magnetical and true Azimuth together,
taken in the Afternoon and counted froin the North, the magnetical
exceed the true, their Difference is the Variation Eaſterly ; but if
the true exceed the magnetical, their Difference is the Variation
For in the 5th, 7th, and 11th Figures, where c reprelents the
Sun, N O the true Azimuth, and no the magnetical, it is manifeſt
that N n the Difference between NO the true Azimuth, and
no the inagtetic, is the Variation Eaſterly, for becauſe the Arch
N O is leſs than n , the magnetic North muſt Iye'to the Eaſtward
of the true North; wherefore if the true Azimuth was 106 deg. 54
min. North Weſtward, and the magnetic Azimuth 120 deg. 30min.
the ſame way, and it be required to find how much, and which
way the Variation is.
N 120 30 Weſt
Take the true Azimuth
N 106 54 Weſt
Weſterly.
Einar
Remains the true Variation
13 36 Eaſt
But if as in Fig. the 6th, 8th, and 10th, where No the true A-
zimuth exceeds n @ the magnetical Azimuth, it is plain that their
Dif
396
Methods for finding
Difference Nn will be the Variation Weſterly; for the Arch no
is leſs than NC, and conſequently the magnetic North point n
will lye to the Weſtward of the true North point N: wherefore if the
true Azimuth N O be 70 deg. 44 min. North Eaſtward, and the
magnetical Azimuth be found by Obſervation to be 64 deg. 48 min.
Weſt, and it be required to find the Variation, and which way it
· is.
From the true Azimuth No
N 70 44 Weſt
Take the magnetical Azimuth
N 64 48 Weſt
There will remain the true Variation
i n
-5 56 Weſt
72
20
10. If in comparing the Azimuths together, they are found to
be contrary, that is, if one be Eaſt and the other Weft, their Sum
will be the Variation, which is Eaſterly if the Obſervation be made
in the Afternoon, but Weſterly if the Obſervation be made in the
Forcnoon.
Fig. 13.
For in the adjacent Figure, where
º repreſents the Sun, so the true
Azimuth Weiterly, and so the
magnetic Azimuth Eaſt, it is mani-
feft, that Nnthe Arch of Variation,
equal to Ss, equal to the Sum of the
E
W
true Azimuth So, and the magnetic
Azimuth s 0, is Eaſterly ; for the
e
Arch ES is leſs than a Semicircle,
and conſequently the magnetical
North point muſt lye to the
Right-
S
hand, or to the Eaſtward of the true
North point N: wherefore if the Sun by Obſervation is found to
appear 6 deg: 20 min. to the Eaſt of the Meridian, when by Cal-
culation he is found to be 10 deg. 20 min, to the Weſt, and it be
cequired to find the Variation, and how much it is.
To the magnetic Azimuth
S 6 20 Eaſt
Add the true Azimuth
S 10 20 Weſt
Their Sum will be the true Variation
-16
40 Eaſt
But if as in the adjacent Figure, where repreſents the Sun, so
the true Azimuth Eaſterly, and so the magnetic Azimuth Weſt
,
it
:
the Variation of the Compaſs.
397
Fig. 14
it is evident, that Nn the Arch of the Variation equal to Ss, e-
qual to the Sum of so the true Azimuth, and so the magnetic,
is Wefterly, for the Arch nw ,
and conſequently the Arch nw.S is lels
than a Semicircle, and conſequently
N
the magnetic North point lyes to the
Weſtward of the true North point ;
wherefore, if the magnetic Azimuth
by Obſervation be found to be 4 deg.
56 min. to the Weſtward of the Me- W
F
ridian when the truc Azimuth is
W
5 deg. 20 min. to the Eaſt, and it be
required to find the Variation, and
how much it is.
S
To the magnetical Azimuth
Add the true Azimuth
S4 56 Weſt
SS
20 Eaſt
The Sum will be the true Variation
JO
16 Weſt
And univerſally, if in comparing the magnetical and true Azimuths
together they are found to be the ſame way, that is, both Eaſt or
both Weſt, their Difference is the Variation : but if they are con-
trary, that is, if one be Eaſt and the other Weſt, their Sum is the
Variation, and to find which way it is.
Having drawn a Circle to repreſent the Compaſs or Horizon, fet
off the true Azimuth from the North or South points towards the
Right-hand if it be Eaſterly, or Morning Azimuth, but towards
the Left-hand if it be Weſterly, and it gives the true Place of the
Sun ; from whence ſet off the magnetic Azimuth the contrary way
to its Title, that is, if the magnetic Azimuth be Eaſterly let it off
Weſterly, but if it bę Weſterly ſet it off Eaſterly, and it gives the
magnetic North or South points.
Having fixed upon the point o to repreſent the Sun, ſet off the
true Azimuth upwards towards the North if it be counted from the
North, or downwards toward the South if it be reckoned from the
South; and towards the Right or Left, according as it is Eaſtward
or Weſtward, and it gives the North or South point of the Horizon,
and from the ſame point o ſer off the magnetic either upwards or
Fff
down-
Or,
I
*
:
398
Methods for finding
downwards, and rowards the Right or Left, according as it is rec-
koned from the North or South, either towards the Eaſt or towards
the Weſt, and it gives the magnetic North or South; now if the mag-
netic North point lyes to the Right-hand of the true North point,
the Variation is Eaſt; but if it lyes to the Left-hand of the true
North, the Variation is Weft.
Suppoſe for Example, the magnetic Azimuth x0, ſee the 9th Fi-
gure, was 86 deg. 40 min. North Eaſtward, and the true Azimuth
NO was 98 deg. 10 min. North Eaſt ; now becauſe they are both
the ſame way their Difference is deg. 30 mio. will be the Variation,
and to find whether it be Eaſt or Weſt, having pitched upon the
point o to repreſent the Sun, ſet oft' firſt 98 deg. 10 min from o
upwards to N, and it gives the true North pojnt, and from the ſame
point o, ſer off the magnetical Azimuth 86 deg. 40 min. the ſame
way, and it gives the magnetical North point n; now becauſe the
magnetical North point n is ſhort of the true North point N, and
therefore lyes to the Right-hand of it, the Variation is East
;
and
the lame Method is to be uſed in all other Caſes.
It inſtead of counting the point of Sun Riſing or Setting, from the
Eaſt or Weſt point of the Horizon, it be numbred from the North
or South, theſe Laws which relate to the Azimuths, will obtain in
the Amplitude, and all the varictys that can happen in boch, will
fall under the laſt General Rule, which for its great Simplicity
and Univerlality, ought to be prefer'd before the reſt ; being in it
ſelf fufficient to anſwer all the purpoſes, tho' to gratify the inquiſi-
tive Reader I have inſerted them all.
As the Compaſs is certainly the moſt uſeful Inſtrument the Mari-
ners have, ſo a Knowledge of its Properties is certainly of the great-
eſt Benefit.
It is now above 140 Years ſince it was firſt diſcovered at London,
that the magnetic Needle formed an Angle with the true Meridian,
at which time it pointed 114 degrees to the Eaſtward of the true
North point of the Horizon, but how much more it might have
done before that time does.not yet appear by any Account, and
from that time the Variation it ſelf has continued to vary, and the
Needle conſtantly approached to the true North point, till in the Year
1666, the Needle it ſelf pointed dire&ly North, and there was no
Variation; and ſince that it has ſtill continued to move in the ſame
Dire&tion, and at preſent is found to vary above 12 degrees Weſter-
ly, and how far further it may move, or what will be the greateſt
Dc-
.
the Variation of the Compaſs.
399
Deviation of it, Time only can diſcover; for till it has once fini-
ſhed its Period, it is next to an Impoſſiblity to aſſign the Law that
itis govern'd by, or fix any real or true Hypothefis for its Solution ;
and as it is very different in different places at the ſame Time, and
obſerves different degrees of Increaſe and Decreaſe, nothing more
at preſent can well be done, but to treaſure up a good Stock of
Obſervacions carefully made, and then it is polible a good Theory
of it may be obtained, which will not only be of great Service to
the Nautical Science, but of ſingular Uſe in many other Arts; and
therefore it is to be hoped, that all thoſe whoſe Bufineſs more par-
ticularly it is to make Obſervations of this kind, will omit 'no fa-
vourable Opportunity wherever they are of ſo doing, ſince by this
Means they will not only ſerve themſelves at preſent, but lay a
Jaſting and fure Foundation for thoſe that come after to build upon.
Gros
Lucu
Section XVII:
Containing an Account of the Methods made Uſe of, for
meaſuring the Ships Way, for correcting the Courſe.
ſteered, by making proper. Allowances for the Leeway
and Variation, together with the way and manner of
working and correcting of Days Works at Sea, &c.
HE Method made Uſe of by our Engliſh Seamen, for mea-
ſuring the Ships Way, or finding how far ſhe runs in any gi-
ven time, is by the help of the Log-line and Half-minute Glats.
The Log is only a flat piece of Wood, which is tyed or faſtened
to a Line which is called the Log-line.
This Line is uſually divided into certain Spaces, whichSpaces are or
ought tobe ſuch a proportional Part of a Nautical Mile (60 of which
are equal to a Degree of the Meridian) as half a Minute, the
Time allowed for the Experiment, is of an Hour,
The Spaces thus divided are called Knots, becauſe at the End of
each Space there is reev d (between the Strands of the Line) a piece
of Twine with Knots in it which ſhew how many Spaces are run out
Fff 2
+
dut-
400
Preparatory Rules
during the Half-minare, from the point of Commencement, and is
as ready a Way as could be thought of, ſince in the darkeſt Night
or in the ſtormieſt Weather, the number of Knots run off the Reel
are readily and eaſily pronounced, by feeling or counting them with
the Fingers only.
Theſe Knots or Spaces commonly begin at the diſtance of about
10 Fathom or 60 Foot from the Log, that ſo the Log when it is
hove over Board, may be out of the Eddy of the Ships Wake before
they begin to count, and for the more ready diſcovering of this point,
there is faſtened at it a piece of Red Rag.
The Log thus prepared, being hove over Board from the Poop,
and the Line veered Out (by the help of a Reel, that turns eaſily and
about which it is wound) as faſt as the Log will carry it away, or
rather as the Ship fails from it, will fhew according to the time of
Veering, how far the Ship has run in any given time, and conſe-
quently her Rate of Sailing.
A Degree of the Meridian (according to Mr. Norwood's Obſer-
vation, with which the French nearly agree) contains 69.545 En-
gliſh Miles, each Mile by the Statute being 1760 Yards, or 5280
Feet.
So that a Degree of the Meridian contains 122400 Yards, or 367200
Feet, whence a Nautical or Sea Mile being the 6o part, muſt contain
6120 Standard Feet, and conſequently the diſtance between Knot
and Knot ought to be 5 1 Feet.
Hence by knowing how many of theſe Knots or Spaces, the Ship
runs in half a Minute, we know how many Miles the fails in an
Hour, fuppoſing her to move with the fame velocity during that
time.
And therefore it is the general Way to beave the Log every Hour,
to know how many Miles ſhe goes, but if the Gale has not been
the ſame during the whole Hour, or if there has been more Sail ſet,
or any Sail handed, that ſo the Ship has run more or leſs in any
part of the Hour, than ſhe did at the time of the Experiment, there
muſt be an Allowance made for it, and which muſt be according to
the Diſcretion of the Artiſt.
And ſometimes when the Ship is before the Wind, and there be
a great Sea ſetting after her, it will bring home théLog, and the
will go faſter than is given by the Log:
And in this caſe it is uſual if there be a very great Sea, to allow
One Mile in Ten, and leſs in Proportion, if the Sea be not ſo great ;
but
1
for keeping a Journal.
401
Feer up-
But for the moſt part the Ship yawes, and makes an Equivalentfor is
And farther conſidering that for the generality the Ships Way is real-
ly more than is given by the Log, and that it is ſafer to have the.
Řeckoning ſomething before the Ship, it may not be amiſs to leave
out the odd Foot between every Knor, as Mr. Norwood has done ร
and make the Space between Knot and Knot to be 50 Foot exa&ly,
tho' our Seamen for the moſt part make the diſtance but 42
on a Suppoſition that 60 Engliſh Miles makes a Deg. of a Great Cir-
cle, and by common Experience has been forced to leſſen the Half-
minute Glaſs almoſt Six Seconds, making it to run but 24 Seconds
nearly ; which in reality is but correcting one Blunder by another;
and it were therefore to be wiſhed that they would make Uſe of
the former Limitation, ſeeing it is in every reſpect as eaſy, and
much more fure and certain. But,
If at any time it ſhould happen, that the Space between Knot
and Knot be too great in Proportion to the Half-minute Glaſs,
then the diſtance given by the Log will be too ſhort ; and on the
contrary, if the Space between Knot and Knot be too little, then
the Diſtance run will be given too much, and to find the true Di-
ſtance in either Caſe, having meaſured the Diſtance between Knot
and Knot; lay,
As the true diſtance so Foot is to the diſtance meaſured, ſo is the
,
Miles of diſtance given by the Log, to the true diſtance in Miles,
that the Ship has run.
Example 1.
A Ship having run at the Rate of 8 Knots and a half in half a
Minute, but mealuring the diſtance between Knot and Knot, I find
it to be 60 Feet, I demand the true diſtance in Miles ?
As so is to 60, fo is 8.5 to 10.2 Miles, the true diſtance run.
Example 2.
Suppole a Ship in half a Minute has run eight Knoes and a half
off the Reel, but meaſuring the diſtance between Knor and Knot
I find it to be but 42 Feet, and it be required to find the crue di-
ſtance run: Say,
As 50 is to 42, fo is 8.5 to.7.14 Miles nearly ; the true diſtance
.
run.
IF
402
Preparatory Rules
If in either of the two former Caſes, the diſtance run in the half
Minute were given in Feet, then divide the diſtance given by 50,
the Feet in one Knot, and the Quotient will give the Miles of di-
Itance run in an Hour. For,
As so to i, ſo is the Feet run, to the Viles of diſtance.
Thus in the 1ſt Example, where the Ship run 5 10 Feet in the half
Minute, if this Number sio be divided by:50, the Quotient 10.2
will be the Miles of diſtance; and the fame Law will hold good in
all other Examples.
Again, if the Glaſs require longer time than 30 Seconds to run,
it will give the diſtance failed too much, and on the contrary, if the
Glaſs run out too ſoon, it will give the diſtance failed too little ;
and to find the true diſtance failed in either of theſe Caſes,
Having meaſured out the time that the Glaſs requires to run out:
Say,
As the Number of Seconds the Glaſs requires to run, is to half a
Minute or 30 Seconds, ſo is the diſtance given by the Log, to the
truc diftance run in that time.
Example 1.
A Ship having run at the Rate of 8 Knots and a half, but exami-
ning the Glaſs, I find it to run 35 Seconds, I demand the true di-
ſtance lailed..
As 35 ſec. is to 30 ſec. ſo is 8.5 knots, to 7.29 miles nearly; the
true diſtance the Ship fails in an Hour.
Example 2.
A Ship by the Log having run at the Rate of 8 | Knors, but ex-
amining my Glaſs I find it to run but 24 Seconds, I demand the true
diſtance failed, and how many Milęs ſhe runs in an Hour.
As 24 ſec. is to 30 ſec. fo is 8.5 Knots, to 10.62 Miles, the Rate
the Ship runs at in an Hour.
And in order to try the Glaſs whether it meaſures our a juft Quan-
cry of Time or not, if there be no Watch nor Clock that vibrates
Seconds, let there be taken a String of 32} Inches long, and at one
End of it ler there be faſtened a Plummer, and let the other End be
hung upon a Nail or Peg, and having put the Pendulum in motion,
count the Number of vibrations that it makes during the time the
Glaſs
for keeping a Journal.
403
Glaſs is running out, and if theſe are found to be 30 in Number, it
thews that the Glaſs is good ; but if they are found to be more or
fewer in Number, the Excels or Defect, will fhew the Error of the
Glaſs.
And tho'it may be objected againſt this Method of Tryal, that
ftrialy ſpeaking, the vibrations are not equal among themlelves, and
that the Pendulum it ſelf will ſoon return to reft, in aſmuch as every
Arch of vibration will be ſtill leſs and leſs; yet the Error is ſo very
ſmall
, that the difference will not amount to any determinate Quan-
tity of Time, as is well known to thoſe who are moſt converſant in
the Buſineſs of Pendulums, and is therefore as good a Method as a-
ny hitherto known. But to return,
If there happen to be an Error both in the Log-line and Half-mi-
nute Glaſs, then the Caſe will be a little more complicate, and will
admit of three varieties ; and inaſınuch as the flower is the Glaſs, the
greater will be the diſtance failed obtained by the Glaſs, and the
contrary; and the greater the Space between Knot and Knot, the
leſs will be the diſtance failed obtained by the Log, it follows;
1. That ii the Glaſs run too long, and the diſtance between Knot
and Knot be too ſhort, that the diſtance given will always be too
great, ſince both Errors give the diſtance too great: Thus for
Example.
Suppoſe a Ship is found to run at the Rate of 8 į Knots, but exo
amining the Glaſs I find it to run 40 Seconds, and meaſuring the
Log-line, I find the Space between Knor and Knot to be but 42 Feet
the true diſtance run, in this Caſe will be 5-355, or s Knots near-
ly : for,
As 40 ſ. to 30 f. fo is 8.5 k. tº 6 375 k.; and
As so f. to 42 f, ſo is 6.375k. to 5.355k. the true diſtance
Or,
As 50 f. to 42 f. fo is 8.5 k. to.7.14 k.
And as 40ſ. to 30 f. fo is 7-14 k.to.5.355 k. the ſame as
the former: Again,
2. If the Glaſs run too faſt, and the Space between Knor and
Knot be too great, then the diſtance given will always be too ſhort,
ſince the Errors ariſing from each of the Cauſes, contribute to leſſen
the true diſtance. Thus for
Example
Suppoſe a Ship is found to run at the Rate of 81 Knots,
but examining the Glaſs I find it to run but 24 Seconds, and mea..
furing
A
404
Preparatory Rules
ſuring the Log-line, I find the Space between Knor and Knot to be
60 Feet, and the true diſtance run be required. Say,
1. As 24f.:301: :8.5 k. : to 10.62 k. and,
2. Assol:: 6of. : : 10.62 k.: to 12.744,' or 12, the true
diſtance the Ship fails in the time. Or,
As so l. : 60f, : : 8.5 k. : to 10.62 k. and again,
As 241.: 301.: : 10.62 k. ito 12.75 k. or 121, the lame as
the former. Again,
3. If the Glaſs be too ſlow, and the diſtance between Knor and
Knot be too great, or if the Glaſs run too faſt, and the Space be-
tween the Knors be too little, inaſmuch as in each of theſe Caſes
the Effects of the Errors are contrary, the diſtance will ſometimes
be too great and ſometimes too little, according as the greater Quan-
tity of the Error is Negative or Pofitive ; thus for
Example.
Suppoſe a Ship is found to run at the Rate of eight Knots and
half a Glaſs, but examining the Glaſs it is found to ſpend 40
Seconds in running out, and by meaſuring the Log-line the diſtance
between Knot and Knot is found to be 60 Feet, and it be required
to find the Ships true diſtance in half a Minute ; ſay,
1. As 401 : 301. : : 8.5 k. : 6.375 k. and,
2. As so f. : 60f.: : 6.375 k. :7.65 k, the true diſtance run
Or,
1. As 50 f.: 60f:: 8.5k. : 10.2 k. and,
2. As 401, : 301. : : 10.2 k. : 7.65 k. the ſame as before
found, leſs than the diſtance given by the Log-line and Glaſs, by
0.85 k. Again, in
Example 2.
Suppoſe a Ship to run at the Rate of 8 Knots in half a Minute,
and upon examining the Glaſs, it is found to require 32 Seconds to
run out, while the Space between Knot and Knot is found to be 60
Feet, and it be required to find the true diſtance failed, it will be
1. As 32 l. : 30 f. : : 8.5 k. : 7.97 k. and again,
2. Assoſ.: 60f.: : 7.97k.. 9.564k. the true diſtance run,
greater than the diſtance given by the Log-line and Half-minute
Glaſs, by 1.064k.
:
But
for keeping a Journal
405
.
:
N
a
.forced to Leeward by the force of the Wind upon her Hull,
But if in either of theſe laſt Examples it happen, that the time
the Glaſs requires to run out, be to the diſtance between Knot
and Knot, as 30 the Seconds in halt a Minute to so, the true
diſtance between Knot and Knot whatſoever be the time of running,
or the diſtance between the Knots, the diſtance given by the Log-
line, will be the true diſtance in Miles, as is abundantly manifeſt
from what has been already ſaid upon this Head.
Lee or Leeward-way, is the way that a Ship glides fideways
from the point of the Compaſs ſhe Capes at, occaſioned by the Force
of the Wind and Surge of the Sea when ſhe lyes to Windward, or
is cloſe hauled : thus for Example, if the Wind be at North, and
the Ship lyes up Eaſt North Eaſt, or
Weſt North Weſt towards e and a
within 6 points of the Wind at N, al-
tho? by the Rudder her Head is kept
up to the Eaſt North Eaſt or Weſt
e.
North Weſt within 6 points of the
d
Wind, yet by reaſon of the Force of W
E
the Wind and Surge of the Sea upon
her, ſhe falls off to Leeward, and
really makes her way good upon the
Ramb-lines cd and cb, and the An-
gle ec d or a cb, in this Caſe formed
S
by the Rumb-line upon which ſhe
endeavours to fail, and the Rumb-line cd or cb upon which ſhe really
makes her way good, is called the Leeway; and if it be equal to one
point, the Ship is ſaid to make one point Leeway, if it be two
points che Ship is ſaid to make two points Leeway, C..
The Quantity of this Angle is very uncertain, inaſmuch as ſome
Ships with the ſame quantity of Sail, and with the fame Gale,
make more Leeway than others; and tis obſervable that thoſe Ships
which draw moſt Water make the le's Leeway; and alſo that the
ſame Ship if ſhe be out of her Trim, makes more Leeway than other-
wiſe ſhe would do; for being not able to go fo faſt a-head, ſhe is
and
the ſetting of the upper Surface of the Sea, and moves in a Courſe
compounded of theſe two, after the ſame manner as a Ship that ſails
in a Current, is compelled to move in a Direction, and with a velocity
compounded of the Force and ſetting of the Wind upon her Sails,
and the drift and ſetting of the Current under foot:
Ggg *
To
400
Prepardtory Rules
To diſcover the Quantity of the Angle ecd or acb, or how much
Lecway a Ship makes, tis uſual to ſet the Ships Wake, and for
that Purpoſe ſome Ships have a quarter of a Compaſs drawn on the
Rail on each Quarter, or if a Ship be near the Land it may be
found after the manner taught in Page 213, 214, c. but the Al-
lowances generally made, are as follows,
1. If a Ship be cloſe hauld, has all her Sails fet, the Water ſmooth
and a moderate Gale of Wird, ſhe then makes little or no Leeway.
2. If it blow fo freſh as to cauſe the ſmall Sails to be handed, iis
uſual to allow one point.
3.
If it blow fo hard that the Top-fails muſt be cloſe reeft, the
Ship then makes about two points Leeway.
4. If one Top-fail muſt be handed, tis proper to allow two points
three quarters, or three points.
5. When both Top-fails muſt be handed, then allow about four
points Leeway.
6. When it blows ſo hard as to occaſion the Fore Courſe to be
handed, then allow between s į points and 6 points.
7. When both Main and Fore Courſes muſt be handed, then 6 or
6 points muſt be allowed for her Leeway.
8. When the Mizen is handed and the Ship is trying a Hull, ſhe
then makes her way good about one point before the Beam, that is
about 7 points Leeway.
And tho'theſe Rules here delivered, are fuch as are generally e-
ſteemed and approved of, yer, inaſmuch as the Leeway depends in a
great meaſure upon the Mould and Trim
of the Ship, the ſmoothnels
of the Water or the Sea running high, I ſhall recommend the un-
experienced Navigator when on Board, to get the Afliſtance of ſome
skilful Sailor, with whoſe help theſe Inſtructions and a little Expe-
rience, his Judgment will be ſo well fortified, that he will ſeldom
fail of making a good Allowance.
The Courſe ſteered and Leeway being given, the moſt natural and
eaſy way to find the true Courſe is, by ſuppoſing your Face turned
directly to Windward ; and if the Ship have her Larboard Tacks on
Board, to count the Leeway from the Courſe ſteered toward the
Right-hand, bue if the Starboard Tacks be on Board, then to count
it from the Courſe ſteered towards the Left-hand ; thus for Example,
fuppole the Wind at North by Eaft, and the Ship lyes up within lix
points of the Wind, with her Larboard Tacks on Board, making one
point Leeway, it is plain that in this caſe her Courſe Steered is
Eaſt
for keeping a Journal.
407
E
C
Eaſt by North, and her true Courſe is Eaſt : after the ſame man
per if the Wind were at North North Weſt, and the Ship lay up
within o į points of the Wind, with her Starboard Tacks on Board,
making 1 į point Leeway, it is maniteſt; that in this caſe the true
Courſe is Weſt South Weſt ; and the ſame Rule muſt be obſerved in
all Caſes whatſoever.
But it any one cannot fo readily conceive this way, let him draw
a Circle as 'NESW to repreſent the Compaſs, and let him ſet off
the Wind, ſuppoſe North North Weſt as in the laſt Example, from N
to a, and draw the Line ca, which will repreient the Courſe of the
Winds, and becauſe the Ship makes her way good within o ź points
of the Wind, with her Starboard Tacks
onBoard, ſet off o from a towards W,
to b and draw the Line cb, this will
a N
repreſent the Weſt half South Rumb-
line, and the Courſe upon which the
Ship failed according to the dire&tion
of the Rudder, from whence ſet of
W
E
the Leeway až point, from the Wind
b
towards s to d, and draw the Line
d
cd, this will repreſent the Weſt South
Weſt Rumb-line, and the true Courſe
of the Ship; and after the ſame man-
S
ner may the true Courle of the Ship
be found in any other Caſe.
The Variation of the Compaſs (as has been ſhewn in the former
Sextion) is an Arch of the Horizon, intercepted between the magne-
tic and true Meridian, which if the Needle point to the Eaſtward,
Or to the Right-hand of the true North point, it is called Eaſt Va-
riation ; but if the Needle point to the Weſtward or to the Left-
hand of the true North point, it is called Weſt Variation.
So that it the Variation of the Compaſs be one point Weſt, then
the Norch point of the Card or Needle, will coincide with the true
North by Weſt point of the Horizon, the North by Weſt point of
the Card with the North North Weſt point of the Horizon, the
Eaſt, South and Weſt points of the Card with the Eaſt by North,
South by Eaſt, and Weſt by South points of the Horizon, &c. and
on the contrary,
If the Variation be one point Eaſt, then the North point of the
Card will coincide with the true North by Eal point of the Hori-
Ggg 2*
zon,
du
408
Preparatory Rules
.
zon, the North by Eaſt point of the Card with the North North
Eaſt point of the Horizon, and the Eaſt, South, and Weſt points of
the Compaſs, with the Eaſt by South, South by Weſt, and Weſt
by North points of the Horizon, &c.
Whence it is manifeſt, that if a Ship fail North by the Compaſs,
and the Variation be one point Weſt, the true Courſe is North by
Weſt, it ſhe ſail North by Weſt by the Compaſs, her true Courſe is
North North Weſt ; and if ſhe fail either Weſt, South, or Eaſt by
the Compaſs, her true Courſe will be Weſt by South, South by Eaſt,
or Eaſt by North; and on the contrary,
If a Ship ſail North by the Compaſs, and the Variation be one
point Eaſt, then the Ships true Courſe is North by Eaſt; if ſhe fail
North by Eaſt by the Compaſs, her true Courſe is North North
Eaſt; and if ſhe fail either Eaſt, South, or Weſt, by the Compaſs,
her true Courſe will be accordingly Eaſt by South, South by Weſt,
or Weſt by North, 60.
And the ſame Law will hold good let the Quantity of the Vari-
ation be what it will.
From whence we may draw this General Method, for finding the
true Courſe ; the Courle ſteered, and the Variation being given : and
that is to ſuppoſe your Self to ſtand in the Center of the Compafs
with your Face towards the point of the Compaſs upon which the
Ship is ſteered, and if the Variation be Eaſterly, count the Quanti-
ty from the Courſe ſteered, towards the Right-hand; but if the Va-
riation be Weſterly, count the Quantity towards the Left-hand, and
the point of the Compaſs thus found, will be the trueCourſe ſteered ;
thus for Example, fuppofe the Courſe ſteered be North North Weſt
halt Weſt, and the Variation be one point Weſterly, then the true
Courſe will be North Weſt by North half Weſt, one point more
towards the Left-hand, or nearer to the Weſt, but if the Va-
riation had been one point Eaſterly, then the true Courſe had been
North by Weſt half Weſt, one point more towards the Right-hand,
or nearer to the North ; and the ſame Method muſt be obſerved in
all other Caſes whatſoever.
And notwithſtanding the greateſt Care imaginable be taken, in cor.
re&ting the Ships Courſe, by making proper Allowances for the Varia-
tion and Leeway, yet by reaſon of the many Accidents that attend
a Sbip in one Days running, ſuch as the different Rates at which
a Ship runs, between the times of heaving the Log, the want of a
due Care at the Helm, by not keeping her ſteady, but letting her
yaw
....
for keeping a Fournal.
409
ܪ
yaw or fall off, ſuddain Storms, when no Account can be kept, ac-
cidental Currents under Foot, &c. the Latitude by Dead Reckon-
ing is very often found to differ from the Latitude obtained by Ob-
ſervation (the only certain datum that Mariners have to depend up-
on) and therefore whenever this is found to happen, Care muſt be
taken that proper Corrections be made in the Courſe, Dillance, Dif-
ference of Longitude, or wherever it may be judged neceſſary, and
how that may be done comes next to be fhewn. And,
1. If a Ship fail under the Meridian, and the Latitude obtained
by Obſervation differs from the Latitude by Dead Reckoning, there
is an Error either in the Courſe, in the Diſtance, or in both.
1. If the Courſe be true the Error will be in the Diſtance, and
will be ever equal to the Difference between the Latitude obtained
by Obſervation and that given by Dead Reckoning. For Example,
Suppoſe a Ship in the Latitude of 40 deg. 40 min. North, after
having failed due North for the Space of 24 Hours, at the Rate of
6 Knots an Hour, is found by Obſervation to be in the Latitude of
42 deg. 52 min. North, it is manifeſt, that in this Caſe the Error
in the Diſtance failed is 'Twelve Miles.
2. If the Diſtance failed be true, then the Error will be in the
Courſe; and this will occaſion an Error in the Departure, and to
corrc& both we have given, the true Difference of Latitude and Di.
ſtance failed, whence by the 3d Caſe of Plain Sailing, the true Courſe
and Departure may be readily found. Suppoſe for Example,
A Ship from the Latitude of 40 deg. 40 min. North, having fail-
ed due North by the Compaſs for the Space of 24 Hours, at the
Rate of 6 Knots an Hour, is found by Obſervation to be in the
Latitude of 42 deg. 52 min. and it be required to find the true
Courſe and Departure ; having found the true Difference of Lati-
inde 2 deg. 12 min. equal to 132 Miles, it will be by the 3d Caſe
of Plain Sailing,
1. As the true Diſtance 144 Miles, to the true Difference of La-
titude 132 Miles, fo is the Radius, to the Co-line of the true Courſe
23 deg. 33 min.
min. And again,
2. As the Radius, to the true Diſtance failed 144 Miles, ſo is the
Sine of the true Courſe 23 deg. 33 min. to the true Departure 52.74
Miles, whence it appears that the true Courſe is North 23 deg.
33 min. Eaſt, or North 23 deg. 33 min. Weſt, and the true Depar-
ture 52.74 Miles.
3. But
2
410
Preparatory Rules
3. But if neither the Courſe nor Diſtance failed be true, there will
be an Error in both; and this will likewiſe caule an Error in the
Departure, and the readieſt way to approximate in this caſe as near
the Trurh as may be, is to find an Arithmetical Mean between
the true Difference of Latitude and Diſtance by Eſtimation, and
with this and the true Difference of Latitude, find the true Courſe
and Departure; thus for Exanple,
Suppoſe a Ship from the Latitude of 40 deg. 40 min. North, ha.
ving failed due North for the Space of 24 Hours, and at the Rate
of 6 Knots an Hour, and after that by Obſervation is found to be
in the Latitude of 42 deg. 52 min. North, and there bcio; Reaſon
to ſuſpect that the Error in Latitude ariſes from an Error in both
Courſe and Diſtance, and it be required to find the true Courſe, &c.
Having found the Mean between the true Difference of Latitude
132, and the Diſtance by Eſtimation 144, equal to 138, it will be,
1. As true Dilt. : Rad. : : true Diff. Lat. : Co-ſine of true Courſe,
That is, As 138 : 90.00: :132 : CS, 16.57. And,
2. As the Rad. : true Diſt. :: S. Courſe : true Departure;
That is, As.90.co :
138 : : S. 16.57 : 40.23 Miles.
Whence it appears, that the true Courſe is either North 16 deg.
57 min. Eaſt, or North 16 deg. 57 min. Weſt, and the true Depar-
ture 40.23 Miles.
The ſmaller the Angle of the Courſe, the leſſer is che Exceſs of
the Diſtance failed above the Difference of Latitude, and conſequent-
ly when the Angle of the Courſe is ſmall, that is when a Ship
fails near the Meridian, a ſmall Error in the Courſe will occaſion lie-
tle or no Error in the Difference of Latitude, provided the Diſtance
failed be true, and therefore when a Ship fails near the Meridian,
and the Error in the Latitude be conſiderable, it is for the moſt part
judged to proceed from an Error in the Diſtance failed, and accounted
for accordingly ; ſince it would require too great an Error in the
Courſe, to cauſe a conſiderable Error in the Latitude; thus in the
2d Example, to correct an Error of 12 Miles of Latitude in 144
Miles
run, it will require an Error of above two points in the Courſe, which
cannot well happen if due Care be taken at the Helm, and proper
Allowances be made for Lceway and Variation, and therefore it is
moſt reaſonable to conclude, that the Error aroſe chiefly from a Fault
in the Diſtance run, and that the Courſe had little or no Share in
it ; and hence it is, that when the direct Courſe is within two points
of the Meridian, that the Error in Latitude is chiefly attributed to
an
+
for keeping a Journal.
411
an Error in the Diſtance run, and accounted for accordingly.; as
ih all be ſhewn hereafter.
2. If a Ship fail due Eaſt or Weſt, and the Latitude by Account
differs from the Latitude by Obſervation, the Error muft ariſe from
the Courſe; fince no Error in Diſtance can occaſion an Error in La-
titude, and conſequently to find the true Courſe and Departure we
have given as before, the true Difference of Latitude, and Diſtance
run, whence the true Courſe and Diſtance may be found, by the 3d
Caſe of Plain Sailing ; thus for Example,
Suppoſe a Ship from the Latitude of 40 deg. 40 min. North, af-
ter having failed due Eaſt or Weſt for the Space of 24 Hours, at the
Rate of 5 Knots an Hour, and then by Obſervation is found to be
in the Latitude ot 41 deg. 35 min. North, and it be required to find
the true Courſe and Departure made.
Having found the true Difference of Latitude 55 - Miles, it will
be by the 3d Caſe of Plain Sailing.
1. As true Dift. : true Dift. of Lat. : : Rad.: c s, true Courſe,
that is, As 144 :55: : 90.00 :CS, 22.27 and again,
As Rad. : Diſt. : : S. Courſe : Depart. that is,
As 90.00 : 144 : : S. 22,27 : 133.1 Miles.
Whence it appears, that the true Courſe is 67 deg. 33 min. from
the Meridian, and that the true Departure is 133.1 Miles.
The greater the Angle of the Courſe, the nearer does the Depar-
ture and Diſtance run, approach to an Equality ; and inaſmuch as
an Error in the Diſtance can occaſion no Error in the Courſe, and
little or no Error in the Departure; and thar a conſiderable Error
in the Courſe can cauſe no great Error in the Departure, hence it is
that when the Courſe is within two points of the Eaſt or Weft, that
the Departure in this caſe is taken for the true Departure, and by
the help of that and the true Difference of Latitude, is the true
Courſe and Diſtance failed found.
3. If a Ship fail upon any Courſe between the Meridian and Pa-
rallel of Eaſt or Weſt, and the Latitude by Account differs from the
Latitude deduced from Oblervation, this Error in Latitude will
cauſe an Error in the Departure ; and arifes either from an Error in
the Courle, in the Diſtance, or in both. And,
1. If the Error be in the Diſtance, tho' the Quantity be not known,
yet it the ſame 'run regularly thro' the whole Work, it will cauſe
an Error in the Difference of Latitude and Departure, but not in the
direct Courfe ; wherefore, if the Latitude can be obtained by Ob.
ler
3
412
Preparatory Rules
fervation, the true Diſtance and Departure may be eaſily had, and
conſequently the Error it ſelf in the Diſtance.
For having found the true Difference of Latitude from the Lati-
tude of the Place the Ship was in the Day before, and the Latitude
obtained by Obſervation. Say,
As the Difference of Latitude by Account,
To the true Difference of Latitude ;
So is the Departure by Dead Reckoning,
Toʻthe true Departure.
And ſo is the dire& Diſtance by Dead Reckoning,
To the true direct Diſtance.
Or,
So is any particular Diſtance run,
To the true correſpondent Diſtance
Suppoſe for Example, that a Ship from the Latitude of 40 deg.
40 min. North, after having ſailed upon ſeveral Courſes for the Space
of 24 Hours, is found by Account to have made 85.05 Miles of
Northing, when the Latitude by Dead Reckoning will be
42.05
North, and 75.75 Miles of Weſting, whence the direct Courſe will
be found to be North 4r deg. 35 min. Weſt, and the direct Diſtance
114.1 Miles, but by a good Obſervation I find the Ship to be in
the Latitude of 42 deg 18 min. North, and knowing the Error
to proceed from a Fault in the ſeveral Diſtances run, I would know
che true Departure, &c.
Having found the true Difference of Latitude 98 Miles, from the
Latitude the Ship was in the Day before, and the Latitude obtained
by Obſervation it will be,
As the Comp. Dif. Lat. 85.05, to the true Dif. of Lat. 98 ; fo is
the Comp. Dep 75.75, to the true Dep. 87.28 : And ſo is the
Comp. Diſt. 114.01, to the true Diſtance 131.47.
So that the true Departure is 87.28 Miles, the true dire& Courſe
is North 41 deg 35 min. Weſt, and the true dire& Diſtance failed
131.47 Miles.
2. If there be an Error in the Courſe, tho' the Quantity be
not known, yet if it be regular, that is, if it run equally the
ſame, and the ſame way throughout the whole Days Work, the di-
rect Diſtance will notwithſtanding be true ; and the dire& Courſe
will differ from the true Courſe, the ſame in Quantity and the ſame
way; wherefore, it the Latitude can be obtained by Obfervation,
the trueCourſe and Departure may eaſily be had, and conſequently
the Error in the Courſe it felf.
Ha-
Preparatory Rules
413
i
A
For having found the true Difference of Latitude, by the help of
the Latitude firſt given and the Latitude obtained by Obſervation,
it will be by the 3d Cafe of Plain Sailing,
As the true Diſtance failed,
To the t ue Difference of Latitude;
So is the Radius,
To the Co-line of the true Courſe. And again,
As the Radius,
To the Sine of the true Courſe ;
So is the true dire& Diſtance failed,
To the true Departure.
Suppoſe for Example, that a Ship from the Latitude of 44 deg.
30 min. North, after having ſailed upon ſeveral Courles for the Space
of 24 Hours, is found by Account to have made 87.3 Miles of
Northing, and 51.9 Miles of Weſting, whence the direct Courſe is
North 23 deg. 55 min. Weſt, and the direct Diſtance 127.98 Miles,
but by a good Obſervation I find the Ship to be in the Latitude of
45 deg. 50 min. North, and knowing the Error to be in the Courſe
only, I demand the true Courſe and Departure.
Having found the true Difference of Latitude 8o it will be, as the
true Diſt. 127-98, to the true Dift. of Lat. 80; ro is the Rad. to
the Co-fine of the true Courſe 51° 12'. And, as the Rad. to the
Sine of the crue Courſe 51° 12', ſo is the true Dift. 127.98, to the
true Departure 99 51. So that
The true Courſe is North 5 i deg. 12 min. Weſt, and the true Di-
parture 99.51 Miles.
The Reaſon of all which is very evident from the Properties of
Similar Triangles, and the common Method of conſtru&ing of Tra-
verſes:
In either of the former Caſes, if the Error in the Courſe or Di-
ſtance be irregular, the whole Days Work muſt be Wrought over a-
gain; and the proper Corre&ions muſt be made in their proper pla-
ces.
3. In comparing the Latitude by Dead Reckoning, with the La-
titude by Obſervation, (all proper Allowances having been made
for Leeway, Currents, known Errors in the Courſe and Diſance,
c.) they are found ſtill to diſagree, there is ſome unknown Error ei-
ther in the Courſe, or in the Diftance, or in both; and in order to
come as near the Truth as may be in this Cafe, obferve,
Hhh *
1. Tha
.
§
414
Preparatory Rules
1. That if the Courſes are for the moſt part near the Meridian,
and that the direct Courſe be within two, or two points and a half of
the Meridian, that the Error (for the Reaſons given in the 2 10th
Page) proceeds from ſome Fault in the Diſtance run, whence to find
the true Courſe, Diſtance, and Departure.
Having found the true Difference of Latitude fay,
As the Difference of Latitude by Dead Reckoning,
Is to the true Difference of Latitude made,
So is the Departure by Dead Reckoning,
To the true Departure.
Whence the true Courſe and direct Diſtance failed may be found
by the 4th Caſe of Plain Sailing.
Example
A Ship from the Latitude of 48 deg. 20 min. North, having failed
upon ſeveral Courſes for the Space of 24 Hours, is found by Dead-
Reckoning to be in the Latitude of 45 deg. 52 min. North, and
to have made 61.4 Miles of Eaſting, but by a good Obſervation ſhe
is found to be in the Latitude of 45 deg. 40 min. North, I demand
the true Courſe, Diſtance failed, and Departure.
Having found the true Difference of
Latitude A B equal to 160 Miles, it
will be, as A b the Difference of Lati-
tude by Dead-Reckoning 148, to AB
the true Difference of Latitude 160,
ſo is bc the Departure by Dead-Rec-
koning 61.4, to B C the truc Depar-
trure 66 4 Miles
b
C
160X61.4
For,
66.4
B
148
Whence by the 4th Caſe of Plain
Sailing, the true Courſe will be found
to be South 22 deg. 32 Eaſt, or South South Eaſt nearly, and the
true direa Diſtance 173.2 Miles.
2. If the Courſes are for the moſt part near the Parallel of Eaſt
and Weſt, and the dire&Courſe be
within st and 6 points of the Me-
ridian, and tho' there may be an Error both in Courſe and Diſtance,
yet the Departure will be very nearly true, (for the Reaſons given
in Page 411) and thence by the help of this and the true Diffe-
C
renco
for keeping a Journal.
415
3.
rence of Latitude, may the true Courſe and dire& Diſtance be
readily found, by the 4th Caſe of Plain Sailing.
Example.
Suppoſe a Ship from the Latitude of 43 deg. so min. North, af-
ter having failed upon ſeveral Courles for the Space of 24 Hours, is
found by Dead Reckoning to be in the Latitude of 42 deg. 45 min.
North, and to have made 160 Miles Weſting, but by a good Ob-
fervation the Ship is found to be in the Latitude of 42 deg. 35 min.,
North, I demand the true Courſe and direa Diſtance failed.
Having found the true Difference of Latitude 75 Miles, by the
help of this and the Departure 160 Miles, the true Courfe by the
4th Caſe of Plain Sailing will be found to be South 64 deg. 53 Weſt,
or Weſt South Weſt 4 Southerly nearly; and the true Diſtance 176.7
Miles.
If thé Courſes are for the moſt part near the middle of the Qua-
drant, and the dire&t Courſe be within three and ſix points of the
Meridian, the Error may be partly in the Courſe, and partly in the
Diſtance, and this will cauſe an Error both in the Difference of Lati-
tude and Departure ; and conſequently in the dire& Courſe and Di-
ſtance made that Day, and to correctas near as poʻlible in this caſe.
Having found the true Difference of Latitude, by the help of
it and the direct Diſtance by Dead Reckoning, find a new Depar-
cure, the Arithmetical Mean between this and the Departure given
by Dead Reckoning, will be very nearly equal to the true Depar-
ture, whence the true Courſe and Diſtance may be readily found :
or if greater Exa&neſs be required, by the help of the Departure by
Dead Reckoning and the true Difference of Latitude, find a new Di-
ſtance and direct Courſe, and the Mean between this Courſe or Di-
itance and the Courſe. or Diſtance obtained by the true Difference of
Latitude and Departure laſt found, will give a new Courſe of Di-
ſtance, ſufficiently true for all Nautical Uſes; and may therefore
without tenſible Error be taken for the true Courſe or Diſtance,
whence the true Departure, 6c may be readily found; or having
found a new Departure, by the help of the true Difference of Lati-
tude and Courſe, the middle between this and the former Depar-
ture, will give a new Departure ; whence the true Courſe, ce may
be obtained; the Reaſon of all which is ſo evident from what has
been ſaid before, that it needs no Explanation. So that
Hh h 2
*
Sup-
:
?
---
3
416
Prepardtory Rules for keeping a fournal.
Suppoſing a Ship from the Latitude of 45 deg. 40 min. North,
after having ſailed upon leveral Courſes between the South and Eaſt
for the Space of 24 Hours, is found by Dead Reckoning, to be in
the Lacitude of 43 ueg. 40 min. North, and to have made 116 Miles
of Eaſting, but by Obſervation to be in the Latitude of 43 deg. so
min. her true direct Courſe by the former Method of Inveſtigation
will be found to be South 47 deg. os min. Eaſt, and her true dire&t
Diſtance 161:55 Miles.
And if it ſhould happen thar the Error ariſing from any one of
the aformentioned Cauſes, fhould remain undiſcovered for more than
one Day, ſo ſoon as it is found out the whole Work muſt be correct-
ed, and each Day muſt have its proportional Allowance.
The true Difference of Latitude and Departure being thus obtain-
ed, the Latitude the Ship is in may be readily found, by the ad Def
.
of Section the it, the dire& Courſe and Diſtance the Ship has failed
by the 4th Caſe of the 2d S:tion, her Alteration, or Difference of
Longitude, by the 8th Caſe of Mercator's Sniling ; alſo her Bearing
and Diſtance from her intended Port, after the manner taught in the
2d Example of Traverſe Sailing
Alſo by compounding the ſeveral Differences of Latitude and De-
partures made every 24 Hours, after two or more Days run, in the
ſame manner as we compound the ſeveral Differences of Latitude and
Departures, anſwering tocach ſingle Courſe and Diſtance failed, may
the Difference of Latitude and Departure between the Ship and her
firſt Port be found, and thence her Bearing and Diſtance from the Place
from whence the took her Departure, by the ſame 41h Caſe of Plain
Sailing
And for a General Example to the preceding Rules, I have added
a Specimen of Six Days Work, taken off from the Log-Board, with
the Form of the Log-Book, and the Way and Manner of working the
ſame, by making all proper Allowances where requiſite, and which
contain within themſelves as great a Variety as can well happen in
fo Lort a Space of Time.
THE
!
bi
:
Saxaaxx
ac
THE
LOG-BOOK
With the Form of keeping
and Manner of working of
Days Works at Sea.
>
4
SOURCERROXOX
4
418
The Log-Book, with the Form of keeping
H.KH. K. Courſes.
H.KH. K.
Winds.
Obſervations and Acci-
dents, ) March 12, 1721.
North.
2
3
ws
Moderate Gales and
fair Weather, at Four this
Afternoon I took
my
De-
parture from the Lizard in
the Latitude of so deg. 00
min. North, it bearing
North North Eaſt, Diſ-
tance five Leagues.
NNE
4
5
6
6
6
SW by W
3
I
7
81 6
I
91 7
SW by S
Eaſt.
10 7
II
1217
I
I
:.
TheGale freſhening and
being under all our Sails.
Il 8
2) 8
3
8
SS W
Eaſt by South
1
1
41 8
1
5
8
6 8
1
The Variation I reckon
to be one point Weſterly.
South Weſt.
ENE
8
8! 8
91 8
I
I
I
SW by S
rol 8
II 8
128
I
I
1. For
and manner of working correcting of Days Works
. 419
Courſes Corre&.' Dift.
North.
South.
Eaſt,
Weſt.
South Weſt
32
22.6
22.6
SSW
63.5
58.6
24.3
South by Weſt 48.5
47.6
9.5
:
SW by S
25 5
21 1
14.1,
150.0
70.51
1. For the direct Courſe, ſay by the 4th Caſe of Plain Sailing,
As Differ. Lat. : Depart. : : Rad. : T. Courſe.
That is, As 150 : 70.5 1 : : 90.00 : t, 25.11 2.
I
2. For the dire&t Diſtance, ſay by the ad Caſe of Plain Sailing,
As the Rad. : Secant of the Courſe :: Dif. Lat. : Diſtance.
That is, As 90.00: Secant of 25° 11' 1 ::150 : 165:7.
3. For the Difference of Longitude it will be,
As the prop. Diff. of Lat. : Merid. Differ. of Latit. : : Depart. :
Difference of Longitude. That is,
As 150 : 227.6: : 70.5 : 107.05
So that this Day I have made my Way good South 25 deg. 11
min. { Weſt, Diſtance 105.7 Miles, my Difference of Longitude be-
ing 107.05 Miles Welt, or i deg: 47 min. Weſt, and Latitude by
Account 47.30 min. North.
At Noon the Lizard bore from me North 25 deg. It min. Eaft,
Diſtance 165.7 Miles.
The
420
The Log-Book, spith the Form of keeping
HK H.K.
Courſes
Winds.
Obſervations and Acci-
dents, o March 13, 1721
I
1
8
2! 8
SW by s
ENE
1
I
The firſt part moderate
Gales and fair Weather:
The middle and latter
part freſh Gales and clou-
dy.
1
SSW
41 8
58
I
61 8
-
I
Took in the Stay-fails
7 8
8 8
9
8
South by Wen ESE
1
10 8
II
8
1217
SSW
South Eaſt
To the beſt of my Judg.
ment we make one point)
Leeway.
17
27
31 7
South Weſt
ESE
I
416
5 6
6 6
I
W NW
South Weſt
Variation as before
one point Weſt
Tack'd to the Southward
South WSW
lol 6
6l
And
and Manner of working e correcting of Days Works. 421
Courſes Corre&. Dift.
North.
South
Eaſt,
Weſt.
SSW
46.5
42.91
17.8
South by Weſt 41.5
40.7
8.1
South
24.
24
South West
20.
14.1
14.1
W NW
18.
6.8
16.6
SSE
24.
22.2
92
6.8
143.91
9.2
56.6
6.8
9.2
137.11
474
And proceeding after the ſame manner as in the former Day-
Work, we ſhall find the direct Courſe to be South 19 deg. 04 min.
Weſt, Diſtance 145:1 Miles, the Latitude the Ship is in by Accouns
45 deg. 12 min. 5 North, and the Difference of Longitude made
68.67 Miles, or i deg. 8 min. Weſt ; whence to find the Bearing
and Diſtance from the Lizard, it will be by the 4th Caſe of Plain
Sailing, as the Sum of the two Differences of Latitude (becauſe
they are both Southerly)or whole Southing, to the Sum of the two
Departures (becauſe they are both Weſterly) or whole Wefing ; ſo
is the Radius to the Tangent of the Courſe.
That is; As 287.11 : 117.9: : 90.00 : t, 22.191.
And again, As Rad. : Sec Courſe : : Diff
. Lat. : Diſtance. That is,
As 90 00 : Sec. 22.191 :: 287.11 : 310.4. So that
At Noon the Lizard bore North 22 deg. 19 min. - Eaſt, Diſtance
310.4 Miles.
Iii *
The
422
The Log-Book, with the Form of keeping
H KH. K.
Courles.
Winds.
Obſervations and Acci-
dents March the 14 1721
I
South
WSW
6
3
The firſt
part freſh
Gales and cloudy Wea-
ther, the middle and lat-
ter part hard Gales and
[qually with Rain
moment
6
SSE
South Weft
4
5
6
6
I
7 5
85
95
I
South Eaſt
SS E
Tack'd and took a Reef in each Top-fail
I
Parametrelerinin
WSW
South
105
IIS
.
I 5
2! 4
3 4
Clofe Reeft the Top-fails
Weft. SSW
Making in (my Opini-
on) two points Leeway
I
1
1
I
I
Took in the Fore Top-fail
SSW is
Making about 2 Lee-
way.
1
71 4
8 4
94
Variation 1 point Weſt
Handed M.Top fail
Making (to the beſt of
my Judgment) 4 points
South by Wef Leeway
Lat. per Obſerv.44° 46'N
1
103
113
1 2 3
I
Ву
T
and Manner of working & corre&ting of Days Works 423
Courſes Correct. Dift.
North.
South
Eaſt
Weſt
SSE
18
16.6
6.8
South Eaſt
18
12.7
12.7
ESE
16.5
6.3
15.2
WSW
20
7.6
18.5
Weit by North
18
3.6
17.7.
W NW &W
16
5.4
15.1
NW by W
10.5
5.81
8.7
14.83
43.2
34.77
60.0
14.81
34:7
Diff. Lat.
28.31 Departure 25,24
By proceeding after the ſame manner as in the laſt Days Work,
I find that I have made my way good South 41 deg: 35 min. Weſt,
Diſtance 38.03 Miles, my Difference of Longitude being 35.7 Miles,
and Latitude by Account 44 deg. 44 min. . So that,
At Noon the Lizard bore from me North 24 deg. 24 min. Eaſt,
Diſtance 340.5 Miles.
Iii 2
For
424
The Log-Book, with the Form of keeping
HK H.K.
Courſes.
Winds.
Obfervations and Acci-
dents, March 1 15,1721.
I
Weft.
South by Weſt
3
23
3 3
I
For the moſt part ſtrong
Gales of Wind and ſqual-
ly Weather, with Rain
and Hail.
1
1
1
NW by W South Weſt
41 3
513
63
I
I
7 3
I Reeft Courſes
Leeway about 4 1 points
83
9 3
NW by N
WSW
Variation i point Weſt
10) 3
1113
1213
3
2 3
31 3
Wore Ship
South by Eaſt
ml
41 3
51 3
ola 2
Violent Squalls
Leeway about si points
Handed Fore Sail
terrible Storm of Wind
712
8] 2
9 2
Leeway about 6 points
Handed Main Sail Lying a Hull, Leeway
about 7 points.
1
IO
2
II
2
Handed Mi-zen Sail
121 I
I
Lat.by Obferv. 45*25'N.
This
+
and Manner of working e correcting of Days Works. 425
Courfes Correct. Diſt.,
North
South.
Eaſt
Weſt
NW by W
10.5
5.81
8.7
NNW
10.5
9.7
4.0
NNW halt W
6
5.3
2.8
North half Eaſt
I 2
I1.9
1:2
i
lalzla 15 lalo
ES E half E
I 2
3:5
II.S
E by S half E
7.5
0.73
7.5
Eaſt half North
6
0.6
6.0
Eaſt
1.5
1.5
33:31
4.2.4
27.7
15.5
I
4.2
IS.S
29:1
Departure
12:2
Error
Sii
T. Depart. 17:3
This Day I have made my Way good, Allowance being made for
the Error in Latitude, North 23 deg. 35 min. Eaſt, Diſtance 42 66
Miles, my Difference of Longitude being 17.2 Miles Eaſt, and true
Latitude 45 deg. 25 min, North.
At Noon the Lizard bore from me North 24 deg. 36 min. Eaſt,
Diſtance 302.4 Miles.
The
.
426
The Log-Book, with the Formof keeping
H.K H. K.
Courſes.
Winds.
Obfervations and Acci-
dents, 9 March 16, 1721.
I
I
South by Eaſt, WSW
2) I
1
3 I
The firſt part ſtrong
Gales of Wind, with rio-
lent Squalls: The middle
hard Gales, the latter ve-
ry freſh Gales, but more
moderate Weather.
I
4 I
511
61 2
I
Set the Mizon Leew.6; pts
margar
2
8
2
91 2
Set M: Sail Leew. så prs.
Variation i point Weſt
102
I
2
I
12
2
I
-
1
up Sº
off S. E.
WSW
2
3
4
5
6 3
up SSEoÁ'SEbye South Weſt
Set Top-fail
SSE. half E South Weſt
We drove about two
Miles an Hour, making a-
bout s points Leeway.
71 3
8 3
91 3
Leeway about 41 points
Let the Reeflout of Courſes
Leeway about 4 points.
I
- 에
​I
Set M. Top-fail
10 3
II 4
I
Leeway about 3 points.
Latitude by Obſervati-
On 45 deg. 32 min., North
I 2
This
x
and manner of working es correcting of Days Works.
427
co
Courfes Correê. Dift.
North.
South.
Eaſt.
Weſt.
Eaſt by North
7.5
I.S
7.37
1
Eaſt half North
8.
0.8
8.0
Eaſt South
7.5
0.31
7.4
E by N half N
4.
1.2
3.8
Eaſt
1.5
15.0
Eaſt half South
7.
0.7
7.0
1
ES E half E
9.
2.6
8.6
3.5
361
57.2
3.5 2
On
This Twenty-four Hours I have made my Way good North 84
deg. oo min. Eaſt, Diſtance 57.5 Miles, my Difference of Lon-
girude being 81.3 Miles, or or deg. 21 min. Eaſt, and the Lati-
tude by Obſervation 45 deg. 36 min. North.
At Noon the Lizard bore from me North 14 deg. 23 min. Eaſt,
Diſtance 276.7 Miles.
a
This
#
428
The Log-Book, with the Form of keeping
HK H.K
Courſes
Winds.
Obſervations and Acci-
dents, 5 March 13, 1721
South
WSW
Il 41
2 4
3) 4
I
The firſt part freſh
Gales and fair Weather:
The middle and latrer
part moderate and hazy.
I
arra
I
South by Weft Weft by South
44
54
6 4
I
I
Set Fore-top fail
reservedelemmer
7:5
Leeway about 2 points
85
9. S
Let the Reefs outoftheT.S.
Leeway about 1 point
OS
I 16
I
2
6
I 6
24 6
3
6
South Well
W NW
The Error in my Lati
tude I judge to be occafi-
oned by a ſwelling Sea,
ſetting N.E. by E., at the
Rate of about of a Mile
an Hour.
!
41 6
5
67
NNE
Variation I pointWeſt
.
717
817
91 8
SW by W.
lo 8
8
Latitude by Oblervati
121 8
on 43 deg: 47 min. North
This' way of menticning the Leeway allowed in each courſe, and the Quantity of
the Variation, tho' it be a thing not uſually practiſed, yet I can't but think it is very
necesſary, fince by thisMeans a Perſon is capable, if need be, of over-looking a Whole
Voyage, and correcting any Miftake that may have ſipt in, during the Whole Time;
which atherwiſe he can't well do.
This
and Manner of working & correcting of Days Works 429
Courſes Correct. Diſt.
North.
South
Eaſt
Weſt
: 7.
1
2
South aſt
13.5
9.51
9.5
SE by S
13.5
11,2
7.5
SSE
20
18.5
7.6
-j. ww
South by Eaſt
18
17.7
3.6
SouthSouth Weſt
24
22.2
9.2.
SW by S
22.5
18.7
12.5
South Weſt
32
22.6
22.6
I
Diff. Latit.
120,4
28 21
44:3
Error
10. I
28,2
T.Diff Lat. 110.31 Departure 16.0;
.
Error
15.0
T. Dep
1.0?
This Twenty-four Hours I have made my Way good, proper
Allowances being made for all Impediments, South oo deg. 32 min.
Weſt, Diſtance 110.01 Miles, my Difference of Longitude being
1.47 Miles, and the Latitude I am in 43 deg. 48 min. North.
At Noon the Lizard bore from me North 1o deg29 min. Eaſt,
Diſtance 38 35 Miles.
Kkk *
The
>
430
The Solution of the former Days Work.
586
The Method here made Uſe of for finding the Difference of
Longitude, &c. from the Sum of the ſeveral Departures made in
the Space of 24 Hours, tho'it be generally practiſed, comes very
near the Truth in ſmall Runnings, and anſwers in a great meaſure
all the Purpoſes of Navigation, yer inaſmuch as it is certainly Er-
roneous, as has been plainly made appear in the 3ood Page, it were
much to be wiſhed that our Seamen would be ſo far prevailed upon, as
to work every Days Work according to the ſtrick Rules of Mercator's
Sailing, after the Method taught in the ſame Page, it being very
eaſy, and requiring very little more Time than: the common
Method by Plain Sailing, and to encourage the diligent Navigator
in the purſuit of this Method, I ſhall give an Example of the firſt
Days Work, after the manner ſhewn in the 3ood Page, with the
Solution of the other five Days Work, after the ſame way.
Latitude
50.00
34745
IA Diff. Lar: 22.6
35.0 35.0
49.37.4 3439.5
89.71 3.72
48.38.7 3349.77
47-6
71.43
14.2
47.51.12
3278.3
31.4
:47:30
3246.9
In the firſt Column you have the ſeveral Latitudes anſwering to
each Courſe and Diſtance ſeperately, obtained by the help of the
Latitude firſt given, and the ſeveral Diferences of Latitude made,
in the żd Colum you have the Meridional Parts anſwering to each, in
the 3d Column you have the Meridional Differences of Latitude, and
in the 4th Column you have the correſpondent Differences of Lon-
gitudes, taken out of the common Traverſe Table by Infpe&ion; all
which may be performed with great Eaſe, in much leſs than a quarter
of an Hours Time.
And after ſo caſy a Way has been paved, it will be very ſurpri-
ſing, if People ſhould ſtill go on to make Uſe of tentative Methods,
and amuſe themſelves with Departure, &c. which ſerves for
no other Purpoſe as I know of, but to confound their Heads, and
keep them in ignorance ; and if thoſe to whom the Care of young
Beginners are committed, would but encourage and promote thisWay
of Solution, it would foon be univerſally pra&iſed.
A
1
I
21.12
21.0
by Mercator's. Sailing.
431
1
t
Courſes Corrected. Diſtance. Difference of Latit.
South Weſt
South South Weit
South by Weſt
S. W. by S.
Difference of Latit. Difference of Lon it
North South. Eaſt Welt
3.2
22.6
35.0
63.5
58.61
37.2
47.6
25 5
21,1
2 I O
Ditt. of Latit.
I50.0
48.5
14.
Dif.Longl
107.4
To find the dire&t Courſe it will be by the 2d Problem of Mercators Sailing,
As the Merid. Diff of Lat. : Diff. of Long. : : Radius: T. Courſe,
That is, As 227.6 : 107 4
: : 90.00 : t, 25°15'.
To find the direct Diſtance, it will be by the ſame Problem,
As the Radius: Secant Courſe : : proper Diff. Latit. : Diſtance.
As 90.00 : Sec. 25° 15' : :
:165.9; whence
it appears that my direct Courſe is South 25 deg. 24 min Welt, Diſtance
failed 165,9 Miles, Difference of Longitude made 107 4 Miles, or
Miles, or i deg.
47 min. 7 tenths Weſt, and Latitude 47 deg 30 min. ſo that
At Noon the Lizard bore from me N. 25° 15'E. Diſtance 165 , Miles,
150
Courſes Corrected. Diſtance
46.5
Difference of Latit. Difference of Longit.
North South East Weſt
42.91
26.1
40 7
24.
41.51
II 8
I
South South Weft
South by Welt
South
South Weſt
Weit North Weſt
South South Eaſt
24.
20.
14. I
10,2
18.
6.8
23-7
24.
2 2.2
6.8
143.91
13.1
13.2
818
68
13.2
Diff, of Latit 137.15
Diff. Long 78.7
The direct Courſe made this Day is South 19 deg. 04 min. Weft, the
Diſtance 145.2 Miles, the Latitude the Ship is in is 45 deg, 12 min. 8
tenths North, the Difference of Longitude 68.7 Miles, or i deg. 8 min,
7 tenths Weft.
At Noon the Lizard bore N. 22 deg. 29 min. E. Diſtance 3 10.8 Miles
Kkk * 2
The
.
*
1
432
The Solution, &c.
Courſes Corrected. Diſtance.
Difference of Latit.
North South
Weit
16.6
Difference of Longit.
Eaſt
9.71
I 1.9
21.3
18.
127
6.3
South South Eaft 18.
South East
Eaſt South Eaſt
16.5
West South Weſt
Welt by North
18.
W.N W&W.
16.
North Weſt by Wert! 10.5
2.
7.6.
25.8
3.6
25.21
21.2
5.4
5.84
14.8
I 2.5
84.72
49.
43.2
14.8
28.34
Diff. Latit
Diff. Long
49.
3573
The direct Courſe made this Day is S. 43 deg. 15 min. W. the direct
Diſtance is 38.93 Miles, the Latitude the Ship is in is 44 deg. 46 min. N.
and the Difference of Longitude made 35.7 Miles Weſterly.
At Noon the Lizard bore N 24 deg. 33 min, E. diſtance 345.2 Miles.
4
Courſes Corrected. Diſtance.
Difference of Longit.
Eaft
Weſt
105
Difference of Latit
North South
5.81
97
5.3
I 69
10.5
6.
I 2.5
5.6
2
40
I 2.
North Weſtby Weſt
North North Weſt
N, N. W. W
North half Eaſt
E S. E. half E.
E. by S half E.
Eaſt halt North
Eaſt
17
16.9
12.
3.5
0.72
7:
9.9
0.6
8.0
1.5
2 I
4.2
38.6
2
22 I
4.2
22.1
$
29. I
15
50
Diff.Long.
Error
T.DLon.
22 I
The direct Courſe made this Day is N. 20 deg. 55 min. E the direct Di-
Atance 41.75 Miles, the Latitude the Ship is in is 45 deg. 25 min. N. and
the Difference of Longitude made 22.1 Eaſt.
At Noon the Lizard bore N. 24 deg. 55 min, E. Diſtance 303.2 Miles.
The
:
by Mercator's Sailing.
433
el
1.2
Difference of Latitude. Difference of Longit.
Courſes Corrected Diſtance
North. | South.
Eaſt. Weft.
Eaſt by North
7.5
11.5
East half North 8
0.8
10.9
Eaſt & South 7.5
0.31
10.9
E. by N. half N.
4.
5.8
Eaſt
1.5
Eaſt half South
0.7
9.9
E.S. E. half E.
9.
3.5
9.0!
Diff. Long
9.5
Departure. 01,
The direct Courſe made this Day is N. 83 deg: 04 min. E the direct Di-
ſtance is 58 Miles, the Latitude the Ship is in is 45 deg 32 min. N the
Difference of Longitude made is 82.3 Miles, or i deg. 22 min. 3 tenths E.
At the Noon the Lizard bore N. 15 deg. o7 min. E Diſtance 277.6 Miles.
21.4
lill
y.
26
I 2.1
82.3
E
II 2
1
20.
2 2.2
Courſes Corrected. Diſtance.
Difference of Latitude. ¡Difference of Longitude
North, South. Eaſt Weſt
South Eaſt
13.5
y.sk 13.6
SE by S
19.5
10.71
South South East
18.5 10.7
South by Eaſt
18,
177
5.0
South South Weſt
24.
12,8
S. W by S.
22.5
175
South Weit.
22.6
31.0
Difference of Latit,
120.45
40. I
Error
40.1
Difference of Latit. | 110.41
Error
Diff. Long
18.7
32.
613
-:'.
10.
21.2
2 I I
00.1
The direct Courſe made this Day is South oo deg. 22 min. Weft, the
direct Diſtance 109 Miles, the Latitude I am itt is 43 deg. 43 min. North,
and Difference of Longitude made p.r Miles..
At Noon the Lizard bore N. 11 deg. oz min. E. Diſtance 384 1 Miles.
The
434
Preparatory Rules
The Difference of Latitude and true Difference of Longitude be-
ing obtained, after the former manner, the true Courſe and direct
Diſtance may be readily found, by the 2d Problem of Mercator's Sail.
ing, as has been thewn in the preceding Example.
And by adding or ſubftra&ing the Difference of Longitude made
each Day, to or from the whole Difference of Longitude, that is,
the Difference of Longitude between the Place from whence the
Ship took her Departure, and the Place ſhe was in the former Day
at Noon, (according as the Cale requires) it will give the Difference
of Longitude between the Ship and her firſt Port.
With this Difference of Longitude, and the Latitude of the Place
from whence the Ship took her Departure, and the Place fhe is in,
may her true Bearing and Diſtance from her laſt Land be found, by
the ad Caſe of Mercator's Sailing ; and laſtly with the Difference of
Longitude between the Place the Ship is in and the Place the is bound
to, and the Latitudes of thoſe cwo Places, may her direct Bearing
and Diſtance from that Place be determined, by the ſame Caſe.
And in keeping an Account after this way, if any Errors ſhould
ariſe from any of the aforementioned Cauſes, they are to be correc-
red after the fame way, and by the ſame Method as is heretofore
Taught ; only inſtead of the common or proper Difference of Latitude,
you muſt uſe the Meridional.
The Rules and Inſtructions here delivered being well underſtood,
nothing more remains to be done to compleat this part, but to give
the induſtrious Navigator a few General Directions, how to proceed
from the Beginning, and ſhew him after what manner the whole
Work muſt be Journalized. And
1. He ought to meaſure his Log-line, and ſee if it be of a juſt
Length, it not, to be ſure to make proper Allowances for it.
2. He ought to meaſure his Half-minute Glaſs, not only at the
beginning of his Voyage, but as often as he fhall he judge it conve-
nient, or have juſt Reaſon to luſpe& that it does not keep juſt Time.
27
3. AS
.
for keeping a Journal.
435
3. As often as he has Opportunity, he ought to take the Azin
muth and Amplitude of the Sun, not only to pronounce the Varia-
tion, but to find whether or no, the Compaſs it ſelf be not faulty;
or be not diverted from its proper Direction, by ſome accidental
Cauſe, which may very often happen.
4. He ought to omit no Opportunity of finding the Latitude from
Obſervations made, either of the Sun or Stars, this being the chief
Thing to be depended upon, and the only certain Datum by which
a Reckoning is confirmed or re&tified.
5. No Opportunity ought to be negle&ed of trying of Currents,
inaſmuch as in the ſame place the Setting and Drift of the Current
has often been found to differ.
6. Having taken care to correct each Courſe, by making proper
Allowances for the Leeway and Variation (if there be any) ſet
down each correct Courſe and its correſpondent Diſtance onc
under another, after the manner ſhewn in the Log-Book, and if the
Ship has failed in a Current or Tides-way, or in a ſwelling Sca,
that ſets to Leeward, the Sercing and Drift of the Current, Tide,
c. muſt be orderly ſet down as another Courſe and Diſtance, and
worked accordingly; and the whole Days Work being finiſhed, if
the Latitude by Dead Reckoning or Account, agree with the Lati-
tude by Obſervation, then is the Reckoning confirmed, and the place
of the Ship truly determined; but if they are found to difagree,
then as the Phrafe is, the Ship hath out run her Reckoning, or the
Reckoning hath out run the Ship; and there is ſome Error either in
the Courſe, or in the Diſtance run, or in both ; and how theſe may
be re&ified, and proper Allowances or Corrections made, has been
lufficiently thewn already.
The Errors (if there be any) being adjuſted, and the Days Work
finiſhed, the true Latitude of the Place the Ship is in, her Difference of
Longitude made, her true Bearing and Diſtance from the Place from
whence ſhe took her Departure, and the moſt remarkable Tranſac-
tions of the Day muſt be diſpoſed of, after the manner following,
(which is ſo very plain that it needs no Explanation) and which is
called the Form and Manner of keeping a Fournal.
A
#S
At Noon the
tal
A JOURNAL from the Lizard towards Barbadoes, in the Ship Good-
Succeſs, A. B. Commander.
Weekl Minto, Mon.
Winds.
Diret Pift. Latitude Whole Diff Bearing and Diſt. Remarkable Obſervations and Ac
Days Tiar. Days !
Courſe. Miles Corret. Long, made. from the laſt! and
cid nts.
)
March.
I 2 403 W by W
25,15 W 1165.447.30 1° 47'. W
AderateGales and fair Weather,
1722.
SW by s
Lizard bore N.25lat Four this Afternoon, I took my
SSW and
15 Eaſt, Diſtance Departure from the Lizard, bear.
South West
1659 Miles. ing North North Eaſt 5 Leagues.
13
At S W by S $ 19,4 W 145.145.127.2.5675 At Noon the Li The firit part of the Day mode.
SS W
pard bore N. 22. rate Gales and fair Weather, the
Couth Weſt and
29 Eaſt, diſtant middle and latter part freſa Gales
WN W
310-8 Miles.
and cloudy.
14
At South S43.15 W 38.931 44.46
43.15 3.3154
At Noon the Li The firltpart freth Gáles and
SSE
kard bore N. 24 cloudy Weather, the middle and
South Eaſt and
33 Eaſt, diſtant latter part hard Gales, and ſqually
WSW
with Rain.
2
15
At Welt N 20.55 E (+1.745.25
3.097?
At Noontheli. For the moſt part Itrong Gales of
NW by W
0
zard bore N. 24. Wind and ſqually Weather, with
NW by N
55 Eaſt, diſtant Rain and Hail, at Six a terrible
and S by E
303.2 Miles. Storın.
외
​At S by E N 83.04 E 58.
49.32
1.4713
At Noon the Li The is part it:ong Gales of wind.
O
pard bore N. 15. with violentSquals, the middle hard
07 Eaſt, diſtant Gales, the latter part freſh Gales
277.6 Miles and more inoderate Weather.
17 At South So. 22 Weſt 109. 43.43 1.4715 At Noon the Li The first part freil Gaies ang
South by Weſt
pard bore N. 11. fair Weather, the middle and latter
outh Weſt and
23 Eaſt, diſtant part moderate and hazy.
W by W
284.1 Miles.
And to diſtinguiſh the Latitudes obtained by Obſervation from thoſe deduced by Calculation, I have
marked thoſe that have been obtained by Obſervation with the Character .
345,2 Milës.
+0
1
Of the Equation of Time.
437
The chief Bufineſs of keeping a Journal being only to exhibit the
Place the Ship is in, or was in, at any time propoſed, at one View,
together with ſuch remarkable Accidents and Occurrences as are
worthy of Note, to incumber it with Northings, Southings, c.
Latitude by Obſervation and Dead Reckoning, Meridional Diſtance,
Gc. can ſerve to no other Purpoſe in my Opinion, but to perplex
and confound it ; and is the proper Buſineſs of the Log-Bonk, and
which ought always to go along with it, and to which Book for
ſuch Information, the Inſpector ought to be referred.
The Rules here delivered are I believe, as true and more general
than any that have appeared abroad, and I may venture to ſay, that
there is ſcarce any Occurrence that can happen, in the Theory or
the Art of Navigation ; but what has been taken Notice of and ex-
plained in ſome place or other, and therefore there is no room left
to doubt, but that if the Learner will make Uſe of what Helps are
here afforded him, he will rarely want Afliſtance; and be able at all
Times to give a good Account of the Ships Place. And laſtly,
As the keeping of a good Account of the Ships Way, and being
ready at all times to pronounce where ſhe is, is not only the chief
End of the Nautical Science, (but next under God) the only thing up-
on which the Lives and Safetys of all the People concerned intirely
depend ; no Pains ought to be thought too great, nor any Care
wanting to accompliſh it, ſince one unlucky Miſtake, which too
often happens, may render them incapable of ever rectifying it after-
wards, or of telling how it came to paſs, and from which fatal Ac-
cident for the future, I pray God to deliver every Perſon concerned.
no
Section XIV.
$
Concerning the Equation of Time, or the inequality of the
Solar Days.
Ime is a part of the Duration, meaſured out and aſſigned by
Motion, which when it Flows on equally and uniformly, is
called Abſolute Time, to diſtinguiſh it from the Relative and Appa-
rent Time, which is meaſured out by the Motion of the Sun, and
L11*
Flows
438
of the Equation of Time
Slows unequally, and is divided into Parts called Years, Months,
Weeks, Days, c.
An Equatorial Day, is that ſpace of Time that elaples, while the
the Earth makes one intire Revolution about her Axis ; or which is
the ſame thing, while the Heavens being carried about, the ſame
point of the Equator that was at firſt under the Meridian, returns
again to the ſame Meridian; and inaſınuch as the Revolutions of the
Earth about her Axis, or of the Primum Mobile or ficit movcable
Orb, are conſtantly equal among themſelves, and the ſame in every
point of the Orb (at leaſt the ineqnality is ſo ſmall, if any there be,
that it has eſcaped the Knowledge of the moſt ſtrict Scrutators
into the Laws of Motion of the heavenly Bodies) there therefore are
taken for Standards, or meaſurers out of equal Portions of Time, to
compare the linequal Meaſures or Portions of Time with
A Sydereal Day, is that Space of Time that Flows, while the Earth
inakes an intire Revolution to the ſame fixed Star, or which is the
faine thing, while the Heavens being carried about the ſame star
that was before under the Meridian, returns to the ſame Meridian
again : And inaſmuch as the Stars by reaſon of the Rece:lion of the
Equino&ial Points ſeem to move on Eaſterly, at the Rare of so Se-
conds a Year, and conſequently at the Rate of 8'"' 12'' 48:47',
Oc. a Day it follows, that while the Earth makes one intire Revolu-
tion about her Axis, the Star is moved on forward in the Order of
Signs 8'"' 121 489 47', &c. which anſwering in Time to 32 519
15",&c. it is manifeit, that the Equatorial Day is lefer than the Sy-
dereal Day by 32'"' SI 15,66. wherefore, if the Equatorial Day
be put equal to Twenty-four Hours, the Sydereal Day will be 24h.
oo min. oo fec. 00'' 32+ SIS 155, 6c.
The Solar or, Natural Day, ſo called to diſtinguiſh it from
the Artificial Day, which is oppoſed to Night, is that Space of
Time that elapſes while the Earth makes an intire Revolution to the
Sun, or which is the ſame thing, while the Heavens being carried
about the Sun returns to the ſame Meridian it was under at the
Noon before ; and inaſmuch as the Earth is carried on in her Orb
Eaſterly, according to the Order of the Signs, or which is the ſame
thing, the Sun feems to move on in the Ecliptic from Weſt to Eaſt
during the ſame time, it is manifeſt that the Solar Day is longer than
the Equatorial Day, by a Portion of the Equator that anſwers to
the apparent Morion of the Sun in the Ecliptic, in the mean while.
The
or the inequality of the Solar Days. 439
...
The Earth finiſhes her annual Revolution, and returns to the ſelf
fame Place in her Orb from whence ſhe began to move, or which is
the ſame thing the Sun compleats his Courſe thro'the Ecliptic, (and
which is called a Tropical Year, becauſe after it is finiſhed all the
Seaſons return again in the ſame Order) in 365 Days 5 hours 48
minutes, 57 leconds, whence the diurnal mcan Motion will be
found to be 59 min. 08 ſec. 19", 43+ 47' 21", &c. equai in
time to 3 inin. 56 ſec. 33''' 18'55' 9', &c. whence it follows, that
if the Equatorial Day be put equal to 24 Hours, the mean So-
lar Day will be 24h. 03 m. 50 f. 33'' 18' 555 09', &c. Eut if
the vean Solar Day be put equal to 24 Hours, then the Equatorial
Day will be 23 h. som. 41. OS" 2618so?, oc, and the Si-
derea! Day but 23 h. 56 m. 04 f. 05'' 59' 04' 42', &c.
It the Axis of the Earth ſtood at Right-angles to the Plain of the
Orb in which ſhe moved, and the Sun were placed exactly in the
Center of the ſame Orb, then the mean and apparent Solar Day
would be the ſame, and the Difference between the Equatorial and
Solar Day would be conſtantly equal.
But becauſe of the Inclination of the Axis of the Earth to the Plain
of her Orb, the Arch of the Equator paſſing thro the Meridian, is
not always equal to the correſpondent Arch of the Ecliptic, which
paſſes thro the ſame in the ſame time, but is ſometimes bigger and
Sometimes leſs, hence it is that the diurnal Arches of the Equator
which paſs over the Meridian are unequal, and hence ariſes one Cauſe
of the inequality of the Solar Days.
For in the adjacent Figure, let PSA
P
repreſent a Quadrant of the Solftitial Co-
Jure, P the Pole, Ar the Radius of the
Equator, and reso a Quadrant of the E-
cliptic, r the Equino&ial point or place
of the Sun in the firſt point of Aries, when
he is under the Meridian of a certain Place
upon a certain Day, O the place of the
Sun under the Meridian of the ſame Place
the next Day at Noon, then will ro re-
preſent the diurnal Motion of the the Sun, and if thro' the point
0, the Circle of Right Aſcenſion POR be drawn, the Arch of the
Equator r R will repreſent the correſponding Right Aſcenſion; or
Arch of the Equator Culminating together with the Sun, and which
muſt be neceſſarily lels in this caſe than the diurnal mean Motion of
LIT 2
the
la
R
*
..
440
Of the Equation of Time
11
the Sun, inaſmuch as it is one of the containing Sides of a Righe-
angled Spherical Triangle, of which the Arch of Longitude rois
the Hypothenuſe, let us ſuppoſe for Example, the Arch of Longi-
tude r o equal to the Sun's diurnal mean Motion, equal to oo deg.
59 min. 08 ſec. then by the ift Problem of Section the 4th of Part the
Sth, the correſpondent Arch of Right Aſcenſion r R will be found
to be equal to oo deg. 54 min. 13 ſec. leſs than the correſpondent
Arch of Longitude by 4 min. 55 ſec. equal in Time to oo deg. 19
min. 40 ſec. and ſo much in this caſe is the Apparent or Solar Day
leſs than the mean Solar Day, and conſequently the Equation thence
reſulting muſt be Negative, and being therefore ſubftracted from the
apparent Time, will give the Mean.
Let us now ſuppoſe the Sun to be at the diſtance of 89 deg. 00
min. 52 ſec. from the Equino&ial point Aries, that is ſhort of the
Soifitial Colure 59 min. 08 ſec. the diurnal mean Motion ; the cor-
reſpondent Right Aſcenſion will be found to be 88 deg. 55 min. 31
fec: leſs than the Arch of Longitude 89 deg. 00 min. 52 ſec. by
s min. 21 ſec. equal in Time to 2 1 ſeconds 24 thirds, the Equati-
on in this caſe ſtill to be ſubſtracted from the apparent Time, to ob-
tain the Mean; and by comparing the Arch of Right Aſcenſion 54
min. 13 fec. contained between the firſt point of Aries, and the point
anſwering to 59 min. 08 fec. with the difference between the Right
Afcenfion 88 deg. 55 min. 31 fec. and 90 degrees, anſwering to
the Arches of Longitude of 89 deg. oo min. 52 ſec and 90 degrees,
equal to i deg. 4 min. 29 ſec. we ſhall find that the Arch of the E-
quator that tranſits over the Meridian, while the Sun travels over
his diurnal Arch of mean Motion 59 min. 08 fec. at the End of the
firſt Quadrant, is greater than the correſpondent Arch of the Equa-
tor that tranſits over the Meridian, while the Sun performs his di-
urnal mean Motion at the Beginning of the Quadrant, by 10 min.
16 ſec. equal in Time to 41 ſeconds 15 thirds; whence it is mani-
felt, that altho' the Revolutions of the Equator and its ſimilar Parts
are iſochronaly yet, tho' the Sun fhould move over equal Arches of
the Ecliptic in equal Times, he will return to the Meridian with un-
equal Portions of the Equator ; and confequently for this Cauſe on-
ly, the Solar Days would be unequal, and the diurnal Differences
turied into Tine will give the Equation, which muſt be ſubſtrated
in the firſt and third Quadrants of the Ecliptic, that is while the Sun
is paffing from the firſt point of Aries to the firſt point of Cancer, and
from the firſt point of Libra to the firſt point of Capricorn ; becaule
in
or the inequality of the Solar Days.
441
in each of the Quadrants the Arches of Right Aſcenſion are always
leſs than the correſpondent Arches of Longitude.
But becauſe in the ad and 4th Quadrants of the Ecliptic, (as has
been ſhewn elſewhere) the Arches of Right Aſcenſion counted from
the firſt point of Aries, are greater than the correſpondent Arches
of Longitude, it follows in this caſe that the apparent Solar Days
are ſhorter than the mean Solar Days, and that while the Sun is go-
ing from the firſt point of Cancer to the firſt point of Libra, and from
the firſt point of Capricorn to the firſt point of Aries, the Equacion
or Difference between the Longitude and Right Aſcenſion thence
reſulting, muſt be added to the apparent Time to obtain the Mean.
And inaſinuch as the Sun it ſelf is not placed in the Center of the
Earths Orb, but at the diſtance from it of 1692 ſuch Parts as the
mean Diſtance of the Sun from the Earth is roocoo, and that as the
Earth is carried about in her annual Orb, the Line connecting the
Centers of the Sun and Earth, deſcribe cqual Arches or Portions of
the Ecliptic Surface in equal Times, it neceſſarily follows, that the
Arches deſcribed in equal Times will be unequal, and conſequently
that the diurnal Arches of the Ecliptic that the Sun leems to move
over, will be greater at one time than at another, and greateſt when
the Earth is upon her Perihelion, or when ſhe is the neareſt to the
Sun ; which in the preſent Age happens when the Sun appears in a-
bout 8 deg. of Capricorn, or upon the 19th of December, and leaſt
when the Earth is upon the Aphelion, or when ſhe is fartheſt off from
the Sun, and which happens when the Sun appears in about 8 de-
grees of Cancer, or upon the 19th of June, and hence ariſes another
Cauſe of the inequality of the Solar Day, as will be more evident
from the following Figure,
Where A BPN repreſents the great Orb, in which the Earth is
annually carried about the Sun, C its Center, A the Aphelion,
or Place of the Earth at the Noon of that Day wherf the Earth is
in the Aphelion, B the place of the Earth the next Day at Noon,
AL a given Veridian of the Earth, the Arch A B or Angle ACB
the mean Motion of the Earth, between the Noon of the given
Day and the middle Noon of the Day following, L a point in the
given Meridian dire&ly oppoſite to the Sun, which while the Earth
moves forward in her Orbit from A to B, is whirled about by the
diurnal circumvolution of the Earth from L thro o in the firſt place
A, to d in the ſecond place B, during which time the Earth has
made an intire Revolution about her Axis, inaſmuch as the Meridi-
1
an
442
Of the Equation of Time
an B d in the ſecond Scituation B, is become parallel to A L, its
Poſition the Day before when it was in the point A, but the appa-
rent Noon does not happen till the ſame point by the Revolution
of the Earth about her Axis is carried to é dire&tly oppoſite to the
Sun, who governs the civil Day; and hence it is manifeſt, that the
apparent Solar Day is longer than the Equatorial Day by the Archi
BA, equal to the Angle d Be (for the Lines B d and À O are pa-
rallel) the apparent otion of the Earth, or the diurnal Motion of
the Sun in the Ecliptic converted into Time:
But the mean Solar Day is not compleated, neither will the mean
Noon happen, till while the Earth being ſtill carried on forward in
her Orb, the point e which is now directly oppoſite to the Sun, is
carried to f, directly oppoſite to the Center of the Orb C; and inaf-
much as the Revolutions of the Earth about her Axis are equal and
uniform and conſtantly the ſame, the returns of the fame in Conjun-
dion with the Center of the Orb, muſt be conſtantly equal among
themſelves ; and hence it is evident, that the mean Solar Day excecds
the Equatorial Day by the Arch df, equal to the Angle B C A, the
diurnal mean Motion converted into Time, and that the ſame mean
P
In
k
Я
d.
B
A
Day
2
or the inequality of the Solar Days. 443
Day exceeds the Solar Day by the Arch e f, equal to the Difference
between the mean and apparent Motion of the Sun, equal to the
Angle O B C, the Equation of the Orb; this therefore converted
into Time and fubftracted from the Length of the mean Day, will
give the Length of the Solar Day, and conſequently being ſubſtrac-
ted from the apparent Time, will give the mean or equal time, and
the ſame Law obtains in whatſoever part of the Semicircle BAN,
the Earth is found, and which is called the firſt Semicircle of Ano.
maly.
Let us now ſuppoſe the Earth in her Perihelion point at P, and
Pm a given Meridian, upon which the Sun appears upon the Noon
of the Day when fie is in that point, as the Earth is carried on
from P towards A when her Center is got to N where the Meridi-
an Ng becomes parallel to Pm, its Poſicion the Day before, the
Earth has made one intire Revolution about her Axis, and the Equa-
torial Day is compleated, but the mean Noon will not happen,
neither will the mean Solar Day be compleated till the Earth fill
roaling forward in her Orb; the Point g by her Revolution about her
Axis is got to h, when it is directly oppoſite to the Center of the Or-
bit C; and that is when the Point g has moved over the Arch gh,
the Meaſure of the Angle g Nk, equal to the Angle PC N, equal in
Quantity to the Arch P N, the diurnal mean Motion, but the So-
lar or apparent Noon does no: happen ; neither is the Solar or ap-
parent Day finiſh’d; till while the Earth being ſtill carry'd for-
ward, by her revolving Motion, the Point h has mov'd over the
Arch k, where it is directly oppoſite to the Sun, at O; whence it is
manifeſt that the Solar or apparent Day, in this Cafe, is greater
than the Equatorial Day, by the fame Space of Time that the
Earth requir'd to move orer the Arch gk, the Meaſure of the An-
gle g Nk, equal to the Angle PON, ihe apparent Motion of
the Sun in the ſame Time ; and that the ſame Solar, or apparent
Day, exceeds the mean Day by a Space of Time, equal to That
that the Earth requires to move over the Arch h k, equal to the
Difference between the mean and true Motion of the Sun, equal to
the Equation of the Orb: This therefore converted into Time, and
added to the Length of the mean Day, will give the Length of
the apparent Day, in this point, and conſequently being added to
the apparent Time, will give the mean or equal Time ; for the
mean Noon is Antecedent to the apparent Noon, and the ſame
Law obtains in whatſoever part of the Semicircle PNA the Earth
is found, and which is called the ſecond Semicircle of Anomaly.
In
}
444
Of the Equation of Time
In accounting for this latter part of the Incquality of the Solar
Days, the Orb in which the Earth moves has been ſuppoſed circu-
lar, and that ſhe in her Revolution round it deſcribes equal Angles
about the Center in equal Times, and tho neither of theſe really
obrain, yet he that knows how great an Analogy there is between
the Circle and its inſcrib'd Ellipfis, how little the Earths Orb dit-
fers from a Circle, and underſtands the Nature of the Keplerian A-
reas, will readily perceive that the ſame way of reaſoning will hold
good in the Ecliptic Orb, and that in accounting for it in one you
account for it in the other.
Hence it is abundantly manifeſt, tha: the Equation of Time, or
Inequality of the Solar Day, ariſes from two Caules, viz the In-
clinaciun of the Earths Axis to the Plain of the Ecliptic, and the
diſtance of the Sun from the Center of the Orb, and is a Compound
of the Difference between the Longitude and Right Aſcenſion of
the Sun, and the Difference of the mean and true Place, and when
theſe two parts are both Negative, or both Poſitive, their Sum
makes the abſolute Equation of Days, but when one is Pofitive and
the other Negative, their Difference is equal to the Equation; and
if the greater of the two parts is Negative, the Equation is Nega-
tive, but if the greater of the two Parts is Poſitive, the Equation is
Poſitive : And herice we are taught a direct way to Inveſtigate the
abſolute Equation of Days, at any time propoſed.
Let it therefore be required to find the abſolute Equation of Time
for the 7th of May 1721.
On the 7th of May 1721 at Noon mean Time, the Sun is in Tau-
yus 27 deg. 34 min. 07 ſec. when his Right Aſcenſion by the ift
Problem of Se&tion the 4th, of Part the sth, will be found to be 55
deg. 17 min. 13 ſec.
Hence their Difference 2 deg. 16 min 54 ſec. equal in Time to
9 min. 7 rec. 36 thirds will be one part of the Equation of Time,
which becauſe the apparent Noon preceeds the mean Noon, muſt
be ſubſtracted from the apparent Time, to find the mean Time, or
muſt be added to the mean Time, to obtain the apparent.
The mean Anomaly is 10 Signs 1.8 deg. 13 min. os ſec. hence
the Equation of the Orb will be found to be i deg. 16 min. 18 ſec.
(and how this may be found, the Reader may ſee in the following Sec-
tion) equal in Time to 5 min. s fec. 12, which becaule the mean
Noon precedes the apparent Noon, muſt be added to the apparent
Tiine, to obtain the mean Time, but ſubſtracted from the mean Time
to find the apparent.
And
IT:
or the inequality of the Solar Days. 445
1
And becauſe the two Component Parts of the Equation are con-
ţrary one to the other, that is one Negative and the other Poſitive,
their Difference 4 min. 2 lec. 24 thirds, will be the ablolute Equa-
tion of Time, which becauſe the greateſt of the two Component
Parts of the Equation is Negative, muſt be added to the mean Time,
to obtain the appareat Time, but fubtracted from the apparent
Time, to find the mean Time, and after the ſame manner may the
abſolute Equation of Days for any other Time be readily found.
And for the more expeditious obtaining of theſe two Parts of the
Equation, the Two following Tables are calculated, the firſt of
which ſhows that part of the Equation that ariſes from the excen-
tricity of the Sun, and is the ſame with the abſolute Equation of
the Orb reduced into Time, and is ever equal to the Difference be-
tween the mean and true Place of the Sun, the Second Chews that
part of the Equation, that ariſes from the Inclination of the Earths
Axis to the Plane of her Orb, or from the Obliquity of the Ecliptic,
to the Equator; and is ever equal to the Difference between the
Sun's Place and the correſpondent Right Aſcenſion, both of which
Parts muſt be compounded together, according to their reſpeđive
Titles, to obtain the true and abſolute Equation of Time.
4
.
Mmm
The
1
146
Of the Equation of Time
The EQUATIONS of the Apparent Tim E.
Subſtraa from the Apparent
if the Suns Mean Anamo-
ly be.
Subſtract from the
Apparent.
1
0
2 3 4
lilllllllllll
5
n
al
8 M:71
1
1
1
O
0.6
j
2
I)
:
\
o 0.00 3.486.3917.4516.473.57.30
10.0813.545.4317.4516.4313:50125
20.16 1.025.477.456.3913,42 28
310.244.00/5.507.456.3513.3527
410.324.15 6.5417 456.3 3.28 2Ć
510.40/4.22 5.587.446.263.2025
60 484.2617.01 7.44 6.2113.13 24
70.554.351.057.436.16 3.0523
81.034'427.0817.426.11|2:57 23
91.11 4.447•117.416.062.50 21
101194.5417.147.4016.01 2.42120
11 1.275.00 7.1 77.385.5% 2.34.15
121.3515.06|7.1917:375.51|2.2018
1311.425.427.2 27.35 5.45.12.18117
14/1.505-187.2417.3415.402.11110
151.58 4.247.27 7:32 5.341.03.15
1612.7615.30 7.297.34.15.21.5514
17 2.13)50357.317.285.221.4713
182.215.4117:33 7.235.1611.35 12
192.28 5.4017:35 7.235.10 1.3011
2012.3615.5217.3617.20.04.1.2210
21/2.435.711-3817.174.581.14
22 2.51 6.0:17.397.1914.5c1.008
2312.5 9 16.077.4017.12 1.450.58 7
243.066.127.417.09|4638191 6
2513:1716.117.427.0+302415
2013.2" 16.217.437.02 7.2510:33) 4
273.276.267.446.58 7.1$6.25 3
28 3.34 6.317.446,5511.110.16 2
2013.416.34, .456.511.0400S
30 3.4816.3917456.473.575.000
9.
8 1.7 6
Add to the Apparent Time.
8.24 8.453
0.2.0 8.34 8.35129
0.40 8.44 8.2412
31 0.591 8.531 8.1327
4 1.199.02 8.01.26
51 1.391 9.10 7.48 24
61.581 9.171 73424
7) 2.18] 9.24 7.2012,
8 2.37) 9.30 7.00 25
91 2561 9.35 6.50 21
103.15 9.45| 6.352
3.341 9.441 6.1816
123,521 9481 0,021
131 4.10 9.50 9.4411
14) 4.281 9.57 5:27/11
151 4.40 9.531 5.081.
i 5.231 9.54 4.50114
17 5.20 9.54 4.311
18 5.371 9.5 4011/12
195.53 9.50 3.520
2016.091 9.491 3.321
21 6.25 9.401 3.119
221 6 40 9.4.2 2.51
:3 6.549.39 2.30 71
24 7.09 9.31 2.0.
257,239.25 1.4815
26 7:30) 9.191 1.26
4
277.481 9.12 1.05 3
18 8.01 8:04 0.431 2
9
8.12 855 0.21
8.23 8.45 0.00
H me mu
Nive
Add to the Appar.
By
I
j
10
o
er the inequality of the Solar Days.
447
.
By the Help of theſe Two Tables, may the abſolute Equation of Time
be had by Inſpection, the Place of the Sun being given, and his mean Ano-
maly, and how theſe may be obtained may be ſeen in the following Se&tion,
for by entering the Firſt Table with the mean Anomaly, and ſeeking the
Siga at the Top or Bottom, and the Degree on the Side, in the common
Area, by making Proportion for the odd Minutes and Seconds, you will
have that Part of the Equation that belongs to it with its proper Title; and
by entering the Second Table with the Sun's Place, ſeeking the given Sign
at the Top or Botcom, and the Degrees of that Sign on the Right or Left
Hand Columns on the Side, and making proper Allowances for the odd Mi-
nutes and Seconds, you will have that part of the Equation that ariſes from
the Difference between the Sun's Place and his Right Aſcenſion, with its
proper Title, theſe two Parts being compounded together according to their
Example
, it were required to find the abſolute Equation of Time for the
7th of May 1721, when the Sun is in Taurus 27 deg. 34 min. 07 (ec..his
mean Anomaly at that Time being 10 Signs 18 deg. 13 min os ſec.
Entering the Firſt Table therefore with the mean Anomaly, I find to Signe
at the Bottom ; and right againſt 18 Degrees, the next leſs whole Degrees
in the Right-hand Column, in the common Area, I find 5 min. 6 ſec. and
againſt 19 Degrees, the next greater Degree to the mean Anomaly, in the
ſame Column of 10 Signs, I find 5 min. o fec. Then I ſay, as i deg. or 60
min. is to 6 lec, the Difference between 5 min. 6 ſec, and s min. o ſec.
ſo is 13 min, os ſec. the Exceſs of the mean Anomaly above 10 Signs 18 m.
to i ſec. the Part Proportional, this ſubtracted from the Equation 5 min.
o ſec. anſwering to io Signs 18 deg. becauſe the Equation diminiſhes, will
give s min. 5 ſec. for the proper Equation anſwering to to Signs 18 deg.
13 min. cs ſec of mean Anomaly, which by the Title is to be added to the
apparent Time to find the mean.
Entering again the Second Table with the Sun's Place in Taure:s 27 deg.
34 min, 07 ſec. I find Taurus in the Head, and right againſt 27 deg in the
Left-hand Column, in the common Area I find 9 min. 12 ſec. and againſt
28 degrees, the next greateſt Degree to the Sun's Place, I find 9 min. 4 ſec.
Then I ſay, as i deg. is to 8 fec, the Difference between 9 min 13 ſec. and
9 min. 5 ſec. fo is 34 min. 07 ſec. the Exceſs of the Sun's Place above 27 d.
to 4 ſec. nearly; this taken from 9 min. 12 ſec. the Equation anſwering
to 27 deg. becauſe the Equation decreaſes, will leave 9 min. o7 ſec. í, for
the Equation anſwering to 8 27 deg. 34 min. o7 ſec. and which according
toits i'itle muſt be ſubtracted from the apparent Time to diſcover the mean;
and conſequently their
Difference 4 deg. 2 min. will be the abſolute Equa-
tion of Time, and the ſame as was before determined.
After the ſame manner may the abſolute Equation for any Day be found,
and after this manner are the following Tables of Equation of Days com-
puted,
Mmm 2
.:
448
Tables of the Equation of Time.
V
#
1
I/
A *
S*
1
come
28|14.47 0.351 1.4813.50 1.32 4.341 5.00 2.40 12.50 15.57 6.581 7.2828/
A TABLE for the Equating of Time for the Year 1724,
being the Billextile or Leap. Tear ; to be applied to the Appa.
rent time, according to the reſpe&ive Titles.
1724
Jan. Feb. Mar. (Apri.. May. Fune. July. Aug. Sept. fotob Niv. Dec,
N
ATA A A* S S* A
S
S S
18.40 14.49 10.01 0.42 4.04 0-401 4.55 4.30 4.741 3:3415.30 5:34 !
9.09 14.489.44 0.26 4.05 0.34 5.07 4.20 4.25.13.48115.211 5.05 2
9 32 14.40 9.270.10 4.00 0.22 5.14 4.05] 4.45 1401 15.11 4.303
49.5414.44 9.115.05| 4.06 0.14 5020 3.5
5•26 3.515.0614.14115.00 4.064
0.1914.41 8.52 0.20 4.05 A03 5.2 3.47 3:27 14.26 14.48 3.36 5
C10.30 14.37 8.34 0.351 4.041 01c 5.3 3.351 5.48|14:38 4.30 3.00!
10,50 14.32 8.16 0:49 4.02 0,29 5.36 3.22 6.09|14.45
| 4.23 2.35
111,15114 27 7.58 1,031 4.00 0.421 5.41 3.00 6.30 14.99 14.79 2.06 )
Sil.33.14.21 7.40 1.10 3.57 0.55 5.45 2.541 6.51115,04 13.541 1.369
101.514.14 7.211 1.29 3.54 1.07 5.42.44 7.12.5.14 -3.39 1.05
11:2.07.14.00 7.02 1,42 3.50 1.2015.51 2.20 7.33115:28 13:23 0 361
12|12-23 13.5 16.43 1.541 3.45 1.33 5.54 2.11 7.54.15.30 13.06 0.0012
13|12:38|13.4%) 6.24 206 3.40 1.46 5.5 1.55 8.1415.4: 112.48 A2413
1412.5213.39 6.05 2.17 3.35 1.59 5.57 1.39 8. 415.45 12°291 0.5414
1413.00 13.24) 3.40 2.27 3.29 2.11 5.57 1.2318 54115:54: 2.09 1.245
1:3 19.113.19 5.27 2.371 3.23 2.235.501.00 9.14 16.00 11.491 1.53
111 3.31 13.08 5ice 2.46 3.16 2.35 5:55 0.451 9.34 16,0411.28 2.2217
113.42 12.571 4.452,55 3.09 2.47 5.531 0.31 8.54 16.07 11.07 2.51
19|13.5212.45 4.30 2.03 3.01 2.591 5.51 0.13110.54 16.09 10.45 3.2059
2014.01 12.33 4.12 3.11 2.52 3.01 5.481 So510.33 16.10 10.22 3.4920
21|14.1012.20 3.54 3.19 2.43 3.23 5.45| 0.23|10.51116.11 9.58 4.18 2!
2214.18 12.00 3.30 3.26 2.34 3.34 5.41 0.41 11.09 16.12 9.34 4.46 22
2314.29 11.52 3.183.33 2,25 3.45 5.361 1.091.21 16.12 9.09 5.14123
24114.31111.37) 3.00 3.39 2.15 3.56 5:34 1.20 11.44116.11 8.43 5.4124
2914.36 11.22 2.42 3.44 2.05 4.06 5:25 1.40 12.01|16.09) 8.17 5.0825
2014-4 11.07 2 24 3 481 1.541 4.16 5.192.00 12,1816.00 7.511 6.35
27 14.44 10.5i 2.00 3.52 1.43 4.25 5.13 2.20 12.3416.02 7.25 7.022?
29.14.49.10.18) 1.31 3.59-1.211 4:43 4.5813.01 13.05 15.511 6.30 7.5329
30 14:52 1,14 4.02 1,10 4.51 4.45) 3.2213.20 15.45 0.02 8.1729
3114.50
0.58
4.401 3.431
15:38 8.40131
10
18
9,581
A
A
Tables of the Equation of Time.
449
A TABLE for the. Equating of Time for the Year 1729,
being the Firſt Year after Leap-Tear ; to be applied to the Ap-
parent Time, according to the reſpective Titles.
1725.
fan. Feb. Mar. April May June July Aug. Sept. O&tob Wov. Dec.
1
11
u
A*
1
IC
1
A A' A
S S * A A* S S SI S *
19.03 14.47 10.03 0.461 4.02 0.471 4.581.4.321 359113.30 15.32 5:41
21 9.20 14.45 9.40 0.30 4.04 0.35 socc 4.22 4.20 1344115.23 5.12 2
31 9.48 14.431 9.29 0.1414 05 0.231 5.13 4.11 4.41 13.58119.13 4.43 3
410.09 14.40 9.12 S.02 4.00 0.11 5.2 4.00 5.02 14.11 15.02 4.14 4
510.30-14.308.5410,1714.25
4.24 A01 5.20 3.491 5.23114.23 14.5 | 3.44
610.59114.311 8.300.31 4040 013 5.311 3:37 5:44 14:35 14.391 3.14
7.11.09 14.20 8.1 | 10:454.0 0,20 5.30 3.241 6.05.14.47 14.202.4417
12.27 14.20 8.00 0.591 4.06 0.34 5:41 3.11
0.30 5:11 3011 6.26 14.5814.17 2.141 8
11.45|14.131 7.42 1.13 3.57) 0.521 5.45 2.7 6.47|15.08 13.5 1.4419
1012.02 14.05 7.24 1.26 3.54 1.05 5.48 2.43
3.54 1.05 5.48 2.43 7.28 15.18113.43 1,1
1112.18 13.581 7.05 1.391 3:51118 5:51 2,28
1 185.511 2,281 7.28115.2713.20 0.4411
12 12.3413.49 6.40 151 3.17 1.319.531 2.13
2.131 7.48115.3513.05 0.14 1:
13 1 2.48 13.40 6.27 2.72 3.19 1.445.54 1.591 8 og 15.42 12.52 A 1713
1413.02 13.30 6.08 2.13 3.3 1.575.55 1.13
13 3.34 1.575.55 1.45 8.20 15.4812.33 0.47/14
1513.15113.20 5.49 2.241 3.30 2.05 5.551 1.25 8.49 15.53.12.14 1,1715
1613.27 13.09 5.30 235 3.23 2.211 5:55 1.089.09 15.58 11.54 1.47
1713.33 12.50 5.11 2.45 310 2.335-541 0.51 9.2916.0211.33 2.1717
18/13.48 12.464.52 2.54 3.09 2.45 5.52 0.34) 9.49.16.66 11.12 2.47118
1913.5812.34 4.341 3.02 3001 2.57 5.50 1.16 10.09 16.05 10.50 3.1019
2014.07.12.211 4.16 3.10 2.53
4.16 3.102.531 3.05 5.481 S.02 10.28 16.11.10.28 3.45120
2114.15 12.07 3.58 3.18 2.44 3.21 5.45 0.20 10.45 16.12.10.05 4.1427
2214.22 11.531 3.40 3.25 2.35 3.39 5.41 0.38 11.05 16:13 9.46| 4.4222
2314.28 11.381 3.22 3.32 2,20 3,43 5:30 0.57 11.23 16.12 9.15.07|23
244.33|11.23 3.04 3.38 2.16 3.44 5.31 1.17.11.41 16.11 8.50 5.3624
254.38 11.08 2.40 3.43 2.06 4.04 5.25 1.37 11.5816.09 8,24 6.0325
2014.42 10.53 2.28 3.47 1.561 4.14 5.191 1.57 11.15 16 07 7.98 6.30 26
27 14.45 10.37 2.10 3.51 1.451 4.24 5.13 2.1712.31 16.04 7.32 6.57127
28 14.47 10.20 1.52 3.54 1.34 4.33 5.06 2.37 12.46115,59 7:05 7:23 28
29 14:48
1.35 3.57 1.231 4.42 4.58 2.5813.01 15:53 6.38 7.4829
30 14.49 1•18 4.00 1.11 4.40 4.50 3.19113.16 15.47 6.10 & 1235
31|14.48
4.411 3.39 15.404
8.35131
3:1
1.021
0.59
450
Tables of the Equation of Time.
A TABLE for the Equating of Time for the Year 1726,
being the Second Year after Leap-Year ; to be applied to the
Apparent time, according to the refpe&ive Titles.
1726..
Fan. Feb. Mar. April May. Fune. Fuly. Aug. Sept. Poob Nov. Dec.
7
#
#
A *
A*
I(
A А A
SS* A
S Ś S S*
18.5814.4810.08 0.50 4.02 051 4.50 4:34 3.54 13.2715.34) 5:48
1 9.21 14.47 9.51 0.34 4,04 0.38 $.04 4.24 4.15 13.4.15.20 5.1912
3 9.43114.451 9.34 0.78 4.05 0.26) 5.11 4.14) 4.36 13.55|15.10 4.50 31
410.05|14.42 9.171 0,02 4.05 0.14 9:18] 4.03) 4.57414.03 15.05 4.211 4
5.10.2014.381 9.00 $.13 4.04 0.01 5.2413.52 5.1814.21454 3:51 5
(10,45|14:33) 8.42 0.281 4.03 A12 5.30 3.40 5.39|14.32 14.42 3.81
211.04 14.28 8.24 0.42 4.02 0,251 5.35 3.27 6.00|14.4314.30 2.5117
$111.2214 22 8.051 0.564.00 0.381 5.43.131 6.21 14.5 14.15 2.21 &
$111.4 114.167.471.103:58 0.54 56441 2.50 6.4215,03 14.01 1.5il
11.57 14.091 7.25 1.23 3.50 1.04 5.47 2.457.03.15.13 13:40 1.2.10
1112,141 14.01 7.11 1,36 3:53 1.175.49 2.31 7.24 15:27:3:30 0.511!
14 12.30-13.521 6.52 1.481 3.49 1.30 5.50 2.16 7.44.15.31 3.13 0.21112
13 12.44 13.431 6.33 2.00 3.44 1.43 5.53 2.01 8.04.15.35 12.55 Aog|13
12.58113:34 6.15 2.11 3.38 1.50 5.54 1.451 8.24.15.4312'37) 0:39/14
1513.11 13.24 5.50 2.21 3.32 2.085-55 1.20 8.44 15.51 2.18 1.0915
1013.23 13.13] 5.37 2.31 3.25 2.20 5.55 1.12 9.04/15.50 11.59 1.39/10
17 13.3513.08 5.162,41 3.18 2.32 5,54 0.55 9.2416.00 11.39 2.0,17
1813.40 12.5C5.01|2.51 3.11 2.44 5.52 0.3 9.4416,04.11.18 2.38118
15|13.50|12.371 4.42 3.00 3.04 2.56 5.55 0.2c 10.04.16.07 10.50 3.0719
2014.0512.25 4.23 3.08 2.56 3.08 5.48 0.0210.2316.09 10.33 3.3626
21 14.1312.12.4.04 3.15 2.47 3.19 5.45 S 1511.42 16.1.7 10.10 4.0521
22 14.20/11,58 3.45 3.22 2.38 3.30 5.44 0.33|11.01.16.13 9.461 4.3422
2314.26.11.43 3.27 2.29 2.29 3.411 5.38 0.5211.19 16.12 9.21 5.02 23
2414.32 11.28 3.09 3.35 2.19.3.5,2 5,33 1,1211.37|16.11 8.50 5.2924
2514.37 11.12 2.513 41 2.09 4.02 8.21 1.32 11.5416.09) 8.30 5.5025
20|14.41|10.58 2:33 3.46 1.59 4.12 5.22 1.5212.01 16.00 8.04 6.2326
27 14.44 10.44 2.15 3.50 1.48 4.22 5.15 2.1212.27 16,031 7.38 6.5027
28|14.40 10.25 1.57 3.54 1,371 4:30 5.08 2.32 12.43 15.59 7.10 7.1628
2914.48 1.401 3.57 1.264 40 5.01 2.58 12.58115.54 6.:41 7.4125
30|14.49
1,234.00 1,14 4:43 4:53 3.143.131548 6.16 8.053
131|14.49
1.00
1.03 4.441 3:34
15.41
8.2013:
A
.
Tables of the Equation of Time.
451
A TABLE for the Equating of Time for the Year 1727,
being the Third Year after Leap-lear ; to be applied to the Ap-
parent Time, according to the reſpe&ive Titles.
1727
Jan.
Feb. Mar. April May June July Aug. Sept. 1080b Wov, Dec.
1 VI
II IN
1
A *
obovala
13.12.41 13.45
A A A
S S* A A * S S S S*
18.52|14.49 10.12 0.54 4.05 0.531 4.541 4.30 3.50 13.2315391 5:54 1
29.15 14.47 9:59 0.38) 4.04 0.41 5.02 4.27) 4.11 13:38|15.28 5.25 2
31 9:37 14.45 9.38 0.22 405 0.381 5.091 4.171 4.32 13.52|15.10 4.501 3
9.59 14.42 0.21 0.061 405 0.16.5.17 4.09 4.53 14.05 15.08 4.2714
510.2014.389.04 5.09 4.04 004 5.231 3.531 5.14 14.18 14:57 3.58
610.4114.34 8.40 0.244.03) AC9 5.591 3.41 9.35 14.30 14.4) 3.2916
741.00 14.29 8.2 0.38) 4.02 0.22 5.35 3.29 5:55 14.41|14.321 2.59
811118114.24) 8.10 0.52 4.00 -0.35 5:39) 3.16 6.16 14.52 14.19 2.29 $
941.30 14.18 7.52 1.063.58 0.481 5.43 3.03 6.37 15.02.14.051.59
JO11.53.14.11 7.34) 1.19 3:50 1.01 5.40 2.49 6.58115.12.13,56 1.210
1:12.15 .
14.031 7.1, 1:32 3.53 1.13 5.49 2.34 7.1915.21 13.34 0.5911
1212.2013.54! 6.50 1.45 3.49
3.49 1.2
1.20 5.51 2.19 7.40 15:29f13.17) 0.29112
6 39 1.57 3:45
1.57 3.451 139 5.53 2.04
139 5.53 2.04 8.00 15.37'3.00 AOL13
14/12.;5.13.30 6.19 2.09) 3.40
6.19 2.09 3.40 1052: 5.541 1.481 8.20 15.44 12.42 0.31.14
15:3. 8113.20 6.00 2.20 3.34 2.05 5.55 1.32 8.40 15.591 2.23 1.0215
13.41 13.15 5.41 2.30 3.27 2.17) 5.55 1.159.00 15.55|12,0. 1.32116
3 13:32 13:04 5.232.40 3.20 2.29 5.54 0.59 9.20 16.00 11.44 2.02.17
18
13.431..531 5.05 2.49 3.13 2.411 5:53) 0.42 9:40 16.0411.23 2.32.18
1943.53112.41 4.46] 2.58 3.06 2.531 5.51 0.25 9.59 10.07111.0 3.0119
2.4.02 12.28) 4.271 3.07 2.58 3.05 5.44 0.08 10.18 16.09 10.38 3.3920
21:4.10 12.15 4.08 3.15 2.50 3.17 5.46 5.10 10.37 16.11 -0.15 3.5920
2214.18 12.01 3.50 3.22 2.41 3.28 5.43 0.29 10.55
16.12 9.51 4.28/22
23 14:25 11.47 2.32 3.29 2.32 3.341 5.39 0.48 11.13 16.12 9.274.50 23
24:4.3111.32 3.14 3.35 2.22 3.50 5.34 1.07 11.31 16.11 9.02 5.23 24
23/14.30 11.17 2.56 3.40 2.12 4.60 5.28 1.27.11.49 16.09 8.30 5.50/25
2014.40 11.01 2.37 3.45 2.021 4. 1 5.22 1.48 12.06 16 97 8,13 6.1726
27 14.43 10.45 2.19 3.49] 151 4.21 5.15 2.08 12.22 16.04 7.44 0.427
2314.45 10.29 2.01 3.531 1:40 4.311 5.08 2.28 12.3810,00 7.17| 7.0928
1.44 3:57 1 29-4,3 5.01 2.48 12:53 15.551 6.50 7.34|29
30 14:48
1.27 -4.001.117 4.471 4.53 3.99 13.08. 15:49 6.2 7.59130
4.451. 3.29 15.431
823121
$
23|14.47
31|14.49
1.11!
1.05
452
Of the Equation of Time
As a well regulated Clock will always meaſure out equal Time,
ſo if it be truly adjuſted to the Length of the mean Solar Day, and
ſer to the mean Time, the Difference of Time ſhewn by this Clock
and a good Sun-Dial (which always meaſures out Apparent Time)
will be always equal to the ſeveral Equational Numbers in the pre-
ceding Table: thus for Example, on the 7th of May, when accord-
ing to the former Calculation the Apparent Time is more than the
Mean Time by 4 min. 2 ſec. if when it is 12 h, oo min. oo ſec by
the Sun-Dial, the Clock be ſet to 11 h. 55 min. 58 ſec. or which is
the ſame thing, when it is 4 min. 02 ſec. after 12 by the Sun-Dial,
the Index of the Clock is made to point to 12, if the Clock keep e-
qual time on the nexe Day at Noon, when it is 12 of the Clock by
the Sun-Dial it will be 11 h. 56 min. oo fec. by che Clock, or which
is the ſame thing, when it is 12 by the Clock, it will be 12 h. 04
min. co fec by the Sun-Dial, and conſequently the Apparent Solar
Day at this time is leſs than the Mean Solar Day by 2 ſec. the next
Day after this, or the 9th of May, when it is 12 by the Clock it will
be but 12 h. 03 min. 57 ſec. by the Sun-Dial, and conſequently the
Solar Day continues leſs than the Mean Day by 2 ſec. { in Quanti-
and after the ſame manner the Solar Days will continue to di-
miniſh in Quantity, and the Apparent Noon approximate to the
Mean Noon, till on the 5th of June following, the Apparent and
Mean Noon will happen at the ſame part of abſolute Time nearly,
and the Hour Index will point at 12 at the ſame time that the Sha-
dow upon the Sun-Dial talls upon the ſame Hour, after this the Ap-
parent Solar Day will ſtill continue to ſhorten, and the Mean Noon
will preceed the Apparent Noon till about the 15th of July follow-
ing, when the Mean Noon will preceed the Apparent Noon 5 min.
55 lec, and conſequently when the Hour Index of the Clock points
to 12, the Shadow upon the Sun-Dial will fall upon 11 h. 54 min.
os ſec. after this the Solar Days will begin to lengthen ; and the
Apparent Noon will approach to the Mean Noor till the 20th of
Auguſt following, when they will both happen at the ſame time;
after this the Apparent Noon will preceed the True Noon, till a-
bout the 22d of Odober, when the Clock will be too ſlow for the
Sun-Dial by 16 min. 13 ſec. which is the greateſt Variation; after
which it will return again towards the Apparent Noon, always keep-
ing the ſame Difference as is expreſſed in the former Table.
Hence the Apparent Time being given the Mean Time is eaſily
had, or the Mean Time being given, the Apparent Time anſwering
to it is readily found.
ty;
Buc
or the inequality of the Solar Days.
453
But inaſmuch as the Tropical Year conſiſts but of 365 Days,
Ś Hours, 48 Minutes, and 57 Seconds; hence it is that after
every 4 Years the Sun returns to the ſame point in the Ecliptic 44m.
12 l. nearly, ſooner than he did before ; hence and from the Præ-
ceſſion of the Equino&ial points it is that the abſolute Equation of
Time for the ſame Day of the Julian-Year, is in a conſtant Flux, and
continually Changes, and thoʻthe Alteration or Difference be buç
ſmall,yet to render the Table more compleat,I have added the follow-
ing Table, (the Conſtruâion of which is very obvious, from what has
been ſaid upon this Head) which ſhews the Variation of the Equa-
tion to every Day in the Year for 100 Years to come; whence the
true and abſolute Equation for any Day in the intermediate Time
may be readily found, to the greateſt Degree of Exadneſs.
4
Nnn
#
.
:
rerit
..
454
Tables of the Equation of Time.
I
.
il
0.031
0.0
0.12 0.13
...
0.01
0.081 0.12 0.081 0.01
A TABLE thewing the Variation of the Equation of Time,
to every Day in the Julian Tear, for the next Hundred Years.
Fan. Feb Mar April May June July Aug. Sept. 10406 Nov. Tec.
A Si S S LS $ A A A A A
11 0.03 0.19 0.10 0.11 0.08 0.2 0.20 0.12 0.13 0.03 0.08!
20.03 0.10 0.16 0.10 0.00 0.234 0.19 0.03 0.12 0 13 0.03 0.081 2
31 0.02 211 0.16 0.09.200.29 0.15
0.02! 0.12 0.13
0.071 3
4) 0.02 0.111.0.16 0.08) oog 0,23 0.18 0.01 0.13 00 0.07
4
51 0.01
0.12 0.16 A07 0.1CA20,18 $.00 0.12 0.13 0.0: 2,0615
.oil
0 0.01 0.12 0.16 0.070n| 0.23 0.19
0.12 0.13 0.01 0.03
7| 0.00 ol 3 0.16 Oo6| 0.151.27 0.17 0.01 012 0.1 3 0.0.1c 6 7
8 S.ool 0.13 0.16 0.06 0.19 0.23 0.16 0.02 0.12 0.13 0.01 0.00 :
9 0.00 0.14 0.16 0.05 0.13 0.23 0.161.002 0.12
0131 0.01 0,06
10 0.01
0.14 0.16 0.05 0.13 0.25 0.15 0.03 0.12 0.12! 0.01 0.0310
U 0.01 0.14 0.16 0.04 2.14 023 0.15 0.03 0.12 0,14 0.0
0,12 0.0 0.0 IT
121 0.02 0.15 0.16 0.04 0.15 0.23 0'14 3.041 0,12 0,12 Soc 0.0612
13 0.02 0.15 0.16 0.03 0.16 0.23 0.14 0.04 0.12 0.11 0.0 A0013
14 0.02 0.15 0.16 0.03 0.17 0*231 2,13 0.05
0.05 0.12 poll 0.01 0.0614
15 0.03 0.16 0.16 0.03 0.17 0'2 0.17 0.75 0.1 0.01 0.0615
16 0.04 0.16 0.10 0.02 0.10 0.23 0.12 0.00 0.12 0.10 0.02 0.0016
17 0.04 0.16 0.19 0.02 0.18 0.23 0.11 0.06 0.12 0.097 0.02 0.0617
0.03 0'17 0.15 0.01 0.19 0.23 0.11 0.071 0.12 0.09 0.03 0.0018
0.05 6.17 0.151001 C190.231 Oslo 0.07 0.12 0.0 | 0.03
0.02 0.0019
2C 0.06 0.17 0.15 5.00 0.19 0,23 0.04 0.08
21 0.001 0.18 0.14 0.01 0.20 0.23 0.09 A08 0.12 0.07 0.04 1.0521
221 0.071 0.18 0.14 0.02 0.20 0.23 0.08 0.09 0.12 0.07 0.05 0.0522
23] 0.071 0.18 0.14 0.02 0.20 0·22 008 0.10
0,12 0.071 0.05 0.0523
24 0.07 0.18 0.13 0.03 0.21 0.212.07 0.1
0.12 0.00 0.05 0.05124
25 0.08 0.18 0.13 0.03 0.211..22 0.37 0.11 0.121 0.060..6 0.0425
26 0.081.0.17.0.13 0.04 0.21 0.21 0.061 0.11
0.12 0.05 0.00 0.0426
271 0.08 0.17 0.17 0.04 0.22 0.21 0.06 0.17 0.12 0.05 0.06 0.0427
28 0.09 0.17 0.12 0.05 3.22 0.21 0.05 0,11 0.12 0.04 0,07 0.04/28
290.090.17 0.12 0.06 0.22 0.24 0.05 0.12 0.12 0.04 0.07 0.04/29
301 0.09
0.11 0.06 0.23 0.20 0.04 0.12 0.13 0030.07 0.0330
3 ilgio
0.111 0.07 0.231 2,041 0.12
0.02 0.03131
0.12
8 0.01
0912
The
of the Equation of Time
A
I
Sol.
The Uſe of this Table is ſo very eafy, that it needs little Explana-
tion ; for by entering the Table with the given Time, and finding
the Month in the Top, and the Day in the Right or Left-hand Co-
lumn, in the common Area, you will have the Variation in 100
Years; and by allowing proportionally for any leſler Time, you will
have the true Variation, which being added to, or ſubſtracted from
the Tabular Equation, according to its reſpective Title, will give
the true and abſolute Équation of Time, to the Time propoſed.
Thus for Example, Suppoſe it were required to find the abſolute
Equation of Time for the 7th of May 1770, this being the 2d Year
after Leap-Tear, and 44 Years after the Year 1726, 'I find the pro-
portional Alteration to be s : Seconds to be ſubſtracted, this theret
fore taken from 4 m. 02 f. the Equation anſwering to the 7sh of
May 1726, being the Second Year after the Leap-Tear, will leave 3 m.
for the true and abſolute Equation of Time for the 7th of
May 1770; and after the lame manner may the true and abſolute E-
quation of Time for any other Day be readily found.
For the ready obtaining of the true and abfolute Equation of Time
from the Sun's Place only, and which in many Caſes is very uſeful,
I have computed the following Table, which ſhews the Équation
correſponding to every Degree of the Ecliptic, which is conſtructed
after the ſame manner as the former Tables are, and built upon the
ſame Principles, and may be readily compounded of the two Tables
in Page the 446th without much Difficulty: And tho' this Table
is fubje&t-to vary in proceſs of Time, upon Account of the Motion
of the Sun's Apogæon, which changes its Place and goes Eaſterly,
with the Fixed Stars occaſioned by the Præceffion of the Equi
noxes, yet inaſmuch as the ſame Degree of Anomaly keeps near
ly in the fame Degree of the Ecliptic for upwards of Fifty Years,
the Difference will be very ſmall; but how much it is; and which
way it muſt be applied, may be eaſily gathered from what has been
already faid.
.
..
Nnt
*
456
Tables of the Equation of Time.
A TAB L E hewing the Equation of Time, anſwering roe-
very Sign and Degree of the Ecliptic for the Year 1726 ; to
be applied to the Apparent Time, according to the ſeveral
Titles.
A
go
V
1
1x
mio
Si.
A i A
Degrees lo
W.15
m
S
A
А
S
S
A
A
Degrees.
1
Ý
)
1
11
1
1.10
2
.
C.1?
1
T
7.4
3:55.0 5.51 2.15 7.4 15:3;|| 3.3 1.031 617.c.
7.21 1.231 3.5 1.23 5.52 153 8.04.15.43.14 0.34119314,20
7.09 1.36 3.471 1,37 5,53 1.12 8.251517412.521 COS 12,0954.13
36.431 1.49 3.40 1.501 5,53 1.25 2.46|15.5 112.3,
A24 12'25111.05 3
41 6.24 2.01 3.37 2.03 5.531 1.089.00115.57 112:21 0.541 204013.57) 4
56,05 2.12 3.31 2.16 5:53 0.51 9.26|15.0 12.01 1.2 12.54113.495
6 5.40 2.22 3.24 2.291 5.52 0.381 9.40|10.1111.40 1.5 3:07 13.11
5.27) 2.3 2.17 2.42 5.50 6.14 10.04|150|11.19 2.22 13.21 13.3417
5.08 2.42 3.45 2.54 548 5.05 10.24 16.15 10.57 2.5" 13.33 13.20 &
4.49 2.54 3.01 306 5:45 0-24 10.14 13.1 10.35 3.25 13.4 13.7919
14.21 :
2.5 3.18 5.411 0.41 11.72 16,1
3.18!3.53.12.9716
4.12 3.11 2.43 3.39 9.371 1.04.11.2016.11 9.481 4.1 4.02112.4511
3,54 3.19 23:3-421 5:32 1.24*1*3*16.1 6.24 4.4
4.10 12.33/14
1: 335 3.26 2 23 3:53 5.26 1.44111.50|16.05 8.59 5.11 14.17.12.2013
14 3.10 3.33) 2.19 4.041.5.19 2.04 12.13 16.04 8.34 53+14.24 12.06.14
12.58 3.9 2.01 4,141 5.02.2412.2y 6,09 808 tuus 14.3 11.5215
2.40 3.75 1.51 4.24 5.03 2.44.12.415.57:42 6.3 1435 11.3710
172.22 3.54 1.467331 4.55 3.0513.) 15.577.166.50|14.40(11.2217
12.04) 3.54 1.25 -42 4.40 3.20 13:1.15.176.497.2014.44 110718
1.40 3:57) 1.17.4.511 4.37 3:47:3:3 5.40 6.22 7,461 4.46 10'5219
1.25 4.00 1.0 4.59 4.27.4.29.3.4 15:32 5.548.10 14.4810:3620
(1|1.1 4.02 0.51 5.071.4.164.31 13.5:<.23 5.20 8,35|14.49|10.2021
20.54 4.04) 0.31 5:14 4.051 4.52114.1 1750' 44.58 8.50|14.49 10.03332
23 0.37 4.05 0.25 5.21 3.531 5.14 14.2. 15.04 4.30 9.18|14.49 9.4623
-4 0.211 4851 0.12 5.27 3.40 5.504.30 14.531 401 0:
44.489.28 24
240.5 4:05 A01 5132 3:27 50581447 14.41 3 3210.? 4.40.9.1125
20 S.11 4.05 0.1415.37 3.14 6.19 14.5 14.281 3:02 10,22 14:43 8.5326
27 0,26 4.04 0.27 5.42 3.00 6.40 15.0,1.4.162.3-110.4 14.40 8:3527
2810.41 4.02 0.411 5.40 2.451 7.01|15.10 13.02 2.0211.0 114.36 8.17/28
293.50 3.59 0.55 5.49 2.30 7.22152 3:47 1.3211,1|14.311 7059/29
1.101 3,551 1.091s. || 2151 7.4;115:3511 3.311 1.0311.30 14.26 7.40130
ein
Hence
of the inequality of the Solar Days. 457
Hence it is abundantly manifeſt, that the Solar or Apparent Days
or Time that elapſes from the Moment the Sun leaves the Meridi.
an of any Place, till it returns again to the ſame Meridian, and con-
ſequently their ſimilar Parts, such as Hours, Minute:, 6c are un-
equal; and cnfequently the Times meaſured out by the Motion of
the Sun, are unequal among themlelves, and different from chat
which is called Abfolute Time, which Flows equally and unitormly,
anu al ways at the ſame Rate; which by the Aſtronomers is called
the Irve and Equal Time ; according to which all the Cæleſtial
Morioos are regulated and ſettled, and to which the Tables of the
incan Motion of the Sun and all the Planets are adapted. And in-
ali ch as the mean Motion of the Sun and the Planets are found
to be equal and uniform, contantly deſcribing equal Areas, by a
Linc conn.cting their Centers and the Center of their Orbs in equal
Times: Hence it is that Aſtronomers have pitched upon a mean Solar
Day, butween the greateſt and leaſt, as a Standard to compare
their mein Morion with, and to which the Cæleſtial Mocions are.
a djuſted; ſo that by calculating the Time when any norable Appear.
ance will happen, ſuch as the Beginning or End of a Solar or Lunar
Eclipſe, or of any of Jupiters Satellites, &c. by theſe Numbers and;
reducing that Time to che Solar Time, we have the Apparent Time
when thcle Appearances become viſible ; and on the contrary, by
obferving the Apparent Time when any of theſe Appearances hap-
pen, and reducing it to the mean Time, we are then enabled io
calculate by the Tables, when ſuch Appearance will happen ;. and
by comparing obſerved Appearances with thoſe deduced from Cal-
culation, we are able to pronounce the Errors of the Tables, and:
correct them as Occaſion requires: and this is the conſtant Buſineſs
and Practice of all Aſtronomers, and the Method made Ure of to
re&ify the Motions of the heavenly Bodies. And hence it appears
of what Conſequence the Knowledge of the Equation of Days is in
Aſtronomy, it being one of the Chief Pillars upon which all its Pre-
diâions are founded ; and this was perfe&tly unknown to the Anci-
ent Aſtronomers, and one principal Cauſe of the Errors in their Num-
bers, and never rightly adjuſted, till undertaken by our great Aſtro-
nomer the late Mr. F LAM S T E AD, about the Year 1067, (as it
is Publiſhed by Dr. Wallis at the End of Horrox's Remains, in the
Year 167%) whoſe principal End in all his Studies, was to improve
real uſeful Knowledge, by clearing up thoſe Points that were
before
4.
458
of the Equation of Time
before Obſcure or Imperfe&, and laying a lafting Foundation for
the Improvement of true Science, and to whom fucceeding
Generations will be greatly indebted.
The Tables here delivered have been all calculated de novo by my
Self, and not copied from Books, without enquiring whether they
are right or wrong, exa&ly computed or not, or whether they agree
or diſagree one with another; a thing very much pra&iſed of late
Days, and to be found in moſt of the Tables now in Uſe.
ETIZEAZAAZAAAA2228
Section XX.
Containing the Theory of the Sun; and the Conſtruction
and Uſe of the Solar Tables.
Every one who is but meanly skill'd in the Science of Aſtrono-
my, knows that it conſiſts of two General Parts; the Firſt and
Principal of which is employed in determining the Species of the
Orbs in which the Planets move, their Magnitudes, Periods and Diſtan-
ees from each other, whence proper Tables are made, by the Help of
which their respective Places in their Orbs, their Poſitions, Con-
jun&ions, various Afpects, c. may be readily pronounced at any
given Time and accounted for.
The Second and no leſs uſeful Part, is principally concerned in
explaining the Origin Cauſes and Diviſions of the ſeveral Circles of
the Sphere, which are made Uſe of in accounting for the ſeveral
Appearances in the Heavens, ariGng from the Diurnal Motion of
the Sun, or the Rotation of the Earth about her Axis; and in the
Solution of the Chief and Primary Problems ariſing from it, ſuch as
to determine the Time of the Rifing and Setting of the Planets and
fixed Stars, their Amplitudes, Azimuths, and the Hours of the Day
or Night, from their obſerved Altitudes, G. and how this may
be done, from the Sun's Place being firſt known or given, has been
amply fhewn in the Fifth Part of this work, itfremains therefore new
ceffary, in order to render this part of the Science compleat, to thew
how
Of the Theory of the Sun.
459
ity
how the Sun's Placelit ſelf may be obtained at any given Time by
Calculation, and how the Tables themſelves by which it is generally
computed, are conſtructed or made.
And the firſt thing neceſſary to be known, is the exact Length of
the Solar Year, or the Time that the Sun requires to move thro' the
Ecliptic ; and this is readily obtained by comparing the Times to-
gether, when the Sun ſhall be found by Obſervation to poſſeſs the
ſame point of chc Ecliptic
How the Place of the Sun in the Heavens, or the point of the E-
cliptic that he poſſeſſes at any Time may be found from his Meridional
Altitude taken with a good Quadrant,whence and from the Latitude of
the Place of Obſervation, his Declination, Right Aſcenſion and Place,
may be readily deduced, or from his Dittance meaſured from ſome Pla-
net, ſuch as Venus in the Day, and the diſtance of that Planer from
ſome fixed Star obſerved the following Night, or from the Difference
of Times between the Tranfits of the Sun, and ſome fixed Stars -
ver the Meridian, meaſured by a well regulated Clock; whence the
Right Aſcenſion of the Sun, and thence his Declination and Place
in the Ecliptic may be found, has been explained at large, in the
3d and 4th Aſtronomical Seations ; and by comparing theſe Places of
the Sun thus deduced from Obſervations, and their correſpondent
Times together, may the Length of the Solar Year be determined.
But the Meihods uſually made Uſe or by Aſtronomers, is by find
ing and comparing the true Times of the Sun's Ingreſs into either of
the Equino&ial Points.
For in aſmuch as the Ecliptic cuts the Equator in two oppoſite
points, the Sun will twice every Year appear in the Equator or Equi-
noctial Circle; and this will happen when by his apparent Motion
he arrives at the Interfection of theſe two Circles, and the exact
Time of this Appearance may be thus determined.
At that Moment of Time the Sun appears in the Equator, he
then enters into the firſt point of Aries or Libra, and his diſtance
from either Pole, is exactly equal to a Quadrant or 90 Degrees : So
that if at that time he happens to be upon the Meridian of any Place
of his. Diſtance from the Zenith ot that Place, at that time will be
equal to the Latitude of the Place of Obſervation; and conſequently
if at any time the Meridian Altitude of the Sun be found by Obler-
varion, to be equal to the Complement of the Latitude of the Place,
the Sun ar that Moment, or at the Noon of that Day, enters into
one of the Equinoctial points.
Buc
460
of the Theory of the Sun.
But becauſe the Sun is continually moving forward in the Eclip-
tic Eaſtward, and at the Rate of one Degree nearly a Day, as will
appear hereafter, he makes no ſtay in the Equino&ial Point, but in
moving forward, at the ſame Moment he arrives in either Equino&ial
point, at that ſame Moment he begins to quit it; and tho the Day
the Sun enters the Equino&ial point be called the Equino&ial Day,
becaule the Days and Nights are nearly equal, yet unleſs the Sun
enters the Equinoctial Point at the Noon of that Day, the above-
mentioned Method will not take place, and therefore in this Cale
to determine the exact Moment of the Sun's Ingreſs, it is uſual to
take tlie Meridional Altitude of the Sun, either upon the Equino&i-
a! Day, or as near it as poſſible ; and the Difference between the
Attitude and the Complement of the Latitude of the Place of Ob-
fervation, will give the Declination of the Sun, or his Diſtance from
the Equator at that time. Again, the next Day, or as ſoon as poſ-
libe, obſerve again the Sun's Meridian Altitude, and find his Decli-
nation as before, if theſe two Declinations when compared toge-
ther, are of different kinds, that is one North and the otber South,
the Equinox has happened fome time between the two Obſervations;
but if they be found to be both of the ſame Kind, that is both North
or both South, and the Declination at the time of the latter Obſerva-
rion be leſs than the former Declination, the Sun has not entered
into, or is ſhort of -he Equino&ial Point; but if the Declinations
are found to increaſe, that is, if the latter be greater than the former,
the Sun is paſs'd it, and in order o find the exact Time of the
Ingreſs,
Having from chefe Declinations thus deduced from Obſervations,
and the greateſt Declination, found the correſpondent Longitudes,
or Diſtances of the Sun from the next Equinoctial point, according
to che Method caught in the ad Caſe of the ift Problem, of Section
the 3d of Part the 5th,
Say, as the Sum of theſe two Longitudes, or Diſtance of the
Sun from the Equino&ial Point, if the Declinations are of diffe-
rent Kinds, otherwiſe fay, as the Difference between the Longi-
cudes thus deduced, is to the Space of Time elapſed between the
two Obſervations, ſo is the Diſtance of the Sun from the Equino&ial
point at the Time of the firſt Obſervation, to a proportional part
of Tine, which it the two Declinations are o a contrary Kind, that
is one North and the other South ; or when they are of the ſame
Kind, that is both North or both South, and the Declination at
the
aleria
Of the 'Theory of the Sun.
461
time of the Second Obſervation be leſs than the Declination at the
time of the Firſt Obſervation, being added to the time when the
Firſt Obfervation was made, will give the exact time when the E-.
quinox happened : But it the Declinations are both the ſame way,
ápd the Declination at the time of the Second Obſervation, be
greater than the Declination at the time of the Firſt Obſervation,
that is, if the Declinations are found to increaſe, the 4th propor-
cionable Number laſt found, or proportionable part of Time, being
ſubſtracted from the time of the Firſt Obſervation, will leave the
time of the Ingreſs: The Realon of which is very obvious to the
intelligent and skillful Reader.
By the fame Method of reaſoning, may the true time of the Equi-
nox be found, from the Difference of the Right Aſcendion of the Sun,
obtained from the Difference between the times of the Tranſits of
the Sun and ſome known Fixed Star over the Meridian of any Place.
And, after the ſame manner, by the help of proper Obſerva-
ation, may the time of the Sun's Ingreſs into the ſame Equino&ial
Point be determined for the next, or any ſucceeding Year ; and
by comparing theſe times together, may the Length of the Solar
Year (the principal Foundation upon which the Solar Tables are
built) be determined.
As an Error of one Minute in the time of the Equinox, when
compared with an Obſervation made at the diſtance of one Year,
will cauſe an Error of 60 Min.or one Hour, in producing the time of
the Equinox at the diſtance of 60 Years, ſo the fame Error of one
Minute in the Equinox, when compared with an Obſervation made
at the diſtance of 60 Years, will cauſe but an Error of Second
in one Period; and this is the Reaſon why Aſtronomers when they
are about to ſettle the Periods of the Sun, or of any of the Planets,
or fixed Stars, and account for the Inequality in their Motions,
compare the moſt diftant Obſervations that they can meet with,
ſince a ſmall Error in the Obſervation in that Caſe, will be broke
into fo many ſmall Parts, that the Difference from the Truth, will
in a manner vaniſh.
According to the Greenwich Obſervations in the Year 1681, the
Vernal Equinox happened on the 9th of March, at vih. 41m. 13 f.
mean Time, when the mean Motion of the Sun from the faine E-
quinox" was 11 S. 28 deg. 04 min. 56 ſec. and in the Year 1701,
the ſame Equinox was celebrated on the 9th of March, at 8 h: 2 m.
22 f. mean. Time, when the Sun's mean Motion was II S. 28 deg.
Ooo *
05 min.
!
462
Of the Equation of the Sun
9
os min. 50 lec. ſo that in 30 Julian Years or 7305 Days, wanting
3 h 38 min. 51 ſec. or in 7304 Days 20 h. 21 min 9 ſec. the Sun's
Motion was 20 Revolutions over and above o S. oo deg, oo min.
fec. whence by the common Rule of Proportion it will be found,
that if 20 Rev.00 S. oo deg, oo min. 09 ſec. are performed in 7304
Days 20 h: 21 min. 9 ſec, that one Revolution will be performed in
365 Days 5 h. 48 min. 57 ſec. c. which is the true Length of
the Solar Year, or ſpace of Time that the Sun conſumes in finiſhing
his Courſe thro' the Ecliptic ; that is, till he return to the ſelf fame
Place in the Ecliptic from whence he began to move; ſo that ac-
cording to the former Computation, if the Sun enters into the firſt
point of Aries on the 9th of March, in the Year 1701, at 8h. 02m.
22 f, in the Year 1702, according to mean Motion he will enter a-
gain into the ſame point on the 9th of March, at 13 h. 51 m. 19 f:
This Period of 365 Days 5 h. 48 min. 57 ſec. is likewiſe called
the Tropical Year, becauſe during this Space of Time all the vari-
ety of Seaſons are celebrated ; and after it is finiſhed they return a-
gain in the ſelf ſame Order, and all the Conſequences that ariſe
from the Acceſs or Receſs of the Sun, to or from the Earth, will
happen during that time ; and after it is paſt, they will return a-
Astronomers by comparing the Places of the Fixed Stars deduced
from Obſervations made at diſtant times, have diſcovered that
there is a Præceſſion of the Equinoxes, or Receſſion or Motion of
the Equinoctial points in Antecedentia, or to the Weſtward, and
contrary to the Order of the Signs, by which the points of Inter-
ſedion do conſtantly move back, at the Rate of a Degree in 72 Years,
and do as it were meet the Sun: So that the Sun will annually ar-
rive at the Interſection before he has compleated his Courſe, and ar-
rived ac the ſame abſolute point of Space, or to the ſame Fixed Star
that he was in Conjun&ion with before ; and conſequently in the
Space of 20 Years this retrograde Motion will amount to 16 min.
40
fec. whence it comes to paſs that in the Space of 20 Julian Years,
or 7305 Days wanting 3 h 38 min 51 ſec there will be but 19
Rev. 11 S. 29 deg 43 min. 29 ſec. of the Sun to the ſame Fixed
Stars, and conſequently each ſingle Revolution will be performed in
thie Space of 365 Days 6 h. 9 m. 14 ſec. and which is called the A-
nonaliſtical Periodical or Sydereal Tear ; inaſmuch as in that Space of
Time the Earch makes an Inter-Revolution round her Orb, and the
Sun exhibits all his various Aſpects to each Fixed Star
Buc
gain a new
:
of the Theory of the Sun: 463
But inaſmuch as the Sun performs his Revolution with regard to
the Ecliptic in 365 Days sh. 48 min. 57 ſec. he will be found to
move over 59 min. 08 tec. 19 thirds of the ſame Great Circle in
one Natural Day; for by the common Rule of Proportion, if in
365 Days 5 h. 48 m. 57 f. the Sun moves 12 Signs or 360 Degrees,
the whole Circumterence of the Ecliptic, in 24 Hours he will move
59 min. 08 fec. 19 thirds, or 59 min. 08 fec. nearly ; and this is
called the Mean Motion of the Sun for one Day, and in the fol-
lowing Table is the Number placed aginſt the 1ſt of January ; and
by the ſame Method of Reaſoning he will be found to move over
i deg. 58 inin. 16 ſec. 38 thirds, or i deg. 58 min. 17 ſec. nearly,
which is called the Mean Motion of the Sun for two Days, and
ſtands againſt the 2d Day of January, in three Days he will move
over 2 deg 57 miri. 24 ſec. 57 thirds, or 2 deg. 57 min. 25 fec.
which is called his Mean Motion for three Days, and is the ameNum-
ber that is placed againſt the 3d of January, &c. and in 365 Days
or one common Year current, he will have moved over 11 S. 29deg.
45 min. 40 ſec. and which is called the Mean Motion for one Year
current, and ſtands againſt the 31ſt of December, and againſt Num-
bers in the Table of Years current.
By the ſameMethod of Inveſtigation he will be found to move over
the 24th part of 52 min 8 ſec. 19 thirds, or 9 min. 27 ſec. 5o tirds,
or 2 min. 28 fec. nearly in 1 Hour, which is called the Mean Motion
of the Sun for one Hour, and which is the Number placed againſt
one Hour in the Table of Mean Motion for Hours, &c. in two
Hours he will move over 4 min. 55 ſec. 40 thirds, or 4 min. soſec.
nearly, and which is the Mean Motion for two Hours, and ſtands
againſt two Hours in the Table of Mean Morion for two Hours, &c.
and in one Minute Or the góth part of an Hour, he will move over
the 6th part of oz' 27" so'" or 2'27''' 50+, or. 2" 28'"' nearly;
and which is the Mean Motion for one Minute of Time, &c. and
in one Second of Time, or the both part of a Minute, he will move
over the both part of 2" 27"" 50*, or 2 274 503 which is the
Mean Motion of the Sun for one Second of Time
The Mean Motion for one common Year or 365 Days, has been
found to be 11 S. 29 deg. 45 min. 40 ſec. this therefore being
doubled, will give one Rev. and i S. 29 deg. 31 min. 20 fec.
over, which Surplus is called the Mean Motion for two Years, and
which ſtands againſt two Years in the Table, and being trebled, will
give a Rev. and i S. 29 deg. i7 min. oo fec. over, which Surplus
O 002
is
30
464
Of the Theory of the Sun
29
is called the Mean Motion for 3 Years.; and becauſe every fourth
Year conſiſts of 366 Days, therefore to determine the Mean Motion
for 4 Years, you muſt to the Mean Motion for 3 Years, viz. 11 S.
29 deg. oo min. add the Motion for one Year, equal to 11 S. 29
deg. 45 min. 40 ſec. and over and above the Morion for one Day,
equal to 59 min. 08 ſec. &c. and the Sum o S. o deg. : min. 49
fec. will be the Mean Motion for four Years; and is the Number
that ſtands againſt four years in the Table of Mean Motion for the
intermediate Years; and by adding the Motion of one Year 11 S. 29
deg. 45 min. 40 ſec. to the Mean Motion for four years equal to
oo S. oo deg. ol min. 49 fec. you will have u S. 29 deg. 47min.
fec. the Mean Motion for five Years, &c. and by doubling,
trebling, and quadrupling &c. the Mean Motion for four Years,
you will have oo Soo deg o3 min. 37 ſec. the Mean Motion for.
8. Years, oo S. oo deg. 05 min. 26 ſec. the Mean Motion for in
Years, oo S. oo deg 07 min 15 lec: the Mean Morion for 16 Years
OC. and oo S. oo deg og min 4 ſec. the Mean Motion for 20
Years ; this again being doubled, trebled, &c. will give oo S. 00
deg. 18 min. 8 ſec. &c. the Mean Motion for 40 Years, os. o deg.
min. 12 ſec. the Mean Motion for 60 Years, and oo S. oo deg 27
mjo 20 ſec the Mean Motion for o0 Years; this again being
doubled, trebled, Oc will give oo S. or deg. 30 min. 40 ſec. the
Mean Motion for 200 Years, oo S. 02 deg 16 min 0 ſec. the
Mean Motion for 3.00 Years, Gc and oo S. 07 deg. 33 min. 20
fec. the Mean Motion for 1000 Years ; this again being doubled,
trebled, c. will give 00 $. 15. deg. 6 min. 40 ſec the Mean Motion
for 2000 Years, oo S. 22 deg. 40 min. oo ſec. the Mean Norion
for 3000. Years, c. and on S. 07 deg. 46 min 40 ſec. the Mean
Motion for 5000 Years, oc.
And inaſmuch as the common Julian-Tear is leſs than the Tropical
Year by 5 h. 48 min. 57 ſec. hence it is, that every 4th Year 1 Day
is added to the Year by reckoning the 6th Day of the Kalends of
March twice, and inſerting a Day into the Kalendar next after the
241h Day of February, making thereby two 24ths of February, and.
the month to conſiſt of 29 Days (of which a more ample Account
thall be given in its due Place) and becauſe in the Tables of Mean
Motions for Months, &c. the Month of February conſiſts but of 28
Days; hence it is, that in computing the Place of the Sun from theſe
Tables in the Leap-Tears, after the 28th of February, the Aftronomers
add the Motion of one Day to the Sum of the commion Mean Mo-
tion
..
Of the Theory of the Sun. . 465
tion, or which comes to the ſame thing, they add one Day to the
time given, and account the if Day of March for the agth of Febru-
ary, and make Uſe of the Mean Motion that ſtands in the Tables of
Mean Motion, againſt the 'A. Day of March, for the 29tb or Febru-
ary current
In like manner they account the ad Day of Marab for the 'A Day
of March, and make uſe of the Mean Motion that Aands againſt the
ad Day of March in the Tables of Mean Motion for the Day of
March current ; and proceeding after the lame manner; till when-they
come to the laſt Day in the Year, or the 31ſt Day of December they
add to the Tabular Mean Morion againſt the 31ſt of December, the
Mean Motion that ſtands againſt the firſt Day of January, or the
Mean Motion for one.common Day, and this Sum will be the pro-
per Mean Motion for thar Day; and which being added to the Radix
of Mean Motion for the current Year, it will give the Radix of
Mean Motion for the ſucceeding Year; after which all things go on
in the ſame Order, and the Mean Morions are made Uſe of in the
ſame manner as before
The Mean Morion of the Sun from the firſt point of Aries for the
arch, 1701, at 08 h. 2 min. 22 ſec. Mean Time has been
found to be ' 1 S. 28 deg. 5 min. 5 ſec from this therefore taking
away the middle Motion of the San for 68 Days, 8 h. 02. min. 22
ſec the Quantity of Time between the 9th of March at 08 h.
• 2 min 22 ſec. and the 3 1$7 Day of the preceding December equal
to 02 S. 07 deg. 21 min. 15 fec. the Remainder 09 S.o deg, 43
min. 56 ſec. will be the Mean Motion of the Sun from the firſt point
of Aries for the 31ſt Day of December, 1700; and is called the Radix
of Mean Motion for the Year 1701, and is the Number that ſtands
againſt the Year 1701, in the firſt Table, or Table of the Earths
Mean Motion from the Vernal Equinox, for the Years ſince CHRIST:
And if to this Mean Motion 9 S 20 deg. 43 min. so ſec. be added
US 29 deg. 45 min. 40 ſec, the Mean Motion for one common
Year, the Sum 9.S. 20 deg. 29 min. 30 ſec.' will be the Mean Mo-
tion of the Sun from the firſt point of Aries, for the 31ſt Day of De-
cember 702, or the Radix of Mean Morion for the Year 1702 : In
like manner, if to the Radix of Mean Motion for the Year 1701,
viz. 9 S. 20 deg. 43 min. so ſec. be added oo Soo deg. 9 min.
4 ſec the Mean Motion for 20 common Years, the Sum 99. 20 deg.
$2 min 54 ſec. will be the Mean Motion of the Sun for the 311 of
December 1720, at Noon, Mean Time; or the Radix of Mean Mo-
tion
9th of
466
of the Theory of the Sun.
ز
tion for the Year 1721. And again, if to the Radix of Mean Mo.
tion for the Year 1721, viz. o S 20 deg. 52 min. 54 fec. be added
the ſame Motion for 20 Years, viz oo S.oo deg. 09 min. 04 fec.
the Sum 9 S. 2 I deg. ol ini, 58 ſec. will be the Mean Motion of
the Sun for the 3 1/1 of Decerrber, 1740; or tlie Radix of Nean vo-
tion for the Year 1741, and is the Number that ſtands againſt 1741
in the firſt Table; and if to the Radix:9 S. 20 deg. 43 min. so ſec.
the Radix of Mean i otion for the Year 1701, be added oo S. 00
deg. 45 min. 20 ſec the Mean Motion for a Hundred Years, the
Sun 9S. 21 deg: 29min. Is ſec. will be the Mean Motion of the Sun
from the Vernal Equinox for the 31ſt Day of December at Noon, in the
Year 1800, or the Radix of Mean Motion for the Year 1801, d.
Again, if from 9 S. 20 deg. 43 min. so ſec. the Radix ot Mean
Morion for the Year 1701, be taken away oo S. oo deg. 09 min.
04
fec. the middle Motion for 20 Years, the Remainder 9 S. co deg.
34 min. 46 fcc will be the Radix of Mean Motion for the Year
1681 j
and if from the Radix of Mean Motion for the Year 1681.
9 S. 20 deg. 34 min. 40 ſec. be taken away o S.. o deg: 9 min. 4 fec.
the Motion for 20 Years, the Remainder 9 S. 20 deg 25 min. 42 ſec.
will be the Radix of Mean Morion for the Year 1661, 6c . And
laitly, if from the Radix of Mean Motion for the Year 1701, 9 S
20 deg 43 min. 5o fec. be taken away oo S. 12 deg. so min. 46
ſec, the Mean Motion for 1700 Yearsthe Remainder 9 S. 07 deg.
53 min. 02 ſec. will be the Mean Motion of the Sun from the Vera ·
nal Equinox, on the 31ſt of December at Noon, immediately prece-
ding the commencement of the Christian Era, or the Radix of Mean
Motion for the firft current Year of CHRIST.
After the ſame manner may the Radix of Mean Motion for any 0-
ther Year be computed, and conſequently the Mean Motion of the Sun
from the Vernal Equinox for any Time paſt or to come, be readily
Calculated, by adding or ſubftra&ting to or from, the Radix of vean
Motion for the given Year, theSum of the Mean Motions for the Sur-
plus, or the Defeet of Time, as will be exemplified hereafter; and
after this manner were the following Tables of Mean Motion con-
Itructed.
Ard inaſmuch as all theſe Mean Motions are adjofted to Mean or
Equal Time, it is neceſſary if the Time given be Apparent Time,
that it be forft redaced to Mean Time, according to the Rules laid
down in the former Section.
1
IE
of the Theory of the Sun
1
407
If the Sun were placed exa&ly in the Center of the Earth's Orb,
and the Motion of the Sun round the Earth were equal, that is if
the Earth deſcribed equal Angles round the Sun in equal Time, then
the Sun's Apparent Morion in the Ecliptic would be always e-
quable and equal at all Times to the Mean Motion before Inveſtiga
ted, and the Sun would move forward each Day at the Rate of 59'
8'' 19'', and proportionably for a a greater or Icffer Time, whence
the Place of the Sun being known at any certain Time fixed, his
Place at any other Time aſſigned, would be very readily computed,
by adding or ſubftrading of the Mean Morion anſwering to th: In-
terval between the two given Times.
But Aſtronomers by comparing together the Places of the Sun dedu-
ced from Obſervation, have found that the Apparent Motion of the
Sun thro’the Ecliptic is unequal, and that he moves ſwifter thro'fonc
parts of the Ecliptic than he does thro’ other; that his Apparent
diurnal Motion is ſometimes 61 min. nearly, and at other times
ſcarce 57m. and that he is longer by 7 Days nearly, in travelling from
the firſt point of Aries thro' the Norchern half of the Ecliptic, till
he arrives at the firſt point of Libra, than he is in travelling from the
firſt point of Libra thro’the Southern half of ir, till he enters again
into the firſt point of Aries ; for in the Year 1701, the Vernal Equi-
nox happened on the gtń of March, at 8 h 2 min. 32 ſec. P.M when
the Autumnal following did not happen till 11th of September follow-
ing at zoh. 04 m. 271. P.M. Mean Time, and wlrich ought to have
happened on the 8th Day of the ſame.Month September, at 10 h. 58
min. 55 ſec. P.M -lean Time, preciſely, if the Apparcnt votion of
the Sun were exaatly equal ; morever it is found by meafuring the
Apparent Diameter of the Sun by a good Micrometer, that when he is
near the 8th Degree of Cancer, and which happens about the 18th
of June, at which time he is ſaid to be in Apoge, his Appa-
rene Seinidiameter is about 15' 0", and that when he is near the
8th Degree of Capricorn, and which happens about the 18th Day of
December, and when he is ſaid to be in Perige, his Apparent, Semi-
diainerer is somnin, 23 ſec (which Apparent Semidiameters muſt have
been equal, if the sun had been placed in the Center of the annual
Orb) whence it follows, (inaſmuch as the Apparent Semidiameters
are reciprocally as the reſpective Diſtances) that the greateſt Distance
of the Sun from the Earth, is to its leaſt Diſtance, as 983 to 950:
or as 101707 to 98253; whence the Diſtance of the Sun froin the
Center of the Orb will be 1707 fuch Parts as the Mean Diſtance of
the
408
Of the Theory of the Sun.
the Sun from the Earth is 100000, and which is therefore called the
Excentricity ; but Mr. Flamſtead by comparing ſeveral correct Places
of the Sun together, deduced from other Obſervations, has found
the Excentricity to be but 1692 of the ſame Parts.
The ancient Aſtronomers who allowed. no other Motion in the Hea-
vens but what were circular and equal, thar they might Account for
theſe Inequalities in the Sun's Motion, and adjuſt the ſeveral Quan-
ticies of ic in ſeveral Parts of the Orb, ſuppoſed the Sun co move
-round the Earth, or the Earth round the San (for iras the ſame thing
which we ſuppoſe to move, and which we'll fuppoſe to ſtand ſtill) in a
circular Orb, but Excentrical, that is, whoſe Center was at ſome
Diſtance from the Center of the Eclipric, in which they placed the
Sun or the Earth, and that this circular Orb was deſcribed by an e-
qual Motion; in ſuch manner, that a Line or Ray drawn from the
Center of the Orb, to the Sun or Earth, did deſcribe equal Angles
in equal Times ; and therefore by having the Excentricity or Ďir-
tance of the Sun from the Center of the Orb, and his can Diſtance
from the Earth, as alſo the Diſtance of the Sun from the Apogeon
or Perigeon, they could readily compute by the help of the Dočtrine
of Plain Trigonometry, the Equation of the Orb, or the Difference
between the mean and True Place, in whatſoever part of the Orb
the Earth was ſcituated, and thence by knowing the Mean votion of
the Sun from the firſt point of Aries they could compute the true
Place of the Sun in the Ecliptic, and his Diſtance from either Equi-
noctial point.
But the Great Kepler by comparing the Obſervations of the Fa-
mous Tycko together, has diſcovered that Mars was not carried round
the Sun in a Circular but in an Elliptic Orb, and that the Sun was
placed in one of the Foci of that Ellipſes; and that in inoving round
the Sun his Motion was ſo regulated, that a Ray or Line drawa
from the Sun to the Planet did deſcribe an Elliptic Area or Space,
always proportionable to the time that the Planet moved; and ha-
ving found the fame Law to obtain in all other Planets, he concluded
it to be but Rational, that the Earth ſhould obſerve the ſame law,
and be carried likewiſe in an Elliptic Orb; and this having been con-
firmed by all the Obſervations made fince his Time, there is no
room left to doubt of the Truth of it; and therefore I ſhall proceed
to thew in the next Place how theſe Innequalities in the Sun's Mo-
tion may be accounted, for, and how the proper Equation, or Prof-
thapherefis, for adjuſting of theſe Inequalities may be computed upon
this Principle only. And,
if IF
>
܀
469
Of the 'Theory of the Sun.
1. If the Earth be carried about the Sun in an Elliptic Orb, in
one of whoſe Foci the Sun is placed, and in moving deſcribes Areas
every where proportional to the times, by a Line connecting the
Centers of the Sun and Earth ; I ſay that the velocity of the Earth
in every point of the Orb ſhall be reciprocally, as a Line drawn from
the Center of the Sun, perpendicular to the Tangebt to the point
given.
Let A 1 EP repreſent the Earths
Orb, S the Sun placed in one of its
Foci, I and E two points in the
I
Orb, in which the Earth is ſuppo-
ſed to be at two different Times, I
ſay that the velocity in T is to the
velocity in E, as the Line Sq drawn
perpendicular to Eq, a Tangent to
the Orb in the point E, to the Line
SQ, drawn perpendicular to TQ,
a perpendicular to the Orb in the
point 7.
BA6
t
Let It and Ee be two Arches
of the Orb deſcribed by the Earth,
1 오
​i in equal, buc infinitely ſmall parts of
Time, and draw the Lines St,
Se, and becauſe the Areas deſcribed
are ever proportional to the Times the Triangles STt and see
inaſmuch as they are deſcribed in equal Times are equal, but the
Triangle STt is equal to half sexTt, and the Triangle SE is
equal to half SqxEe, wherefore, half sexTt=half SqxEe ; and con-
ſequently Tt: Eé : : iSq : SQ; or as Sq: Sl; and ſince
the Revolutions of the Earth in the points T and E are as the Lines
Tt and E e, deſcribed in the ſame time it follows that the velocity
of the Earth in the point T, is to the velocity in the Point E, as
Sq to se
Hence it follows, that the velocity is leaſt in the Aphelion, and
greateſt in the Perihelion, and that while the Earth is moving from
the Perihelion to the Aphelion, ſhe conſtantly Nackens her -Motion,
till having pafled the Aphelion, the on the contrary begins to quick-
en her Pace, till the arrives again at the Perihelion, when it again
becomes the greateſt, 66.
2 The
:
Рpp
**
470
Of the Theory of the Sun.
+
2. The fame things being ſuppoſed as before, I ſay that the An-
gle with the Earth ſeen from the Sun, will appear to deſcribe in a
Imall Particle of Time, will be every where reciprocally in a Dupli-
cate Proportion of her diſtance from che Sun.
About the Center S at the diſtance of St and Se, deſcribe the
finall Arches + B and eb, and make Sc equal to. Se, and draw the
Small Arch cd, and the angular velocity in e will be to the angular ve-
locity in t, as the ſmall Arch e b is to the ſmallArchc d, but the Pro-
portion of eb to cd is compounded of the Proportion of eb tot B, and
of + B tood, and becauſe the Triangles e S E and t.SZ are equal,
ob will be to B, as St to Se, and becauſe the Arches + B and cd
are ſimilar, t B is to cd as St to.Sc, or as St to Se, wherefore the
Proportion of eb tocd, is compounded of the Proportion of St to Se,
and of St to Se, and conſequently the angular velocity at e, is to the
angular velocity at t, as the Square of St to the Square of Se; that
is reciprocally as the Square of the Diſtances.
This increaſe, and decreaſe
of the angular velocity, may
perhaps be better underſtood
by comparing the true Motion
of the Earth in her Orb, with
the Motion of a Body ſuppoſed
to move, with an equable and
uniform Motion round the Sun
as a Center
T
A
D
CA
ma
ON
Z
In the adjacent Figure there-
fore, let AQPR repreſent the
Orbit of the Earth, in one of
whole Foci as S, the Sun is
placed, and let ÅP be the lon-
P
geft Diameter, and Q R the
E
ſhorteft, and SB equal to SF,
a mcan Proportional between
the two Diameters.
Abour Sasa Center, with the diſtance of SB, deſcribe the Circle
DBHR, this will be equal in Area to the Ellipſis AQPR (by the
1.5th and 16th Ar. of seats the ad of Part the ist, in Page 83) and
let us fuppofe a Body to move over the Circumference of this Cir-
dle, in the ſame time as the Center of the Earth deſcribes the Peri-
phery
12
:
471
of the Theory of the Sun.
phery of the Ellipſis, and when the Earth is in the Aphelion A;
jer the circulating Body be in E, the Line of the Abades, the
Motion of this Body will repreſent the equal or middle Motion of
the Earth, and the Body in its Morion round the Sun S, will de-
Icribe Areas or Seators of Circles, which are proportional to the
Times, and equal to the Elliptic Arcas that che Earth deſcribes ac
the ſame time.
Let Esm be the Angle or equal Motion that the Body moving
round the Sun at S has deſcribed, proportional to any given Time,
and take the Elliptic Area AST equal to the Secor Esm, and then
the true Place of the Earth in her Orb will be in the point T, and
the Angle mSD equal to the Difference between the mean Motion of
the Earth, and its true Motion will be the Equation or Proſtha-
phärelis, and conſequently the Area AEDT, and which muſt be e-
qual to the sector DSm, will be ever proportional to the Proſtha-
phærefis, and the ſame will happen in whatſoever part of the Orb
the point T be taken ; whence it follows,
1. That the greater this Area is, the greater will be the Proſta-
phæreſis or, Equation, or Difference between the mean and true
Place of the Earth, till when this Area comes to be the greateſt,
the Equation or Proſthaphæreſis will be the greateſt allo; and this
will happen in the point B, the common Interſection of the Ellipfis
and the Circle, for when the Earth deſcends farther too, the Equa-
tion becomes proportional to the Difference between the Areas
ABE and BI 0; or to the Area 10 BH.
For let G be the Place of the Body moving uniformly in the Cir-
cle when the Earth is at o, and the Secor ESG will be equal to the
Elliptic Area A So and taking away the common Spaces, there
will remain the Area ABE or HBP leſs the Area B 10, equal to.
the Area Lo PH, equal to the Sector GSH or the Equation.
2. That in the Perihelion and Aphelion, the middle and true Mo-
sion of the Earth are the ſame, and confequently the Equation or
Profthaphæreſis vaniſhes; for the Semicircle EBK is equal to the
Semi-Ellipſis AQP; but
3. That after the Earth departs from the Periphelion P, its Mo-
tion is conſtantly quickes, and it goes before the Body, moving e-
qually with the mean Motion, and conſequently the true Place will
be in conſequence of the Mean Place, and the Equation will be poo
Gtive as in the former Semicircle, in moving from the Aphelion
its Motion was conſtantly lower, and the Earth followed the
PPP 2
Ilower Body
472
Of the Theory of the Suni
diminiſhes, till it arrives
Body moving with an uniform Motion equally with the mean Mo.
tion, and conſequently the true place will be in Antecedence of the
mean Place, and conſequently the Equation or Proſthaphæreſis will be
Negative: For let the Angle KSZ be proportional to the Time,
take the Area PSx equal to the Sector KSZ, and x will be the Place
of the Earth in her Orb, in conſequence of the mean Place Z, for
the Angle PSx is greater than the Angle KSZ, and the Area KPxy
will be equal to the Sector 2Sy, and conſequently the Angle Z S y,
the Equation in this Semicircle will be poſitive, as in the former
Semicircle the Apgle GSI or the Equation was negative ; and
this Equation ZSy will be greateſt : likewife in the point F,
the common Interfe&tion of the Ellipfis and the Circle, for there the
Area KPxy will be the greateſt.
It has been already proved that in the point A the velocity of the
Earth is leaſt, and that in deſcending from thence to the Perihelion,
its velocity will conſtantly increaſe, but will be ſtill leſs than the
mean velocity till it arrives at B, when the Circle and Ellipfis cut
each other, and when it becomes juſt equal to the mean : For when
the Earth is in B, let the Body moving with the uniform equal Mo-
tion be in the point I, and the Areas deſcribed round the Sun in the
fame infinitely ſmall Time be the ſmall Triangles qSB and u ST, and
which muſt therefore be equal, and conſequently rBxBS is equal
to ulxTS ; and becauſe SI and SB are equal, rB and I will be e-
qual, and the Angle qSB will be equal to the Angle u SI, and con-
ſequently at the point B, the angular and mean velocity are
cual.
But as the Earth moves from B, and approaches nearer to the Pe.
rihelion, the true velocity grows bigger than the mean velocity, and
as it comes conſtantly nearer to the Sun its velocity will conſtantly
increaſe, till it comes to the Perihelion P, where it is greateſt of
all, becauſe its diſtance. SP from the Sun is the leaſt; but the Earth
departing from thence and alcending to the Aphelion, leaves the
Body that proceeds conltantly with a uniform Motion behind it,
but as it receeds farther from Ithe Sun its velocity decreaſes, but is
ſtill bigger than the mean velocity, till it comes to the point F the
common Interfe&ion of the Ellipſis and the Circle, when the angu-
lar velocity of thc Earth is juſt equal to the mean angular velocity,
and alţer it has paſſed that point its velocity becomes leſs than
when it is the leaſt of all, its Diſtance from the Sun at that time be-
ing the greateſt
Irie
Of the Theory of the Sun
473
par:
Inaſmuch therefore as the Earth in her annual Motion round the
Sun, is governed by the cqual and uniform deſcription of Areas,
which increaſe and decreaſe uniformly with the time, it is impoſſi-
ble that ſhe can every where move with the ſame uniform velocity,
but it muſt be conſtantly changed ; and that in every different
of her Orb the will acquire different Degrees of velocity, and
therefore to determine her true Place at any given Time, we muſt
find the Pofition of a Right Line as ST, which paſſing thro' one of
the Foci S, of the Ellipſis
AQPR, will cut off a Tri-
Jineal Asea AST, deſcribed
А.
by its Motion, to which
the whole Area of the El-
lipſis ſhall have the ſame
T
Proportion that the Peri-
F
odical Time of the Earth
has to any other given Time,
which Poſition being found,
we ſhall ha' e the Place ofB
Q
D
the Earth, at the given
R
point of Time.
t
ܪ:܀
This Problem was firſt
S
propoſed by Kepler, after
he had discovered the Law
of uniform Areas, and tho'
P.
he never was Maſter of a
direa way of reſolving it as
he himſelf expreſly tells us,
yet the Method that he made Uſe of in calculating the Profthapham
refis, and computing the Places of the Planets as it is very eaſy to
be underſtood, ſo it is very expeditious and certain in the Practice,
and when the Excentricity is ſmall, as is the preſent Cafe, it is not.
much infcrior to the direct Methods now in Úſe.
In the preceding Figure, Let ATQPR repreſent the Ellipſis thac
the Center of the Earth deſcribes in her annual Motion round the
Sun, S the Sun placed in one of the Foci, C the Center of the Orb,
and CS the Excentricity, Athe Aphelioror point in the Orb where
the Earth is at her greateſt Diſtance from the Sun, and P the Peri-
helion, or point where Ibre is at the neareſt diſtance, then will be
the
474
of the Theory of the Sun.
*
the greateſt Diſtance of the Earth from the Sun, and PS the leaſt
Diſtance, and CP equal to CA the mean Diſtance.
Let I repreſent the Place of the Earth in her Orb, at any given
point of Time, and draw the Line S7, this, therefore is called the
Generating and Defcribing Ray, which while the Center of the
Earth is carried from Athro T, , P, and R, to A again, will de-
Scribe che whole Elliptical Area ALPR and A.
On the Center at the diſtance of CA or CP, deſcribe the cir-
cumſcribing Circle ABPD, and thro the point I or the Place of the
Earth, draw the Line 1 F, perpendicular to AP, and produce it
till it ineet the Circumference of the Circle in the point t, and draw
the Lines : S, C, and TS, and at Right-angles to AP draw the
Line BD, then will QR be the forceſt or conjugare Diameter of
the Ellipſis.
It has bcen demonſtrated in the 15th and 18th Articles of Setion
the ad of Part the it, in Page 85, that any Segment of a Circle as
At F, is to the correſpondent Segment ATF of the Ellipſis, as the
circular Ordinate Ft is to the correſpondent elliptical Ordinate
I F; and in the 16th Prop. of Seation the ist, of Part the 1st in
Page 44, it has been proved that the Area of the Triangle + SF is
to the Area of the Triangle TSF, as the fame Line + F is to the Line
TF; wherefore by Cor. the sth of Prop. the 15th, of Se£tion the ift
of Part the ift, the circular Area At S will be to the elliptic Area
-47S, and conſequently the whole circular Area will be to the whole
elliptical Area as the Ordinate + F to the Ordinate TF; that is
by the 15th Article of Section the 2d, of Part the if in Page 85, as
the Semidiameter of the Circle BC, to the Semi-conjugate Diame-
ter QC of the Ellipſis, or as the Diameter of the Circle BD to the
conjugate Diameter Q R of the Ellipſis; that is becauſe AP and BD
are equal, as the longeſt Diameter PA of the Ellipſis, to the ſhor-
seſt QR; and the fame Law will obtain in whatſoever part of the
Periphery of the Ellipſis the point I be taken : Hence the A-
rea of any Trilineal Space in the Ellipfis, ſuch as A S.Z being
known, the Area of the correſpondent Trilineal Space in the Cir-
cle, ſuch as A Se is known allo; and on the contrary, the Area of
any Portion of a Circle being known, fuch as A St, the Area of the
correſpondent Portion of the Elliplis, ſuch as ASI are given :
And hence if we have a Method of drawing thro' the point b &
Line as St that ſhall cut off an Area of the Circle as A $t, which
All be to the Arca of the whole Circle in a given Proportion, it
will
#
Of the Theory of the Sun.
475
on
will be easy to cut off an Area of the Ellipſis as AST, which ſhall
have the ſame Proportion to the whole Ellipfis, viz. by letting fall
from the point t in the Circumference of the Circle, a Perpendicular
the Axis AP, which will cut the Ellipfis in the point I required,
to which draw the Line ST, and it will divide the Ellipſis in the.
given Proportion; ſo thar 'l will be the true Place of the Planet
or Earth, for if two Bodies 7 and. t ſet out from the point A ac
the ſame time, one of which t moved thro'che Circumference of the
Circle, while the other I deſcribed by its Motion the Periphery of
th: Ellipfis when the Body t arrives at in the Circumference of the
Circle the Body I will arrive at T in the Periphery of the Ellipſis,
and this will happen whereſoever the Line 1 Fbe drawn.
The Trilineal Space AST is called the mean Anomaly, and is c-
ver proportional to the mean Motion before inveſtigated ; ſo that if
the carrying Ray S7 deſcribe the whole Elliptic Area in 365 Days
5 h 48 min. 57 fec the fame Ray will deſcribe an Area as AST
in one Day or 24 Hours, equal to 59. min. 08 fec: 19 thirds, ſuch
Farts, as the whole. Elliptic Area is 360 deg. oo min. oo fec. oo
thirds, in two Days, it will deſcribe an Area of i deg. 58 min. 16
ſec: 38 thirds of the ſame parts; and ſo proportionally for a great-
er or leſler Space of Time.
And as the Trilineal Space AST is called the mean Anomaly;
the Angie AST which the ſame Ray deſcribes in the ſame Time is
called the co-equate Anomaly, and is ever equal to the viſible or
true Motion of the Sun in the Ecliptic, and the Dift:rence between
thele two Anomalys is called the Equation or Proſthaphæreſis, which
is Negative, while the Earth is paffing from A thro T to P, or thro
the firſt Semicircle of Anomaly, and therefore being ſubftracted from
the mean Motion of the Sun will give its truc Place; but during
the time the Earth is going from P to A, and which is called the ſe-
cond Semicircle of Anomaly, it is Poſitive, and
being therefore ad-
ded to the mean Motion, it will give the true Place.
Kepler, who knew no direct Method of finding the true or co-
equa'e Anomaly from the mean Anomaly firſt given, madeUſe of the
following Contrivance for calculating the Tables of Profthaphärelis.
1. Having affumed the Arch At which is called the Anomaly of
the Excentric, and which is equal to the Area of the Se&or A St,
in ſuch parts as the Area of the whole Circle is 360 Degrees, &c.
at pleaſure, to this he added the Area of the Triangle Cis, in the
famc parts, and which is found by multiplying the Baſe C S by half
of 1 F, and the Sum will give the whole Area or Superficies A St,
whick
Š
476
Of the Theory of the Sun
which is called the Competent mean Anomaly, which will be to
the true mean Anomaly or Superficies AS, as the longeſt Dia-
meter of the Ellipfis A P to the fhorteſt QR
2. To find the true or co-equate Anomaly or Angle AST corre-
ſponding to the mean Anomaly, or Area AST before found, in the
Triangle t SC are given, Ct the mean Diſtance of the Sun from the
Earth, CS the Excentricity, and the external Angle t CA the Ano-
maly of the Excentric at firſt affumed, whence by the fourth Cafe
of Oblique-angled Plane Iriangles, having found the Angle A St, it
will be, as the longeſt Diameter of the Ellipfis AP to the ſhorteſt
QR, ſo is the Tangent of the Angle AS t laſt found, to the Tangent
of the Angle AS1, the correſpondent co-equate or true Anomaly,
and the Difference between this and the mean Anomaly before found,
will be the Proſthaphæreſis, or Equation anſwering to the mean A-
nomaly before found; and having thus found the mean and co.
equate Anomaly, and the correſponding Equation to every Degree
of the Anomaly of the Excentric, he could readily find the Equations
or Proſthaphæreſis anſwering to every whole Degree of Anomaly. of
each Semicrcle, by the common Rule of Proportion, as will appear
by the following Examples.
It has been thewn that the Excentricity CS is 1692 ſuch Parts
as the mean Diſtance is 100000, whence the greateſt Diſtance of the
Sun from the Earth SA will be 101692, the leaſt Diſtance SP 98308
and the Semiconjugate Diameter 999857 of the ſame Parts.
In the 11th Article of Se&tion the ad, of Part the 1st in Page the
84th, it has been ſhown, that the Area of any Circle is to the Square
of the Radius, as the Circumference of the fame Circle is to its Di-
ameter : Hence the Radius CA of the Circle CABPD will be found
to be 10.70476 ſuch parts as the intire Area is 360; and the Ex-
centricity CS 1811244 of the ſame parts, theſe things being pre-
miſed.
Let it be required to find the mean and co-equal Anomaly, the
Anomaly of the Excentric being aſſumed equal to I deg oo min
oo fec. oo thirds, "And.
iſt It will be, as the Radius : to the Sine of the Anomaly of the
Excentric: : ſo is the mean Diſtance CA, in luch parts as the Area
of the Circle is 360 Degrees : to F in the ſame parts. That is,
.
As
.
Of the Theory of the Sun
477
-9.27:4318
As the Radius-
10,0000000
To the Sine of the Angle t CA 1.00.00
So is the Logarithm of CA 10.70476
8.2418553
1.0295705
To the Logarithm oft F :1868236
2d, Becauſe t Fx CS is equal to the Area of the Triangle + C S.
To thc Logarithin oft F.:868 36 -- --9.271438
Add the Logarithm of į CS ·0905622. 8.9569469
The Sum
8.2283787
Will be the Logarithm of .01691916, the Area of the Triangle
tCS, in ſuch Parts as the whole Area is 360 ; this therefore added to
the Area of the Sedor AC t equal to 1000.00, willgive 1.01691916
for the Area of the Superficies AS t. or Competent Anomaly.
3d, Becauſe the Area of the Triangle TFS is to the Area of the
Triangle + SF, as IF to + F; or as BC to QC, it will be,
As the Mean Diſtance CA 100000
5.0000000
4.9999378
To the Semi-Conjugare Diameter CQ.99985.7–
So is the Competent Anomaly ASt 1.01691916 -0.0072863
To the mean Anomaly AST 1.016773
-0.0073241
Equal to 1 deg i min. oo fec. , and to find the correſpondent co-
equate Anomaly, or Angle AST.
I. In the Triangle t CS are given, Ct the mean Diſtance equal
to 100000 and CS the Excentricity 1692, and the Angle tС A equal
to i deg, oo min. oo ſec. whence to find the Anglet SA, it will be
by the 4th Caſe of Oblique-angled Plane Triangles,
As citos: Ct-CS:: t,<CA:t, <s'tSA-S+C; that is,
becauſe Ct+t S=SA, and C1-CS=SP, As SA: SP::t, <CA
:t <stSA-StC; that is,
....
Q99
As
;
478
Of the Theory of the Sun.
As the greateſt Dift. of the Sun from the Earth 10 692 5.0072868
To the leaſt Diſtance 98308
-4 9925889
So is the T. of half the Anomaly of the Excentric 00.30 7.9408584
To the Tangent of the Angle + SA-SC 00:29.00 7.9261605
Which being therefore added to oo deg: 30 min. oo ſec. half the A-
nomaly of the Excentric, will give oo deg. 59 min. oo ſec. for
the Angle + SĂ; whence to find the Angle l'SA, or Cocquate A-
nomaly, it will be,
As CA: CQ: :1, SA: t,<TSA: that is, .
: <+:
As the mean Diſtance CA 100000 --
5.0000000
To the Semi-Conjugate Diameter CQ 99985.7 4.9999378
So is the Tangent of the Anglet SA 00.59.00 8.2346208
To the Tangent of the Angle TAS 00:58.591 8.2345586
Or true Anomaly, which being taken from the mean Anomaly
i deg. o! min. .00 ſec. į will leave oo deg. 02 min. or fec. nearly,
for the Equation or Proſthaphæreſis anſwering to deg: 1 min. 00
fec ; of mean Anomaly.
After the ſame manner, and by the like Method of Inveſtigation,
if the Anomaly of the Excentric or Arch At be aſſumed equal
TQ 02 deg. oo min. ou fèc the Area of the Triangle SC will be
found equal to 03383316, and conſequently the Area of the Trili-
neal Space AST or mean Anomaly, will be found equal to 2.033541,
equal to 2 deg. 2. min. I ſec.; alſo the Angle AŠt will be found e-
qual to 1 deg. 58 min. oo ſec. and the Co-equate Anomaly AST
i deg: 57 min. 59 ſec. and conſequently the Equation or Proktira-
phærefis oo deg, 04 min. oz fec:
Whence to find the Equation or Proſthaphzrefis anſwering to
i deg. Oo min. oo lec. of miean Anomaly, it will be, as or deg.
I min. oo fec. , the Difference between the two mean Anomalies
or deg. oi min. oo fec. , and o 2 deg. 02 min. 01 ſec. to oo deg.
02 min. or fec..the Difference between oo deg. 02 min. ol ſec.
and oo deg. 04 min oa les. the two correſpondent Equations, fo is
oo deg:
}
of the Theory of the Sun.
479
.
oo deg. oi min. oo ſec. 5, the Exceſs of the given mcan Anomaly
above or deg. oo min. oo ſec. to oo deg. oo min. 02 ſec. the pro-
portional Part: which therefore taken from oo deg. 02 min. oi ſec.
the Equation or Profthaphæreſis anlwering to or deg oi min. oo ſec.
i of mean Anomaly, will leave oo deg. i min. 59 Iec. for the abſo-
lute Equation or Proſthaphæreſis anſwering to i deg, of mean Ano-
maly: and ſo is 59 min. nearly, the Defea of the leaſt mean Anoma-
ly, to 2 deg. oo min. oo ſec. to 1 min. 58 dec. nearly, which there-
fore added to 2 min. I ſec. the Equation anſwering to i deg. I min.
oo ſec. of mean Anomaly, will give oo deg. 03 min. 59 ſec. for the
Equation anſwering to 2 deg. 00 min. oo ſec. of mean Anomaly.
In like manner the mean Anomaly anſwering to 3 deg. oo min.
oo ſec. of the Anomaly of the Excentric will be found to be oz
deg. 3 min. ol ſec. the co-equate or true Anomaly 2 deg 56 min.
59 ſec. the correſpondent Equation oo deg. 06 min. 02 ſec. and the
Equation or Proſthaphæreſis anſwering to 3 deg. oo min. oo ſec. of
mean Anomaly, will be oo deg. os min. 58 ſec. and after the fams
manner may thc Equation or Profthaphæreſis anſwering to every de-
gree of mean Anomaly, in either Semicircle be inveſtigated, and the
Tables of Equations conſtru&ed.
How opcroſe ſoever this Method may appear at the firſt view,
yet if it be put in Pradice, it will be found to be very eaſy and expe-
ditious: For, becauſe CS, CA and CQ conſtantly remain the ſame,
if to the Logarithm of t F in ſuch Parts as the Area of the Circle
or Ellipfis is 360 deg. oo min. oo ſec. be added the Logarithm of
half the Excentricity in the ſame. Parts, the Natural Number an-
ſwering to their Sum increaſed by the Anomaly of the Excentric,
will give the Competent mean Anomaly, or Area A St, and if
from the Logarithm of this Space be taken the Difference between
the Logarithms of C A and cl, the Remainder will be the Loga-
rithm of the mean Anomaly.
Again, becauſe SA and SP conſtantly remain the ſame, if from the
Tangent of half the Anomaly of theExcentric, be taken the Difference
between the Logarithms of the greateſt and leaſt Diſtances, the Rc-
mainder will be the Tangent of an Arch to which adding half the
Anomaly of the Excentric, we ſhall have the Competent co-egugc.
Anomaly A St; and laſtly, if from the Tangent of this Arch be
taken away the Difference between the Logarithms of CA and C&
the Remainder will be the Tangent of the crue Anomaly. or Angle
AST,
Q49 ?
Thus
#
480
Of the Theory of the Sun.
Thus in the former Example, where the Anomaly of the Excentric
was afljmed equal to 1° 0'00" if to 9: 714318. the Logarithrn of
t F, in fuck Parts as the whole Area is 360, be added the conſtant
Number 8.9569469 the Logarithm of half CS, the Sum 8.2283787,
will be the Logarithm of 016919265, which added to i degree,
will give 1.0169:9265 for the competent Anomaly: Laſtly, if from
O 007 863 che Logarithm of this Number, be taken away the con-
ftant Quantity 627, the Difference between ş:ooooooo, and the
Logarithm of AC, and 4 9999378 the Logarithm of ce, the Rc-
mainder 0.0072241 will be the Logarithm of 1.016773, or deg.
o! min oo fec. , the correſpondent mean Anomaly.
Again, if from 7.9408584 the Tangent of oo deg. 30 min. on ſec.
half the Anomaly of the Excentric be taken away the conſtant Num-
ber 0.0146979, the Diffcrence between 3.0072868 the Logarithm of
101692 the greateſt Diſtance, and 49925889 the Logarithm of 98308
the leaſt diſtance, the Remainder 7.926:605, will be the Tangent
of oo deg. 29 min. oo ſec which being added to oo deg. 30 min.
oo lec. half of the Anomaly of the Extentric, the Sum oo deg. 59
min. oo ſec. will be the Angle A St; and if from 82346208, the
Tangent of this Angle, be taken 622 a conſtant Quantity, and the
Difference between the Logarithms of CA and CQ, the Remainder
8.2345586, will be the Tangent of oo deg. 58 min 59 ſec. ! the
cosequate Anomaly or Angle AST"required.
After this manner may Tables of Proſthaphæreles for any or
all of the Planets be conſtru&ęd, their Excentricities or Diſtance
of the Sun from the Center of their Orbs being firſt known or given
This Way of computing thë central Equation of the Planets tho'
it be very eaſy, and every way adapted to the Uſes in Astronomy.
for, which it was firſt invented, yet the Aſtronomers of former times
would by no means allow of it, as being not Geometrical; and
obje&ted againſt the Hypotheſis it ſelf, becauſe they knew of no
direct way of finding the true Anomaly from the mean, and therefore.
hey ſubſtituted other. Hypotheſes in the room, as the Reader may fee,
if he will conſult the Treatiſes wrote about that time by Ward, Bul-
linldus and others, but theſe being no ways conſentaneous to the Mo-
tions of the heavenly Bodies, and incapable of accounting for the Laws
that they are found to obſerve, Geometers ſince that time have ſet
themſelves about finding out ſome dire& way of inveſtigating the
true or co-eguate Anomaly from the mean, in which they have hap-
pily fucceeded ; and among the ſeveral Methods that have been pro-
poled,
*
of the Theory of the Sun
481
*
..
+
f
1
4
h
9
poſed, I ſhall make uſe of that which is given by Sir Iſaac Newron,
in his Philo. Nat. Princip. Mathm. which is eaſy to be underſtood,
and very apt for Prađice.
In the adjacent Figure, let
ATP repreſent a part of the
Earths Orb, AQNP the cir-
А.
cumſcribing Circle, C the Cen-
ter, and S the Sun placed in one
of the Foci, then will CS be the
A
Excentricity
Thro I the Place of the Earth,
let TH be drawn perpendicular
T
to the Axis AP, meeting with
the Circle in the point l, then
will the Area of the Trilineal
Space AS, be to the Area
of the whole Circle, as the time
in which the moving Ray deſcri-
I n
bed the Space Asl, to the
whole Pericdical time.
From thro' the Center C draw the Line 2c, and from S the
Sun let fall the perpendicular $ F upon e C produced, if needful
to F, then if the Arch of the Circle A N be taken proportional to
the time thar the Earth deſcribes the Space ASQ, I ſay that the
Right-line. SF will be equal to the Arch QN
For the Area of the Sector is equal to half QC multiplied into AQ,
and the Area of the Triangle SQC is equal to half QC multiplied into
ŞF, but the Triangle sec+the Secor ACQ, is equal to the Tri-
lineal Space ASC, wherefore the Area ASQ 'is equal to half QCx
AltiQ.CxSF, but becauſe halt QC is a conſtant Quantity, the
Area ASQ will be always proportional to the Arch A +the
Right line 5F, when the Earth defcends from the Aphelion to the
Perihelion, or while ſhe is moving thro the firſt Semicircle of Ano
maly, but while ſhe is aſcending from the Perihelion to the Apheli-
on, or while ſhe is moving thro the ſecond Semicircle of Anomaly,
the Area PSq is equal to the Se&tor PCq--"the Triangle CSq, and
conſequently the Arca sq will be proportional to the Arch P92
the Right-line Sf, wherefore if we take the Arch-A N or Pn pro-
por-
*
41
.
>
482
of the Theory of the Swx*
portional to the time AQ+SF will be equal to 4 N, and pq
Sf will be equal to Pn, for then AN and P * will be to the whole
Periodical time, as the Spaces ASQ and P Sq, are to the whole
Area, and conſequently proportional to the time.
Hence if the Arch AQ be known, the Arch & Nequal to the
Right line CF may be readily found, and conſequently the
Arch AN which is ever proportional to the time and equal to the
mean Anomaly, and hence the Anomaly of the Excentric being
known, we are caught a direct way of finding out the correſpondent
mean Anomaly.
In Sec. 2d of Part the if it has been ſhewn, that the Circumfe.
rence of every Circle is to its Diameter, as 3.1415926535+ to 1,
wherefore if z6o be divided by 6.283185+, the Quotient 57.29578
will be the Length of an Arch of the Circumference of a Circle e-
qual to the Radius, expreſſed in Degrees and Decimal Parts of a
Degree: Let QC be to $ C, as the Number 57.29578 to a fourth
Number, this will give us the Length of an Arch equal to SC the
Excentricity in Degrees and Decimal Parts, and let this Arch be
called B, and becauſe SC is to SF as the Radius to the
Sine of the Angle SCF or AC fay as thc Radius to the Sine of
the Angle ACQ, ſo is the Arch B to a fourth proportional, and
this will give us an Arch of the Periphery equal to the Right-line
SF, and becauſe SF is equal to l N, we all have an Arch AN,
proportional to the time, and equal to the mean Anomaly.
The Excentricity of the Earth is 1691 fucb Parts as the mean
Diſtance is 100000, whence to find the Length of SC in ſuch Parts
as the whole Circumference is 360, it will be,
As the meanDiſtance 10.000
5.0000000
To the Excentricity 1692
3.3284004
So is 57.29578+, the Length of the Radius 1.7581936
4
:
To the Length of an Arch cqual to SC. 9694447 9.9865230-
And becauſe the Radius is equal to 10.0000000, if to the Loga.
rithmic Sinc of any Anomaly of the Excentric, be added the conſtant
Logarithm of B, equal to 9:9865230, we all have the Logarithm
of the
corcelpondent mean Anomaly, ſo that if it were required to find
the mean Anomaly, the Anomaly of the Excentric being 29. deg: 35
mio.
3
Of the I beory of the Sun
483
LE. l9; that is, becauſe CE is nearly equal to q PPP99,
min. 20 ſec. to the Logarithm Sine of 29 deg. 31 min. 20 ſec.
9.6925620, add the conſtant Logarithm of B 9.9865230, and the
Sum 9.6791594 will be the Logarithm of .477705 equal to 28 min.
39 ſec. 45 thirds or 28 min. 40 fec. which therefore added to 29 deg.
31 min, oo lec. the Anomaly of the Excentric, will give 30 deg.
00 min. oo ſec. for the competent mean Anoinály : And hence we
are taught an eafier and ſhorter Way, than that which Kepler
has given, to find the competent mean Anomaly from the
Anomaly of the Excentric firſt given ; and hence we are taught a
dire&t Method of finding the Anomaly of the Excentric, and thence
the co-equate
or true Anomaly, from any given mean Anomaly.
For becauſe the Arch AN
which is equal to AQUQ N, is
ever equal to the mean Anomaly,
A
and conſtantly proportional to the
true, if at any time the Arch
AN and the Area ASQ are pro:
H
portional each to the time, and
I take N P equal to SF, the
9
point P will fall upon the point
R, but if the Area AS 2 be P P
not exactly proportional to the
time, the point P will fall çi.
N
thor above or below the point
R, according as the Area AsQ
is bigger or lefler than the
Truth; let the true Area there-
fore bé AS9, and upon C q let
fall the perpendicular SE, which
by what has been already thewo
is equal to Nq, and therefore SE-SF or SF-SE, that is nearly
the Line LE is equal to qP=QP-29, or 29-QP, now when
the Angle QCq is very ſmall, the Arch 2q will differ but little
from a ſtreight Line, and conſequently we ſhall have CE:Cq::
CE: C2: :P-Qq: Qq; wherefore CE+cq:cq: : (På
Qq+Q4=) QP : 09; and after the ſame way when Bq is leſs
than a Quadrant, cq-CE: Cq: : QP: 09, but when the Earth
is near the Aphélion or Perihelion, CE will be nearly equal to
CS
A
.:
484
Of tbe Theory of the Sun
R
2
K
C.5, and CQ+CE nearly equal to CS, and therefore QP
: 09 :: AS: AC, when the Arch Aq is leſs than a Quil
drant, but when Pq is leſs than 2 Quudrant QP:Q::5B:SC.
Make CS : CQ:: Rthe Radius to a Lin:1, and then CSxL=
(SxL
CQxR, and Co=
C alſo lor Rad. : Cc-five AC as SC:
CF or C E nearly, then RxCE=Co-ſine AC QXSC, and CE=
Co-fine ACQxC,
whence we have hy Subftitution,
R
S Cx Co-fine ACOUST, CSxL
As QP:Qq::
ard
R
CS
by dividing the ſecond and third Terms by we ſhall have QP
:Qq::L+Co.fine of ACQ : L when A Q is leſs than a Quadrant,
but it it be greater than a Quadrant, QP : 09::L-Co tine ice
:L; ſo that if the Arch Á Q or Anomaly of the Excentric be afiu-
med bigger or leſs than the Truth, we ſhall find the Arth 9,
whick being added to, or ſubftracted from AQ, according as the
Cafe happens, will give us an Area ASQ nearly propcrtional to
the time, and if inltead of A Q we take A q and proceed after the
ſame manner, we ſhall have a new Qq, and conſequently another
AQ, nearer to the Truth than the former, and by this means we
fhall conſtantly approach to the true Arch, till the Difference be
leſs than any given Quantity.
Let it therefore be required to find the co-equate Anomaly and
Profthaphærefis, the mean Anomaly being ſuppoſed 1 Degree
Take AQ the Anomaly of the Excentric equal to 58 min. oo ſec.
equal to .966666, &c. then to 8.2271335 the Logarithm Sine of 58
min. oo ſec. add 9:9865230 the conſtant Logarithm of CS, in ſuch
Parts as the Circumference is 360 Degrees, and the Natural Num.
ber .01635523 anſwering to their Sum 8.2136565, will be equal
to NP, this therefore taken from the mean Anomaly A N equal
10 1,0000000, wiil leave AP equal to 0.98364477, from which
therefore taking away A Q affumed equal to ,9666666, &c. becauſe
the point P falls below the Point Qi and the Remainder 0.01697
811, will be equal to PQ; wherefore ſay, as L+Co-fine of ACO
to L, that is, as 60.1015 126 to 59.101655, fo is P Q equal to
0.01697811, to Qq, equal to 0.0166955, which being therefore
**
added
Of the Theory of the Sun. 485
added to AQ equal.co.9.66666,6c. will give ng cqual to .98336216
equal to o deg. 59 min. oo fec. 02 thirds, for the true Anomaly of
the Excentric, or true Length of the Arch AQ:
The Excellency of this Method will.better appear by aſſuming
the Anomaly of the Excentric different from what it was before,
and wide from the Truth, whereby the Render will readily fee,
that by applying the proper Correction, the Arch AQ firſt aſſumed,
will conſtantly give the Truth, till the Difference will become in-
ſenſible.
Let us now aſſume the Anomaly of the Excentric AQ in the for-
mer Example, where the mean Anomaly was given i deg oo min.
oo ſec. equal to so min. oo ſec. equal to 8.333333, 6c Decimal
Parts, then to the Logarithmic Sine of so min. o fec. equal to 8.16
26808, add the conſtant Logarithm of CS equal tu 9.9865230, the
Sum 8 1492038, the Logarithm of .0140991 will be the Length of
NP, this therefore taken from AN equal to 1.0000000, will leave
09859005 for the Length of AP ; froni which taking away AQ equal
to .8333336, we ſhall have 0.1525672 for the Arch PQ; where-
fore as 1.7788856 the Logarithm of 60.1015492 to 1.7715955 the
conſtant Logarithm of 59.101655, ſo is 9.1834612 the Logarithm
of QP, equal to .1525672, to 9.1701711 the Logarithm of 1500
276, the Length of the Arch 29, this therefore added to 8.333333
the Length of the Arch AQ, will give .9833609, equal to oo
deg. 59 min. oo ſec. 06 thirds for the Length of the Arch cq, the
true Anomaly of the Excentric different from the Anomaly of the
Excentric found by the former Calculation but 4'thirds of a Degree.
The Anomaly of the Excentric being thus determined to be o deg.
59 min. oo ſec. the true or coequate Anomaly will be found after
the manner taught in the Keplerian Method of Inveſtigation in Page
477, to be 58 min. oi fec. and conſequently the Equation or
Proſthaphæreſis oo deg. or min. 59 ſec.
Let us now ſuppoſe the mean Anomaly AN to be 2 deg. oo min.
oo fec. and take A l equal to 1 deg 58 min. oo ſec. then to its
Logarithmic Sine 8.5355228, add the conſtant Logarithm of CS ·
equal to 9.9865230, the Sum 8.5220458 will be the Logarithm of
NP, equal to 0.0332695, this taken from AN equal to 2.0000000,
will leave 1.9667305 for the Length of AP, whence Qq will be
found to be 0.00006, and conſequently the Anomaly of the Excentric
will be or deg. 58 min. oo ſec. 12 thirds; hence the true or coc-
Rrr *
quats
486
of the Theory of the Sun
quate Apomaly will be found to be or deg. 56 min. or fec. , and
conſequently the Equation, or Proſthaphærefis oo deg. 03 min. 58
foc, .
Again, if we take the mean Anomaly AN equal to cz deg.
oo min. oo ſec. and A L equal to o2 deg. 57 min. oo ſec, we ſhall
find NP equal to 0.0498920, AP equal to 2:950108, Qq equal to
0.0001, and conſequently the Anomaly of the Excentric will be
found to be o 2 deg 57 min. oi fec. whence the true or coequate
Anomaly will be found to be a deg. 54 min. 03 ſec. and conſequenc-
ly the Equation or Proſthaphærefis oo deg. 05 min. 58 ſec.
And after the lame manner may the coequate Anomaly and Pro
ſthapliære ſis be found to any Degree of mean Anomaly.
Hence the Place of the Aphelion being known, we are taught a
dire&Method of finding the true Place of the Sun in the Ecliptic at any
time propoſed. According to the Obſervations of Mr. Flamſteed, the
mean Motion of the Apogæum, or point in the Earth's Orb where
the Sun is removed at the greateſt diſtance from the Earth, and when
his apparent Semidiameter is the leaſt, from the firſt point of Aries
in the Year 1681, at the time of the Vernal Equinox was 03 S. 07
deg. 23 min, 42 ſec. and in the Year 1701, at the time when the
fame Equinox happened, the mean Motion of the fame point from
the firſt point of Aries was 3 S. 7 deg. 44. min. 42 ſec. ſo that in 20
Julian Years nearly, the Motion of the Sun's Apoge was 21 min.
o lec. and deducing from thence 4 min. 20 ſec. for the Motion of the
Apoge, in refpe& to the Fixed Stars, during that Space of time there
will remain i6 min. 40 ſec. for the Motion of the Apoge in the
fame time whence the Annual mean Motion will be found to be 50
Seconds.
Hence the mean Motion of the Apoge for two Years will be found
to be 2 min. 06 ſec. for Three Years o3 min. 09 ſec. and for Four
Years 4 min. 12 fec. for 40 Years 42 min. oo fec. and for 100 Years
i deg. 45 min. oo ſec. &c. for half a Year or 182 Days nearly, it
will be but 31 ſec. for Three Months or 91 Days nearly, it will
be but 15 ſec. , and for every 7 Days and a half it be will about
one Second.
The mean Motion of the Apoge at the time of the Vernal Equinox
in the Year 1701, was c3 S 07 deg. 44 min. 42 ſec. from which
fubſtraating 12 Seconds for the Motion of the Apoge between that
time and the laſt of December immediately preceeding, we fhall have
03 S.
Of the Theory of the Sun
487
'
03
S. 07 deg. 44 min. 30 ſec. for the mean Motion of the Apoge
from the firſt point of Aries on the laſt Day of December in the Year
1700, which is the Number placed againſt the Year 1701 in the firſt
Table, under the Column Entituled The mean Motions of the Earths
Perihelion from the first point of Aries.
Again, if to the mean Motion of the Apoge be added 25 min.
oo lec. the Motion for 20 Years, we ſhall have 3 S. 8 deg. os min.
30 ſec. for the mean Motion of the Apoge for the Year 1721: And
if to the fame mean Motion of the Apoge for the Year 1701 cqual
to 3 S. 07 deg 44 min. 30 ſec. be added 1 deg. 45 min. oo fec.
the mean Motion of the Apoge for 100 Years, we ihall have
03
S
09 deg. 29 min. 30 ſec. for the mean Motion of the Apoge for the
laſt Day of December in the Year 1800, and which ſtands againſt
the Year 1801 in the ſame Table, &c.
Again, as by ſubſtracting 21 min. oo ſec. the mean Motion of the
Apoge in 20 Years, from 3 S. 7 deg. 44 min. 30 ſec. the rnean Motion
of the Apoge for the laſt Day of December 1701, we ſhall have the
mean Motion of the Apoge for the laſt Day of December 1681, equal
to 3 S. 07 deg23 min. 30 ſec. ſo by ſubſtracting 29 deg. 45 min.
oo fec. the mean Motion of the Apoge for 1700 Years from 3 S.
deg. 44 min 30 ſec. we ſhall have 2 S. 7 deg. 59 min 30 ſec. for
the mean Motion of the Apoge from the firſt point of Aries, for the
laſt Day of December immediately preceding the commencement of
the Chriſtian Æra.
After this manner may the true Place of the Aphelion for a-
ny time propoſed be readily computed, and after this manner was
the Table of mean Motion of the Apoge conſtructed.
Let it now be required to find the true Place of the Sun in che E-
cliptic on the 7th of May, in the Year 1720, at Noon mean time.
The mean Motion of the Sun from the firſt point of Aries on the
31ſt Day of December 1719, at Noon mean time was 9 S 20 deg.
08 min. 03 ſec. to this therefore adding 04 S. 06 deg. 09 min 46
ſec. the mean Motion of the Sun for 128 Days, equal to the Space
of time between the 31st of December and the 7th of May following,
and which is found by laying, as 365 Days 5 h 48 min. 57 ſec. is
to 128 Days, fo.12 S. oo deg. oo min. co fecto a fourth Pro-
portional, and the Sum 01 S. 26 deg. 17 min. 49 ſec. will be the
mean Motion of the Sun from the firſt point of Aries at the time gi-
Rrr 2
The
S. 07
Ven.
1.
488
Of the Theory of the Sun.
The mean Place of the Apoge on the 31ſt of December 1919 at
Noon is 03 S. 08 deg. 04 min. 27 lec. to this therefore adding
22 ſec. the mean Motion of the Apoge for 128 Days, and the Sum
3 S. os deg. 04 min. 49 fec. will be the mean Motion of the Apoge
at the time propoſed, this therefore taken away from 1 S. 26 deg.
17 min. 49 ſec. the mean Motion of the Sun from the firſt point of
Aries, will leave 10 S 18 deg. 13 min. oo ſec. for the Sun's mea!
Anomaly, whoſe Supplement 1 S. 11 deg. 47 min. oo ſec. will be
the diſtance of the Sun from the Apoge.
Let us now ſuppoſe the Anomaly of the Excentric AQ to be 40
deg. 30 min. oo lec to its Logarithmic Sine 9.812544.4 therefore
adding 9 9865230 the conſtant Logarithm of C S, the Sum 9.799
C674 will be the Logarithm of NP, equal to .6296039, this there-
fore taken from AN equal to 41.7833333, &c. equal to 41 deg. 47
min. oo fec the Apogeon Diſtance, will leave A Pequal to 41.153
7-94, this therefore leſlened by Al, aſſumed equal to 40 deg. 05
min. will leave .6537294 for the Length of PQ, whence Qq will
be found equal to .6440480, and A 2 equal to 4!.144048, equal
10. deg. 08 min 38 ſec.
Hence according to the Rules delivered in Page 481, 482,) the
true or coequate Anomaly will be found to be 10 $ 19 deg. 29 min.
18 ſec. and which therefore being added to the true Place of the A-
phelion 35. 8 deg. 04 min. -49 fec. will give or S. 27 deg. 34 min.
07
ſec. for the true diſtance of the Sun from the firſt point of Aries,
and confequently the Sun's Place in the Ecliptic is Taurus 27 deg.
34 min. 7 ſec
The mean Anomaly was 10 S. 18 deg. 13 min. oo ſec. this there-
fore taken from the true or coequace Anomaly laſt found, will leave
1 deg. 16 min. 18 fec. for the Equation or Proſthaphæreſis, this there
fore added to oi S. 26 deg. 19 min. 49 ſec. the mean Motion of
the Sun from the Vernal Equinox, becauſe the Sun is in the ſecond
Semicircle of Anomaly, will give 1 S: 27 deg. 34 min of ſec. for
the Sun's true diſtance from the firſt point of Aries, or Taurus 27 deg.
34 min. 07 ſec. for his place in the Ecliptic as before determined;
and after this manner may the Place of the Sun in the Ecliptic be de-
termined for any time propoſed, without the help of the Solar Tables
firſt made or computed.
The Length of the Vector ST ſee the Figure in Page 473, is e-
ver equal to the diſtance of the Earth from the Sun, and to find it
we
Of the Theory of the Sun
489
we have given firſt, in the Triangle St F, the Angle t SF the com-
petent Anomaly, the Angle + CS the Anomaly of the Excentric, and
the Side Ct the mean diſtance of the Earth from the Sun, whence
to find St it will be by the ad Axiom of Plane Trigonometry.
As S. t SC: S. <t CS::Ct:+C; that is,
As the Sine of the competent Anomaly, to the Sine of the Ano-
maly of the Excentric, ſo is the mean Diſtance, to the competent
Diſtance + C, and becauſe,
The R : SE::St,<tCF:Ct,
And as R:SF: :St<TEF: 1C; it will be,
As S, <t SF:St, <TSF: : St:ST That is,
As the Secant of the competent Anomaly to the Secant of the co-
equate or true Anomaly, ſo is the diſtance laſt found, to the true
diſtance of the Earth from the Sun.
Let it therefore be required to find the diſtance of the Sun from
the Earth in the former Example, viz. on the 7th of May 1720 at
Noon, when the Anomaly of the Excentric was 41 deg. o8 min. 39
ſec. the competent Anomaly, or Angle + SF 40 deg. 30 min 51
ſec. , and the coequate Anomaly 40 deg. 30 min 37 ſec. and
firſt,
As the Sine of the Anglet SC 40.30 5 IẢ
9:8126720
To the Sine of the Anglet CS 41:08.30
9.8181968
So is the mean Diſtance Ct 1000000
5.0000000
To tlie Difrance Ct 10128.8
5.0055248
10-1190482
Which being obtained ſay,
As the Secant of the Angle + SF 40-30.511-
To the Secant of the Angle TSF 40:30:37
So is the Diſtance of St laſt found 101278.-
10.1 1902II
- 5.0055248
To the true diſtance SF1012740
nen
5.0054977
Whence
490
Of the Theory of the Sun.
*
:
Wheoce it appears that the diſtance of the Sun from the Earth
at that time was rol 2.71.8 fuch Parts as the mean Diſtance was
100000, and its Logarithm 5.0054879, when the Logarithm of the
mean Diſtance was 5,0000000
Kepler in his Commentary upon the Planet Mars, gives us the
following Rule for finding the Diftance of the Planet from the Sun,
from the Anomaly of the Excentric only, viz.
As the Radius, to the Cc-line of the Anomaly of the Excentric,
ſo is the Excentricity to a fourth Number; which he calls the Li.
bration, which being added to the mean Diſtance while the Sun is
moving thro the upper Semicircle, that is, when the mean Anomaly
is greater than Nine Signs and leſs than Three Signs, but fubftracted
from the mean Diſtance while the Sun is moving thro the lower Semi-
circle, that is when the mean Anomaly is greater than Three Signs
and leſs than Nine Signs, will give the Diſtance of the Planer from
the Sun ; thus in the Triangle CSE (ſee the Figure in the next Page)
where CS repreſents the Excentricity, and the Angle SCE equal to
IC A the Anomaly of the Excentric it will be
As R:S. <CSE: : CS : CE,
Now + C+CE while the Planet is moving from R thro A to
will give t E=TS, but i C=CN-CE while the Planet is moving
from thro P to Ř, will give EN equal to S m, the diſtance of the
Planet from the Sun, ſo that in the former Example when the Ex-
centricy is 1692, and the Anomaly of the Excentric 41 deg. 08
min. 39 ſec. to find the diſtance of the Sun from the Earth it will
be,
As the Radius
10.0000090
To the Co-fine of 41.08.39
So is the Logarithm of 1692
9.8768275
3.2284004
Tathe Logarithm of 1274.1 mm
3.1052279
Which becauſe the Sun is in the upper Semicircle, or which is
the fame 'thing, becauſe the mean Anomaly is greater than Nine
Signs and leſs than Threes added to 1.00000 will give 101274 for
the
of the Theory of the Sun
491
the diſtance of the Sun from the Earth at the time propoſed :
And the Reaſon of this Law is founded upon this, that if from S
the Place of the Sun upon any Ray C or CN produced (if need
bel a Perpendicular as SE be drawn, that te will be always e-
qual to TS, and NF will be equal to S m, whereſoever the points
T or m be taken.
А.
I
TY
F
a
R
B
a
m
:
N
S
P
For in the adjacent Figure, if ATQPMRA repreſent the Earths
Orh, A the Aphelion and P the Perihelion, Š the Place of the
Sun, and T the true Place of the Earth, SA the greateſt Diſtance,
SP the leaſt Diſtance, SQ equal to CA the mean Diſtance, and
TS the true Diſtance; in this caſe of the Planet from the Sun, I
ſay if from S the Place of the Sun, a Line as S E be drawn perpen-
dicular to t, till it meet t C produced in E, that t E will be e.
qual to TS.
For B Cq:C Qq : : t Fq:T F9, by the 15th Article in Page
And BCq: BCq-Ce: : t Fq: t F9-TF9 (by the 17th Cor.
of Prop. the 15th,
Wherefore BCq: t Fq: :C Sq: t Fq-TFq by the 13th Cor. and
Again, Ctq: t Fq: : CSq: S Eq, by Prop. the reth;
Therefore s Eq=r Fq-TF9,
And conſequently tEq=t Sq=) + Fat-FS4 (-S Eq=) - Tq
+T F=TS.
Wherefore, t EST Sq.
After
the 85th,
8th Prop:
--
492
Of the Theory of the Sun
After the ſame manner it may be proved that E N will be equal
to Sm, wherefore, &c. as was to be demonſtrated.
After the ſame manner may the Geocentric Place of any of the
Planets and its Distance from the Sun be inveſtigated, whence and
from the Inclination of its Orb, may the Geocentric Place, and the
true Diſtance of the Planet from the Earth be inveſtigated, without
the help of Tables by the common Rules of Trigonometry, but of
this as well as of tome other Matters relating to the Theory of the
Planets Motions, I ſhall trear of at large in a Diſcourſe by it felt;
if God ſpare me Life and Health.
NEW
@
---
.
RI...
NÉ W
SOLAR TABLES,
Deduced from the
Flamſteedian Obſervations,
Made at the
...
Royal Obſervatory in Greenwich-Park:
AND
Conſtructed after the mariner Taught in this
SECTION.
T
Sss
:: 215
1
oilasi: 01bbA
What
O
I
2
5
Signs
W
o
Ilo
210
O!
034. 4217 08
9
IOI
Ili
1614 0216 477 45103913
3.214 1916 $417 4516
147 4010 0112
+94 The Equations of Apparent Time
Subſtract from the Apparent if the Sun's # Subltract from the
Mean Anomaly be
Apparent.
3 4
r218 mil Il
0013 .
486 3917 4516 47 3 57 30 || 00 00 8 248_45130
083 556 437 4516 433 50129 10 2018
Ilo 2018 34 8 35 29
312 247 ogló $02 456 353 827/3/0 5918 538 2327
303 28
40
50_404__2216 387_446_2613 2023|| 511 3919 102_4825
60 484 297 h 441
4486 213
13|24|| 61 5819 1717 3424
70
5514 3517 0517 43
6 03 05123
05|23|| 72 18
9 247 2023
8/1
74216112
572211 812 3719 307 0622
114 487147
574116 00 02 92.5ol9 35 5021
1914 547
0113_43201103_159_402_3520
2715 (017 1717 3815 562 349|113 349 446 18 19
455
107 1917 375 $1122618||123 5219 486 02118
13
4,245
1217 2211
7 3515 .4512
18171314 1019 505 44/17
44 SOS187 245 341 140 144_1819 $215, 2716
15
1585
2417_2717
325 342_035|154 4019 535 0815
0°15 307 2917 3015 18 5514105 0319 544 50114
135
3517 3717 285 221 4713 1715 2019 544 3113
182 215
4117 3317 2515 3912/185 379 534 11 12
a 285
4017 3517 235
10
3. iiti9l5 5319 513 52711
3015_527_367_.2015 04
04f1_12010206 099 493_33140
4115
5717 387 1714 5811409||2 116 259 4613 I
5 116 027 397 1514 5011 06|08||225 409 4212 511 8
0717 407 1214 450 5807|236 549 382 307
127 417 0914 380 4900247 0919 3112 09
153 13
1316 1617
1617 427 ost4 zip_4105|1257 2319 251_48_3
263
2016 2017
2117 4317 14 250 3304||2017 30 9 191 26
2716 2017
446 584 180 35032714 4863 121
283 3416 307 446 5514 I DO 2 16 02 1288 019 040 43 2
293 416 3417 456 5114
5114 040 0801||29,8 1218 55
03 486 3917_456 4713 570 0000308_238 450 000
8
7
6
* me motherlig
mo
Add to the Apparent Time. Add to the Aphelion.
I 2
16/2
172
161
192
102
21/2
222
23/2 5816
$86
2413 006
273 2716
211 I
5510
Ir
10
9
&
495
ST
S.
I
1681
II
A Table of the Earth's Mean Motion, the Place of
the Perchèlion, &c.
Tears. M. Motion of The Perihelion Common Mean Motion The Peribelion
the Earth from from the Vernal Tears of the com-
and Flxcd
Currentltbt Equinox.
Equinox.
pleat Tears.
Stars,
Tears. S. 9 1
Tears
S
I 9.07.5 2.50 2.07.59.30
1 1.29.45.40 000.01.03
1501
9.19.1 2.50 3.04.14.30
2
II.29.31.20 0.00.02.06
1581 9.1949.23 3.05.38.30
3 11.29.17.00 0.00.03.09
1601 9.19.58.30 3.05.59.30
4 11.00.01.49
0.00.04.12
1621
9.20.07.34
3.06.20.30
5 11.29.47.29
0.00.05.15
1641
9.20.16.38
3.06.41.30
11.29 33.09 0.00.06.18
1661
9.20.25,42 3 07.02.30
7 11.29.18.49 0.00'07.21
9.20.34.46
3.07.23.30
OQ'00-03:37 0.02.08.24
1701 9.20.43.50 3.07.44.30
9 11,29:49.18
0.00.09.27
1721 9.20.5 2.54
3.08.05,30 10 11.29.34.58
0.00.10.30
22
9.20.38.34 3.08.06:33
11.29.20.38
0.00.11.33
23 9.20.24'10 3.08.07.37 12 00.00.05.26 0.00:1 2.36
24 9.20.09.54
3.08.08.39 13 11.29.51.06
0.00.13:39
25 9.20.54.43 3.08.09.42 14 II.29.36:47 0.80.14.42
26
9.20.40.23 3.08.10.45
IS II.29.22.27 0,00.15.45
27
9.70.25.03 3.08.11.48
16 00.00.07.15 0.00, 16,48
28
9.20.11.43
3.08.12.51
17 11.29'52.55 0.00.17.51
29 2.20.56.31 3.08.13.54
18
11.29.38.35 0.00.18.54
30 9.20.42.12
3.08.14.57
19 11.29.24.15
0.00,19:57
31 9.20 27.52
3.08.16.00
0.00.09:04 0.00,21,00
22 9,20.13.32 3.08.17.03 40 0.00.1808
0,00.42.00
33 9.20.58.20 3.08.18.00
6.
0.00.27.12
0.01.03.60
3.4 9:20.44.00
3.08.19.09
80 0.00.36.16 0.01.24.co
35 9.20.29.41 3.08.2012
100
0.00.45.20 0101.45.09
920*15.21 3.08.21.15 200 0.01.30 40 0.03.30.00
9.20.00.09 3.08.22.18
300 0.02.1 6.00
0.05.19.00
9.20.45 49 3.08.23.21
400 0,03.01.20
0:07.00.00
39 9.20.31.29 3.08.24.24
0.03.16.40
9.20.17.09 3.08.25.27
600 0.0.4.32.00 0.10.30.00
1741 9.21.01.58 3.08,26,30
700 0.05.17.20 0.12.15.pp
1761 9.21.11.02 3,08.47.33
17* tti20.04.
80 6:09.92 40 1 19an.co
3.09.08.30
900 0.06.48.00
0.15.45.00
181
9.21.2.910 3.09.29,30 1000 0.07.33.20 0.17.30.00
9.21.38.14
3.09.50.30 2010 0.15.06.40 1.05.00.00
1811
9.21.47.18
3.10.11.30 3000 0.22.14.00 1.22.30.00
1861
9.21.56.22
3.10.32.30 40no 1.00.13.20 2.10.00,00
gelbaslamowlssolowanlowa
20
3
30
37
38
mmm
500
0.08.45.00
40
1821
Sss 2
496
A Table of the Earths Mean Motions
iDays
T
1
S. d. ſ
1
January.
Eartbs Mean
Motion
9. d.
100 59 08 OC
201 58 17 oC
33 02 57 25 oC
42 03 56 33 011
504 55 42 01
J05 54 5o 01
7
006
53 58 01
8? 07 53 07 101
g
508 52 15 01
12,0951 23 02
12° 10 50 32 02
IP!I 49 4002
13" 12 48 48 02
14? 13 47 57 02
14? 14, 47 C5 Da
1610 15 46 13
1712 16 45 22 0
o 17, 44 3003
15 18 43 3803
2017 19 42 47 103
21 20 41:55.03
2113 21 41 03 04
233 22 40 12.04
24 23 39.2004
25 24-38 28 104
2013 25 37 3704
2712 26 36.454
28%) 27 35 5305
29 28 35 0204
30% 29 34 1025
111 00 33 18 10
February
March
Earths Mean
Earths Mean
Motion.
Motion.
S. d.
i oi 32 27 1 29 08 2016
1 02 31 35 or
2 00 07 28 lic
1 03 30 43 oC
2 01 06 3610
1 04 29 52 56 2 02 05 45 IC
1 OS 29 OG loc 2 03 04330
I 06 28 08 15€ 2 04 04 OIII
1 07 27 1650 2 OS D3 10 11
I 08.26 2527
2 00 02 180
1.09 25 3307
2 07 OI 2611
I 10 24 41 27
2 08. 00 35 111
III 23 5017
2 08 59 4312
| 12 22 58 2 09 58 si 12
I 13 22 06
2 10°58 0012)
07
I 14 21 19 08
2 11 57 0812
I 15 20 23 08
211256 1612
[ 16 19 3' 118
2 '13 55 2512
| 17 18 40 138 2 14 54 33 13
[ 18 17 48681 - 2 15.53.4113
1 19 16 568 2 16 52 50|131
I 20 16 04.18 2:17 51 5813
1 21 15 13 109
2 18:51 06:13
! 22 14 2159
2 1950 15141
! 13 13.30 69
12.20049 2314
1 7412 38 69 212 21 48 3114
( 25 11 46
p୨
2 22 47 40 14
I 26 10 55109
2 23 46 48 14
27 10 03 10 112 24 45 56115
I 28 09 I 10 2 25 45 05 15
2:26 44 13 15
2 27 43 2115
2 28:42 20115
April
Earths Mean
Motion.
D. d. II
2 29 41 38115
3 CO 40 46116
3 of 39 55116
302 39 0316
303 38 11 16
3 04 37 2010
3 og 36 27/16
3o6 35 36117
3.07 34 4517
308 33-5317
3 Cg 33 0117
3 10 32 10117
3.11 31 1818
3.12 30 2618
3.13 29 34.18
3 14 28 18
3.1927:51
3. 16. 27 Oo
18
3 17 26 08.19
3 18 25 1619
3 19 24 24 19
3 20 23 33119
3 21 22 41 19
3 22 21 4920
3.23 20 5820
3 24 20 06 20
3 25 19 14 /20
3 26 18 23 20
3. 27 17 31 20
3.28 16 3921
43
18
..
Š
497
to every Day in the Year.
|Days/
l
304 27. 50
May
Earths Mean
Motion.
S, d. 1
13 29 35 4821
24 OO 14 5421
31+ 01 14 04121
41+ 02 13 1321
514 03 12 21 21
614 04 11 29 22
74 05 10 38122
814 06 09 46122
94 07 08 5422
104 08 08 03/22
11/4 og 07 11 22
124 10: 06 15 23
13+ II 05 28123
1414 12 04 36/23
1514 13 03 44123
104 1502 53123
1714 15 02 01 24
1814 16 01 09 24
1914 17 00 18:24
2017 17 59 2624
20 + 18 58 3:4124
22 + 19 57 49 24
2317 20 56 5125
2416 21 55 59125
254 2255 08125
2014 23 54 1625
274 24 53 24 25
28 4 25 52 3325
29 4 26 51 41/26
3114- 28 49 $8/26
June
Earths Mean
Motion,
S. d.
4 29 49 06/26
soo 48 14 26
SOI 47 23 26
5. 02 46 31127
5 03 45 39127
5 04 44 48127
5.05. 43 5627
5.06.43 04/27
507 421327
5 08 41 21128
5.09 40 29 28
5 10 39.38.28
5.11 38 46 28
5 12 37 54128
5 13.37.03/28
5 14 36,1129
5 15 35 19|29
5 16 34 28/29
517 33 30125
518 32 44/29
5 19 31 53124
5 20 31 013
5. 21 30 0934
5 22 29 1835
5 23 28 26130
5.24 27 34134
5 25 26 43/31
5 26 25 5033
5 27 24 5931
5 28 24 08131
July
Auguſt
Earths Mean
Earths Mean
Motion,
Motion.
S d. !
S. d. ]
05 29.23 1631
06 29 56 34137
06 20 22 24131
107 no 55 42137
26.01 21 32/32
07 01 54 51137
06 02 20 4032
07 02 53 59137
06 03 19 4932
07 03 53 07137
26 04 18 57132
07 04 52 1637
06 05 18 0632 07 05 51 24138
0600 17 14132
07 06 50 3238
06 07 15 22 33 07 07 49 41138
06 03 15 3133
07 08 48 49138
06 09 14 3933 07 09 47 57138
06 10 13 47133
107 10 47 06138
06 IL 12 56133 07 FL 46 14:39
06 12 12 04133 27 12 45 2239
06 13 11 1234 137 1-3 344 31139
26 14 10 2134 07 14 43 39 39
06 15 C9 29/34 07 15 42 473,
06 16 08 37 34
07 16 41 5639
20 17 07 46 341 07 17 41 0440
0618 0654134
07 18 40 1240
06 19 060235 07 19 39 2140
06 20 OS 1135
07 20 38 29140
106 21 04 19135
27 21 37 37 40
06 22 03 2735
07 22 36.4640
06 2302 30135
07 23: 35.5441
06 24 01 44/36 07 24 35 0241
0625 00 5236
107 29 34 1141
06 26 00 01 36 27 26 33 '941
06 26 59.0430
27 27 32 2341
26 27 58 1736 107 28 31 3642
0628 57 26136
107 29 30 44142
.
498
A Table of the Earths Mean Motions.
Days 1
October.
Earths
Mean
Motion,
5. d.
)
ob con alawon
1
September.
Earths Mean
Motian.
S, d.
8 00 29 52 12
8 01 29 01112
3
8 02 28 OS
8 03 27 171+2
8
04 26 30173
8
05 25 341+3
7
8 06 24 4243
88 07 23 51113
8 08 22 59113
1018 09:22 07144
8 10 21 16 14
12/8 II 20 24144
1318 12 19 32 44
14 8 13 18 41 44
15
8
14 17 4944
8 15 16 57145
17
8 16 16 05145
1818 17 15 14145
19
8 18 14 22145
2018 19 13 30145
2018 20 12 39145
2218 21 11 47 46
2318 22 10 5514
24
8
23 10 04146
2518 24 09 12 46
2018 25 08 20 40
2718 26 07 29146
288 27 26 37 47
29
8 28 05 45147
3018 29 04 54/47
311
November
Earths Mean
Motion,
S d. ?
10 00 37 20153
10 01 36 29153
10 02 35 37153
10 33 34 45153
10 04 3354153
10 OS 33 02153
10 06 32 10154
10 07 31 1954
10 08 30 27/54
1009_29 3354
10 10 28 44154
10 II 27 52154
IO 12 27 C0155
IO 13 26 09155
10 14 25 17155
10 15 24 25155
10 16 23 34155
19 17 22 421551
10 18 21 501501
10 19 20 5913
10 20 20 07 56
10 21 19 15159
10 22 18 24156
10 23 17 32157
10 24 16.40157
10 25 15 49157
10 26 14 57157
10 27 14 0557
10 28 13 13158
10 29 12 22158
9 00 04 02 47
901 03 1047
9 02 02 19 47
9 03 OI 27118
934 00 3540
9 04 59 44148
905 58 5248
06 58 00148
9.07 57 08|48
908 56 1749
909 55 2549
900 54 34/49
911 53 42149
9 12 52 50149
913 51 59 49
914 51 07150
915 50 1550
9 16 49 24150
917 46 3250
9 18 47 40155
9 19 46 49154
9 20 45 579
921 45 OS 15
9 22 44 14151
923 43 225_1
9 24 42 3951
925 41 39151
9 26 40 47151
9 27 39 5555
9 28 39 04152
19 29 35 12152
December
Eartbs Mean
Motion
S.a.
NI 20 11 30 58
II OI 10 38158
II
1 02 09 47158
II C3 08 55159
11 04 080359
II OS 07 1259
11 06 06 22159
11 07 og 2859
II 08 04 3759
11 09 03 4760
11 10 02 5360
II II 02 0260
II 12 01 J0/60
ir 13 00 18160
11 13 59 27 63
11 14:58 35161
IL 15 57 4361
11 16 56 5261
II 17 56 0061
55
119 54 17,61
11 20 53 2562
11 21 52 3362
II 22 51 42 62
11 23 50 5062
II 24 49 5862
11 25 49 07162
11 26 48 15163
11 27 47 2363
11 28 46 3263
11 29 45 40631
168
II 18
0861
.
:
e are
.
499
A Table of the Earths Mean
Motions, to Hours and Parts
of an Hour
O
H.
III iv
iv
00
3011
to o2
310
28
C4 5 6
07 24
13 55
16 23
18 51
21
23
28 42
TO Donal
2
310 07
40 09
09 51
so I 2 19
60
14 47
17 IS
80
19 43
90 22
11
rola
06
38
24 38
27 06
O2
ilo
120
1310
1410
In the Leap-Year after
February add a Day to
the given time, and its
Motion to the Motion col-
letted.
431
48 25
321
330
19
3410
47
3511 26
14
361
3710
31 IC
3811
33 38
391 36
4011
34
411
41
421
43 29
45
44 I 48
4511
50 53
4611 53 21
4711 $ 49
480 58 16
4912
502 03 12
5112 05 40
5212 08 08
532
10 36
542 13 03
512 15
15:31
56* 17
17 59
172 20:27
25 55
5912
22 23
6ef2 25 50
29 34
02
34 32
36 58
1610
39 25
710
41 53
1810
44 21
1910 46 49
2010
49 17
53 45
54 13
230 56 40
3
oo 44
2110
2210 54
2410 59
59 08
4
261
2711
7
23
GI 36
04 04
06 32
09 OO
2911
11 2711
10ft. 1551
2811
5812
.
50g A Table of the Equations of the Earths Orb.
I
2
man
1311 Sign)
Subſtraat
Sign o
3
4
.
5
0 00 00 00 00 57 0701 39.4101 56 1901 41 4800 59 15130
100 or 59100 58 5for 40 4201 56 20 or 40 48.00 57 2629
200 03 5901 00 3301 41 41101 56 1901 39 40 00 55 37 28
3 00 05 5801 02 150I 42 3901 56 1601 38 410 53 48127
400 07 5701 03 550I 43 35 01 56 1201 37 35100 51 57 26
500 09 5601 25 3501 44 2901 56 050136 27 00:50 0525
600 1155 or 0? 1401 45 2101 55 5601 35 17:00 18 13/24
7.00 135301 08 501 46 1101 55 451 34 0500 40 1923
doo 155101 10 2701 47 door 55 315 32 5200 44 2522
900 17 49 05 12 OIO 47 460 55 1501 31 3700 42 3021
1000 19 47 01 13 3501 48 31 01 54 58 01 30 2000 40 34/20
1100 21 4501 15 07 01 49 1301 54 3801 29 Oloo 38 3719
I 2100 23 4201 16.380 49 5301 54 1601 27 4100 30 4018
13.00 25 38or 18 07 01 50 3201 53 5201 26 19 00 34 41117
1400 27 3501 19 3601 51 09 01 53 26 24 55100 32 42/16
1500 29 3001 21 03101 51 44 or 52 58101_23 3000 30 4315
1600 31 25/01 22 2801 52 1701 52 27/01 22 03 00 28 4314
17 00 33 20 01 23 511 52 4801 SI 55 01 10 3 sloo 26 4213
1800 35 14/01 25 1401 53 1601 SI 2001 19 05 00 24 41112
19 00 37 081 26 350r 53 43 01 50 4401 17 33100 22 3911
2000 39 010 27 5501 54 0701 so051 16 00 00 20 37 10
2100 40 5301 29 13 0 1 54 3001 49 24/01 14 2600 18 3419
22100 42 4401 30 2901 54 soor 48 4201 12 5000 16
8
31
2300 44 350r 31 4301 55 ogor 47 571 ir 1300 14 28
24100 46 250r 32 5001 55 25101 47 11 09 3400 12 24.5
2500 18 1401 34 0801 55 3901 46 2201 07 5400 10 211
2600 50 0201 35. 1801 55 5II 45 31 or 06 13 00 08 1714
2700 si 500 36 2601 56 oipi 44 3801 04 31 00 06 13 3
2850 53 3501 37 3301 56 ogor 43 43 01 02 47 00 04 09 2
2.930 55 2201 38 3801 56 15101 42 4701 O1 02 00 02 04 1
39357 0701 39 4101 56 19 01 41 4800 59 15 00 00.00
୨
7
6
Add
O
IO
The Uſe of the following Table
301
By the help of the Tables thus conſtructed, may the Place of the
Sun in the Ecliptic at any time be eaſily and readily computed, by
obſerving the following Directions.
1. In the firſt Table
Entituled A Table of the Mean Motions, &c.
ſeek the given Year in the firſt Column on the Left-hand, if you
find it not, take the mean Motion anſwering to the next leſs Year,
and to it add the mean Motion for the Reſidue of Years; as alſo the
mean Motions anſwering to the Day of the Month, Hour, Minute,
c. in the firſt, lecond and third Tables, and the Sum will give the
mean Motion of the Sun from the firſt point of Aries, or mean Place
of the Sun ; alſo at the ſame time take out the mean Motions of the
Perihelion ſtanding againſt the reſpective Years, Months, Days,
Hours, &c. add theſe together, and the Sum will give the mean
Longitude of the Perihelion, or its Diſtance from the firſt point of
Aries.
2. Subſtra& the Longitude of the Perihelion from the mean Mo-
tion, the Remainder is the mean Anomaly.
3. With the mean Anomaly thus found, enter the 4th Table,
Entituled A Table of the Equations of the Earths Orbit, and finding
the Sign in the Top and the Degree in the firſtColumn, on the Left-
hand, if it be leſs Ithan fix Signs, but if it be more than fix Signs
then ſeek the Sign at the Bortom, and the Degree in the utmoſt Co-
lumn on the Right-hand, againſt it in the common Area cake out
the Equation, by making Proportions where requifite ; this, if the
inean Anomaly be leſs than 6 Signs ſubſtracted from, but if it be more
tban 6 Signs added to the mean Motion, will give the Sun's true
Place in the Ecliptic.
Let it therefore be required to find the Place of the Sun on the 7th
of May 1720, at Noon mean time
1. In the firſt Table againſt the Year 1701, the next leſs Year to
the given Year I find 9 S. 20 deg 43 min. 49 ſec, for the Sun's mean
Motion or radical Place, to this therefore adding 11 S. 29 deg. 24
min. 14 ſec, the mean Motion for the 19 reniaining Years, 4S. 6deg.
9 min. 46 ſec. the mean Motion ftanding againſt the 8th of May in the
2d Table, becauſe the Year propoſed is Leap-Tear; and the Sum $.
26 deg: 17 min. 49 ſec. will be the mean Motion of the Sun, or his
mean Place at the time propoſed ; alſo in the firſt Table in the 2d
Column againſt the given Year I find 03 S. 7 deg: 44 min. 30 ſec.
for the radical Place of the Perihelion, and to it adding oo S. oo d.
Tot *
19 min,
502 Rules for computing the Sun's Place.
19 min. 57 ſec. the mean Motion for 19 Ycars, alſo po S. oo deg.
oo min. 22 ſec. the mean Motion ſtanding againſt the 8th of May in
the third Table, the Sum 3 S. 08 deg: 04 min. 49 ſec. will be the
mean Motion or the mean Place of the Perihelion, which being taken
from 1 S 26 deg. 17 min. 49 lec, the mean Motion before found,
will leave 10 S. 18 deg. 13 min. oo ſec. for the mean Anomaly.
Entering therefore the fourth Table with 10 S. 18 deg. 13 min. oo
fec. the mean Anomaly, in the Column over 10 Sigos, and right a-
gainſt 18 deg. (the next leſs whole Degree) in the Right-hand
Column, I find i deg. 16 min 38 ſec. and againſt 19 degrees the
next greater whole degree in the ſame Column, I find i deg. 15 min.
07
fec. then I ſay, as 60 min. the Difference betwcen 108. 18 deg.
oo min. oo fec. and 10 S. 19 deg. oo min. oo ſec. of mean Anoma-
ly, to į min. 31 ſec. the Difference between i deg. 16 min. 38 ſec.
and i deg. 15 min. 07 ſec. the correſpondent Equations, ſo is 13
min. oo fec. the Exceſs of the given mean Anomaly above 10 S.
18 deg. oo min. oo ſec. to 20 ſec. fere, this becauſe the mean A-
nomaly anſwering to 10 S. 19 deg. po min oo ſec. is leſs than the
mean Anomaly anſwering to 10 S. 18 deg, oo min. oo fec. ſub-
ſtraded from í deg. 16 min. 38 ſec. the Equation anſwering to 10
S. 18 deg. oo min. oo ſec. will leare i deg. 16 min. 18 ſec. the E-
quation anſwering to 10 S. 18 deg. 13 min. oo !ec. the proper mean
Anomaly, this therefore added to the mean Motion or S. 26
deg. 17 min. 49 ſec. becauſe the mean Anomaly is more than fix
Signs, will give 1 S. 27 deg. 34 min. 07 ſec. for the true Place of
the Sun, or his Diſtance from the firſt point of Aries.
2. Again Let it be required to find the Sun's Place on the 24th of
June 1722, at 6h. 24 min.so ſec. P.M. the Operacion is as follows.
And
A
Rules for computing the Sun's Place
503
M. M. Sur M.M. Perihelion
S.
S.
1721 9.20.52.51
III.29.45.40
June 24 5.22.29.18
6h.
14.47
24m.
59
so"
2
3.08.05.17
50
24
3.08.06.31 M. M. Perihelion
3 13:23:37. M.M Sun
0.05.17 06
Mean Anomaly.
mean Motion Sun 3.13.23:37
Equation add 0 10.28
Sun's Place.
3:13,34.05'or. 13° 34' 5"
!!
And proceeding after the ſame way, may the Place of the Sun
be computed to any time given.
The times to which the Tables are fitted are mean times, and
therefore if the time given be Apparent time, it muſt firſt be reduced
into mean cime, according to the Rules laid down in the former
Se&tion; or elſe proper Allowance for the Motion during the Interval
muſt be made.
The San's Place being thus obtained, his Right Afcenfion from
either Equino&ial Point, and his Declination from the Equinoctial
it lelf may be had by Inſpe&ion, in the following Tables to every
two Minutes of the Quadrant ; the firſt of which ſhews the Declina-
tion of the Sun, or of any point of the Ecliptis, to every two Mi-
nutes of the Ecliptic, and the ſecond his Right Aſcenſion to every
two Minutes of the firſt Quadrant in the Ecliptic, whence his
Right Aſcenſion may be had to every two Minutes in the 'other 3
Quadrants of the Ecliptic, by Addition or Subſtraction only, the
greateſ Declinacion being fixed according to the accurate Obſerva-
tions of Mr. Flamsteed at 23 deg. 29 min.
For by finding the Sign che Sun is in, and the Degree of that
Sign in the Top, and the Minutes in the Left-hand Column, if the
Sun be in the firſt or third Quadrant of the Eclipe c, but if he be
in the ad or 4th Quadrants of the Ecliptic, by finding the Sign the
Sun is in, and the Degree of that Sign at the Bottom, and the Mi-
Itt 2*
nutes
504
The uſe of the following Table
4
nuces in the Right-hand Column in the common Area, you will have
the correſpondent Declination ; thus if the Sun be ſuppoſed to be in
22 degrees of Taurus or 22 degrees of Scorpio, by ſearching for the
Sign in the Head, and the Minutes in the Right-hand Column in the
common Area, you will find 19 deg. 38 min. 44 ſec. which is the
correſpondent Declination, and which is North, if the Sun be in
26 degrees of Taurus, but South, if he be in 26 degrees of Scorpio.
In like manier, if the Sun had been in 4 degrees of Leo or in de-
grees of quarius, then by finding the Sign and the degree of that
Sign in the Botcom, and the minute in the Right-hand Column, we
fall find in the common. Area 19 deg. 38 min. 44 lec, for the pro-
per Declination, which is North, if the Sun be in Leo which is in the
third Quadrant; but Sout, if the Sun be in Aquarius, which is the
furth and laſt Quadrant.
But if the Sun's Place does not conſiſt of any even Number of
minutes, then a proportionable part muſt be allowed forche Exceſs
or Defeet,
Thus for Example, ſuppoſe it were required to find the Declinati-
on anſwering to 27 deg. 33 min. 57 ſec of Taurus, becauſe this
Number cannot be cxactly found, I ſeek for the Declination anſwer
ing to the next leſs Tabular Number, viz. 27 deg: 32 min. oo ſec.
and find it to be 19 deg. 38 min. 44 ſec. after that I ſeek for the
next greater Tabular Number, viz. 27 deg. 34 min. 08 ſec. and
find the Declination correſponding to it to be 19 deg. 39 min. 11 ſec.
then I ſay as 2 minutes or 120 ſeconds, the Difference between 27
deg. 32 min. oo fec and 27 deg. 34 min. oo ſec. to 27 ſec. the Dif-
ference between the correſpondent Tabular Declinations, ſo is i min
57 ſec the Exceſs of the given Place above 27 deg. 32 min. oo ſec.
to 26 ſec. nearly, which being added to 19 deg 38 min. 44 ſec. the
Declination correſponding to 27 deg. 32 min. oo fec. becauſe the
Declinations are increaſing will give 19 deg. 39 min. 10 ſec. for the
Declination anſwering to 27 deg 33 min. 57 fec: or ſo is 03 ſec.
the Defeat of the given Declination 27 deg 33 min, 57 ſec. to 27
deg. 34 min. oo ſec. to or fec. nearly, which therefore taken from
29 deg. 39 min. 11 ſec. the Declination anſwering to 27 deg. 34
min. oo ſec will teave 19 deg. 39 min. 10 ſec. for the proper De-
chination, and which is North, becauſe the Sun is in the firſt Qua-
drant of the Ecliptic; and after this manner may the Declination
anſwering to any point in the Ecliptic be found.
This
Rules for computating the Sun's Place sos.
This being well underſtood there is no need of Exemplifying the
Uſe of the 2d Table, ſince as it is made to every two Minutes of
the Quadrant, the correſpondent Right Aſcenfion to every point in
the Eclipric may be found after the fame manner as the Declinacion
was in the former Example, mutatis mutandis; but inalmuch as this
Table is only made to the firſt Quadrant of the Ecliptic, to prevent
the ſwelling of the Book too much, it may not be amils to fhew
how it may be continued, or made to ſerve for any of the other
three Quadrants, for as the ſame Quantity of Right Aſcenſion
anſwers to the ſame correſpondent Longitude reckoned from the face
Equino&ial point, if the Sun be in the ad Quadrant of the Ecliptic
the Right Aſcenſion found muſt be ſubſtracted from 180 Degrees,
if in the 3d Quadrant it muſt be added to 180 Degrees; and laſtly,
if he be in the fourth and laſt Quadrant it muſt be ſubſtracted from
360 Degrees, and the Quantity thus produced will be the pro-
per Right Aſcention, that is becaule the 27th deg: 33 min. 57 ſec.
of Iaurus or Scorpio, or the ad deg. 26 min. 3 ſec. of Leo or Aquarius
are each of them equi-diſtant from the two Equinoctial points, the
correſpondent Right Afcenfions will be found to be 55 deg. 17 min.
02 ſec. and 235 deg. 17 min. 03 fec. or 124 deg. 42 min. 58 ſec
and 304 deg 42 min 58 ſec. as is abundantly manifeſt.
?
Here follows
A Correct TABLE, Shewing the De-
clination to every Two Minutes of the Ecliptic ; the
greateſt Declination or Obliquity being fixed, at 23 deg.
29 min.
lioo
& TABLE Thewing the Declination to every two
Minutes of the Ecliptic.
Vanden
Vi and
and more
Y and more on
rände
rinden
r and more
2
3
4
Minitos!
5
Minutes!
1
-000
O
00 03
i
23 5947 Po 11 4291.35 3+|59 25
200 00 40 24 4010 43 3721 12 2001 36 22 22 00 1358
too oi 300 25 310) 49 29 13 1701 37 02 01 Olio
02 2+10) 26 18 50 12 14 0;lor 37 5:2 01 48154
8 03 130) 27 0 | 510 14 5310 38 44|02 02
! 00 02 5410 27 50 51 44 31 15 40 39 2202 03 23150
Wo 04 4700 28 470 52 30 16 2801 40 % 2 3 4
04 (!14
400 os 3;1:0 29 3020 53232 17 1601 41 oi
2 0 56179
10100 C6
23
30 30 17 02 54 11 18 40 41 5;
2 05 45144
8.07 07 110, 31 Oslo 545420 18 5701 42 4232 06 347
fro 07 581) 31 530 55 4 119 353[ 43 3102 07 214
:200 08 46) 324110 JO 34,3 20 277 44 102 0d 0330
24 00 09
3400 33 28 10 57 2201 21 12 45 02 08 5736
2.50) 34 160 58 Ioi 22 OPI 45 542 0944134
lepoitico 350430 58 sobor a2 slow
35 0400 58 58o1 12 50l 40 42 10 31 32
Joo II SE 20 35 522959 41 2? 3.92 47 22 11 2016
1200 12 45 10 35 401 00 31 24 201 43 12 12 0; 2.
Puloo 13 33.00 37 2801 01 2101 25 1401 49 05102 12 5512
16 oo 14 21 0 38 15 02 osor 26 ollol 49 5272 13 43 24
8
10 15 os po 39 oct 02 50101 26 490 50 9002 14 34 22
ooms 5700
39 51 91 03 441 27 37 01 31 2822 15 18 25
20 16 44
10 40 3901 04 210 28 25
01 521,02 16 05lit
1400 17 320 41 27 IOS 19 01 29 1201 53 03 02 16 5311
ic
00 18 2100 42 14
00 42 141 06 071 30 0001 53 50122 17.4114
-8
oc 19 of po 43 01 I CO 55 30 4801 54 3702 18 28112
0 19 5600 4?_sopr 07 4301 31 35 01 55_262 19 161
200 20 440 44 37|07 08 3095 32 230! 56 1402
56 14/02 20 03 8
140 21 320 45 25ør 09 18for 33 TOP 57 02 02 20 511
:
50 22 100 40 13DI 10 0601 33 5901 57 4402 21 394
idoo 23 07 0 47 0 10 5451 34 4601 58 3702 22 26 2
1000 23 550 47 4501 11 4201 35 34,01 59 2552 23 14 o
29
28
27
26
25
24
N and up and * in and * m and *
me and Hm and #
al
16
...
A TABLE lhewing the Declinution to cvery two
507
Minutes of the Ecliptic.
r and 2
y and
r and a
r and be
Vand
Tant
6
8
7
10
9
Minutes!
Minutesi
N
1 2 1
boniemen
2
1
26
22
24
02 23
0? 10 15133 34 2
3 58 0
2$ 6
32 24 02 02 47 49103 1 323 35 143 58514 22 25158
32 24 49132 48 36 -3 12 2023 26 013 59 374 33 1256
32 29 37102 49 24°3 13 07 03 36 414 00 244 24 OC50
6
n2 26 24 2 50 103 13 5423 37 3514 OD U14 24 4753
2 27 12 02 50 5923 14 4203 38 2:14 01 5424
25 345
02 27 591 2.SI 4033 15 29.03 39 104 02 404 26 2048
2 28 4 12 52 3403 16 103 39 5:14 03 33124 27 08116
2 29 35 -2 521P3 17 00103 40 45194 04 2 4 27 55141
16
02 30 2202 54 0913 17 51103 4.1 324 05 084 28 4242
18
12 31 1 2 54 502_18 3903 42 1524 05 5014 29 29 40
62 31 57 2 55 431-3 19 2003 43 04 06 4:14 30 1630
-2 32 45 -2 56 3003 20 1303 43 514 07 3 4 31 03136
32 33 32 22 57 18 03 21 03 44 41 94 08 17124 31 5034
26
2 34 2002 58 osoz 21 483 45 284 09 004 32 3722
2
2 35 07102 58 521-3 22 263 4.6 15 14 09 524 33.221?(
BC
128
32 35 55112 594 03 23 233 47 034 10 3904 34
32
32 36 423 20 273 24 11-3 47 50 4 11 2604 34 5826
31
32 37 3
3 OT 1523 24 583 48 37104 12 13°4 35 452
2 38 1733 02 02103 25 453 49 274 13 OC|P4 36 32/22
22 39 053 02 53 26 33103 50 1214 13.47114 37 1820
02 39 54°3 03 3703 27 2003 50 5914 14 3404 38 518
42
-2 40 4013 C4 25 23 28 o8l03 51 4624 15 2174 38 5216
44
32 41 2&3 05. 12103 28 5503 52 33 4 16 084 39 40114
22 42 1513 06 OCO3 29 4233 53 21604 16 564 40 2712
02 43 og 03 06 47103 30 30 3 54 0824 17 43114 41 1410
50
02 43 503 67 35|93 31 1703 54 5524 18 394 42 038
52
12 44 383 08 2293 32 04103 55 4204 19 17 24 42 486
154
02 45 2603 09 0903 32 5203 56 29104 20 04 24 43 354
52 46 14 03 09 5703 33 3903 57 15 04 20 51124 44 22 2
6c >2 47 013 10 433 34 26 03 58 034 21 38 24 45 08.
|
21
19
18
up and X 1 mand im and * T and * mye and * men and 4
24
36
38
1
46
48
56
5.8
11
23
22
20
1
508
A TABLE beping the Leclination to every two
Minutes of the Ecliptic
hand
rander
rando
Y and more on
rant
raut are not
18
Minutesi
I 2
14
15
16
17
1
:6 21 2010 44 31
2
04 45 0805 08 3 +05 31 5605 55 106 18 2206 41 27660
204 45 595 09 2105 32 4+05 si SO 19 08 05 42 12/18
424 45 43105 10 6805 33 2:05 So 4405 19 54 00 42 5255
624 47 30 5 10 5505 34 2525 57 35 23 4005 43 451:4
8714 48 105 II 4205 35 005 581;
14 40025 12_2805 35 48 05 59 0 - 22 1306 45 1752
120L 49 5095 13 1505 36 3405 59 4 6 22 59 06 46 0748
1424 50 3705 14 02105 37 21 00 00 3
6 23.4506 46 4845
1004 si 251'5 14 48 05 38 07 06 or 2
06 24 31106 47 24144
1804 52 11 05 15 3505 38 51106 02 OS -6 25 1706 48 2042
2004 52 5825 16 2205 39 41 36 02 55
06 26 03 06 49 0614°
22/04 53 4,0; 17 08los 40 2705 03 4:136 26 496 49 5238
2401 44 31105 17 55.05 41 1336 04 28 36 27 36100 50 38135
2024 55 2805 18 42 05 42 o 38 of 14101 28 22 36 51 2.34
2804 56 os 5 19 29.05 42 4006 06 oc 06 29 02 05 52 1032
3004 56 saos_2015'Os 43 3,56
06 4105 29 546 52 533
32104 57 39 05 21 02 05 44 10 05 07 2305 30 4005 53 41128
34 34 38 2005 21 58105 45 06 00 08 2 26 31 26 36 54 27 26
3604 59 135 22 35 05 45 S, 06 09 005 32 136 55 1824
38105 oo ocos 23 22 05 45 356 09 5:06 32 59-6 55 592
40105 00 4715_24 0805 47 2106 IV 346 33 4605 564
4205 or 335 24 55 05 48 1726 11 2905 34 3126 37 318
+415 02 24°5 25 42 05 48 59 6 12 12 06 35 1706 58 17/16
46105 03 0,5 26 2805 49 456 12 581 636 0305 59 0314
48-5 04 54'5 27 1505 50 3206 13 43 15 36 49 26 59 4912
$c|p5 04 41105 28 02.05 si 1905 14 356
37 35127 05 3510
52105 05 275 28 4805.52 0536 15 116 38 211.57 Os 20
8
$4135 06 145 29 35105 52 51 16 03/26 39 0817 02 00 6
lolos 07 Oil05 30 22 05 53 36 26 16 4926 39 5527 52 54
805, oz 4805 31 085 54 2556 17 335 40 4127 03 38
sclos 08 3405 31 50105 55 12 26 18 22 36 41 27107 04 24
16
15
14
me and #
up
mp and H
me and * me and H
17
1.4
12
1
in and *
and #
14 TABLE powing the Declinution to every two 509
दा
20
21
22
Minuteslº
O
Minutes! 100
$207 24 12 07 46 5808 29 30|08 32 068 54 27 9 16 421 &
Minutes of the Ecliptic.
rand
r and are in and more r and rand more
T and
18
19
23
clový 04 297 27 1527 So oolo8 12 37 08 35 of 18 57 2760
2 >7 05 17 28 107 50 45 8 13 2 203 35 58 58 U1
4127 05 5517 28 46 07 SI 3158 14 0708 36 3508 58 55 56
( 07 06 41107 39 32 07 52 16 8 14 52108 37 24 18 59 4034
627 07 2747 30 717 53 01 8 15 378 38 059 00 2452
197 08 12137 30 0207 53 468 16 2 28 38 5:19 01 0950
12.07. 08 5837 31 4807 54 32408 17 07/08 39 349 01 53.48
1407 09 4427 32 34607 55 1708 17 528 40 isloo 02 38.46
1627 10 37 33 1907 56 v2 08 18 37 8 41 049 03 23144
1807 1 15107 34 0507 56 47/08 19 22 8 41 451 9 04 07142
2017 12 017 34 507 57 33108 20 07 08 42 3319 04 5240
22/07 12 4727 35 36 27 58 18108.20 52/08 43 18.9 05 3650
227 13 32107 36 21107 59 0308 21 37 8 44 02 09 06 21 36
26-7 14 1807 37 07107 59 4808 22 2208 44 47| 9 07 05134
281 15 04.07 37 52108 00 34/08 23 07 08 45 329 07 4932
3402 15 507 38 3838 OL 1908 23 52108 46.179 08 34/30
32 27 16 35107 39 23|08 02 04/08 24 3798 47 09 og 18/28
340% ry 21107 40 og 08 02 5008 25 228 47 46 29 10 03/26
3427 18 07 07 40 5408 03 3508 26 0758 48 3 19 10 4724
3837 18 527 41 4008 04 2008 26 52108 49 159 11 3122
15.7.19 3807 42 2508 05 oslo8 27 37|08 49 599 12_1620
4207 20 2407 43 1108 05 5108 28 228 50 4459 13 oC 18
4137 21 0907 43 508 06 3608 29 07 08 51 291-9 13 4516
4027 21 55107 44 42|08 07 2158 29 528 52 139 14 291.
48137 22 41107 45 27108 08 668 30 368 52 589. IS 1312
Sco7 23 2707_46 1308 08 si 08 31 21108 53 4309 15 581
5497 24 58107 47 4458 10 2108 32 578 55 129 17 2616
15607 25 44107 48 29/08 11 08 33 36/08 55 5229 18 11 41
$807 26 29 07 49 15 08 11 5.108 34 21 08-50 42 09 18 55) 2
17 27 15 07 5o oco8 12 37108 35 05/28 57 27129 19 39
9
8
7
6
m and X and Xl ngand X 19 and Xm and * 12 and X
Uu.u
611
II
10
510 A TABLE pewing the Declinution to every two
Minutes!
Minutesla
0
10Q
15.419 38*4710 00 4210 22 2810 44 04111 Os 2510 25 44
Minutes of the Ecliptic.
Tant el r and everyone rand randa
r andere rand on
24
26
27
28
29
919 39'09 41 4:1101? 3710 25 22 10.46 51 08 2 5
209 2) 2209 42 27/10 04 20 10 26 0510 47 351 09 02158
4 9 21 0809 43 iclio 05 010 26 4810 48 22 11 09 45 156
619 21 5209 43 5410 05 4810 27 3110 49 0511 10 2854
& 09 22 3609 44 3 10 06.31 10 28 1510 49 48" IL 1052
119 23 2009 45.2210 07 15 10 28 5810 50 3111 in 53150
1219 24 0509 46 C610 07 5910 29 4110 SI 141 12 35+8
14 09 24 4909 46 5 10 08:4210 30 2410 51 57 11 13 18176
1619 25 3309 47 3210 09 2610 31 08 10 52 3511 14 01 +4
18/09 26 17/09 48 17 10 10 0910 31 5110 $3 221 [ !4 431+2
209. 27. 0109 49 01 10 10 5310 32 3410 54 OSII 15 26/10
22/09 27 45109 49 45 10 11 3610 33 1710 54 481 16 08178
241 9 28 29 49 50 29/10 12 20 10 34 OiI0 55 31 11 16 5136
26 9 29 14109 51 1310 13 02 10 34 44.10 56 1411 17 334
229 29 58109 st 57 10 13 47 10 35 27 10 50 sen 18 1032
34129 30 4209:52 4:10 14 3110 36 1010 57 3911 18 58130
32109 31 26.09 53 2410 15 14 10 36 5310 58 22 U 19 44 28
349 32 1009 54.0&f10 15 57 10 37 3610 59 051 20 2326
3609 32 5499 54 52110 16 41 10 38 1910 59 4811 21 0524
389 33 38|09 55 360 17 24110 39 0211 00 310 21 4 22
40 09 34 2209 56 110 18 08 10 39 451 01 130 22 3420
4209 35 C6 09 57 og 10 18 510 40 2811 0 560 23 13
4.3935 5009 57 47 10 19 3510 41 11 02 3911 2355 16
4409 36 3409 58 31 10 20 1810 41 5411 03 22 24 3:14
4819 37 18109 59 15 10 21 01 10 42 371 04 om 25. 2012
$29.38 0309 59 5810 21 45 10 43 2011 04 411 26 0910
8
5409 39 31 rooi 2610 23 1210 44 47 106 III 27 2;
5699.40 15 10 02 09 10 23 5510 45 3061 06 5411 28 OS
529 40 5910 02 5:10 24 3910 46 13 11 07 37 11 28 51
61
9 41 4310 03 37 10 25 2210 46. 5611.08 2011 29 34
5
4
3
30
me and Him and Him and #lm and
m and X and *
1
18
18
6
11
2
I
1
A TABLE ſewing the Declination to every two
501
***
Minutes of the Ecliptic
8 and m
8 and in 8 and mi 8 and in 1 % and in 18 and in
30
3
4
5
1
2
O
Minutesi ON
Minutes): 1
Opt 29 34 11 So 3012 11 26 12 32 05 12 52 3113 12 4460
211 30 161 si 1712 12 07 12 32 412 53 11 13 13 2558
411 30 5811 SE 5912.12 4912 33 272 53 5213, 14 05156
611 31 4111 52 4112 13 39!! 34 0812 $4 3313 14 4554
811 32 23 11 53 2312 14 12 12 34 4912 55 13 13 15 25.52
1011 33 051 54 0512 14 5312
35 39 12 55 5413 16 0510
12 11 33 47 11 54 4612 15 3512 36 012 56 35 13.16.4545
14'11 34 29 11 55 28 12 16 10 12 36 5212 57 1513 17 26 46
1611 35 nu 56 1012 16 57 12 37 3312 57 5613 18 0644
1811 35 5311 56 5212 17 39 12 38 1412 58 3013 18 41 42
2011 36
35 11 57 34:2 18 2012 38 3512 59 1713 19 2010
22 11 37 171 38 1512 19 0112 39 36112 59 57 13 20 0038
2411 37 591 58 57 12 19 43 12 40 17 13 00 3813 20 46 36
2611 38 41 11 59 3912 20 2412 40 5813 01 1813 21 26134
281 39 24 12 00 20 12 21 0512 41 39 13 01 5913 22 06132
3011 40 06 12 01 02 12 21 40 12 42 1913 02 3913 22 4630
32 49 48 12 oi 4422 28 12 43 20 3 03 23 23 2628
3411 41 30 12 02 25112 23 0912 43 41113 03 00 13 24 06/26
3911 42 12 12 03'07 12 23 5112 44 223 04 4013 24 4524
138
21 42 54 12 03 49 12 24 32 12 45 0313 05 2113 25 26122
401 43 36 12 04 3112 25 1312 45 4313 06 0113 26 06 20
4211 44 1812 OS 12 12 25 5412 46 2413 C6 42 13 26 4618
4411 45 0012 o5 54/12 26 35 12 47 0513 07 2213 27 2616
40.11 45 4212 06 3612 27 17.1.2 47 4613 08 o213 28 00114
4811 46 24 12 07 17 12 27 5812 48 2713 08 4313 28 4612
501 47 06 12 07 59 12 28 39 12 49 0713 09 23 13 29 2610
5211
47 4812 08 40 12 29 20 12 49 48113 10 0313 30 06 8
54111 48 30 12 09 22/12 30 01 12 50 29113 10 4413 30 406
15611 49 12 12 10 03 12 30 42 12 51 09 13 11 24113 31 2014
58
11 49 54 12 10 44/12 31 2412 51 5913 1 2 0413 32 06 3
"I 50 35 12 11 26 12 32 0512 52 3113 12 44113 32 460
29
28
27
26
25
N and a and my
N and a
U u U2
60
24
1
N and
N and w
N and
*
512
A TABLE jbewing the Declination to every two
Minutes of the Eclipric.
8 and in X and m 8 and mix and m8 and m
7
8
9
♡ and in
6
Minutes!
IO
Minutes
013 32 4913 52 32 14 12 0514_30_2414 30 2915 09 1860
213 33 25 13 53 11 14 12 44 14 32 0214 SI 0715 09 5558
4.13 34 0413 53 514 13 23 14 32 40114 51 4515 10 3250
613 34 44 13 54 30 14 14 02 14 33 19 14 52 2215 II 09 541
813 35 2413 55 ng 14 14 41 14 33 57 14 53 00 15 11 47152
1013 36 0413 55 4814 15 20 14 34_3514 53 38 15 12 24 50
1213 36 43 13 56 27114 15 39 14 35 14 14 14 16 15 13 01 48
1413 37 23 13 57 0714 16 3714 35 5214 54 53/15 13 39146
J013 38 0313 57 46 14 17 10 14 36 30114 55, 3415 14 1644
1813 38 4713 58 25|14 17 5414 37 0914 56 08 15 14 5342
2013 39 2213 59 0414 18 32 14 37_4714 56 4915 15 31.49
22 13 40 2213 59 4314 19 11 14 38 25 14 57 24 15 16 08138
2413 40 41 14 00 23114 19 5914 39 0314 58 0215 16 45136
2013 41 21 14 OI 024 202914 39 4914 58 3915 17 22 34
2813 42. 0014 01 414 21 0714 40 20 14 59 1715 17 5932
3013 42 4014 02 2014 21 40 14 40 5814 59 5515 18 30 30
3213 43 2014 02 5914 22 25 14 41 3615 DO 3215 19 13/28
3413 43 5914 03 3914 23 03114 42 1415 01 10 15 19 5020
B63 44 39 4 04 374 33 414 42 25 đi 48 20 2824
3813 45 1814 04 50 14 24 20 14 43 30 15 02 2515 21 05 22
40113 45 5814 05 3514 24 5914 44 0815 03 03 15 21 4220
4213 46 37 14 06 5414 25 3814 44 4015 03 40 15 22 1918)
4413 47 17 14 06 5314 26 1914 45 2415 04 1815 22 5016
4613 47 50 14 07 32 14 26 55114 46 02 15 04 5515 23 33114
4813 48 3514 08 11 14 27 3314 46 40 15 05 3315 24 10/12
501349 1514 08 50 14 28 1214_47_1815 06 10 15 24 4710
5213 49 54114 09 29 14 28 50 14 47 50 15 06 4815 25 241 8
15413 50 34/14 10 08 14 29 39 14 48 34115 07 2515 26 01 6
15 613 SI 13/14 10 42 14 30 0714 49 12115 08 02 15 26 38 4
5813 51 52114 11 26 14 30 45 14 49 50 15 08 40 15 27 15 2
60113 52. 3214 12 0514 31 24 14 50 29115 09 1815 27 51
23
19
18
Qand w and an
2 and the
Q and
n and more on
N and more on
21
20
IT
Å TABLE fewing the Declination to every two $13
Minutes
12
1.
Minutes 1
.
15 44 57416 03 Oc16
Minutes of the Ecliptic.
KX and m 8 and m 8 and m8 and m y and mi 8 and m
13
14
IS
16
17
Q
19.27 5115 46 116 04 1210
21 58 16 39 26116 55 3760
215
28 28/45 46 4016 04 48:16 22 32 16 40 00 16 57 8
415 29 08 15 47 2216 05 2316 23 0716 40 3516 57 45156
615 29 42 15 47 5816 05 5916 23 4216 41 10 16 58 1954
815 30 i8fis 48 35116 3416 24 17|16 41 4416 58 5352
1915 30 55115 49 16 07 1016 24 5216 42 19 16 59 27150
1215 31 3445 49 4:16 07 46116 25 20116.42 53117 oo 01 48
14/15 32 ogl's so 2316 08 22 16 26 0 16 43 2817 co 3546
1015 32 4615 51 16 08 5716 26 38 16 44 0317 01 og 14
1815 33 22 15 51 34116 09 33.16 27 1? 46 44 37117 01 43 42
2015 33 5915 52 11 16 10 09 16 27 4916 45 1217 02 17 49
22 15 34 30 15 52 40 16 10 44 16 28 2416 45 4617 02 51138
2410 35 12 15 53 246 11 20 16 28 59116 46 2017 03 25136
20115 35 45 15 54 00 16 11 55116 29 34 16 46 55117 03 5934
5 35 2015 54 3015 12 ?1116 30 0916 47 25117. 04 33 32
315 37 0:15 55 12 16 13 6115 30 44 16 48 0117 os 06 30
32.5 37 39 15 55 13 13 4216 31 116 48 38 17 og 4-20
3
15 38' 1615 56 24 16 14 1716 31 5:116 49 1217 06 14 26
15 38 5415 57 oC 16 14 53116 32 2816 49 47:7 06 48 24
15 39 29115 57.36 16 15 2816 33 09 16 go 2117 07 21 22
is 40 og 15 58 12 6 16 0416 33 3846 50 513 07 5520
42
15 40 4215 58 416 16 3916 34' 13116 51 30 17 C8 2910
4915 41 1915 59 2115 17 15 16 34 4216 52 04 17 09 03/16
4,6
15 4155116 00 OC 6 17 5016 35 2316 52 36/17 09 36114
48
15 42 32116 00 36 16 18 2516 35 57 16 53 1217 10 10 12
50
15 43 od 16 or 12116 19 01 16 36 32 16 53 46.17. 10 44 10
5.2
15 43 44 16 01 40 16 19 3616 37 07116 54 21117 II 171.83
$415 44 21116 02 24 16 20 1116 37 426 54 55117 11 516
$6
58
15 45 34.16 03 30 16 21 2216 38 51116 56 0314712 58 2
60
15 46 10 16 04 1216 21 58 16 39 26 16 56 37 17 13 3110
17
15
14
13
N and more on l and one and a lot and see then and more Su and me
2
1
3.6
38
1
16
12
1514 A TABLE Thewing the Declination to every two
20
21
Minures ) i
Minutesla
Q
50 18 27 1818 42 271241
Minutes of the Ecliptic.
8 and m 8 and in 8 and m 8 and mi 8 and my and in
18
19
22
23
17 13 30 17-30 0717 46 25118 02 22 18 18 03 18 33 24
347 14 0517 30 4017 46 57118 02 5518 18 34/18 33 5158
417 14 38 17 31 13 17 47 25118 03 26 18 19 018 34 2516
617 15 12 17 31 4617 48 01/18 03 58 18 19 36 18 34 5 154
8 17 15 45 17 32 1817 48 34 18 04 30 18 20 07|18 35 25152
017 16 18 17 32 5147 49 (018 O5 0218 20 318 35 5110
12 17 16 5217 33 2147 49 3818 05 3318 21 0918 36 2048
14 7 17 2517 33 5717 50 1018 06 05 18 21 418 36 56
1017 17 5817 34 30 17 50 42 18 06 36 18 22 11 118 37 20 +4
1817 18 3217.35 0212 51 1.,118 07 08 18 22 42 18 37 5€ 142
2017 19 OS 17 35 3517 51 401
8
07 39
18 23 12 18 38 26 +
2217 19 38 17 36 0817 52 it; 18 08 1
08 11 18 23 4:18 38 5:13
24 17 20 1217 35 40 17 52 5118 08 42 18 24 14 18 39 27136
2017 20 4517 37 13 17 53 2318 09 13/18 24 45.8 39 51134
28 17 21 18 17 37 46|17 53 5518 09 45 18 25 1618 40 2,132
397 21 5117 38 1917 54 2718 10 1618_25_4718 40 şi o
3217 22 2517 33 51117 54 5818 10 4718 26 17 18 41 27/28
34.12 22 5817 39 2312 55 30118 19118 26 4:18 41 57 26
0218
38 17 24 0417 40 25 17 50 34 18 12 24 18 27 4918 42 5722
46117 24 371 7 41 0117 57 0618 12 5218 28 19118 43 2720
4217 25 1017 41 3317 57 38 18 13 2318 28 50 18 43 57/18
4417 25 4317.42 0617 58 10 18 13 5518 29 2018 44 2416
40 17 26 16 17 42 38117 58 4218 14 26 18 29 51118 44 50 14
8:17 26 4917 43 17 59.1318: 14. 5718 30 2 118 45. 202
17 27 22 17 4.3 4317 59 4518 15 2818 30 5218.45 Solo
5217 27 5517 44 1518 Co 1718 IS 5918 31 2218 46 208
15417 28 2817.44 4818 og 4918 16 3018 31 58 46 546
5617 29 01 17 45 2018 01 2018 17 01 18 32 22118 47 25 4
5817 29 34 17 45 52118 or 5218 17 32 18 32 54118 47 55
60117 30 0717 46 25118 02 23 18 18 0318 33 24 18 48 25
6
my
and many
N and more
on and on
N and more
:
II
IC
9
8
7
anderen
2 and
N and
JA TABLE jbewing the Declination to every two
515
:
Minutes of the Ecliptio
8 and in and m 8 and m8 and an
and m
Y and m
28
24
26
25
27
Minutes!
29
o
Minutes on
18
20/18 54 4
17 48 25 19 03 0519 17 25 19 31 2119 45 0319 58 2260
218 40 5419 03 34 19 17 5419 31 5:19.45 3019 58.4858
48 49 2419 04 0319 18 2219 32 219 45 5719 59 1456
618 49 5419 04 32 19 18 5019 32 4819 40 2419 59 40184
81.8 50 231 905 01 19 19 19 19 33 1119 46 51 20 00 06 52
1918 50 519 Os 3019 19 47 19 33 4:19.47 18 20 00 32.52
1218 SI 2219 05 59 19 20 1519 34 10 19 47 44 20 00 58.48
1418 51 5219 06 28119 20 4319 34 3819 48 11 20 01 2446
1018 52 24 906 5219 21 111935 og 19 48 38 20 of 5044
18
52 519 67 2019 21 4019 35 33 19 49 05 20 02 10/12
2.8 53 21
9 07 5419 22 0819 30 oc 19 49 32 20 02 42 4:0
2218 53 59 08 239 22 30119 36 2¢19 49 5820 03 08];8
2418 54 19 08 5219 23 0419 36 519 50 2520 03 34 36
9 09 2119 23 3219 37 22 19 50 5520 04 034
2811855 9 09 4919 24 0019 37 5419 SE 18/20 04 2532
318 55 481 9 10 1819.24 2419 38 15 19 51 4520 04 51 30
32118 56 1119 10 47 19 24 16 9 38 419 52 11/20 og 1628
18
40119 il 1519 25 24 19 39 1119 52 38/20 OS 42 26
13618 57 11 4419 25 5219 39 39 19 53 6420 06 08124
3818 37 45 9 12 13 19 26 2019 40 C619 53 3120 06 34 12
40118 58
9 12 471920 1919_403319 53 5720 07 00 20
42 584111913 1019 27 959 41 oC 19 54 23120 07 25118
18
59. 1219 13 3819 27 43 19 41 279 54 520.07 S1116
4018 59 49 19 14 07 19 28 119 41 5419 55 1620 08 17|14
4819.00 11 9 14 35/19 28 319 42 2.955 4320 08 4212
SC 19 CO 4219 15 04/19 29 07 19 42 48 19 56 og 20 09 0810
:
$19 01 0919 15 32 19 29 341 9 43 1519 56 35 20 09 3:8
5449 OI 3819 16 0 19 30 02 19 43 42 19 57 0220 09.596
59 19:02 0719 16 29 19 3 30 19 44 09119 57 29|20 10 25 4
18
19 02 30 19 16 5719 30 $819 44 319 57 58/20 10 592
6819 03 05 19 i 2519 31 2519 45 03/19 58 2 20 11 15 o
4
3
30
N and everything
82 and
N and more on
0 X
3411
56
3
F
2
I
N and
12 and more on
2 and
516
A TABLE Jewing the Declination to every two
Minutes
I
2
4
Minutes
1
58
14820 21 1920 33 3520 45 38 20 56 5721 08 04/21 18 471
Minutes of the Ecliptic
I and 7 II and I II and II and I II and 4
II and
30
.3
5
2011 15:20 23 5 20 36 oC 20 47 420 59 1321 10 1660
2/20 11 4120 24 84 20 36 2420 48 11 20 59.3521 30 37
4/20 12 06 20 24 38120 36 4820 48 34 20 59 5:21 10 5856
620 12 32 20 25 03 20 37 12 20 48 5 21 00 2021 ili 4
8
20 12 57 20 35 27 20 37 2012 20 21 00 4221 IT 41152
1920 13 2310 25 52120 38 og 20 49 43 21 OI
LOL 03121 12 0:57
1220 13 46 20 26 17 20 38 2420.500
ei ol 27 21 12 24/48
1420 14 14|20 26 41 20 38 40120 50 2921 01 45121 12 4646
16 20 14 3920 27 06 20 39.3220 50 521 02 11 21 13 07 44
18 20 15 0420 27 3020 39 3922 5 1526 02 3421 13 29142
2020 IS 2320 27 5520 39 5 20 11 321 02 5621 13 5040
22 20 15 5420 28 20 20 40 22 20 52 0121 03 1821 14 1138
24 20 16 18 20 28 44 20 40 4520.52 2. 21 03 4921 14 3336
20 20 16 4520 29 og 20 41 0920 52 421 04 02 21 14 5434
28 20 17 1020 29 32/20 41 3320 53 41 04 2521 15 15 32
30 20 17 35 20 29 57 20 41 520 53 3341 04 47 21 15 3730
32 20 18 oC 20 30 21 20 42 20 20 53 5021 05 09 21 15 5:28
34 20 18 25/20 30 4620 42 43 20 54 19 21 05 31 21 16 1926
36 20 18 50 20 31 1 20 43 07 20 54 4121 05 5321 16 40 24
38 20 19 15 20 31 34/20 43 320 55 0421 06 1521 17 02 42
40120.19 10 20 31 5820 43 54 20 55 27 21 06 2621 17 23
4220 20 05|20 32 22 20 44 18 20 55 5021 06 58 21 17 44
44/20 20 3420 32 47 20 44 4° 20 JO 1221 07 20/21 18 0516
4620 20 55 20 33 11 20 45 og 20 56 3521 07 4221 18 2014
12
5020 21 4520 33 5920_45 $120 57_2041
08 26 21 19 0810
8
52.20 22 09/20 34 23 20 46 14 20 57 4321 08 48/21 19 29
5420 22 34 20 34 47/20 46 38 20 58 05 21 09 10 21 19 596
5020 22 5920 35 11 20 47 01 20 58 24 21 09 32 21 20 11 4
2
58 20 23 2420 35 35 20.47 2420 58 57 21 09 5421 20 32
60 20 23 5020 36 0020 47 48/20 59 13 I 10 16 21 20 5310
29
28
27
26
25
24
-1
Wand
pue $
w and 5 w and go and ve
1
20
18
-
ment
Wo and
Correct Table,
Shewing the
RIGHT ASCENSION
To every Two Minutes of the Firſt Quadrant of the
ECLIPTIC
The greateſt DecLINATION being
23 deg. 29 min. which by common Addition or
Subſtraction, according to the Rules delivered
in Page 505, may be made to ſerve for the pa
ther Three Quadrants.
Y yy*
526
A TABLE ſheiving the Right Aſcenſion anſwering
to every two Minutes of the firſt Quadrant of the Ecliptic.
r
r т
r т
hr
no
}
.
Minuteslo
I
2
3
4
5
1
o
Minutes 1 1
2
I 2
20
09 00 Oloo 55 0201 So 0402_45 0703 40 10/04 35 15e
00 Ol sopo 57 50 51 5402 46 5703 42 00 04 37 052
4
00 03 40100 58 +2 01 53 4402 48 4703 43 5004 38 554
do os 30 01 00 3201 55 3402 50 37103 45 400440 4516
00
20 07 20131 02 2201 57 2402 52 2703 47 30134 42 35 €
co 09 140 04 12 01 59 1502 54 17123_492104 44 2610
00 il QUOI C6 0202 or 0502 50 0703 51 1104 46 10 12
1400 12 500 06 5202 02 5502 57 5703 53 01 04 48 of 14
100 14 401 08 42 02 04 45102 59 48|03 54 5104 49 5616
18
20 16 300 11 3202 06 35123 oi 3803 50 43/04 51 461
90 18 2001 13 22102 08 2503 03 2803 58 3104 53 342
2200 20 OCT 15 03|02 10 1503 05 1804 00.2234 55 272
2400 22 ocor 17 03 02 12 0503 07 08-4 02 1204 57 17124
20-0 23 531 18 53 02 13 55103 08 5804 04 0204 59 07126
2800 25 41 20 43 02 15 4603 10 4804 05 5205 00 584.
30 00 27 3101 22 33 02 17 36 03 12 3804 074205 02_48
3200 29 2101 24 23 02 19 2003 14 2804 09 3205 04 38132
34100 31 11 01 26 13 02 21 1603 16 1997 235 06 2?
3600 33 ollor 28 0332 23 0603 18 04 13 12 05 08 183
38 00 34 5 Tor 29 5302 24 5403 19 5510+ 15 0305 10 0838
4400 30 4101 31 4302 26 4033 21 4904 16 5305 11 5940
42700 38 310 33 3302 28 3003 23 39104 18 4205 13 49 +2
44-0 40 2101 35 23 02 30 2623 25 25 24 20 33105 15 3944
46-0 42 10 37 14 02 32 1623 27 1924 22 2405 17 3440
4800 44 01pi 39 04/02 34 C603 29 0904 24 1425 19 2 48
500 45 50 40 5402_35 50 03 30 590425 04105 21 IC 59
52100 47 4101 42 44 02 37 4703 32 50 04 27 545 22 0152
54100 49 31 01 44 34102 39 3703 34 40 04 29 4405 24 554
5000 51 210 46 2402 41 4703 36 30104 31 3415 26 4156
5850 53 1201 48 14 02 43 3703 38 2004 33 255 28 348
5100 55 0201 So 04102 45 0703 40 10104 35 1505 302260
3
4
5
180 00
180.00+ra, 360.00-T=H.
1
O
1
2
AT ABLE, Jewing the Right Aſcenſion anſwering to 527
every two Minutes of the firſt Quadrant of the Ecliptic.
ԴՐ
r
ՕՐ
no
mo
6
7
8
9
10
II
o
Minutes!
o
Minutes ol~
I
2
6
10
16
05 30 22 06 25 31 07 20 42 08 15 55109 11 11 10 06 30
zlos 32 12/06 27 2107 22 32 08 17 45109 13 0210 08 20
4105 34 0206 29 12 27 24 23108 19 3019 14 5210 10 11 4
blos 35 52.06 31 0207 26 13 08 21 2009 16 4310 12 02
glos 37 42 06 32 52 07 28 04/08 23 17 09 18 33 10 13 53
81
10 05 39 3306 34 4307 29 54 08 25 07 09 20_24/10 15 44
12105 41 2306 36 3307 31 45/08 26 58/09 22 1410 17 35/12
14
05 43 13 06 38 23|07 33 35 08 28 4809 24 0510 19 2614
1605 45 04/06 40 1407 35 2008 30 3909 25 5510 21 17
1805 40 54106 42 0407 37 10 08 32 29/09 27 4010 23 0718
2005 48 440643_5407 39 06 8 34 2009 29 30 10 24 56/20
23105 50 3506 45 45|07 40 5708 36 11/09 31 27/10 26 48/22
2405 52 25|06 47 35107 42 4708 38 01 09 33 1810 28 3924
2605 54 15|06 49 25107 44 3708 39 5 1409 35 0810 30 29126
2805 56 0606 SI 1607 46 2808 41 42 09 36 5910 32 2028
3005 57 5606 53 0607 48 1808 43 3 207 38 5910 34 11/20
3205 59 4606 54 507 50 0908 45 2309 40 41 10 36 0232
3406 01 3706 56 4707 51 59108 47 1409 42 3410 37 53 34
03 27
58 3707 53 50408 49 0409 44 22 10 39 4436
28.06 05 1707 CO 2707 55 4008 50 55109 46 12/10 41 34
40 06 07 0827 02 1807 57_3108 52 4509_48 0310 43 2549
+2106 08 5807 04 0907 59 21/08 54 3609 49 54.10 45 1642
4406 10 48107 05 5908 1108 56 2009 51 4410 47 07144
4606 12 3927 07 5008 oz 0208 58 1709 53 3510 48 57146
4806 14 2907 09 4008 04 5209 oo 07109 55 2650 so 48 48
5006 16 197 li 3008 06 4209 or 57109 57 1710 52 49150
5206 18 1007 13 21108 08 3309 03 4809 59 08 10 54 4052
541
20 00 07 15 11 08 10 23/09 05 3910 00 5810 56 21154
5606 21 5007 17 oro8 12 1409 07 2910 02 4810 58 II SO
58/06 23 4107 18 5208 14 0429 09 2010 04 3911 00 0258
6006 25 31127_20 4208 15 55109 11 111006
06 30 12 OI 53
6
7 1
.:9
180.00-r=m, 180.00+pE, 360.00-14.
3600
06
100
60
53)
8
IO
IL
1
Y yy 2
1528 A TABLE Shewing the Right Aſcenſion anſwering
to every two Minutes of the firſt Quadrant of the Ecliptic.
r
r
т
r т
ԴՐ
I 2
13
14
15
16
17
Minutesiºl
Minutes! 1
0
O
2
56
2
16c
11 01 53111 57 2012 52 5113 48 26 14 44 0615 39 51
11 03 4411 59 1112 54 42 13 50 17 14 46 5715 41 4258
411 05 3512 01 02 12 56 3313 52 08 14 47 4815 43 34
6
11 07 2612 02 5312 58 24 13 53 5914 49 4015 45 264
8
11 09 1062 04 44.13 CO 1513 55 50114 51 31 15 47 17
52
LU 11 0712 06 3513 02 013 57 4114 53 22 15 49 CE 59
12U 12 58 12 08 2613 03 5813 59 3314 SS 14 15 51 0048
1401 14 4912 10 1713 05 4914 OL 2414 57 05 15 52 5240
1611 16 4012 12 08 13 07 4014 03 1514 58 5715 54 4;
1811 18 31 12 13 5913 09 3114 05 07 15 00 4915 56 3
2011 20 21112 15 S013 11 2214 5815 02 49115 58 264
2211 22 1212 17 41 13 13 13 14 o8 49.15. 64 32 16 Co ig
2411 24 0312 19 32113 15 0414 10 4115 06 23 16 02 1.
201 25 5412 21 23 13 16 5514 12 32 15 08 1416 04 02
28
![ 27 45112 23 1413 18 4614 14 2315 10 of 16 05 54
301 29 36 12 25 05 13 20 38 14 16 1515 U 58 16 07 45
3211 31 27 12 20 5613 22 29 14 18 615 13 45116 09 35
3411 33 1812 28 47 13 24 2014 19 57 15 15 4016 11 2426
361 35 og 12 30 38113 26 11 14 21 49115 17 3216 13 224
3811 36 5912 32 29 13 28 02 14 23 40 15 19 23 16 IS I
40111 38 5012 34 2013 29 5314 25 3115 25 15 16 17 0212
4711 40 +112 36 13 31 4514 27 23 15 23 09 16 18 591
4411 42 32 12 38 0213 33 36 14 29 1415 24 5810 19 46 16
4611 44 23 12 39 5413 35 3711 4 31 og 15 26 56 16 22 314
4801 46 1412 41 4513 37 18|14 32 57115 28 41 16 24 3412
SCU 48 0512 43 36 13 39 0914 34 48115 30 32116 26 2!!!
5211 49 5612 45 27 13 41 00 14 36 3915 32 2416 28 13
5411 51 4712 47 1813 42 521 4 38 3115 34 10 16 30 OS
5611 53 38 12 49 09 13 44 43 14 40 2215 36 07116 31 st
5811 55 29 12 SI 0013 46 4414 42 1415 37 51 16 33 4
6CLE 57 2012 52 5113
48 2614
44 0015 39 5 15 35 40
13
15
16
18000-r=19,9780.00+r=, 360.00-r=*
1 2
14
mu
nr.
19
20
21
22
Minuteslo
Minulesio
1
2
10
8 12
2
14
16
20
22
2
10 SO 44117 46 31118 42 3419 38 44 20 35 5921 31 21
A TABLE, Mhewing the Right Aſcenſion anſwering 527
to every two Minutes of the firſt Quadrant of the Ecliptic.
r
r т
" r
18
23
15 35_4017 31 3518 27 36 19 23 4.20 19 5821
16 18
216 37 32 17 33 27 18 29 28 19 25 31 10 21 5021 18 11
4
4 16 39 2417 35 1918 31 2019 27 2420 23 43 21 20 01
616 41 16 17 37 1
17 37 11 18 33 1219 29 21 20 25 3021 21 57
8116 43 07 17 39 0318 35 0419 31 1420 27 29 21 23 so
IcI6 44 59 17 40 55
18 36 56 19 33 0720_29 22 21 25 43
12/11 46 57 42 47/18 38 4919 34 5920 31 132 1 27 35
14.15 48 4217 44 39 18 40 1219 36 52/20 33 0625 29 28
18
1815 52 20 47 48 23 18 44 21 19 4.0 3920 36 51121 33 13
2010 54 18
17 50 1518 46 1914 42 220 38 44 21 35
16 jó id:754078 48
7 52 07:8 48 019 44 2420 40 3721 35 29
6 58.3912 53 5918 50 0319 46 13:0 42 2421 38 52
1519 52 17 55 51118 jI 5519 48 0620 44 2221 40 45
21 42 38
?47 01 45117 57 4318 53 4719 49 5920 46 14
26
17 03 37 12 59 3518 55 3919 51 50 48 07 21 44 3.
(5 29 18 01 oo
18 57 31119 53 4320 50 00|21 45 2 4
7 07 31118 03 1918 59 2219 55° 3520 51 534 48 17
34
3 7 09 19 18 05 11/19 OI
1619 57 2820 53 4521 50 10156
7 11 64118 07 0319 03 og 19 59 21|20 55 3821 5203
* 7 2 12 56.18 08 55119 04 51120 OI 1520 57 3121 53 56
12
7 14 48118 10 4719 06 53 20 03. 0523 59 23 21 55 49 12
+447 10 40 18 12 39 19 08 44 20 04 57 21 01 1021 57 42 44
+6
17 18 32 18 14 3119 10 3720 06 5021 03 09 21 59 3546
48
17 20 24 18 16 23 19 12 30 20 08 4:21 oś 0222 01 28 48
22 16/18 18 1519 14 2320 10 3521 05 54
22 03 2015
5217 24 08 18 20 07 19 16 1520 12 2821 08 47 22 05 14152
5417 26 00 18 22 00 19 18 07 20 14 21 21 10 40 22 07 07 54
56
17 27 51 18 23 52 19 19 59 20 15 13 21 12 33 22 oy ods
58
'7 29 43 18 25 4419 21 51 20 18 0621 14 2622 10
53
17_31_35/18
27 3019 23 4420 19 58/21 16 18 22 12 4660
18
19
23
180.00m=2, 180.00+r=#, 360.00-r=*
IC
2
10
12
38
c
O
----
Sclia
58
50
20
21
22
Maritimes
530 A TABLE (bewing the Right Aſcenſion anſwering to
28
Minutes loi
Minutes
wa habari
23 18
1
2
4022 56 0823 52 47 24 49 3525 46 31 26 43 34127 40 48 101
every two Minutes of the firſt Quadrant of the Ecliptic.
24
25
26
27
29
0122 12. 4623 69 2024 06 0225 02 5225 59 49 26 50 55
2 22 14 3923 11 1324 07 56125 04 4526 Oi 47 26 58 49
422 16 323 13 0024 09 45145 06 39 26 03 37 27 00 44
622 18 20 23 15 OC:4 11 4225 08 33.26 05 31127 02 39
822 20 11 23 16 5344 13 3525 10 27,26 07 20127 04 33
I( 22 22 1
40144 15. 24 25 12 21 20 09 1927 05 27
12 22 24 09 23 20 4424 17 2,25 14 1526 1 1427 08 22
1422 25 50 23 22 3:24 19 1025 16 08:20 13 00127 10 16 14
1622 27 5123 24 2044 21 1025 18 02 26 15 C21254 12 10
18122 29 4423 26 20 24 23 0425 19 56 26 16 50127 14 og
2:22 31 3:23 28 13 5c|
14 24 57 25 21 5026 18 5027 15 55
2222 33 3423 30 C624 26 51125 23 44/26 20 44 27 17 S;
2.
24122 35 2323 32 oc]24 28 4525 25 38|26 22 3827 19.4
2622 37 16 23 33 5324 30 38 25 27 32 26 24 327 2 1 43
28 22 39 0923 35 4624 32. 3225 29 2626 26 2027 23 37
30 22 41 0:23 37 4024 34 2025 31 2026 28 2127 25 32
32 22 43 5623 39 334 36 1925 33 13 20 jo 15127 27 26
34122 44 4523 41 26124 38 1325 35 07|26 32 09127 29 2
14
3022 46 42 23 43 20 24° 40 0725 37 01126 34 03 27 31 1
38122 48 35123 45 14 24 42 0 25 38 5526 35 577 33 os
422 50 2823
47 of 24 42 5425 40 4926 37 517 35 03
+
4222 52 2323 49 01 24 45 48125 42 43 26 39 40127 36 58+2
44 22 54 1523 50 54 24 47 4125 44 37 26 41 40 17 38 53144
16
18
2
20
28
2
6
5
0
48
18
50
-
i4 해
​22 58 0123 54 41124 5 1 2925 48 2526 45 291247 42 42
259 54 23 56 3424 51 2325 50 1926 47 23 27 44 36
52123 01 4723 58 27|24 53 3725 50 1326 49 17127 46 352
5423 03 4124 09 31 24 57 1125 54 0726 51 1247 48 25
56
23 05 3424 02 14 24 59 0425 50 01 26 53 0627 50 1956
23 07 2-24 04 03 25 Co 5825 57 5520 55 0027 52 14,3
60
60
23 09 20/24 (60225 02 52 25 59 4926 56 5527 54 og
25
27
28
29
180.00-rum, 180.00+1=, 360.00-r=#.
SE
24
26
presente
A TABLE, Msewing the Right Aſcenſion anſwering to
531
every two Minutes of the firſt Quadrant of the Ecliptic.
1
8
8
8
४
8
8
O
I
2
3
4
5
Minuteslol
Minutes
0
1 - 1
C27_54 0928 51 3229 49 03130 46 4421 44 33 32 42 3:
2 27 56 03 28 53 27 29 50 5830 48 3931 46 28 32 44 28
427 57 5828 55 22 29 52 5330 50 3431 48 24/32 40 244
627 59 53128 57 17129 54 4930 52 3031 50 2432 48 21 6
&28 or 47/28 59 J11-9 56 4430 54 2 631 52 1032 50 17 8
1:28 03 42 29 Oi 06/29 58 3930 56 2131 54 12132 52 110
12 28 OS 3729 03 01130 00 3430 58 1713 1 56 0032 54 ogliz
1428 07 31.29 04 5630 02 2931 00 1231 58 0332 56 0514
1028 09 26 29 CO 5'30 04 2431 02 037 59 5932 58 01 16
1828 11 21 29 08 4639 06 2031 04 04/32 OI 55132 59 5818
2028 13 1529 10 4110 08 151?! CO 0032 03 51133 01 541?0
22 28 15 10:9 12 3630 1031 07 55 32 05 47133 03 5: 122
24 28 17 05 29 14 31 20 12 C031 09 5032 07 4333 05 47124
26 28 18 59129 16 20130 14 0131 II 4532 09 3933 07 43126
228 20 54129 18 21 30 15 5631 13 4132 11 3533 09 398
38 22 49 29 20 16/?0 17 5231 15 3732 13 31133 11 3530
37128 24 44129 22 01130 19 473: 17 32/32 15 27 33 13 31 32
34 28 26 3929 24 0630 21 4231 19 28/32 17 23:3 15 2734
2428 28 3429 26 01 30 23 38/31 21 24132 19 1933 17 24136
30 28129 27 5630 25 24:1 23 19/32 21 1533 19 21 130
1128 32 23129 29 51139 27 2931 25 15 32 23 11 33 2 1
I10
12/28 34 18 29 31 4 130 29 2531 27 01 32 25 0733 23 131+2
4 8 30 1329 33 42130 31 231 29 06132 27 03133 25 091+1
+428 38 18 29 35 37 30 33 15|21 31 0232 28 5933 27 03/16
448 40 03 29 37 3230 35 11 32 5832 30 5033 29 02 48
8 41 57 29 39 2730 37 06 21 34 5432_32 5233_305
30 5211
$22
43 5229 41 22 30 39 0131 30 5,2 34 4€ 33 32 5413
5428 45 47 29 43 17 30 40 57131 38 46|32 36 4433 34 51154
5628 47 42 29 45 1230 42 5231 40 41132 38 4033 36 47150
528 49 37/29 47 07/30 44 48137 42 37 32 43 3033 38 44 18
6
28 5 1 32 29 49 03130 46 44131 44 33 32 42 3:33 40 4150
3
4
2
180.00-8=, 180.00+8=M, 360.00-8
38
28
50:8
1
2
#
532
A TABLE, ſewing the Right Afcenfion anſwering to
Minutes!
6
10
II
O
12 4134 03 5835 02 21 36 00 54 36 59 36137 58 2938 57 32/30/
every two Minutes of the firſt Quadrant of the Ecliptic.
8
8
४
8
8
ៗ
8
9
37 40 4134 38 59135 37 28136 36 0037 34 55138 33 5.
233 42 37 34 40 5535 39 2036 38 0337 36 52138 35 52
43 44 33/34 42 5235 41 22 36 40 0037 38 50138 37 5
6
33 40 3934 44 45 35 43 19136 41 58.37 40 4€ 38 39 48154
{
33 48 2034 46 46 35 45 1026 43 5537 42 4638 41 40152
3 50 2234 48 4315 47 1326 45 53137 44 44 38 43 44
121,3 52 19 34 50 4 35 49 11 36 47 5137 46 42138 45 45
1433 54 15134 52 3075 51 0836 49 48 37 48 4038 47 4116
16
33 56 12134 56 3335 53 0536 51 46 37 50 3838 49 35+4
18:3 58 934 56 30135 55 02 36 53 4437 52 35138 51 34+2
2034 UC 05124 58 27:5 56 5936 53 41 37 54 33138 53 319
2234 02 01 34 58 24 35 58 5636 53 38 37 56 3138 55 3438
Minutesl 1900
50
48
$
26
IN & 1000
22
42
34
18
16
33 05 5435 04 18 30 02 51137 01 3338 00 2738 59 31
2 3 3 07 5:35 có 1s|36 04 4s|37 03 333 02 2539 01 28:
3434 09 4835 08 12 36 06 4637 os 2938 04 2339 03 27
32 34 11 4435 10 0936 08 4337 07 2638 06 2139 05 25
3433 134 135 12 006 10 4337 09 2438 08 19139 07 23
2634 15 3835 14 03136 12 3837 11 2238 10 17139 09 22
241
38
34 17 34 35 16 oc 36 14 35127 13 1938 12 15139 11 2
+ 34 19 31 35 17 5,116 16 32 37 15 1738 14 13/39 13 18
21 28 35 19 540' 18 3437 17 1538 16 039 15 17
44 34 23 24 35 21 536 20 27 37 19 12 38 14 0939 17 15
+634 25 2135 23 48135 22 24 37 21 1038 16 07139 19 17
34 27 18/35 25 45136 24 24 37 23 0838 22 OS 19 21 11
5934 29 1435 27 42 36 26 1937 25 0638 24 03139 22 1
5234 31 1935 29 39 36 28 1637 27 0438 26 0139 24 09
544 33 08135 31 37/36 30 1437 29 0238 27 5939 27 €
56
4 35 og 35 33 34 36 32 11 37 30 5538 29 5739 29 00
58:+
4 37 02 35 35 31 36 34 08 37 32 5738 31 5539 31 04
34 38 59.5 37 281:6 36 7637 34 55138 33 5439 33 C3
8
1
180.00-8=2, 180.00+8=m, 360.00_=.
48
bu
6
7
0
1.
1
A TABLE Sewing the Right Aſcenſion anſwering
533
to every two Minutes of the firſt Quadrant of the Ecliptic.
8
a
8
8
8
a
I 2
13
14
15
I 6
17
Minutesiol
Minutes
1
2
IC
I 2
IC
20
22
24
20
39 33 03140 32 22 41 31 53 42 31 34 43 31 26 44 31 29c
39 35 01 40 34 21 4.1 33 5242 33 3343 33 26
44 33 29
439 36 59110 36 2041 35 5142 35 33 43 35 204+ 35 344
29 38 58 40 38 1941 37 5042 37 33 43 37 2044 37 306
39 40 56 40 40 18 41 39 451+2 39 32 43.39 26 44 39 348
39 42 5540 42 17 41 41 442 41 3243 41 26 44 41 31110
39 44 5 440 44 1641 43 48 42 43 32 13 43 20 44 43 31 12
39 46 52 40 46 1+41 45 47 42 45 32 +3 45 26 44 45 3114
39 48 50 40 48 13 +1 47 4642 47 31 +3 47 26 44 47 3210
18
39 50 49 40 50 12 41 49 4642 49 3143 49 26 44 49 3218
39 52 47 10 52 11 41 51 4512 51 3043 ST 26 44 51 33120
39 54 46 40 54 1041 53 44 42 53 3443 53 26 14 53 33/22
39 56 45 40 56 0941 55 44 42 55 30 43 55 20 +4 55 34/24
39 58 43 10 58 0841 57 43+2 57 2913 57 2044 55 312
28
40 00 42 11 co 0941 59 42142 59 2943 59 27 44 55 34
30
40 02 411!1 02 0042 01 42 43 01 2944 OI 27/45 OD 352
32
40 04 3941 04 0542 03 41/43 03 29/44 03 2745 03 312
40 06 38141 05 04 42 05 4143 05 2944 05 271+5 05 304
70
08 3:141 08 0342 07 41 43 07 29 44 07 2745 07 343
40 10 35|41 10 02 +2 09 4043 09 2844 09 271-5 09 3 38
40/40 12 3441 12 01142 11 3943 11 28 44 11 2745 11 37110
4240 14 3341 14 01 42 13 39 +3 13 2844 13 271+5 13 34+
441+0 16 3241 16 00:42 15 3043 15 28 44 IS 2145 15. 34
4640 18 31141 17 5942 17 3743 17 2744 17 28 +5 17 39 10
43 20 30 41 19 58 42 19.3743 19 27144 19 28 15 19 39/41
50 40 22 2841 21 57 42 21 36143. 21 27 44 21 28 15 21 4015
5240 24 27 41 23 56 42 23 30 43 23 27 44 23 2245 23 40
5440 25 2641 25 5542 25 36 43 25 27 44 25 281+5 25 41
5640 28 2441 27 5442 27 3543 27 2744 27 245 27 47 56
$840 30 23141 29 5342 29 34 43 29 27/14 29 2¢1 's 27 43153
60
40_32 22 41 31 5342 3 34+39' 20144 31_2815 31 436
13
14
IS
16
180 00-8=m, 180.00+8=m, 360.00-83
34
36
138
48
12
..!
1
17
Z z z
534
A TABLE jbewing the Right Aſcenſion anſwering
to every two Minutes of the firſt Quadrant of the Ecliptic.
8
8
8
a
8
1
18
19
20
21
22
23
Minutesi olm
Minutes
1
2
2
4
6
10
12
1
16
16
18
I
20
22
20
45 3 4346 34 oz|47 32 43 48 33 3949 34 28 so 35 36oc
45 33 4446 34 08 47 34 44 48 35 32 49 36 3050 37 38
+5 35 44 46 36 0947 36 4548 37 3449 38 3250 39 40 4
45 37 4546 38 1047 38 47148 39 3649 40 34150 41 436
8
45 39 45 46 40 1147 40 49 48 41 3849 42 3050 43 45
45 46 4646 42 12 47 42 50 48 43 4049 44 3450 45 43110
45 43 47 46 44 14/47 44 52 48 45 4149 46 4450 47 5
IA
45 45 47 46 46 1547 46 53148 47 43 49 48 41 10 49 53114
45 47 48 46 48 1047 48 5448 49 4549 50 450 51 55
45 49 49 46 so 17147 So 5648 51 4649 52 47 50 53 58
45 51 50 46 52 18 47 52 58 48 53 48 49 54 49 50 56 od 20
5 53 5145 54 19 47 55 oC 18 55 5049 55 550 58 0922
45 55 52 76 56 20 47 57 148 57 5:19 585 lisoo 0624
45 57 5410 58 21 47 59 0348 59 5450 00 5551 02 08126
28
45 59 53 +7 co 22/48 o os 19 01 50 50 03 5751 04 D 28
46 OL S047 02 24/48 ož 06 49 03 571 50 05 Oc151 06 1420
32
46 03 54 +7 04 2548 05 07 49 05 5950 07 02 51 08 1632
3445 05 55 47 06 2048 07 0949 08 0950 09 0451 10 1934
46 07 5647 08 28148 09 10 19 10 03150 I1 07 50 12 2234
46 09 571+7 10 29 48 11 1249 12 05 50 13 0951 14 24
40146 11 58147 12 30148 13. 1449_14 07 10 15 25 16 274.0
4146 13 59 17 14 3148 15 15 +9 16 og 50 17 14 51 18 2042
44/16 15 5947 16 32 48 17 17 49 18 11 50 19 16 51 20 3314
4046 18 0 47 18 33 48 19 19 9 20 13 50 21 1855 22 36 40
48
146 20 01 47 20 35148 21 2049 22 1550 23 21 51 24 3948
$40 22 02 47 22 36 48 23 2249 24 17 50 25 23 51 26 4153
$2146 24 03 +7 24 37 48 25 24 19 26 1950 27 2515 1 28 4452
$440 26 0447 26 39148 27 2549 28 21 30 29 28 51 30 47 54
$646 28 05 17 28 40 48 29, 27149 30 2350 31 30 31 32 4959
30 0047 30 4148 31 29 19 32 25 50 33 33 51 34 5258
6
46 32 07147 32 43 48 33 30 19 34 2850 35 36 51 36 5560
18
19
20
180.00-8=2,180.00+8=m, 360.00-y=
36
ܐ
q
k8
146
67
21.
22
23
A TABLE, Msewing the Right scenſion anſwering to
535
every two Minutes of the firſt Quadrant of the Ecliptic.
8
8
४
a
8
Minutest
24
25
26
27
28
29
.
/
Minutes 11
16 46 l,
t
?
26
51 36 5552 38 253_40 0654 41 $55 44
21 51 38 5852 40 28153 42 0954 44 02 57 46 04150 48 I
4151 41 0152 42 3153 44 1354 46 655 48 0156 50 22
451 43 0452 44 35113 46 17154 48' 0955 50 55 52 270
851 45 07152 46 38153 48 20 54 50 1355 52 10,6 54 311
13151 47 10 52 48 153 50 244 50 17155 54 20 36 56 304
12151 49 12152 50 45153 52 274 54 21155 56 2 56 58 411 2
14:1 51 Is|52 53 4133 54 3454 56 2555 58 29 57 00 4511
161 53 18 52 55 54133 56 3-54 58 39 56 00 33157 02 si
18151 55 21152 56 5-13 58 3855 00 3356 02 327 04 5:18
2011 57 2452 585 114 00 455 02 37156 04 4252 06 59
22132 59 2753 or 01 54 02 45 55 04 41156 06 46 57 09 0422
2452 Os 3053 03 04154 04.49.55 06 45'56 08 51157 11 09/24
52 03 33/53 05 07 54 06 52:5 08 49 50 10 55157 13 14126
2052 05 3053 07 154 08 555 10 5356 13 OC 37 15 18
3152 07 3953 09 1455 110955 12 5750.15 OS 5 7 17 245
32152 09 4253 11 17154 13 04155 15 0 156 17 57 19 292
352 11 45153 13 20 54 15 0855 17 05156 19 1357 21 3.
26
12 13 48153 15 24154 17 1255 19 10 56 21 18157 23 3130
38
52 15 5153 19 2754 19 1055 21 1456 23 2257 25 4.1,
C152 18 543 21 30 14 21 20 55 23 17 56 25 27 17 27 48
1252 19 57 53 21 34154 23 23 55 25 2256 27 32 7 29 522
4 12 22 Col53 23 37 54 25 27 55 27 26 56 29 37107 31 57/+
1152 24 0353 25 40 54 29 31155 29 30 56 31 427 34 026
+352 26 07153 27 4554 29 34135 31.3456 33 47157 36 07148
5052 28 i 53 29 48 54 31 38 55 33 38 56 35 5057 38 1230
$1;2 30 1353 31 5154 33 42155 35 42 56 37 5457 40 17152
94152 32 10 53 33 55 54 35 40155 37 47156 39 59 57 42 21 54
56
52 34 1953 35 5854 37 5455 39 556 42 03/57 44 26156
52 36 22 53 38 02154 39 5455 41 556 44 08157 46 31158
52 38 2553 40 0654 41 58155 44 ods. 46 1357 48 363
24
25
26
27
28
29
18000-83N, 180.00+8=m, 360:00 3.
Zz z 2 *
:
s
5 1.1
11
II
II
II
I
II
II
Minutes 1
O
I
2
4
Minuies
O
O
lo
2
2
4
8
10
I 2
14
10
IS
20
148158 38 38159 41 19100 44 IOT 47 1262 50 2263 53 42 481
1536 A TAB L E, ſhewing the Right Aſcenſion anſwering
to every two Minutes of the firli Quadrant of the Ecliptic.
3
5
57 48 35158 51 09 59 53 52160 56 4461 59 4963 03 01
2157 50 4158 53 1459 55 5760 58 522 or 5563 05 07
457 52 40158 55 19 59 58 01 co 5852 04 0163 07 14
6 57 54 51 58 57 2560 00 0961 03 0462 06 0763 09 21
8 57 56 56 58 59 3160 02 141 05 1052 08 1363 II 27
Ic7 59 01 59 01 36160 04 20161 07 1652 10 1963 13 33
1258 01 00 59 03 41160 06 267 C9 2252 12 2663 15 40
14158 03 11 59 05 4660 08 32161 JI 28 62 14 3263 11
558 05 1659 07 51160 10 3716! 13 3462 16 38)53 19 53
1858 07 21159 09 5760 12 43 61 15 4062 18 4563 22 00
2058 09 26159 12 0260 14 4851 17.4052 20 5163 24. 07
22 58 11 31159 14 07 60 16 53 61 19 5262 22 57/63 26 13122
2458 13 30 59 16 1360 19 0161 21 5852 25 0463 23 2024
26158 15 41159 18 18 60 21 0661 24 0462 27 1063 30 2626
8158 19.4659 20 2460 23 1261 26 1062 29 1763 32 3328
3078 19 52159 22 3060 25 1861 28 1762 31 2463 34 40
32 58 21 57159 24 35100 27 24161 30 2362 33 3053 36 47 32
3458 24 0259 26 4160 29 2961 32 2962 35 3663 38 54
5413.1
3058 26 07 59 28 4660 31 3561 34 3552 37 43 53 41 01
3858 28 12 59 30 51160 33 4161 36 41862 39 4963 43 0838
58 30 17 59 32 50/60 35 4761 38 4762 41 5563_45_15
4258 32 22159 35 0260 37 5361 40 5:62 44 0263 47 2142
44:8 34 27 59 37 07 60 39 59 61 42 5962 46 08163 49 28 44
4058 26 32 59 39 1360 42 0561 45 05 62 48 1563 51 35146
5.8 4° 4359_43 2460 46 1661 491&52 52 28153
55_4950
525 8 42 48 59 45 2960 48 2261 51 2462 54 3463 57 50152
54158 44 53 59 47 3560 50 2861 53 30162 56 41 64 00 02 54
15658 46 58 59 49 40 50 52 3461 55 3652 58 47 64 02 0950
58 58 49 03 59 51 46160 54 4061 57 4263 00 $464 04 16 38
8058 51 09 59 53 5260 56 4661 59 49163 03 01 64 06 2360
4
180.00—1=$, 180.00+8=, 368.00-=V
28
2
505.8
2
I
3
5
1
À TABLE, Newing the Right Aſcenſion anſwering to
537
every two Minutes of the firſt Quadrant of the Ecliptic.
Il
11
TI
Il
TI
II
8
7
9
10
6
Minutes
II
O
Minutes 0 1
nl
6
64 23
4 06 5 09 5466 13 3367 17 2168 21 1869 25 22
2164 08 3005 12 01 06 15 4067 19 2968 23 2669 27 30
464 10 3765 14 08165 17 47167 21 37 6 8 25 3469 29 38
664 12 4465 16 15166 19 5567 23 4568 27 4369 31 47
864 14 5165 18 22 23 02167 25 5268 29 5169 33 5518
1064 16 5865_20 29866_25_1067 28 0068 31 5969 36 03/10
1264 19 0415 22 3756 26 18167 30 08 68 37 07 69 33 T 12
1464 21 1165 24 4466 28 2567 32 1568 36 1569 40 19 14
0464 23 18165 26 51 66 30 3367 34 2368 38 2369 42 27.16
1864 23 2565 28 5966 32 4167 36 3168 40 3169 44 3618
2064_27_3205_31 066 34 4867_38 39 68 42 3969 46 44/20
2264 29 37
65 33 1366 36 55167 40 4768 44 4769 48 5 2 2 2
2464 31 4665 35 2066 39 0367 42 5568 46 559 si 0124
2664 33 53165 37 27|66 41 1067 45 0268 49 0369 53 09:26
2864 36 odlo5 39 3.466 43 1867 47 1068 51 1169 55 1728
3064 38 0765_41 4266_45_2667 49 1868 53 2069 57 2630
3264 40 1465 43 4966 47 33167 51
65 43 4966 47 3367 51 2668 55 2869 59 34/32
3464 42 21165 45 5666 49 4167 53 3468 57 35170 01 42134
3664 44 28055 48 0466 51 4967 55 4268 59 4470 03 5736
864 46 35155 50 1166 53 5667 57 5069 01 5270 05 59138
1064 48 42
65 52 1866 56 0457 59 5869 04 00709 08 08 40
42
64 50 5065 54 2666 58 1268 02 06169 06 08 70 10. 1742
+464 52 57
6S 56 337 00 1968 04 1469 08 1670 12 25 44
4664 55 0465 58 40167 02 2768 06 22169 10 2470 14 3346
4864 57 1156 co 4867 04 3568 08 3.169 12 33170 16 42948
5064 59 1866 02 5567 06 42|68 10 3869_14 4170 18 50 50
$265 or 2536 05 0367 08 50968 12 40 59 16 49.70 20 5852
5465 03 32166 07. 1167 10 5868 14 54159 18 57 70 23 0754
5665 os 3966 09 1867 13 0568 17 029 21 0570 25 1556
5865 07 4636 1 2567 15 13168 19 10 59 23 1370 27 24/58
6065 09 5466 13 3357 17 2168 21 1859 25 2270 29 3360
1
180.00=6, 180.00+I1=1, 360.00--0-1
}
2
8
IO
6
9
II
7
11
1538
A TABLE, Jhewing the Right Aſcenſion anſwering to
every two Minutes of the firſt Quadrant of the Ecliptic.
II
II
Il
II
II
I
12
13
14
15
16
17
Minutes 1 0 1
Minutes of
070 29 337 33 53172 38 1973 42 52174 47 3175 52 16
270 35 4171 36 0172 46 2873 45 0174 49 4075 54 25
4
70 33 5971 38 1072 42 37 73 47 1074 ST 5075 35
670 35 59171 40 1972 44 40173 49 20174 54 0075 58 45
8/70 38 0717 42 277.2 46 55173 SI 29174 54 09176 00 54
1070 40 1771 44 3072 49 0473 53 3874_56_1876
03 04.10
1270 42 25171 46 4572 SI 1473 55 4875 00 28 76 os 14/12
1470 44 33171 48 5472 53 23173 57 57 75 02 37176 07 2314
1670 46 4279 si 0372 55 32174 00 00 75 04 46176 09 33116
1870 48 5171 53 1272 57 4174 02 15175 00 50176 11 43118
20170 50 5971 55 20172 59 5074 04 24175 09 05 76 13 5229
2270 53 07 71 57 2973 01 5974 06 33175 11 15176 16 02 22
24
70 55 1671 59 3873 04 0874 o8 4375 13 2576 18 12/24
2670 57 24172 01 47173 06 1774 10 5275 15 34/76 20 21 26
2870 59 33172 03 5073 08 2674 13 0175 17 4376 22 3128
3071 01 4272 06 0573_10 3574 15 17 19 53 6 24 4130
3278 03 5072 08 1373 12 44 74 17 2075 21 02176 26 5032
3471 05 5972 10 2273 14 53174 19 2975 23 1276 29 0034
3671 08 0872 12 3173 17 03174 21 3975 26 2276 31 1036
871 10 1072 14 4073 19 1274 23. 4875 28 31176 33 2038
4071 12 2572_16_4973 22 2574_25_57175 30 4076 33 3049
4271 14 3 4172 18 5873 23 3074 28 07175 32 50 76 37 40 42
14471 16 4272 21 07173 25 3974 30 1675 34 5976 39 5044
4671 18 5 172 23 1673 27 4874 32 2575 37 09176 42 0046
48,71 21 00172 25 25173 29 57174 34 3575 39 19176 44 0948
15071 23 0872_27_34.73 32 08174 36 4475 41 28/76 46 19 50
52171 25 1772 29 4373 34 17 74 38 5375 43 38176 48 28 5 2
5471 27 2072 31 5273 36 2574 41 0375 45 4876 50 3854
5676 29 35172 34 0173 38 34 74 43 1275 47 57126 52 48156
58171 31.4472 36 1073 40 43174 45 21175 50 0676 54 5758
6071 33 53172 38 1973 42 5274_47_3175_52 1676 37 0760
10176
13
14
I5
16
17
180.00-II=, 180.00+1=7,260.00-W.
I 2
1
A TABLE, Shewing the Right Aſcenſion anſwering
539
to every two Minutes of the firſt Quadrant of the Ecliptic.
II
I
II
п
TI
18
19
20
21
22
23
Minutes 1 07
O
lo lo
Minutes ºn
2
6
>
1477 12
18
76 57. 07178 02 0379 07 03 80 12 081 17 1782 22 30
76 59 1778 04 1379 09 1380 14 13 19 27 82 24 40
77 01 2778 06 2379 11 2380 16 281 21 3782 26 5 11 4
4
77 03 37178 08 33179 13 3380 18 391 23 48 82 29 02
877 05 4778 10 43 79 15 4380 20 4911 25 58 82 31 13
8
10177 07 5778 12 5379 17 5380 22 5081 28 0982 33 23
12 77 10 0678 15 02 79 20 0480 25 1081 30 2082 35 3312
16118 19 1279 22 1480 27 2081 32 31 32 37 44 14
1677 14 2578 19 2279 24 2480 29 30 31 34 41 82 39 $416
1877 10 3578 21 3279 25 34180 31 4 81 36 5182 42 04
2077 18 4578 23 4279 28 4480 3 5 81 39 0182 44 14 20
2277 20 51178 25 32 79 30 54180 36 oool 41 1182 46 25122
2 477 23 05 18 28 0279 33 040 38 1181 43 22 32 48 2024
2 577 25 15178 30 1279 35 1489 4.0 2181 45 3282 50 40 26
28/77 27 2478 32 22 79 37 2483 42 3181 47 4382 52 57128
3077 29 31178 34 32179 39 3580 44 4281 49 5432 55 07/30
3277 31 44 78 36 4279 41 4580 46 521 52 0482 57 17132
3477 33 54 78 38 52 79 43 5530 49 0281 54 14/32 52 28
34
3677 36 04118 41 02179 45 05130 51 13 31 56 25183 01 3936
3877 38 1478 43 12179 48 1580 53 2:31 58 36483 03 49138
4077 40 24178 45 2279 50 2580 55 3332 00_4083 06 0040
42 77 42 34 78 47 32 79 52 3680 57 4482 02 5683 o8 1042
4477 44 44178 49 4279 54 4680 59 5482 05 0783 10 2044
77 46 54 78 51 52 79 50 5831 02 0432 07 1783 12 31 46
4877 49 0378 54 0379 59 0781 04 1582 09 2883 14 42.48
3077 ST 13138 56 1380 of 1781 06 25 82 11 3883 16 53150
5277 53 23 78 58 2380 03 2781 08 3582 13 48183 19 0352
5477 55 3379 00 33 80 oś 37 31 10 4632 15 5983 21 14154
5677 57 4379 02 43180 07 4701 12 5682 18 1083 23 23156
3877 59 53 79 04 5380 09 5181 15 06132 20 2083 25 34558
9978 02 03 79 0703180 12 081 11 17182 22 3083 27 45160
19
180.00-II = $, 180.00+8=, 360.000 =
☆
46
18
2
parent
22
21
23
540
A TABLE ſhewing the Right Aſcenſion anſwering to
TI
II
II
II
II
II
20
Minuteslo
1
Minures 101 Too
I
1
20
22183 51 42
183 32 0684 37 25135 42 46185 48 08187 53 31188 58 58
87 12 1086 17 37107 23 Oc188 28 25189 33 5036
26 26 2017 31 4308 37 08189 42 32/44
3:189 40 55+
every two Minutes of the firſt Quadrant of the Ecliptic.
1
24
25
27
28
29
83 627 45 34 33 0485
38 2486 43 4687 49 1088
543
233 29 5584 35 1485 40 3586 45 5787 51
2088 56 48
683 34 1734 39 365 44 57 86 50 1987 55 42 39 01 076
883 36 2784 41 40 35 47 07186 52 2987 57 52189 03 18
10 63 38 38 84 43 5785 49 1886 54 40 08 00 04189 05 29
1283 40 484 46 v 35 51 2986 36 Sul88 02 1589 07 4
1483 42 5984 48 18 35 53 3986 59 0188 04 25189 09 5014
16 33 46 1c 84. So 398.5 55 5087 Or 1288 06 30 89 12 01
1883 47 2134 52 41 85 58 0187 03 2388 08 47 39 14 12
18
2003 49 3184 54 50 86 to 1187 os 3388 10 58189 16 23
8.4 57 C 86 02 2237 07 4458 13 09189 18 34
2483 53 5334 59 12 86 04 3387 09 5588 15 2089 20 4524
2083 56 0385 or 22186 c6 4387 12 0088 17 3489 22 5512
2803 58 1435 03 3,86 08 5487 14 1788 19 439 25 26
30184 00 25 35 05 4+36 1 0587 16 28/88 21 52189 27 17
3284 02 35 85 07 55 56 13 1537 18 3888 24 07189 29 28 32
3784 04 45 85 10 CO 86 15 24187 20 49 88 26 14189 31 2934
06
38184 09 6685 14 26 86 19 4887 25 1088 30 3589 36 0.13€
+34 II 1785 16 37 186 21 5587 27 21 88 32 40 89 38 ut
42 84 13 2805 18 4885 24 14 37 29 3288 34 579 40 22 +2
3885
46 34 17 4535 23 05 86 28 3189 33 5488 39 1989 44 44
87
59184 22 10 35 27 34 86 32 5,187 38 1588 43 40 39 49 OSEC
5284 24 2185 29 41186 35 02 87 40 2688 45 51139 si 14 52
5484 26 3285 31 5286 37 1487 42 37188 48 02189 53 2714
5434 28 4285 34 0286 39 2487 44 48188 50 1839 55 3856
84 30 53185 36 13 36 41 35137 46 59/88 52 24 89 57 49
60
34 33 0485 38 2436 43 4629 49 1C 38 $4 35193 co oC
25
26
27
28
180 00-II=%, 180.00+8=4, 360.00—1=w.
26
ic
36184
5 6
po
29
24
1
A TABLE ſewing the Declinat ion to every two 517
Minates of the Ecliptic
į and 7 II and II and 71 1 and 7 I aod † | II and f
6
7
8
9
10
II
Minutesla
Minures}
21 20 5321 31 0821 40 5821 50 2421 59 26 22 08 0360
221 21 1421 31 28/21 41 1721 50 4221 59 44122 08.20158
Han 21 3521 31 48|21 41 3021 st 0022 00 02 22 08 37156
621 21 5521 32 07121 41 5521 SI 29 22 00 1822, 08 5454
821 22 1621 32 2727 42 1521 51 3822 00 30 22 09 1052
1021_22 37 21 32 47/21 42 3421 55622 00 5222 09 2650
12/21 22 58 21 33 0721 42 5321 52 14 22 On 1122 09 4248
14,21 23 1821 33 2721 43 12 21 52 3222 01 28|22 09 5946
102 1 23 3921 33 47/21 43 31/21 52 522 or 4622 10 16/44
1821 24 00 21 34 0721 43 5021 53 09 22 02 03|22 10 3242
2021 24 20121 34 2721 44 0921 53 272 2 02 20/22 10 49140
22 21 24 4121 34 47 21 44 2821 53 45 22 02 38 22 11 OS 38
24/21 25 01 21 35 0621 44 4721 54 0422 02 5522 11. 2236
2021 25 22 21 35 26121 45 06 21 54 22 22 03.12 22 11 38 34
2821 25 43 21 35 46/21 45 25/21 54 40122 03 3022 11 5432
3021 26 0321 36 06 21 45 44/21 54 58 22 03 47 22 12 11 30
32 21 26 24 21 36 2521 46 0321 55 1622 04 04/22 12 2728
3421 26 4421 36 45/21 46 22 21 55 3422 04 21 22 12 44/26
3621 27 0421 37 04/21 46 4021 55 5222 07 38122 13 0024
21 27 25 21 37 2421 46 59a1 56 1022 04 55122 13 16 22
4021 27 45 21 37 4421 47 18/21 56 28 22 05 1322 13 3320
+221 28 05 21 38 0321 47 3721 56 4622 05 30/22 13 4918
+421 28 2621 38 2321 47 5521 57 0122 05 4722 14 0516
21 28 46 21 38 42 21 48 14 21 57 2322 06 04/22 14 2114
21 29 05 21 39 0221 48 3321 57 3922 06 2122 14 37 12
5921 29 27 21 39 2121 48 5121 57 57/22 06 3822 14 5310
52
21 29 47 21 39 4121 49 1021 58 15 22 06 5522 15 10 8
54
21 30 07 21 40 0021 49 2821 58 33 22 07 12 22 15 26
56
58
21 30 27 21 40 1921 49 47 21 58 ST 22 07 2922 15 42
21 30 47 21 40 38 21 go og|27 59 09 22 07 4622 15 58
21 31 of 21 40 5821 50 2421 59 20 22 08 0322 16 14
23
19
w and o
to and
W
and stand
Wo and
81
46
4821
60
22
21
20
18
1
SAB
and
XXX
518
A TABLE jbewing the Declination to every two
Minutes of the Ecliptic
I and
II and 7 II and
II and F I and II and I
13
14
15
16
17
I 2
Minutes º
Minutes
O
1622
18
}
22 16 1422 74 0022 31 2022 38 16 22 44 44 22 50 49
2122 16 30 22 24 15 22 31 35
22 38 3022 45 OC 22 51 0158
4/22 16 40 22 24 30 22 31 4922 38 4322 45 12/22 SI 12156
622 17 01 22 24 45 22 32 03 22 38 56/22 45 24 22 51 2454
8/22 17 17|22 25 00 22 32 1822 39 0922 45 36 22 35 52
1022 17 33 22 25 15 22 32 32122 39 23 22 45 48 22 51 47159
12 22 17 49 22 25 30 22 32 4622 39 3622 46 0 22 51 5848
1422 18 0522 25 45 22 33 0022 39 4922 46 1422 52 1046
18 21 22 26 00122 33 14122 40 02/22 46 2522 52 2144
1822 18 3622 26 15 22 33 28|22 40 1522 46 3722 52 3? +2
2022 18 52 22 26 30 22 33 42 22 40 29 22 46 4922 52 444.0
22 22 19 08 22 26 45/22 33 56122 40 4222 47 0222 52 5518
24 22 19 2722 26 59122 34 10122 40 5522 47 14 22 53 0736
20/22 19 322 27 1422 34 2422 41 08 22 47 2622 53 34
28 22 19 5522 27 29 22 34 3822 41 21122 47 38122 53 2932
3022 20 122 27 44 22 34 5222 41 3422 47 5022 53.4130
32/22 20 20 22 27 58 22 35 05 22 41 4722 48 0722 53 51128
41122 28 1322 35 19 22 42 uc 22 48 1422 54 0226
3622 20 57 22 28 28 22 35 33 22 42 13122 48 2622 54 1424
38122 21 1.2122 28 42 22 35 47 22 42 2012 48 38 22 54 21 22
40122 21 3622 28 57.22 36 0022 42 39 22 48 50 22 54 3020
47122 21 4322 29 11 22 36.1422 42 5122 49 02 22 54 4718
44 22 21 58 22 29 2622 36 28 22 43 04 22 49 1422 54 5816
46 22 22 1322 29 4022 36 4122 43 1722.49 2622 55 of 14
4822 22 2922 29 55 22 36 5522 43 30 22 49 38 22 55 2012
50 22 22 44 22 30 09 22 37 09:2 43 42 22 49 5022 55
3110
59122 22 5522 30 2422 37 2222 43 5522 50 02 22 55 420
5422 23 15122 30 38 23 37 39 22 44 0822 50 1422 55 536
5622 23 322 30 52122 37 49 22 44 2122 50 25 22 56 044
5822 23 45122 31 06/22 38 0322 44 3522 50 37 22 56 15 2
6c 12 24 oo 22 21 2022 38 1822 44 48 22 49 22 56 25.
17
16
15
14
13
s y and s W and so
o V and on
34/22 20
$
12
mene
you and
to and
yo and 5
A TABLE jbewing the Declination to every two
او در
Minutes of the Ecliptic.
31 II and 7 II and † | II and
I and F
23
II and I
II and 7
18
19
20
21
22
Minutes 101
Minutes!
2
22 56 25123 cl 38123 06 2023 10 38 23 14 24 23 17 5260
22 56 36123 or 48123 06 2923 10 4023 14 3823 17 5958
22 56 47123 OI 5823 06 3823 10 5423 14 43 23 18 05156
622 56 5823 02 08 23 06 47/23 II 02/23 14 5023 18 1154
822 57 09/23 02 1823 06 56 23 11 1023 14 57123 18 19152
10 22 57.1923 02 2823 07 05 23 11 3823 15 04/23 18 24/50
12 -2 57 30123 02 38|23 07 14/23 il 26 23 15 12 23 18 3048
1422 57 4123 02 47 23 07 2323 11 3423 15 19123 18 36 46
16 22 57 5123 02 57123 07 32 23 11 4223 15 2623 18 42 44
18 22 58 0223 03 06 23 07 41123 11 523 15 33|23 18 48 42
22 58 12 23 03 1523 07 4923 11 5823_15_4023 18 54 40
22 22 58 2323 03 24/23 07 58/23 12 06123 15 47|23 19 0038
24 22 58 33123 03 34/23 08 07 23 12 12 23 15 53123 19 06136
2622 58 44 23 03 43 23 08 1523 12 21 23 15 00 23 19 12134
28 22 58 5423 03 $223 08 2423 12 29123 16 0723 19 18132
3422 59 0423 04 02 23 08 33 23 12 39123 16 1423 19 24130
32/22 59 15 23 04 1123 08 4123 12 44 23 16 21 23 19 34 28
34 22 59 2523 04 21123 08 50/23 12 5223 16 28123 19 30126
3622 59 3523 04 3023 08 5€ 23 13 0023 16 34 23 19 4124
38/22 59 4523 14 39123 09 07123 13 07123 16 4913 19 48/22
422 59 56 23 04 49|23 09 15 23 13 15|23 16 4; 23 19 5220
4223 00 0023 04 5823 09 2323 13 2223 16 5423 19 5978
144 23 00 10 23 05 07 23 29 32123 13 3023 17 0.23 20 05 16
4023 00 2623 Os 17123 09 4023 13 3723 17.0723 20 10 14
4843 00 3023. 05 2623 49 48 23 13 4523 17 14 23 23 1612
1923 00 4623 05 35123 09 5723 13 5223 17_2023 20 2210
52 23 00 5023 05 4423 10 05 23 13 5923 17 27 23 20 278
54 23. OJ 06 23 05 53 23 10 1323 14 0723 17 33 23 20 3316
5623.01 17123 02 23 10 2223 14 54123 17 4923 20 381 4
5823 01 21 23 25 11 23 10 30 23 14 21 23 17 46123 20 43
5023 or 38123 06 20 23 10 3823 14 2923 17 52123 20 48
481 c
8
7
6
ve and
ve and $ W and g y and s Vy and
II
10
9
10
w and
520
& TABLE Shewing the Declinatioł to every two
26
Minutes
Minute's 01
.
320 28 23
Minutes of the Ecliptic.
II and 7
I and and 1 II and I Il and ? I and I
24
25
27
28
29
23 20 48 23 23. 18/22 25 2323 26 523 28 0623 28 4660
223 20 $423 23 23 23 25 2523 26 59123 28 08 23 28 47158
423 30 $9123 23 28123 25 29123 29 0323 28 10 23 28 4850
6123 21 04123 23 33.23 25 33.23 27 05 23 28 1123 28 4954
8 23 21 0923 23 3723 25 36 23 37 08/23 28 13123 28 50 52
10 23 21 1423
2324-23 25_4023 27 10123_28 15 23 28 5150
1 223 31 1923 2; 4023 25 43 23 27 13123 28 10 23 28 51148
14 23 21 2423 23 5023 25 47 23 27 10 23 28 1823 28 5246
1623 21 29 23 23 55|23. 25 5023 27 18 23 28 19 23 28 5344
18 23 21 3523 23 5923 25 5423 27 2023 28 27 23 28 5342
20 23 21 4023 24 03|23 25 5023 27 2323 28 21/23 28 54
22123 21 45|23 24 0723 26 0023 27_25|23 28 24 23 28 5538
24 23 21 5023 24 1123 26 03 23 27 28 23 28 26 23 28 35 36
20 23 21 56 23 24 16 23 16 0023 27 3023 28' 27123 28 56 34
28 23 22 01|23 24 20 23 26 10 23 27 32 23 28 2823 28 56132
3023 22 0023 24 2423 26 13 23 27 35 25 28 2923 28 57120
3223 22 1123 24 2823 26 16 23 27 3823 28 3123 28 5728
34 23 22 17 23 24 3223 26 1923. 27 40123 28 32 23 28 57 26
3023 22 22 23 24 3623 26 21 23 27 4223 28 33/23 i8 58/24
38 23 22 27|23 24 4023 26 2573 27 4423 28 34 23 28 5822
40123
4223 22 37 23 24 4823 26 31 23 27 4823 28 37 23 28 5918
4423 22 42 23 24 52 23 26 3423 27 3023 28 3823 28 5916
40 23 22 47 23 24 5023 26 37 23 27 5223 28 39 23 28 5914
48123 22 51123 24 5923 26 4023 27 5423 28 4023 28 5912
5023_22 5023 21 0323_26_4323 275.623 28 41 23 29 0010
5 223 23 0123 25 0723 26 4623 27 58 23 28 4923 29 008
5423 23 05|23 25 1123 26 49123 18 0023 28 4323 29. 00 6
i5 623 23 1023.25 1423 26 52123 28 0223 28 4423 29 004
58 23 23 14 23 25 1823 26 54/23 28 04 23 28 45 23 29 002
6023. 23 1823 25 22 23 26 5623 28 0623 28 46 23 29 000
5
Vand 5 Wo and 99 vp and on v and $
We ar doo
.
--
4
3
I
o
V
ve and e.
.
537
:
C
Sect. XXI.
of the Orthographic Proje&tion of the Sphere
Onceive the Eye placed at an infinite Diſtance from the Globo,
and at the ſame time a Plane to paſs thro the Center of the
Globe, and to Itand at Right-angles to the Line connecting the Cen-
ters of the Globe and Eye, if from the Eye chus poficed, an infinite
Right-line be imagined to be drawn, thro any point of theCircumfe.
rence of any Circle deſcribed upon the Surface of the Globe, and the
ſameRight-line be carried about theCircumference of the given Circle,
till it return to the Place from whence it began to move; or which
is the farre thing, if from any point in the Circumference of any
Circle deſcribed upon the Surface of the Globe, a Right-line be
imagined to fall perpendicularly upon the given Pline, and that this
Line be carried round the Circumference of the given Circle, con-
ſtantly perpendicular to the Plane and parallel to its ſelf, it will
deſcribe upon the cutting Plane an Orthographic Repreſentation of
that Circle , and after the ſame manner if Rays infinitely long be
imagined to flow from the Eye to the Circumference of every Cir-
cle deſcribed upon the Globe, and theſe Rays be carried about the
Circumference of each refpe&tive Circle till they return to the Place
from whence they began to move, they will trace out upon the gia
ven Plane, what is called an Orthographic Projection of the Sphere.
This Paralleliſm and perpendicularity of the generating or de
ſcribing Ray, is the eſſential and primary Property of Orthographic
Projection, and altho according to the Euclidean Idea of paral-
lel Lines, they can never be conceived to meer if infinitely produced,
yet in order to range this part of Science under the Head of
Perſpedive, and to render it conformable to the ſeveral ways of
repreſenting the Circles of the Sphere upon a Plane, it was abſolutely
neceſſary to define it after this manner; and indeed if we conſider
the iofinite ſmall Inclination of the infinite ſmall Portions of the
incident Rays, intercepted between the Surface of cheSphere and the
Plane, in a Phyfical Senſe; and which is the Senfe in which they are
to be underſtood in this place, the Inclination it ſelf vaniſhes, and
the parallel and perpendicular Property really and actually exiſts.
Аааа*
From
538 of the Orthographic Projection of the Sphere
From the general Definition thus explained it follows,
1. That the common Interſection of the Plane upon which the
Projection is to be formed with the Surface of the Globe, will be
the Circumference of a Circle, which for Diſtinction fake is called
the Periphery of the Plane of the Projection, and that the common
Intersection of the Plane wich the Globe it ſelf will be a Circle,
which for Diſtinction fake is called the Flane of the Projection
For as every Globe may be confidered
as generated by the Rotation of a Semi-
в о E 1 D circle about the Diameter, remaining
fixed as an Axis, in whatſoever part
of it it be cut by a Plane as BN MD.
the common liturfe&tion ATP C Q R
of this pane with the Superficies of
T
C
the Globe repreſented in this Cafe hy
PIAPQR muſt be tle Circumfe-
sence of a Circle as muſt be the cor-
mon Interf.&tion p APQR of the cut-
e ting Plane with the Globe it felf be a
IP
Circle.
a
2. That the Pole of that Circle up-
on whoſe Plane the Sphere is to be
자 ​B M projeEted, will fall in the Center of the
Plane of the Projection.
For becauſe the Arches AP and QP are equal, the infinite Ray Efap
which flows from the Eye at E thro P the Pole of the Circle Apg
which repreſents the Diameter of the Plane Ap2 of the Projection,
muſt fall in the middle point p, by the 8ıb Cor. of the ad Prop of
Part the ift, and conſequently in the Center of ihe Plane of the Pro-
je&tion;
3. That every Great Circle that is perpendicular to the Plane of
the Projection, will be repreſented by a Right-line paſſing thro the
Center and Pole of the Projection, equal to its Diameter and to
the Diameter of the Sphere.
In the preceding Figure, let the lower Semicircle ApQR, be
turned about upon its Axis A2, till it ſtand at Right-angles (o
the Plane of the upper Semicircle APQP, and in this ſcituation
let it reprefent the fame Plane of the Projection, then will AQ
re-
Of the Orthographic Projection of the Sphere 539
repreſent its Diameter, and p the Center of the Projection, and let
À P Q repreſent the upper half of the Circle to be proje&ted.
Inaſmuch therelore as the Eye is ſcituated in the Plane of the Se-
micircle APQp infinitely extended, if the infinite Ray P p flowing
from the Eye at E and extended till it cut the Plane A R l, be
imagined to be carried thro the Semi-periphery AP & of the Cir-
cle to be proje&ed, it will deſcribe by its Paſſage over the Plane
ARQP, the Right-line Ape, equal in Length to the Diameter
of the Circle AP e, and to rhe Diameter of the Sphere APQR,
and which from the general Definition will be its true Repreſen-
tarive.
4. That the Angle that any two great Circles which ſtand at
Right-angles to the Plane of the Proje&ion form with each other,
will be equal to the Angle formed by their Repreſentatives upon
the Plane of the Projection.
For as the Angle formed by any two Circles, is the ſame with the
Inclination of the Planes of thoſe two Circles; and as this is mea-
ſured by the Arch of a Great Circle delcribed at the diſtance of 90
deg, oo min. or a Quadrant, from the angular Point, it follows
from the preceding Corollary, that the Angle formed by the two
Right-lines repreſenting the two Circles, muſt be equal to the An-
gle formed by the Circles themſelves.
5. That all Small Circles that are parallel to any of the pre-
mentioned Great Circles, or which ſtand at Right-angles to the
Plane of the Projection, will be repreſenced by Right-lines in the
Projection, whoſe Dianieter will be equal in Length to the Diame-
ter of the Circle they repreſent.
For if an infinitely extended Ray in the preceding Figure, be car-
ried along the Periphery of the ſmallCircle 1 % C, lying in the ſame
infinitely extended Plane with the Eye, it will trace out upon the
Cutring Plane APQ by its Motion, the Right-line FC for its Re
preſentative, equal în Length to the Right-line TC, the Diameter
of the ſmall Circle to be projected
And inaſmuch as the infinitely extended Ray moves conſtantly in
the Plane of the ſmall Circle TC, and conſequently keeps always
at the diſtance of Cc or T t, from the Repreſentative A2, of the
great Circle to which it is ſuppoſed to be parallel, it follows,
6. That the diſtance of every luch mali Circle in the Proje&ion
from the great Circle to which it is parallel, will be equal to the
Аааа 2.
Sire
*
549 Of the Orthographic Proje&tion of the Sphere
Sine of ſuch ſmall Circles diſtance from the great Circle upon the
Sphere to which it is parallel ; for Cc or. T t the diſtance of the
Repreſentative TC of the ſmall Circle T SC; from Althe Re-
preſentative of the great Circle to which it is parallel, is equal to
the Right Sine of the Arch C Q or It, the diſtance of that ſmall
Circle upon the Globe, from the great Circle AR, to which it
is ſuppoſed to be parallel, and conſequently the Repreſentatives of
theſe ſmall Circles interſect the Periphery of the Plane of the Pro-
jeâion in the ſame points as the finall Circles themſelves interſex it
upon the Globe it felf.
7. That all ſinall Circles parallel to the Plane of the Proje&ion,
will become Circles in the Projection.
For let the Semicircles 1 C and ARQ be erected perpendi-
cularly over the Plane of the Semicircle APQ; now if the infinitely
extended viſual Ray O Tt, paſſing thro the Extremity I of the Di.
ameter 1 C, of the ſmall Circle I 5 C, be ſuppoſed to move round
she Periphery TCP, of the ſame ſmallCircle always parallel to its
felf, and perpendicular io thc:Plane T$CPa, till it return to the Place
from whence it begun to move, it will deſcribe by this Morion a cyl-
lindrical Superficies; and inaſmuch as the Planes TC and ARQ are
parallel the Extremity t of the viſual Ray o it, will trace out upon
the cutting Plane IRC the ſmall Circle trc, equal to the ſmall
Circle IC it repreſents : And after the ſame manner, if infinite-
ly extended viſual Rays be conceived to move round the Periphery
of every other ſmall Circle deſcribed upon the Surface of the Sphere,
parallel to the Periphery of the Plane of the Projection, they will
trace out upon the cutting Plane, Circles parallel and equal to the
ſmall Circles they repreſent,
8. That all theſe ſmall Circles will have one common Center,
viz. the Center and Pole of the Proje&ion.
For becauſe the Arches PC and P T are equal, whereſoever the
points T or C be taken, the infinitely extended Ray paffing from the
Eye at E thro the common Pole at P, perpendicular to the Plane
APQ, will become the Axis of the generated Cylinder, and confe-
quently the Portions pt and pc of the projected: Diameter to, will be
always equal to each other, and conſequently the point p the Repre-
ſentative of their common Pole, will become their common Center.
9. That their Radijor Semidiameters will be ever equal to theSines
of their reſpe&ives diſtance from the Pole, or the Co-fines of their
Elevations above the Plane of the Proje&ion.
For
of the Orthographic Projection of the Sphere* 541
B
AP Q S, it will
For inaſmuch as the viſual Rays E.P p and SC € will be always
parallel or equi-diſtant from each other, pc equal to pt the Radius
or Semidiameter of the proje&ed ſmalr Circle i ro will be ever equal
to ac, the Sine of the Arch PC, the diſtance of the ſmall Circle
I aC from the Pole P, or the Co-fine of ce its Elevation above
the Plane of the Projection ; and the same will happen wherefoevr
the points C or I be taken:
10. That every great Circle that is inclined to the Plane of the
Proje&ion, will be projected into an Ellipfis.
For let AP & Sp repreſent the
Plane of the Projection, and PBCS
any given great Circle that is in-
clined to the Plane of the Projec-
P
tion, at any given Angle Cpc or
BDb, I fay that if an infinitely ex-
D
tended Ray as Cc, be carried along
the Semi-Periphery P BCS of the
A
grear Circle PBC Sp, conſtantly
P.
parallel to it ſelf, and per rendicu-
lar to the Plane of the Projection
the Plane PQSA the curve Line
S
Pbcs, which will be the Scmi-
periphery of an Ellipfis
For from the points B and C in the Semicircle PBCS, ler tall the
Perpendiculars B b and Cc, and from the points b and c where they
interfect the Plane of the Projection APS, draw the Lines b D
and cp, perpendicular to the common Axis PS, and joyn the points
D, B, and p, c, by the Right-lines DB and pc, and becauſe the
Angles Db B and pc Care Right, and the Angles B Db and Cpu
are equal to each other, and to the Inclination of the given great
Circle, the Triangles p Cc and D B b will be ſimilar, wherefore by
Prop. the 18th of Seation the 2d of Part the ist it will be,
As DB: Db::p:pc; and conſequently by Cor. the 13th
of Prop. the 15th, of the fame Section, DB:PC ::06:06;
wherefore by Cor. the 20th of Prop. the 19th of the lame SeEZ. D BQ
:p Cq: :D bq:p0q, but from the Nature of the Circle D Bg:
p Cq:: PDxDS:PpXpS, wherefore D bq: pCq: :P DxDS:
Ppxps, and confequently the curye Line paſſing tiro the points P, b,
1, S, will be a Semi-Ellipfis.
And
542 . Of the Orthographic Projection of the Sphere
A
B
D
%
B
H
And that this is the chiet and primary Property of the Apolonian
or Conic Ellipfis, may be thus demonſtrated.
In the adjacent Figure, let ABC
repreſent the Se&ion of a Cone cut by
A
a Plane thro the Vertex, and Pb c S a
Section of the fame Cone cut with a
P
Plane, ſo as to cut the Sides AC and
AB, in the points P and S, on the ſame
fide of the Vertex A, and thro' the
N
points D and B in the Line PS the
I
common Interledion of theſe twoPlanes,
S
let the Cone becut with two other Planes
EDF and IBN parallel to theBaſe, then
B
will the Triangles PDF and PBN be
ſimilar, and conſequently by Prop the
18th of Seet the ist of Part the ift,
it will be,
As PD: PB : : DF: BN, and becauſe the Triangles SDE and SBI
are ſimilar it will be, as SD : SB::DE:p1; wherefore, as PDX
DS: PBXSB : : DEXDE: BNxBl; and becauſe the Sections thro
the Planes ED.N and IBN are Circles from the Nature of the
Cone, EDXDF=D;bq, and 1BxBN=Boq, wherefore P DxD S:
PBxBS : :D bq: BC9, which was to be demonſtrated.
11. That the Tranſverſe Axis or longeſt Diameter of the Ellipſis
is equal to the Diameter of the great Circle it repreſents, and con-
fequently equal to the Diameter of the Sphere it felf.
For as all great Circles interfect each other at the diſtance of a
Semicircle, the Line that joyns the ewo Interſections together, and
which is the common Iritcrrection of the Planes of the two Cir-
cles, will be ever equal to the Diameter of the Circles themſelves;
thus in the laſt Figure but one, the Line P S which connects the In-
terfe&tions P and Sof the great Circle APQS, upon which the
Sphere is to be projected, and the inclined great Circle P BCS to
be projected, and which in this caſe becomes the Axis of the Ellip-
fis Phc S its Repreſentative upon the Plane of the Proje&ion is the
fame with, and conſequently equal to PS the Diameter of the Sphere
it felf.
12. That the conjugate or ſhorteſt Diameter of every Ellipſis re-
preſenting any greatCircle, is equal to twice the Co-line of its Inclina-
tion
Of the Orthographic Projection of the Sphere 543
tion to the Plane of the Projection, or to the double Sine of its diſtance
from the Pole of the Projection, meaſured in the Arch of the great
Circle that paſſes thro the Pole, and cuts the given Circle at Right-
angles.
For in the Figure belonging to Cor. the 16th it is manifeſt, that po
the Semi-conjugate Diameter of the Ellipſis PALS, repreſenting the
great Circle PBCS, is equal to the Sine of the Angle PCs, the dis-
tance of the great Circle P BCS from the Pole p, meaſured in the
Arch of a great Circle, or the Complement of the Angle Cpc, the .
Inclination of the Planes of the Circle P.BCS to be proje&ed, to
the Plane of the Circle P Q S A upon which it is to be projected.
13. That every ſmall Circle that is inclined to the Plane of the .
Projection, will be projected into, or be repreſented by an Ellipſis.
This is a manifeſt Conſequence of the 10th Cor. and needs no o-
ther Deinonſtration,
14:
That the tranverle Axis or longeſt Diameter of every Ellipfis
repreſenting any imall Circle, will be equal to twice the Sine of that:
ſmall Circles diſtance from its neareſt Pole.
for in the adjacent (Figure, let
APOS repreſent the Plane of the.
R
Projection, A TRCQ a great Circle
inclined to it, Ar 2 its Repreſenta-
tive upon the Plane of the Projection,
and TC the Diameter of a ſmall Cir-
cle to be projected.
A
Q
Inaſmuch therefore as the perpen-
dicular Rays Tt and Cc are paral-
lel and equal, the Right-line. * C, the
Reprefentative of that Diameter upon
the Plane of the Projeäion equal to
S
twice rc is equal to TC, equal to
twice PC. the Sine of the Arch RC, the diſtance of the ſmall Circle
TC from its neareſt Pole R.
15. That this Diameter will always ſtand at Right-angles to the
Repreſentative of that great Circle that paſſes thro the. Pole of the
ſmall Circle, and the Pole of the Periphery of the Proje&ion.
For inaſmuch as the Diameter TC of the ſmall Circle to be pro-
jected, ſtands at Right-angles to the Plane of the Circle Sprp,
which
944
of the Orthographic Proje&tion of the Sphere
which paſſes thro the Poles of the Circle to be projected, and the
Periphery of the Plane of the Projection, and that the infinite Rays
Tt and C c are parallel and equal, it is manifeft, that t C the Re-
preſentative of the Diameter TC of the ſmall Circle to be projec-
ted, will ſtand at Right-angles to the Line SpP, the Repreſenta-
tive of the great Circle Spr PR, conneđing the two reſpective
Poles
And inaſmuch as the conjugate Axis or ſhorteſt Diameter, will
ever form Right-angles with the tranſverſe Axis or longeſt Dia-
meter, it follows;
16. That the conjugate or ſhorteſt Diameter of every Ellipſis re-
preſenting any ſmall Circle inclined to the Plane of the Projection,
will be coincident to, or be projected in that Right-line that is the
Repreſentative of the great Circle that paſſes thro the Poles of the
Periphery of the Plane of the Projetion, and of the Circle to be
projected.
That the conjugare or ſhorteſt Diameter of the Ellipfis repreſent-
ing any ſmall Circle, is equal to the Sum or Difference of the Sines
of the greateſt and leaſt Diſtance of that ſmall Circle from the Pole
of the Projection, or to the Sum or Difference of the Co-fines of the
Elevations of the reſpective points above the Plane of the Proje&ion.
For in the adjacent Figure, if a
repreſent the Pole of the ſmall
P
Circle T°C to be proje&ed, and P
the Pole of the Circle Al, upon
which the ſmall Circle is to be pro-
jected, it is manifeſt (in this caſe
where the Pole of the Periphery of the
A
Pane of the Proje&ion lyes within the
ſmall Circle) that to the projected
conjugated Diameter, is equal to the
Sum of t p and pc, equal to the Sum
S
of T, and Cs, the Simes of T P and
PC, the greaceſt and leaſt diſtances of
the ſmall Circle TC, from the Pole of
the Proje&tion P, or to the Co fine of the Arches AT and QC, the
Elevations of the points T and C, above the Plane of the Projection
AQ.
1
T
If
of the Orthographic Projection of the Sphere 547
PT
T
ز
But if the Pole P, lye without the
Imall Circle, or which is the ſame
thing, if the ſmall Circle lye intirely on
the ſame ſide of the Pole of the Projec-
tion, as in the adjacent Figure ; it is
manifeſt, that to the conjugate Dia-
meter, is equal to the Difference be-
A
tween CS the Sines of the greateſt
Plt
diſtance PC and Tr the Sine of the
leaſt diſtance TP, of the finall Circle
1C to be projected; or to the Diffe.
rence of the Co-lines of QC and QT,
the Elevations of the reſpective Extre-
mities of the ſmall Circle to be pro-
jected. Wherefore
18. If the Sines of the greateſt and leaſt diſtance of the two Ex-
tremities of any ſmall Circle from the Pole of the Projection, be ſec
off either on the ſame or contrary fides of the Center or Pole of the
Projection, in that projected great Circle which paſſes thro the Poles
of the ſmall Circle to be sprojected, and the Pole of the Periphery
of the Plane of the Projection, we ſhall have the two Extremities of
the conjugate Diameter of the ſmall Circle to be projected ; and if
thro the middle of this Line, and at Right-angles to it a Right-line
be drawn equal to the double Sine of the diſtance of the ſmall Circle
from its Pole, we ſhall have the two Extremities of the tranſverſe
Diameter, thro which four points if an Ellipſis be deſcribed, it will
be the Repreſentative of the ſmall Circle to be projected
And inalinuch as the Semi-conjugate Diameter of every great Circleis
equal to the Co-line of its Elevation above the Plane of the Proje&ion,
or to the Sine of its neareſt approach to the Pole of the Projection
it felt, if the Right Sine of its neareſt approach, or the Co-line of
its Elevation above the Plane of the Projection, be ſet off from the
Center or Pole of the Proje&ion p, ſee the Fig. in Page 54 , on ei-
ther Side to c and r, we ſhall have the two Extremities of the conju-
gare Diameter, thro which, and the points P and S, the two Extre-
mities of the tranſverſe Axis PS, if an Ellipſis be deſcribed, it will be
the Reprelentative of the given great Circle in this Proje&ion; and
hence we are taught how to draw the Repreſentative of any great or
Small Circle, upon any given Plane: And that this may the more
evidently appear, I ſhall ſhew how in
Bbbb *
.
Ex-
548 Of the Orthographic Proje&tion of the Sphere
Example 1.
To draw the Hour-circles, or Circles of Right Afcenfion, Parallels
of Declination, Tropics, Ecliptic, &c. Orthographicaliy upon the Plane
of the Solstitial Colure, it being in this caſe ſuppoſed to lye in the
ſame Plane with the Meridian ; and to adapt it to the Latitude of
si deg. 33 min. North.
Let ZONH (in Plate the 4ch) repreſent the Solstitial Colure, upon
whoſe Plane the Sphere is to be projected, then will 26 NH be
the Circumference of a Circle, by the firſt Cor, of the General Defini-
tion, and its Center and the Pole of the Projection, by the ſecond
Corollary of the fame
And becauſe the Eye is ſuppoſed to be placed at an infinite diſtance
from ihe Globe, in that Line that pafles from the Center thro the
vernal Equinox, the common Interſection of the Equator, Ecliptic,
Equinoctial Colure, and in this caſe the Horizon ; and inaſmuch as
thele Circles are all perpendicular to the Plane of the Projection,
they will all be repreſented by Right-lines, by Cor. the 3d, interfcct-
ing each oiber in the Center or Pole of the Proje&tion. Wherefore,
Having drawn the Diameter HrO to repreſent the Horizon,
and at Right-angles to it the Diameter Z Ý N, to repreſent the
Prime Vertical or Circle of Eaſt and Weſt, fer of the Chord of so
deg. 30 min. the height of the Pole at London, or Latitude of the
Place in the Periphery, from 0 to P, and draw che Diameter P S,
this ſhall repreſent the Axis or Six a Clock Hour Circle, or in the
preſent Caſe the Equinoctial Colure, the two Extremities of which
will repreſent the two Poles, the point P the uppermoſt or Northern
Pole, and the point S the lowermoſt or Southern Pole; and if the
Chord of 38 deg. 30 min. the Complement of the Latitude, be ſec
off from 0 to 2 downwards, or from H to A, and the Diameter AQ
be drawn, or which is the fame thing, if che Diameter 4 Q be
drawn at Right-angles to the Axis PS it will repreſent the Equator
in this Projection.
The Ecliptic is a great Circle cutting the Equinoctial in the op-
pofite points of Aries and Libra, and forming an Angle with it of
23
deg. 29 min. equal to the Sun's greateſt Declination; wherefore if
the Chord of 23 deg. 29 min. be fet off in the Solflitial Colure from
A to $, or from 2 to W. and the Diameter or wo be drawn, this
will repreſent the Ecliptic in this Projedion, and where it interſects
the Axis as in r or, it gives the Places of the vernal and autum-
Ral Equinoxes in this Projection.
In-
Of the Orthographic Projection of the Sphere 549
Inaſmuch as the Parallels of Declination, Tropics, Artic and
Antartic Circles are ſmall; Circles, ſtanding at Right-angles to the
Periphery of the Plane of the Projection, they will all be repreſented
by Right-lines, equal in Length to the Diameters of the ſmall Cir-
cles themſelves, by the sth preceding Cor. and at the Sines of their
reſpective diſtances from the greatCircle to which they are parallel;
or which is the ſame thing, their Repreſentatives will interfect the
Solſtitial Colure, in the ſame points in which the Circles themſelves
interſect it upon the Sphere, by the 6th Cor. wherefore if the Chord
of 23 deg: 29 min. the greateſt Declination of the Sun, be ſet oft
in the Solſtitial Colare from A to % and x, and from 2 to C and
w, and the Right-lines SC and x Wo be drawn, or which is the ſame
thing, if at the diſtance of the Sine of 23 deg. 29 min to the
Radius Ar or re, the Right-lines % C and we x, be drawn paral-
lel to Al the Equator, they will be the Repreſentatives of the two
Tropics in this Proje&ion, the Northermoſt or % C the Tropic of
Cancer the Southermoſt or x vp the Tropic of Capricorn ; and if
the Chords of 23 deg. 29 min. the diſtance of the Polar Circle from
the Pole, be ſet off in the Solſtitial Colure from P to a and t, and
from Stoa and t, and the Chord Lines art and ant be drawn; or
which is the ſame thing, it at the diſtance of the Sines of 66
deg. 31 min. the diſtance of the Polar Circles from the Pole, to the
Radius r P, the Lines art and ant be drawn parallel to the Equa-
tor Are, they will be the Repreſentatives of the Polar Circles in
this Projection the Northermoſt; or ar t the Reprclentative of the
Artic Circle, the Southermoft or ant the Antartic Circle: In like
manner, if the Chords of 10, 20, 30, degrees, “c. be ſet of in
the Solftitial Colure from A to 10, to 20, to 30,6 c. and from 2 to jo,
10 20,0 30,6c and the Chord Lines 10, 10, 20, 20, 30, 30, 6c.be
drawn; or if at the diſtance of the S. of 10, 20, 30, deg. 6c Lines as
10 10, 20 20, 30 30,6 c. be drawn parallel to theEquator, they will
be the Repreſentatives of the Parallels of 10, 20, 39, deg &c. of
Declination in this Projection; and after the ſame manner may the
Parallels of Latitude or Parallels to the Ecliptic or be drawn
The Hour Circles, or Circles of Right Aſcenſion, which are great
Circles interſeccing each other in the Poles of the World, and cutting
the Equator at Right-angles, are inclined to the Plane of the Pro-
je&ion, at Angles of 15, 30, 45 deg. c. and conſequently will be
repreſented in the Projection by Ellipſis's, by Cor. the roth, having
their tranſverſe Axis equal to each other, and to the Diameter of the
Bbbb 2 *
Sphere
550
Of the Orthographic Projection of the Sphere
Sphere, by the 11th Cor. and their conjugate or ſhorteſt Diame-
ters equal to the double Co-fine of their reſpective Inclinations to
the Plane of the Projection, or to the double Sine of their reſpective
diſtances from the Pole of the Projection, meaſured in the Arch of
the great Circle that pafles thro the Pole, and cuts the given Circle
at Right angles; wherefore if the Sines of 15, 30, 45, deg. &c. to
the Radius of the Projection re be laid off in the Equator A l,
from r to 7 and 5, 108 and 4, 109 and 3, c. we ſhall have the
conjugate Diameter, &c. 7.5, 8.4, 9.3, &c. of the Ellipſes repreſenting
the ſeveral Hour Circles in this Projection, whole common tranſverſe
Axis is the Axis of the Sphere, or Six a Clock Hour Circle
PrS; wherefore it about the common tranſverſe Axis PřS
and the ſeveral conjugate Diameters 7-5,8 4,9 3,6c. tlie leveral El-
lipſis's P5 S7P, P 8 54 P, P 9 S 3 P, &c. be drawn), we ſhall have
the ſeveral Reprefuncitives of the ſeveral Hour Circles in this
Projection : And among the various Methods made Uie of by the
Conic Writers, for the Conſtruction of an Ellipſis that ſhall paſs thro
the extream points of any two Diameters, tone is more eaſy and ſeems
better adapted to our preſent Buſineſs, than that Method that is
performed by the help of the Line of Sines.
It has been demoſtrated in Page the 5411, that the ſeveral Or-
dinates cb, fe, zw, &c. in any Ellipſis, as ſuppoſe 5 P 7 S, are in
the ſame Proportion to each other, as are their correſpondent Ordi-
nates 10 h, 20.1, 30.2, 6c in the circumſcribed Circle HPOS;
but the Ordinates 10.b, 20.4, 20.w, &c. in the Circle APQ S, are
as the Sines of their reſpective diſtances from their common Pole P,
or as the Co ſines of their reſpective diſtances from the Diameter
of the Circle Al to which they are parallel, to the Radius or
Semidiameter A = of the circumſcribing Circle AP QS; wherefore
theſe ſeveral Ordinates bc, fe, zw, &c in the Ellipfis 5 P7 S, will
be directly as the Sines of their reſpective diſtances from the coinmon
Pole P, or as the Co-fines of their diſtance from the conjugate Dia-
meter 7 r s to which they are all parallel, to the Radius 77, the
Semi conj.:gate Diameter : And inaſmuch as the ſame Law obtains
in every other Ellipfis that can be drawn about the common Axis
PS; hence we are taught a very ready and expeditious Way of de-
ſcribing the Hour Circles in this Proje&tion.
For having made r7 the Semi-conjugate Diameter of the Ellipſis
7's S, equal to the Sine of so deg. oo min. upon the sector, if
the Sines of the reſpeđive Arches P 10, P 20, P 30, Óc. be taken
off from the fame sector, thus extended and laid upon the ſeveral
Oro
Of the Orthographic Projection of the Sphere
551
Ordinates rob, 200, 30 w, and from the points b, e, w, C. CO
c, f, and i, we ſhall have ſo many points thro which the Periphery
of the Ellipſis muſt paſs; thus if the Arches P 10, P 20, P 30, 6c.
be ſuppoſed to be Arches of 80, 70, 60, &c. degrees, if the Sines,
of 80, 70, 60, degrees, bo to the Radius 77, be ſet off
in the reſpective Ordinates 10k, 20 e, 30w, &c. from the
points b, e, w, &c where thoſe Ordinates interſect the tranſverſe
Axis of the Ellipfis, to the points e, f, i, &c. they will give ſo ma-
ny points in the Periphery of the Ellipſis to be deſcribed ; and after
the lame manner may an infinite number of other points be found,
thro which if a Curve Line be drawn, it will give the Repreſenta-
tive of the relpective Hour Circle in this Projectio : Theſe things
being premiſed,
Ler it be required to draw the firſt Hour Circle from the Six a
Clock Hour Circle, that is the four Circle of Seven in the Morn-
ing and Five in the Evening, or of Five in the Morning and Seven
in the Evening, for the ſame Hour Circle in the Projection repreſents
theſe four leveralCircles upon the Globe ; inaſmuch therefore as this
Circle is 5 Hours diſtant from the 12 a Clock Hour Circle, and
confequently its Plane is inclined to the Plane of the Meridian, or
to the Plane in this Caleo the Projection, at an Angle of 75 degrees,
if che Sine of 15 degrees its Complement to the Radius of the Pro-
jection Ar, be laid off in the Equator from to 5 and 7, we ſhall
have the two Extremities 5 and 7 of the conjugate Diameter, and if
th:Semi-conjugate Diameter 77, be made equal to the Sine of 90 d.
oo min. upon the Sečtor, and the Sines of 80, 70, 60. 50, and 40
degrees, at the ſame opening of the Se&tor, be ſet off in the Lines
13b 10, 20 e 20, 30f 30, 401 40, 50050, Gc. drawn thro
the Extremities of the Icveral Arches A10, 1020, 20.30, 30.40,
40.50, doc. in the Periphery of the Plane of the Projection of 10 deo
grees each, from their common Inscríection b, e, h, i, o, with the
tranſverſe Axis of the Ellipſis on each ſide, to the points e and d, f
and g, i and k, m and n, &c. and thro thoſe ſeveral points and the
Poles P and S, a Curve Line be drawn, it will be the Repreſentative
of the Five a Clock, Cc. Hour Circle in this Projection; and after the
fame manner may every other Hour Circle as 8 P 4S, 2P3S,
10 P 2 S, &c. or any Circle of Right Aſcenſion be drawn.
Circles of Cæleſtial Lengitude, inalmuch as they interſect each o-
ther in the Poles of the Ecliptic t anda, and cut the Ecliptic s pivo
which is here repreſented by a Right-line, at. Right-angles, are of
CON***
552 Of the Orthographic Projection of the Sphere
conſequence inclined to the Plane of the Projection, and are t"cre.
fore reprefented by Ellipſis's, whoſe common tranſverſe Axis is the
Axis of the Ecliptict ra, and whoſe conjugate Diameters are equal
to the Co fines of their reſpective Inclinations to the Plane of the
Solſtitial Colure, and conſequently are projected after the ſame man-
ner as the Hour Circles are in the ſame Projection ; thus for Exam-
ple, if it were required to draw the Circle of Longitude paſſing thro
the firſt points of Gemini and Sagittarius, or of Leo and Aquarius,
for each of theſe are repreſented by the ſame Circle of Longi-
tude in this Projc&tion, inaſmuch as this Circle is inclined to the
Plane of the Solſtitial Colure at an Angle of 30 degrees, if the Sine
of 60 degrees to the Radius of the Projection, be ſer off in the E-
cliptic from r or, for both are repreſented here by the ſame point
to I or 11, and to me and f, we ſhall have the two Extremities of
the conjugate Diameter, thro which two points and the Polest and
r of the Ecliptic, or the Extremities of the common tranſverſe Axis
prs, an Ellipfis be drawn after the manner caught in the former
Example, it will be the Repreſentative of the Circle of Longitude
in this Projection ; and becauſe the ſeveral Parallels of Cæleſtial La-
titude are all parallel to the Ecliptic, and conſequently perpendicu-
lar to the Plane of the Projection, they will all be repreſented by
Right-lines in this Proje&tion, and may be drawn if required, after
the ſame manner as the Parallels of Declination are in the fame Pro-
je&ion.
And becauſe the Vertical or Azimuth Circles, interſect each other
in the points 2 and N, repreſenting'the Zenith and Nadir, and cut the
Horizon H r o at Right-angles, and which is here repreſented by
a Right-line, are of conſequence inclined to the Plane of the Meridian
APOS, and are alſo conſequently repreſented here by Ellipfis's
whoſe common tranverſe Axis will be the Right-line z'r N, the
Repreſentative of the Prime Vertical, or Circle of Eaſt and Weſt,
and their ſeveral conjugate Diameters, the Right-lines of the Com-
piement of their ſeveral Inclinations to the Planes of the Meridiani,
or to the Sines of the ſeveral Angles that their reſpective Planes forin
with the Prime Veitical Z r N, and may therefore be projected
after the ſame manner as the Hour Circles and Circles of Longitude
are in the ſame Projection, and inaſmuch as the ſeveral Almican-
thers or Parallels of Altitude are all parallel to the Horizon, and
cut the Prime Vertical at Right-angles, and conſequently are per,
pendicular to the Plane of the Projection, they will all be repreſented
by
of the Orthographic Projection of the Sphere 553
by Right-lines, and are projccted after the ſame manner as the Ciri
cles of Declination, or Parallels of Cæleftial Latitude aru, by ſet
ting off the Sines of their respe&ive Heights above the Horizon, or
the Co-lines of their respective diſtances from the Zenith, in the
Prime Vertical form v to 2, and drawing thro thoſe reſpective points,
Lines parallel to the Horizon, till they incerfect the Periphery of.
the Plane of the Projection.
Havng thus thew how all the Principal, great and ſmall Circles
of the Sphere, may b: drawn upon this plane, I !hall as a farther
Inſtance of the univerfality of the General Rules laid down in
the former Part of this Sciłon, lhtw how all the fame Circles may
be drawn upon the Plane ot the Horizon :. Let it therefore be re.
quired in
Example 2.
To draw the Hour Circles or Circles of Right Aſcenſion, Paral.
lels of Declination, Tropics, Eclipric. &c. Orihographically upon
The Plane of the Horizon, for the Latitude of sideg. 32 min.
North,
In Plate the sth, Let ZrHON repreſent the Horizon, upon
whoſe Plane the Sphere is to be projected, then will its Periphery
ZHEN be the Circumference of a Circle, by the firft Cor. and
:its Center the Pole of the Projection.
And becaufe the Eye is ſuppoled to be placed at an infinite die
ſtance in that Right-line that paffes thro the Zenith and Nadir and
the Center of the Sphere, the Vertical or Azimuth Circlesinaſmuc!
as they interfect each other in the Zenith and Nadir, which in this
Cafe is repreſented by the poinr 2, and cut the Horizon at Right-
angles, will all be reprefenred by Right-lines by Cor. the 3d, in-
terfecting each other in the point z, the Center and Pole of the
Projection, and forming Angles with cach other equal to thoſe that
they repreſent upon the Sphere it ſelf, Wherefore,
Having drawn the Diameter r 2 , to repreſent the Prime Ver-
tical, or Circle of Eaſt and Weſt, at Right-angles to it draw the
Diameter HZN, this will repreſent the Meridian, or Circle of
North and South Azimuth, and if the ſeveral Angles that the feveral
Azimuth Circles form with the Meridian, be laid off in the Peri.
phery of the Plane of the Projectic n, from 11 towards , or from
N towards y, and thro thoſe points and the Center Z, Right-lines
be drawn, they will be the Repreſentatives of to many Azimuths
Z
so
oo
701
30
80
20
69
P
10
im
10
3 R + E
me
8
60
Z
30
B В
H
O
K
th
F
30
20
20
m
31
a
10
101 / 1112
e
13
S
Vs Р
N
554 Of the Orthographic Projection of the Sphere
or Vertical Circies in this Projection ; thns if it were required to
draw a Vertical or Azimuth Circle, that ſhall form an Angle of
30 degrees with the Meridian, fet off 30 degrees in the Periphery
of the Plane of the Projection from H to b, or from N to d, and
draw the Diameter h z d, this will repreſent the Azimuth or Ver-
rical Circle to be drawn, and by proceeding after the ſame manner,
may any other Azimuth or Vertical Circle be drawn, but thef: aré
omitted to avoid Confuſion.
The Arch of the Meridian intercepted between the Zenith of a.
ny Place and the Pole, is equal o the Complement of the La itude
of that Place; wherefore if the Sine of 38 deg 20 min. the Corn.
plement of the given Latitude be ſet off in the Meridian from Z 1o
P, it will give the Repreſentative of the North Pole in this Projec-
rior.
As the Equinoctial is a great Circle deſcribed about the Pole as a
quadrantal diſtance from it, cutting the Horizon or Periphery of the
Plane of the Projection in the oppoſite points of Aries and Libra, and
confequently is inclined to the Plane of the Horizon, at an Angle of
28 deg. 20 min. equal to the Complement of the Laricude of de
Place, it will therefore be projected into an Ellipfis, by the Joth
peceling Corollary. Wherzo e
If the Right Snz of 51 deg. 30 min. or the diſtance of the Equi-
tor from the Zanichi, meaſured in the Arch of ihe Meridian, or the
Complement of the Inclination. he laid off in the Meridian trom Z
to A The contrary way with the Pole point p, it will give one Extre-
mity d of the Semi-conjugate Diameter 2 A of the Ellipfis, about
whích and the tranſverſe Axis r 2 , if the Semi-Ellipfis Y A A
be drawn according to the manner taught in Page the 55114, we ſhall
have the Repreſentative of the upper half of the Equator to be drawn
upon this plane.
And again, as the Six a Clock hour Circle is a great Circle of the
Sphere, that curs the Meridian ar Right-angles paſſing thro the Pole
P, and conſequently inclined to the Plane of the Projection, at an
Angle of 51 deg. 30 min. it will therefore be projected into an
Ellipfis in this Projection, if the Sine of 38 deg. 30 min. the Com-
plement of its Inclination, be ſet off in the Meridian from 2 top
it will give one Extremity p, of the Semi.conjug te Diameter ZP
of the Ellipfis repreſenting this Circle, whoſe tranſverſe Axis will
be the Diameter r 2 , that paſſes thro the Center z, and cuts the
Meridian at Right-angles, about which Diameter, if the Semi-Fl-
lipſis
Of the Orthographic Projection of the Sphere 555.-
lipfis r P be drawn, after the Method taught in the 551 Page,
we ſhall have the Repreſentative of the upper half of the Six a Clock
Hour Circle in this Projection.
And becauſe the reſt of the Hour Circles interſect each other in the
Pole point, and cut the Equator at Right-angles, dividing it into
equal Parts or Portions of 15 Degrees each, if about the common
Semi-conj gate Diameter Z P, and the ſeveral Semi-tranſverſe Dia-
meters Z 1, Z2, Z3, Z4, &c. the ſeveral Semi-Ellipfis's Pi,
2P2, 3
P 3, 4P 4, &c. be drawn after the manner taught in the
former Example we ſhall have theRepreſentatives of the leveral Hour
Circles to be drawn upon this plane, as is abundantly maniteſt from
what has been ſaid.
The Ecliptic is a great Circle of the Sphere inclined to the Plane
of the Equator, at an Angle of 23 deg. 29 min. and conſequently
if the Southern half be Elevated above the Plane, muſt form an An-
gle of 14 deg. 59 min. with the Plane of the Horison, equal to the
Difference between the two Inclinations 38 deg. 28 min. and 23
deg. 29 min but an Angle of 61 deg. $7 min. equal to the Sum of
the two Inclinations 38 deg. 28 min and 23 deg. 29 min. if the Nor-
thern half of the Ecliptic be elevated above the Plane of the Eclip-
tic, and conſequently in either Cafe will be repreſented by a Semi-
Elliplis, whoſe common tranſverſe Diameter will be che Diameter
rŻ, that connects the Interſection of the two Circles together ;
wherefore if the Sines of 75 deg. or min and 28 deg. 03 min. the
Complements of the Inclinacions of the two Semi-Eiliplis's, be lec
off in the Meridian HZ N, trom 2 the Center, to w and s, we
ſhall have the Semi-conjugate Diameters SZW and Zs of the two
Semi-Ellipfis's to be drawn, about which and the common tranſ-
verſe Axis Z, if the Ellipfis's in w and r s be drawn, we
ſhall have the Repreſentative of either half of the Ecliptic upon this
Proje&ion.
Circles of Celeſtial Longitude, which interfe&t cach other in the
Poles of the Ecliptic,and cut the Ecliptic at Right-angles, dividing it
into equal Parts or Portions, are projected after the ſame manner as
the Hour Circles or Circles of Right Afcenfion are ; which interſell
each other in the Poles of the World, and cut the Equator a Right-
angles, but theſe are omitted in this Proje&ion, to avoid Confuſion,
and may be eaſily ſupplied if there be occafion.
Almicanthers, or Parallels of Attitude, inaſmuch as they arei parallel
to the Horizon, or in this Caſo to the Periphery of the Plane of the
C cc c +
Pro-
i
556 of the Orthographic Projection of the Sphere
Projection, will by the 7th Cor. become Circles in the Proje&ion,
will have the Center z, or Pole of the Proje&ion for their common
Center, by the 8th Cor. and their reſpective Radij will be equal to
the Sines of their reſpective diſtances, from the Zenith or Pole of the
Proje&tion 2, or to the Co-fines of their Elevations above the Plane
of the Projection; wherefore if about the Center or the Pole of the
Projection 2, or the Repreſentative of the Zenith, a Circle be de-
fcribed whoſe Radius is equal to the Sine of the diſtance of that Cir-
cle from the Zenith, or to the Co-line of its Elevation above the Plane
of the Horizon, you will have the Repreſentative of that Parallel
of Alticu e upon this Projection ; Thus, if it were required to draw
a Parallel of 40 degrees of Altitude in this Projection, becauſe this
Circle is diſtant from the Zenith so degrees, the Complement of
40 degrees, if about the Center Z, at the diſtance of the Sine of so
deg. a ſmall Circle as 40.40, 40.4", be drawn, it will be the Re
preſentative of the Parallel of 40 degrees of Altitude upon this Plane ;
and the ſame Method muſt be obſerved in drawing of any other Pa-
rallel of Altitude whatſoever.
The Tropics, Polar. Circles, and Parallels of Declination, inaſmuch
as they are parallel to the Equator, will be inclined to the Plane of
the Projection, and will therefore be projected into Ellipſis's, by the
13th Cor. having their conjugate Diameters coincident to the Meri-
dian HZ N, by the 16th Cor. their conjugate or ſhorteſt Diaineters
equal to the Sum or Difference of tbeir greateſt and leaſt diſtances
from the Pole of the Proje&ion, in this Cafe the Zenith 2, by the
17th Cor. and their tranſverſe Axis's or longeſt Diameters, equal to the
doubleCo-ſine of the diſtance of the reſpect ve ſmall Circles diſtance
from its Polez, by the 14thCor. ſo that if it were required to project,
the Artic Circle in this Proje&ion, inaſmuch as it is diſtant from the
North Pole by the Space of 23 deg. 29 min. leſs than the diſtance
between the Pole and the Zenith, equal in our preferi Cafe to 38
deg 28 min, it will fall indirely on the ſame ſide of the Zenith;
wherefore if the Sine of or deg. 57 min. the Sine of 38 deg. 28 min.
, the diſtance of the Pole from the Zenith' and 23 deg. 29 min. the
diſtance of the Artic Circle from the Pole, be ſet of' in the Meridian
from Z to R, we ſhall have one Extremity of the conjugate Diame-
ter, and it the Sine of 14 deg. 59 min. the Difference between 38
deg. 28 min. the diſtance between the two Poles, and 23 deg. 29
min the ſmall Circles diſtance, be laid off on the ſame side of the
Zenith from 2 to be we shall have the other Extremity of the con-
jugate
i
of the Orthographic Proje&tion of the Sphere 557
jugate Diameter at; wherefore if this $be biſected by the Right
line rc, equal in Length to the double Right fine of 23 deg. 29min-
the diſtance of the Artic Circle from the Pole, to the Radius of the
Projection r Z or ZH, we ſhall have the two extream points of the
tranſverſe Diameter, chro which and the two Extremities a and t,
of the conjugats Diameter, if an Ellipfis be drawn, we ſhall have
the Reprelentative of the Artic Circle in this Projcaion.
Again, if the Tropic of Cancer were to be projeEted, becauſe the
diſtance of the Tropic of Cancer from the North or uppermoft
Pole 66 deg. 31 min, is greater than 38 deg. 28 min. the diſtance
of the two Poles of the Horizon and Equator, this Circle will fall
on contrary Sides of the Zenith Z; wherefore if the Sine of
104 deg. 59 min the Sum of 66 deg. 31 min. and 38 deg. 29
min. be let off from Z towards H, it will give ore Extremity of the
conjugate Diameter, and if the Sine of 28 deg. 03 min. the Diffe
rence between 66 deg. 31 min and 38 deg. 29 min be ſet off from
Z the contrary way to %, it will give the other Extremity % of the
conjugate Diameter, ard if this conjugate Diameter be biſfeated by
a Line at Righe-angles, equal to the double Sine of 66 deg. 31 min.
the diſtance of the Tropic from its Pole, we ſhall have the two Ex-
tremities of the tranſverſe Diameter, thro which and the two Ex-
tremities of the conjugate Diameter before determined, if the Peri-
phery of an Ellipfis be drawn, it will be the. Reprelentative of the
Tropic of Cancer in this Projection.
After the ſame manner were the Tropic of Capricon, and the for
veral Parallels of Declination in the Figure drawn.
Parallels of Cæleſtial Latitude or Parallels to the Ecliptic, inaſmuch
as they are inclined to the Plane of the Projection, wil be repreſented
by Ellipfis's in this Projection, and are drawn after the ſame manner
as the Parallels of Declination are in the former Cale, regard being
had to the Poſition of their Pole, and their respective diſtances from
it ; but theſe as well as the Circles of Longitude themſelves are pura
poſely omitted to avoid Confuſion ; eſpecially ſince if there be oco
cafion for them, they may eaſily and readily be done by the Rules al-
keady delivered.
Asithe different Appearances of the ſameCircles of the Sphere upon
different Planes, ariſes only from the different Inclinations of the Planes
of chofe Circles to the Plane of the Circleupon which they are to be
repreſented, ſo the Rules here delivered for drawing the ſeveral Cir-
cles of the Sphere upon the Plane of the Horizon Orthographically, will,
Serve for drawing of the ſame Circles upon the Plane of rhe Equator,
CCCC7
558 Of the Orthographic Proje£fion of the Sphere
it we confider the Vertical or Azimuth Circles, and Parallels of
Altitude, as Hour Circles and Parallels of Declination in the former
Example, and the contrary ; Since the inclination of the Prime Verti-
cal to the Plane of the Equator, is the ſame with the Inclination of
the Six a Clock Hour Circle to the Plane of the Horizon. The ſame
Rules will extend likewile to the drawing of the ſame Circles upon the
Piane of the Ecliptic, if we conſider the Six a Clock Hour Circle
as inclined to the Plane of the Proje&ioni, at an Angle of 66 deg.
31 min. as it really is, inſtead of 38 deg. 32 min. as in the former
Example, and the Inclination of the relt offthe great Circles, and the
diſtances of the Interſections of the ſeveral fmall Circles with the
Meridian, from the Pole of the Ecliptic, in this Cafe the Pole of
the Proje&ion accordingly; and therefore I ſhall omit ſhewing the
particular manner of
drawing them here, upon thoſe two Planes, not
doubting but upon Occafion, the Intelligent Reader will readily and
eaſily perform them himſelf,
Altho every Reprefentation of the Sphere upon the Plane on which
theCircles are projected, according to ſomcknown and eſtabliſh'd Law,
are fufficient to exhibit all the various Triangles form'd upon the
Surface of the Sphere, by the mucual Interſections of the ſeveral
great Circles deſcribed upon it, and account for all the various Ap-
pearances ariſing from the diurnal Motion of the Cæleſtial Bodies,
yet ſuch only are retained for common Uſe, wherсin the Triangles
themſelves are expreſſed to the moſt Advantage, and the Repreſen-
tatives of the Circles themſelves are drawn with the greateſt Eale;
and hence it is that the Stereographic Projections of the Sphere are
preferr'd before all other in common Practice, becauſe the Circles
of the Sphere in general, are projeđed upon all Planes, either into
Right-lines, than which nothing is more ſimple, or into Circles,
which of all Curves are the moſt perfect; and are conſtructed
with the greateſt Eaſe, and that among the many various Reprſen-
tations of the Sphere upon Planes, that upon the Plane of the Meri-
dian is preferr'd before all others; for the Solution of the common
Problems, relating to the Riſing and Setting of the Sun or Stars, Óc.
and that theReprelentation of the ſame Circles upon the Plane of the
Horizon is preferr'd before others for the Solution of Sciotherical Pro-
Hems, and that the Projection of the ſame Circles upon the Plane of
the Equator, for the repreſenting the Cæleftial Images or Constel-
tations, &c; becaufe) the ſeveral Triangles that are requiſite for the
Solution of the various Problems peculiar to each Branch of Science,
1
are
of the Orthographic Projection of the Sphere 559
are exhibited to very grear Advantage in thoſe ſeveral Repreſentaci-
ons, and the Ules for which they are ſeverally deſigned, are per-
formed with the greateſt Eaſe and Certainty: And altho the Ortho-
graphic Repreſentations in general, fall ſhort of theſe, in the conſtruc-
ting of the Problems and their Solutions by the Proje&ions themſelves,
upon account of the Circles that are inclined to the Plane of the Pro-
jection, being repreſented by Ellipfis's, which being not Geometrical
Curves, muſt be drawn per puncta, requiring conſequently more
cime in the Performance than in the drawing of Circles, and being
fubje& to leſs degrees of Accuracy; yer as every particular Proje&ion
of the Sphere, has its particular Advantage above another in the So-
lution of ſome kind of Problems, fo the Representation of the Cir-
cles of the Sphere upon the Plane of the Solftitial Colure, when it
is ſuppoſed to lye under the Meridian, has ics peculiar Excellencies
in giving Solutions to the uſual Problems ariſing from the diurnal
Motion of che Sun, &c. with greater Expedition and Exactneſs in
moſt Caſes, than any other kind of Projection whatſoever will doe,
as will more evidently appear in the Sequel of this Section
From what has been ſaid concerning Orthographic Projection, it is
abundantly manifeſt, that any Part or Portion of a greatCircle that is
projected into a Right-line, which for diſtin&ion Sake is called a
Right Circle, intercepted between the Center and any other point ta-
ken in the ſame proje&ted great Circle, is equal to the Right-fine of
the Arch of that great Circle that it repreſents.; thus in Plate the 4th
it is manifeft, that rw the Repreſentative of the Arch A $ of the
great Circle APON, is equal to D the Right-line of the ſame
Arch A %; and conſequently any Part or Portion of the ſame pro..
jected Right Circle, intercepted between any two given points, is
equal to the Sum of the Sines of the two Arches intercepted bes
tween the Pole of the Projection and each Extremity of the given
Arch, if the Pole lye within the given Arch, but to the Difference
of the Right-fines of the fame two Arches if the Pole of the Plane
of the Projection lye without the given Arch'; or which is the fame
thing, if the given Arch lye intirely on the ſame ſide of the Pole, for
in the Figure in Plate the 4th, the Arch Ww of the projected greaca
Circle PS, is equal to rw trw, the Sum of the Sines of the
Arches t so and Ave, the diſtance of each of the Extremities of the
given Arch % vp, of which the given Right-line Ww is the Repre
lentative, from A (in this Cafe the Pole of the Proje&ion) upon che
Surface of the Sphere, but w7 the Repreſentative of the Arch of the
Peria
560 of the Orthographic Projection of the Sphere
Periphery sa, is equal to the Difference of the Right- fines ro
and rw, of the Arches A a and AS, the diſtances of the Extremi-
ties a and 5 of the Arch l, from the Pole A of the Proje&ion.
And becauſe the ſmall Circle deſcribed upon the Surface of the
Sphere, of which the Right-line sec, is the Repreſentative, in-
alniuch as it is parallel to the Equator, and conſequently with the
Equator cuts off equal Portions of the ſeveral Hour Circles or Cir-
cles of Right Aſcenſion deſcribed upon the Surface of the Sphere :
And inalmuch as the fame happens to every other ſmall Circle that
is parallel to the Equator, or to any other great Circle whatſoever.
it is manifeſt that the ſeveral Parts or Portions AS, Z 7, un
ys, &c. of the ſeveral projected Hour Circles PAS & P755,
Pwr S, &c. contained or intercep-ed between the Diameter A2,
the Repreientative of the Equator, and the Right-line © wc, the
Repreſentative of a ſmall Circle parallel to the Equator, ſuppoſe
the Tropic of Cancer, or of any other Portio is 40 A, m 7Ir,
12.5, &c. contained or intercepted between the Repreſentatives
40 m, 1, n, 40, of any otheriſmallCircle parallel to the Equater, and
conſequently the Portions of any other Number of great Circles,
that cut any proje&ted Right Circle at Right-angles, contained be
tween the projected Right Circle and the Repreſentative of any
ſmall Circle parallel to its ſelf, are equal to each other, and to the
Right-fine of that ſmall Circies diftance from the great Circle 10
which it is parallel ; whence it follows univerſally, that any Part
or Portion of any proje&ed great Circle intercepted between any
two Parallels, is equal to the Sum of the Sines of the diſtances of
thoſe fmall Circles from the great Circle to which they are paral-
lel, if the ſmall Circles lye on contrary ſides of the great Circle to
which they are parallel, but to the Difference of the Right-lines of
the diſtances of each of thoſe ſmall Circles from the great Circle to
which it is parallel, if the ſmall Circies lye on the ſame ſide of the
great Circle to which they are parallel ; thus in Plate the 4th, the
Arch Zz of the projected great Circle PZ2 S, contained or inter-
cepted between the Parallel c and x , is equal to r Wtrw,
Sum of the Sines of the Arches A % and Ax, the diſtances of the
Parallel se and x v, from the great Circle Ae to which they
are parallel, but the Arch Z m of the ſame proje&ed great Circle
PZ Z S contained or intercepted between the Parallels A la and
and A 40, is equal to the Difference of r I and rw, the Sine of the
Arches
y w, the
Of the Orthographic Projection of the Sphere 561
1
Arches A 40 and AS, the diſtances of the Parallels from the
great Circle AL, to which they are parallel. And
Hence we are caught how to lay off any Number of Degrees in
any projected great Circle from any given point, or to meaſure any
Part or Portion of it when projected ; and becauſe every Spherical
Angle is mealured by an Arch of a great Circle deſcribed about
the angular point as a Pole, and intercepged between the two Cie: -
cles that conſtitute the Angle.
Hence we are taught how to conſtru& any Angle Orthographical-
ly, or to meaſure any Angle when projected.
And inaſmuch as any two great Circles which interfed each o-
ther in any given point, formirg thereby with each other any given
Angle, cut off between themſelves, equal Portions of all the finall
Circles that are drawn about the angular point, or point of Inter-
ſedion as a Pole, equal to each other, and to the Arch of the
great Circle the Meaſure of the given Circle: it follows, becauſe ihe
great Circles P 8 S, P7 S, PrS, &c. form equal Angles with each
other, that the Portions xm, ml, I n, nyo &c. and Si, ih, hk,
k , &c. of the Parallels 401 40 and 30 h 30, drawn about the an
gular point P as a Pole, are equal to each other, and to the Arches
87, 7 r r 5, &c. of the great Circle Al, deſcribed about the
angular point P as a Pole, and intercepted between the given great
Circles P 8 S, P7S, Prs, &c.
And again, inaſmuch as the ſeveral Portions al, m6, 6 h, im,
&c. of the ſmall Circle 401, 30 m, &c. are Ordinates in the El-
lipfis, and conſequently are to each other as the correſpondent Or-
dinates 40 l, 30 in, of the circumſcrbing Circle APOS: And in
aſ much as the Ordinates in the Circle ars, are the Sines of their
reſpective diſtances from the Pole of the great Circle to which they
are parallel, it follows that the ſeveral Portions él, ml, bh, and ib,
are as the Sines of their reſpective diſtances from their common Pole
P, to the ſeveral Radij 401, 30 m, &c. and hence we are taught
a ready way to mealure any Part or Portion of a projected ſmall
Circle, that is projected into a Right-line, or to lay off any Num-
ber of Degrees upon it when projected,
AS
aný Part or Portion of a ſmall Circle that is projected into a
Circle, is meaſured after the ſame manner as a common Rectilinear
Angle is, and any Number of degrées laid off upon it, in the ſame
manner as we conſtruct a Plane Angle ; and inaſmuch as any part or
portion of a ſmall Circle that is projeâed into an Ellipfis is meaſu-
red
*
..
.
다
​2
:
6
10
440
-H
A
me
-
L
10
110
2
해
​N
562 of the Orthographic Projection of the Sphere
red, and any Number of Degrecs laid off upon it, after the ſame
manner as we meaſure any Aich, or lay off any Number of degrees
in a projected great Circle inclined to the Plane of the Projection,
regard being had to the tranſverſe Axis and conjugate Diameter of
the given Ellipfis.
Hence we are taught univerſally, how to lay off any Number of
Degrees in a projected ſmallCircle, or to meaſure any Part or Portion
of it when projected-
The great Analogy between the Repreſentatives of the ſeveral
Portions of every projceted great or ſmall Circle, and their corre-
ſpondent Sines, which in great Circles that are either inclined to,
or ſtand at Right-angles to the Plane of the Proje&tion, as well as
in every ſmall Circle that cuts the Periphery of the Plane of the
Projection at Right angles, are ever equal to the Sines of the Arches
they repreſent upon the Surface of the Sphere, and may therefore
be fublticuted in the room of the Arches themſelves, is a peculiar
Property of Orthographic Proje{tion ; and renders the Solution of all
Allronomical Problems, by the Projection of the Sphere upon the
Plane of the Meridian, much more ſimple and eaſy than any other
kind of Proje&tion of the Sphere upon any other Plane can do; as
will more evidently appear from the following Examples.
Let it therefore be required by the figure in Plate the 4th, which
is an Orthographic Projection of the Prolemaic Sphere; ſuppoſing the
Eye placed at an infinite diſtance upon the Line produced from
the Center, thro the vernal Section, the Solſtitial Colure- beirg
under the Meridian, and the whole adapted to the Latitude (ſuppoſe)
of London, which is 5 i deg. 32 min. North, to find (the Sun's place
being firft ſuppoſed to be known or given.)
1. His Right Aſcenſion.
2. His preſent Declination, the greateſt Obliquity of the Eclip-
tic being fixed at 23 deg. 29 min, whence ånd from the Latitude vf
3. His Amplitude, Rifing and Setting, that is how many degrees
he riſes and fers from the true Eaſt or Weft points of the Horizon.
4. His Aſcenſional Difference, or how long be riſes or ſets before
or after the Hour of Six.
5. His height, at the hour of Six.
6. His Azimuth, or on what point of the Compaſs he ſtands,
from the points of Eat and Weft, at the hour of Six.
the Place,
7. His
of the Orthographic Proje&tion of the Sphere 563
.
7. His Height when he appears due Eaſt or Weft.
8. The Time when he appears due Eaſt or Weft.
9. His Height at any Time when he is in the Equator.
10. His Azimuth at any time when he is in the Equator.
11. His Height at any time when he is in any point of the Ecliptic.
12. His Azimuth on the ſame point at any given Hour; and of
theſe in their Order,
And let us ſuppoſe the Sun to be in 8 degrees of Taurus,
and let it be required to find his correſpondent Right Aſcenſion and
Declination
Now becauſe in this caſe the Sun is diſtant from the nextEquinoctial
point 38° oo' if the Right-line of 38° oo' to theRadius of the Proje&ion
Ai, be ſet off in the Ecliptic from r towards % to o, it will give
O for the Place of the Sun in this Proje&ion; and if thro this point
O and the Poles P. and S, a Circle of Right Aſcenſion as POR be
drawn, it will cut off an Arch 1 R, of the Equino&ial equal to the
Right-fine of 35 deg. 37 min. to the Radius of the Proje&tion Ar,
and the Arch Ó R of the Circle of Right Aſcenſion POR, equal to
14 deg. 12 min. will be the correſpondent Right Aſcenſion, which
is found by drawing a Right-line as TO D, thro the Place of the
Sun parallel to Arl, till it cut the Periphery of the Plane of the
Proje&ion in T; for then the Arch of the Meridian TA, will be e-
qual to the Arch of the Ellipfis OR, which may be meaſured by
the Line of Chords.
Bar inaſmuch as the ſmall Circle To D is parallel to the Equator,
it is manifeſt that r D equal to o R equal to TD, is the Sine of
the Arch TA the Meafure of the Arch of the Ellipfis O R, the pre-
fent Declination ; wherefore if the neareſt diſtance of the point o
from the Equator AQ, be applied to a Line of Sines of the ſame
Radius with the Radius of the Proje&ion, it will give at once the
correſpondent Declination.
And inaſmuch as the Quadrant of the Right Aſcenſion POR,
cuts of proportional Arches RT, D, c. from the great Circle
Are, and all Imall Circles as To D parallel to it, it follows,
that O D has the ſame proportion to TD, as the Arch of the Equi-
no&ial FR has to rA; but r R is the Right-line of the Right
Aſcenſion, in the preſent Cafe to the Radius of the Proje&ion r A;
wherefore D will be the Right-line of the ſame Right Aſcenſion to
the Radius TD, wherefore if theArch OD be applied to a Line of Sikes
of the fame Radius with TD, it will give the RightAſcenſion requi-
Dddd *
Icd
A
.
964 of the Orthographic Projection of the Sphere
red: So that having found the point , if the neareſt diſtance of that
point be applied to the Line of Sines upon the Se&tor, and fitted to
the Radius r A, it will give the preſent Declination, and the near-
eft diitance of the fame point from the Axis or Equino&ial Colure,
being applied to a Line of Sines whoſe Radius.is. equal to TD will
give the preſent Right Aſcenſion:
So that in the preſent Caſe having found the point. C, there is no
need of drawing the Ellipfis POR, in order to give a Solution to
the Problem, but only of applying the neareſt diſtance of that point
O to the Equino&ial and Equino&ial Colure, to proper Scales of
Sines, and the proper Requiſites will then be diſcovered.
And in order to adapt the sector to any Scale whatſoever, we have
no more to do, but to take the given Radius as ſuppoſe I D, be-
tween the points of a pair of Compatles, and to open the Sector till
one foot of the Compafles being placed in the 90 Degree point of
one of the Line of Sines, the other foot of the Compaſles will juſt
reach to the 90 Degree point in the Line of Sines upon the other
Leg of the sector, and after this manner is the sector fitted to mea-
ſure any Part or Portion of any great or ſmall Circle
The Sun's Declination being thus determined, let it be required
from thence and the Latitude of the Pace of Obſervation, ſuppoſe
London, which is placed in so deg: 32 min. of Northern Declina-
tion, and to which this Projection is adapted, to find the Amplitude
and Aſcenſional Difference; and that the Arches may be the more
conſpicuous, let us ſuppoſe the Sun to be in the firſt point of Can-
cer, when his Declination is 23 deg. 29 min. North. Wherefore,
Having drawn the ſmall Circle S wc parallel to the Equator, and
to repreſent the parallel of the Sun's diurnal Courſe, where this inter-
ſeats the Horizon as in the point B, it will give the point of the Sun's
riſing or ſetting, and conſequently the diſtance of this point B from
the point r, which repreſents the Eaſt and Weſt points of the Hori-
zon, will be the Righc-line of the Sun's Amplitude ; and being ap-
plied to a Line of Sines whore Radius is ro, it will be found to
give 39 deg. 50 min. the San's Amplitude in the preſent Cafe, and
if the Arch B. w of the Parallel % C, or Parallel of the Sun's diur-
naj Courſe, be meaſured by a Ljæe of Sines, whofe Radius. is equal
to s wor w C, the Semi-Length of the Parallel, it will give 33 deg.
q9 min. equal to 2 hor. 12 min. 36 ſec. for the Aſcenſional Diffe-
rence, or time that the Sun riſes before, and lets after the Hour of
Six. For if chro the point Ba Quadrant of an Ellipfis or Hour-
Circle
Of the Orthographic Projection of the Sphere 505
?
Wenn
Circle be drawn, as PBM, it is manifeſt that rm equal in Value
to w B, the diſtance between the aſcending points r'and m is the
preſent Aſcenſional Difference.
Again, the Sun being ſuppoſed to be in the ſame point of the E-
cliptic as before, and it be regiuired to find the sth and 6th Regui-
ſites, that iş his Height and Azimuth at the Hour of Six, if from
the point w The Interfe&tion of the parallel of the Sun's diurnalCourſe
with the Axis or Six a Clock Hour Circle, the Right-line w C, be
drawn, it will be equal to the Right-line of 150 07' the Sun's Azimuth
at that time from the Eaſt or Weſt points of the Horizon the Radius
C 60, as will a perpendicular Line let fall from the ſame point w
upon
the Horizon H0, give 18 deg, 11 min. for the Sun's Altitude
at that time, if meaſured by a Line of Sines adapted to the Radius of
the Proje&ion, as is very maniteſt, if thro the Pole P and the point
w, a Quadrant of an Ellipſis or Hour Circle as ZW Fbe drawn.
Again, the ſame things being given as before, and it be required to
find the 7th and 8th Requiſites, that is the time when he will ap-
pear due Eaſt or Weſt, and his Height above the Horizon at that
tiinc, if thro the point y the common Interfe&ion of the Parallel of
thie Sun's diurnal Courſe and the Prime Vertical Zr N, a Qua-
drant of an Ellipſis as Prz, be drawn or imagined to be drawn,
fine of 30 deg. 35 min. to the Radius of the Projection « Z for his
Alcitude, and the Arch of the parallel yw, cqual to the Right ſine
min. equal to or h. 20 m. 44 f. to the Radius 6 w
oř wc, for i he time after Six in the Morning, when the sun will
appear due Eaſt, but before Six in the Afternoon when he will ap-
pear due Weſt.
Again," ſuppoſing the Sun to be in the Equator, and it be requi-
red to find the 9th and 10th Requiſites, that is his Height and A-
zimuth at any Hour of that Day, as ſuppoſe at Three in the After-
noon or Nine in the Morning.
If thro the points 9 3, where the 3 or 9 a Clock Hour Circle cuts
the Equator, in this caſe the path of the Sun's diurnalCourſe, and the
Zenith Z a Quadrant of an Ellipfis or Azimuth Circle be drawn, we
ſhall have the Arch 93 k for his Altitude at that time above the
Horizon; and which will be found to be 26 deg. 6 min. by apply.
ing the neareſt diſtance of the point 9 3, from the Horizon, to a Line
of Sines, whoſe Radius is equal to the Radius of the Proje&ion, and
the Arch of the Horizon r k for his Azimuth, from the Prime Vertical
Dddd 2 *
of 20 deg im
at
566
of the Orthographic Projection of the Sphere
I
fame thing, '
at that time equal to 38°04', and which may be found by applying
the Arch of the Horizon r K to a Line of Sines, whoſe Radius is equal
to the Radius of the Projeâion, or by applying the neareſt diſtance
of the point 9 3, from the Prime Vertical Z ? N, to a Line of Sines
whofe Radiusis equal to half the Diameter of the ſmall Circle drawn
thro the point 93, parallel to the Horizon po.
And laſtly, ſuppofing the Sun to be in the firſt point of Cancer as
before, and it be required to find what ſhall be his Height at 40
min. paſt 3 in the Afternoon, or at 20 min. paft 8 in the Morning,
each of which cimes are equally diſtant from Noon, and on what
point of the Compaſs he will appear at either of thoſe times, and
which are theilth and 12th Requiſites.
Having drawn a Quadrant of an Ellipſis as P LR, anſwering to
the given time, where this incerfects the Parallel of the Sun's diurnal
Courſe s C asin L, it will give the Place of the Sun ; or which is the
the Hour from Noon, to the Radius ® w from w to L, it will give
L for the Place of the Sun at that time; and if chro that point L and
the Zenith point Z, a Quadraně of an Ellipſis be drawn, where it
interlects the Horizon as in the point m, it will give r m the Right
fine of 12 deg. 42 min. to the Radius * H,, for the Azimuth from
the Eaft or Weft point of the Horizon, and the Arch Lm of the El-
lipfis 2 Lm equal to 40 degrçes nearly, for the Altitude, which may
be meaſured by drawing a ſmall Circle thro the point L parallel to
the Horizon, till it cut the Meridian, or by applying the Arch of the
Prime Vertical intercepted between that Line and the point: T, to a
Line of Sines çqual to the Radius of the Proje&ion: And by apply-
ing the Portions of the fame Line contained between the point i and
the Prime Vertical, to a Line of Sines whoſe Radius is equal to half
the Length of the ſmall Circle, drawn thro the point L parallel to
the Horizon, we fball have the Azimuth of the Sun at that time from
the Eaſt or Weſt point of the Horizon, without being at the pains
of drawing the Azimuthal Ellipricat Quadrant ZIM.
After the fame manner. may all Allronomical Problems relating to
to the Sun or Stars be reſolved by this Projection, by the help of the
Line of Sines only.
And if any Diameter as Al, each half of it as Ar and als
being firſt divided as the Ling of Sings is, be made to move about the
Center r, and the ſeveral Hour Circles and Parallels be drawn, the
Periphery of the Primiçre Circle being divided into degrees 6.
of the Orthographic Projection of the Sphere $67
si
N
the Proje&ion thus prepared, will readily give Solutions to all Pro-
blems relating to it by Inſpection only ; and being ſo very cafy for
Practice it is uſually paſted upon Boards, and called the Analemma,
and is one of the beſt Contrivances for conveying a juft Idea to the
Mind of the manner how the ſeveral Triangles are formed by the
Interfe&ion of the ſeveral great Çircles deſcribed upon the
Sphere in Plano. And as a farther Inſtance of the Excellency of this
Property of the Orthographic Projection that I have already exemply-
fied, and to thew that Solutions are given by Conſtructions, with
leſs Labour and more Exa&tneſs than by the Stereographic Projetion :
I ſhall give a Solution to one of the moſt uſeful and common Prob-
lems in Astronomy, and that is ſuppoſing it were required to find the
Azimuth of the Sun and the Hour of the Day, the Latitude of the
Place, the Declination of the Sun, and his Altitude being given
Having drawn the Meridian ZONH,
the Horizon HO, the Prime Vertical
Z N, the Equator al as in the Ste-
reographic Proje{tion, draw the ſmall
Circle mn parallei to the Equator, at
the diſtance of the Sun's. Declination,
dia
and the ſinall Circle ab parallel to the
H
Horizon, at the diſtance of the Sine of
the Sun's Altitude, where theſe two in-
terſéet it will give the Place of the Sun,
Band
and se will be the Sine of the Sun's A-
zimuth from the Prime Vertical to the
Radius ac, as will sd be the Sine of
the Hour from Noon to the Radius
md; and after the ſame manner may all Aſtronomical Problems be er-
folved, by the help of the Sines only, without being at the pains-
of drawing of Ellipſis's upon this Principal of Orthographic Projection ;
and inaſmuch as theſe are all perform'd by Right-lines only, hence
it is that Proje&ions of this Kind are called Projections of the Sphere
in Plano, as are the Solutions themſelves, uſually called Solutions by
the Projection of the Sphere in Plang.
How the Triangles are form'd, and how the proper Requiſitos
may be found by Calculation, has been ſufficiently ſhewn in the 24
Volume, and therefore I ſhall proceed to lhew in the next Section, how
all the former Problems may be reſolved from that true Hypothefis that
is grounded upon the Motion of the Earth, it being the moſt con-
fontaneous to Reaſon, Experience, and common Obſervation,
.
568 Of the Solution of Aſtronomical Problems
&
2AAQATALA JSEMARRAZ
Section XXII .
1
Containing a Solution of the chief and primary Problems
in Aſtronomy, according to the ancient Pythago-
rean or Copernican Syſtem of the World.
1
N accounting for the diurnal Phænomena of the heavenly Bodies,
and computing the times when the moſt remarkable Appear-
ances will happen, we have in the preceding Seation, as well as thro-
out the sth Part of the Second Volume, ſuppoled in common with
the reft of the Writers upon this Subje&, that the Earth is fixed im-
moveable in the Center of the Univerſe, and that the Sun as well as
the reſt of the Cæleſtial Bodies, are carried round about this ſlender
Ball of ours once in the Space of Twenty-tour Hours nearly, and
that beſides-this-diurnal Motion of the Sun, he is carried by his own
proper Møtion thro the 'Ecliptic in the Space of 365 Days and 6
Hours nearly, according to the Series or Succeſſion of the Signs from
the Weſt towards the Eaſt, moving at the Rate of one degree nearly
each Day, and contrary to the Motion of the whole Sphere it ſelf,
which is from Eaſt to Weſt; and that inaſmuch as the Axis of the
Ecliptic is inclined to the Axis of the Earth, ar an Angle of 23 deg.
29 min. the Sun in moving from the Vernal Equinox to the Tropic of
Cancer, or from the Autumnal to the Tropic of Capricorn, and which is
performed in the Space of 91 Days nearly; ſeems to deſcribe a Spiral
Line in the Heavens, of ſo many Turns or Windings growing narrow-
er one than another, till he reaches either Tropic; and after that to
go back again in the ſame Spiral Path, till he arrives at the Equino&ial
again, from which Motion of his proceeds the variety of Seaſons
throughout the habitable World, and the Increaſe and Decreaſe
of the Days, in all Places fcituated between the Equator and the
Poles, at different times of the Year
Let us now proceed to thew how the ſame Appearances may be ac-
counted for, and the time when they will happen be determined from
the following Suppoſitions, which are allow'd of to be true by the moſt
skil-
by the Copernican Syſtem 569
skilful Aſtronomers, and have the concurrent Teſtimony of the beſt
Obſervations to ſupport them. And
1. That the Sun is placed in the common Center of Gravity of
this our Planetary Syſtem, having no Motion proper to himſelf be-
fides a Rotation about his ownAxis once in 25 Days 9 Hoursnearly.
2. That the Earth turns round her own Axis once in 24 Hours :
and that this diurnal Motion is always equable, uniform, and the
fame in every point of its Orbit.
3. That beſides its diurnal Motion about the Axis of the Equator,
it is carried round the Sun in a large Path, betwixt the Orbs of Mars
and Venus once in a Year.
4. That the Axis of the Earth during its whole Revolution thro
its Annual Orb, is conſtantly inclined to the Plane of its Orb at che
fame Angle, and is in all points of ic nearly parallel to its ſelf.
5. That the whole Diameter of its Orb, is but as a point in com-
pariſon o the inconceiveable immenſe diſtance of the Fixed Stars.
Theſe Poſtulata being granted,
In Plate the 6th, let r not so repreſent the Annual Orb, in the
Periphery of which the Earths Center is carried about the Sun 0,
placed near the Center once in a Year, according to the ſucceſſion
of the Signs.
Thro o draw the Diameter 79 O, this will repreſent the great
Equirođial Colure, and at Right-angles to it and thro the fame
point , if the Diameter som be drawn, it will be the Repre-
fentative of the great SolſtitialColure in the Heavens. On each of the
four points, and where thele Lines interſee the great Orb defcribe
lo
many
(mall Circlesæt os, theſe will repreſent the Disk and Pofi-
tion of the Earth, at the commencement of the four remarkableSea-
fons of the Year.
And inaſınuch as the Sun illuminates only one half of the Terrer
trial Globe at the fame Moment of Time, (for the Difference ari-
fing from the great Diſproportion between the Magnitudes of the
Sun and Earth is of no Conſideration in this Place) if thro each of
the points e in the four ſeveral Scituations, the Lines set dec, set,
& ec be drawn at Right-angles to the Line OE, there will divide-
the illuminated part of the Disk from the Obſcure, and are there-
fore termed the ſeveral Horizons of the Disk in the four different
Poſitions, and conſequently in what Part foever of the Orb the Earth
is ſcituated, if thro the Center e, a Right-line be drawn perpendi-
cular to the Line conneđing the. Centers of the Sun and Earth, in-
aſmuch
570
of the Solution of Aſtronomical Problems
af much as this Line is the Repreſentative of that Circle, that is the
Bounder or Terminator between Light and Darkneis it becomes the
Horizon of the Disk in that Scituation.
If the Plane of the great Orb be imagined to be extended infinitely
every way, it will form in the Sphere of the Fixed Stars, that Circle
which the Center of the Earth to the Eye placed in the Sun will ſeem
to delcribe or trace out, by her annual Motion round her Orb; this
therefore is called the Great Ecliptic in the Heavens, and inaſmnch
16 the Periphery atos of the ſmall Circle es etc, repreſenting the
Disk of the Earth in every point of the Orb, will lye in the ſame
Plane with the great Ecliptic before determined, this Circle therefore
is called Tbe Ecliptic on the Surface of the Globe, and where there is
no Occafion to conſider the annual Orb, it is called the Ecliptic
ſimply, as is its Center e the Center of the Ecliptic.
And becauſe when the Earth is in the Solftitial points of Cancer and
Capricorn, the Diameter set of the Earths Disk is coincident with
the Diameter of the Orb $ UW, the Repreſentative of the great
Solſtitial Colure in the Heavens, this therefore is called the Solſtitial
Colure upon the Surface of the Earth; and when we have no Occaſion
to conſider the great Orb, it is called the Solstitial Colure ſimply.
In like manner, becauſe when the Earth is ſcituated in the Equinoctial
points of Aries and Libra, the Diameter æe c of the Earths Disk, is
coincident with the Diameter 1 o of the great Orb, the Repre-
ſentative of the great Equino&ial Colure before deſcribed, this Dia-
meter a et of the Earths Disk is called the Equinoctial Colure upon the
Surface of the Globe; and when we have no Occafion to conſider the
annual Orb, it is called the EquinoEtial Colure ſimply.
Conceive the Eye placed in the Superficies ut ine Earth perpen.
dicularly under the point e, and which muſt therefore be the infe-
rior Pole of the Ecliptic, to the Eye thus placed the Northern Pole
of the Globe, or the upper Extremity of the Axis about which the
diuinalRevolutions are made, will appear at p 23 deg. 29min. diſtant
from the Pole of the Ecliptic e, to which if it be joyned by the Line
pe, this will repreſent the conſtant diſtance between the two Poles,
or the Inclination of the Axis of the Globe and Eclipric to each other,
as will the whole Diameter Spezi, the Repreſentative of the Sol-
ſticial Colore be the Repreſentative of the whole Axis it felf; this
therefore is called the Line of the Direttion of the Earths Axis.
Thro P the Pole in each of the fmall Circles Sæt.c, let the
great Circles eps, bpb, P8, 8cc. be drawn forming Angles with
each
by the Copernican Syſtem
571
each other of 15 deg. 30 deg. 45 deg. &c. according to the Di.
rections given in Problem the 4th of Scilion the 2d, of Part the sth,
theſe will repreſent the ſeveral Hour Circles in the ſeveral Scituations
of the Earths Disk, and if spzt repreſent the Meridian, as it does
in the Solſtitial points of Cancer and Capricorn, the great Circle & pc
will repreſent the Six a Clock Hour Circle, &c.
As the Earth turns round upon her Axis, cvery point upon the
Surface therof will deſcribe a Circle; wherefore if about the point p
as a Pole, the great Circle æqc be drawn, after the manner taught in
Problem the 41h, of Section the 2d, of Part the sth, it will be the
Repreſentative of the Equator upon the Earths Disk, or of the Cir-
cle that any Place ſcituated under the Equator ſeems to deſcribe
by the diurnal Revolution of the Earth ; and after the manner taught
in the ſame 4th Problem, if about the ſame point p as a Pole, the ſmall
Circles strc, x nem, be deſcribed one at the diſtance of 66 deg.
31 min. the other at the diſtance of 23 deg. 29 min. they will be the
Repreſentatives one strc of the Tropic of Cancer, the other x nem
of the Artic Circle, or of the Circles that any one place ſcituated un-
der either of thoſe Circles upon the Earths Surface ſeems to deſcribe
upon this Proje&ion. And becauſe the diſtance of any Place upon
the Surface of the Globe, is equal to the Complement of the Lati-
tude of that Place, or of its diſtance from the Équator, if about the
Pole point p, the ſmall Circle l z c be drawn, at the diſtance of 38
deg. 28 min. the Complement of the Latitude of London, it will be
the Repreſentative of that Circle that London will deſcribe by theCir-
cum-Rotation of the Earth upon her own Axis; and univerſally, if
about either Pole a ſmall Circle be deſcribed, at the diſtance of the
Complement of the Latitude of that Place, it will be the Repreſen-
tative of the Circle which thatPlace deſcribes by the diurnal Revoluti-
on of the Earth; and how theſe may be done has been ſhewn at
large in the 4th Problem of Se&tion the ad of Part, the 5th, and elſe-
where.
The ſeveral ſmall Circles thus deſcribed are called the Paths of the
Vertices of thoſe Places to which they refpe&ively belong, and are of
the ſame Uſe in determining the time of the riſing and ſetting of the
Sun and Fixed Stars, in finding their Amplitudes, Azimuths, &c. as
the Parallels of Declination were in the Ptolomaic Proje&tions, deſcri-
bed at large in the 5th Part of this work, as will appear hereafter.
Eeee *
Ву
572
of the Solution of Aſtronomical Problems
By the help of the leveral Repreſentations of the Disk of the Earth
thus prepared, in the ſeveral points of the annual Orb, may all the
commonCæleftialAppearances be eaſily and very naturally accoun-
ted for ; and our felves at one view be inſtructed, how it comes to
paſs why under the Equator the Days and Nights are of an equal
Length all the Year long, why within the Polar Circles the Days at
ſometimes are longer than 24 Hours, and why at the oppoſite
Times of the Year the Nights are as long, and why under each
Pole there is but one Day and one Night during the whole Year, and
why in all other Places the Days are not of the ſame Length at the
ſame time of the Year, and why in the ſame Place the Days differ
in Length at different Times of the Year ; how it comes to paſs that
the Sun approaches to, and receeds from the Vertex of any Places at
different times of the Year, and why ſome Stars are always viſible
when the Sun is ablent, and others never appear, c.
For let us ſuppole the Earth in the firſt point of Libra viewing
the Sun in che firſt point of Aries, its oppofite point in the Ecliptic,
becauſe in this caſe the Line connecting the Centers of the Sun and
Earth coincides with the Equino&ial Colure, the Horizon of the
Disk will coincide with the Solſtitial Colure, in this caſe the Repre-
ſentative of the Axis of the Earth, and confequently the Repreſen-
tatives of all the Paths of the Vertices of every Place deſcribed upon
the Disk of the Earth, and of all others that can be deſcribed,
will be biſected or cut into two equal Parts by the Horizon of the
Disk; one half of which will fall in the illuminated Part of the Disk,
and the other half in the obſcure Part.
And inaſmuch as while the Vertex is tranfiting over the illumi-
nated Part of the Disk, it fees the Sun and it is called Day, and while
it is travelling over the obſcure Part it fees him not, and it is called
Night, it follows, that the obſcure or darkned Parts of the ſeveral
Paths, are equal to the illuminated or diurnal Arches of the ſame,
and conſequently the Days inuft of neceſſity be equal in Length to the
Nights in all Places of the Earth.
Let us now imagine the Earth to have moved from Libra to Ca-
pricorn, the Line of Direction keeping ſtill nearly parallel to its ſelf,
and to the great Solſtitial Colure, becauſe in this Scituation the
Line connecting the Centers of the Sun and Earth, coincides with
the Repreſentative of the great Solſtitial Colure, and conſequently
with the Earths Solſtitial Colure spezto in this caſe, the Axis of
the Earths Equinoctial Colure & ec becomes the Horizon of the Disk,
and
by the Copernican Syſtem.
573
and inaſmuch as the Horizon of the Disk in this caſe touches the
Artic Circle in the point e, it follows, that all Places that lye be-
twixt the two Poles of the Globe and Ecliptic, or within the Artic
Circle, ſee the Sun, and are illuminated during the whole time of
their Revolutions, wherefore the Vertexes of all Places that lye
within that Space, ſee the Sun longer than 24 Hours, and that ſtill
longer, as the Places are leſs diſtant from the Pole but inafmuch
as in this Scituation the Artic Circle it ſelf juft touches the Horizon
of the Disk, it follows, that the Inhabitants under this Circle juſt
ſee the Sun in the Northern Part of the Horizon 90 Degrees from the
Vertex, and conſequently ſo loon as he is paſt it, view him aſcending
again, whence their longeſt Day is equal to juſt 24 Hours, and bea
cauſe the Paths of the Vertexes of all other places that lye without
this Circle, are cut by the Horizon of the Disk into unequalParts, the
greater Part of which lye in the illuminated part of the Disk, and
the leffer in the obſcure, and as theſe Portions ſtill approach to
a Ratio of Equality, till at laſt in the Equator they become equals
Inaſmnch as the Equator and the Horizon of the Disk are great Cir-
cles, and conſequently divide each other into equal Parts, from the
knowo Property of the Sphere, it follows, that the Length of the
Days to all the Northern Inhabitants at this time is leſs than 24
Hours, but greater than the Length of the Night, and that this In-
equality grows leffer the nearer the Places are ſcituated to the E-
quator ; and the farther they are removed from the Artic Circle,
till at laſt under the Equator it ſelf the Days and Nights become
equal 10 each other.
Let us now conceive the Globe to have moved from v to or the
Axis of the Globe or Line of Direction ſtill preſerving its Paralleliſm
to it felf, hecauſe in this Scituation, as well as in its oppofireScituation
the point, the Horizon of the Disk spezt which ſtands at Right-
angles to the Repreſentative of the great EquinoctialColure, coincides
with the Solſtitial Colure, in this caſe the Repreſentative of the
Farths Axis it felf, it follows, that the Paths of the ſeveral Vertic-
es will be all biſested by it, that therefore the enlightened Arches
will become equal to the obſcure ; and conſequently the Days will
become equal to the Nights in all Places of rhe Earth.
And laſtly, if we imagine the Earth to be carried again from A
ries to Capricorn, the Axis ſtill retaining its Paralleliſm, the Ho.
rizon of the Disk æ ec will again become coincident with the Sol-
ftitial Colure, and becauſe the Artic Circle juft touches the Hori-
Eeee 2 *
:
zon
$74
of the Solution of Aſtronomical Problems
zon of the Disk in the point e, the Sun will never appear above the
Horizon, but ſo ſoon as he touches it in the Southern Part, will
immediately begin again to deſcend, and conſequently the Night
will be equal in Lengch to 24 Hours; and inaſmuch as the whole
Artic Circle lies intirely within the obſcure part of the Dísk, the
Nights will be longer than 24 Hours each ; and ſo much the longer
as the Places are removed nearer to the Pole, or the farther from
the Equator; and inaſmuch as the Paths of the Vertexes of all Pla-
ces that lye without the Artic Circle, are cut unequally by the Ho-
rizon of the Disk, the leſſer part of which lying within the illumi-
nated part of the Disk, and the greater within the obſcure part, ma-
king thereby the Nights ſo much longer
than the Days ; juft equal and
contrary to what they were when theEarth was ſcituated in the
op-
poſite part of its Orb at v: And inaſmuch as theſe unequal Se&ions
of the Paths in this Scituation alſo approach nearer to a Ratio of
Equality, till when under the Equator, the no&urnal and diurnal
Arches are equal, it follows, that the Length of the longeſt Night
is lels than 24 Hours, but greater than the Length of the Day; and
that this Inequality likewiſe diminilhes the nearer the Places are fcit-
uated to the Equator or the farther from the Pole, till at laſt the
Days and Nights become equal to fall the Inhabitants that are ſcitu-
ated under the Line.
While the Earth is carried from by w to r, the Northern Pole
will be always in the illuminated part of the Disk, whence a con-
tinual¡Day at that time under the Pole will neceſſarily follow; but
whilſt ſhe is running from r by to it will be conſtantly in the
obſcure part of it, where of Confequence it will be perpetual Night ;
whence it is evident that under the Poles the Day is ever equal to the
Night, and each equal to one half of the whole Year.
And it in any of the intermediate Stations between Libra and Ca-
pricorn, we imagine or cauſe the Earths Disk to be drawn, furniſhed
with the ſeveral great and ſmall Circles, we ſhall manifeſtly ſee that
the illuminated Parts of the Paths will be greater than the obſcure
Parts, and that this continually increaſes as the Earth travels from
Libra when they are equal, toCapricorn, at which Place when ſhe ar-
rives they will be greateſt of all, and the longeſt Days will happen to
all the Northern Inhabitants; and that as the Earth travels from we
to 7 they will ſtill grow lefs and leſs, till at laſt when the Earth
arrives at r, they will again become equal ; and that after this, as
the Earth moves from 1 to $, the obſcure Parts of the Paths will.
CON-
by the Copernican Syſtem
575
continually grow greater than the enlightened, till at laſt when ſhe
arrives at the point %, they will be the greateſt of all, and at which
times the Days will be the ſhorteſt and the Nights the longeſt, to all
the Inhabitants in the Northern Hemiſphere; and that after the Earth
has left this Scituation the Arches themſelves will approach nearer
to a Ratio of Equality, till at laſt when the Earth arrives at again,
they will become mutually equal to each other, and the Days and
Nights will become equal to all the Inhabitants upon the Earth.
And if we imagine the point p to repreſent the Southern Pole of
the World, the ſmall Circle anem to repreſent the Antartic Circle,
the Circle'strc the Tropic of Capricorn, the Circle I z u to repre-
ſent the Path of the Vertex of the Place in 51 deg. 32 min. of South
Latitude, and the whole Circle e Sæt i to repreſent the Southern
half of the Disk, the point to repreſent =, and the points to re-
preſent vo; whatever has been ſaid of Places that lye in the North-
ern Hemiſphere, will hold good of Places that lye in the Southern
Hemiſphere, and conſequently the general Appearances in all Places
on the Earth will be juſtly and truly accounted for.
The general and moſt remarkable Appearances ariſing from the an-
nual Motion of the Earth being thus accounted for, I ſhall next pro-
ceed to ſhew how ſome of the moſt uſeful Problems ariſing from the di:
urnal Motion, may be reſolved upon the ſame Hypotheſis, and in
S
Prob. I.
Let it be required from the Sun's Place, or the Longitude of che
point in the Earths Disk oppoſite to him from the next Equino&ial
point, and the Inclination of the Axis of the Earth to the Plane of
her Orb, and which is fixed at an Angle of 66 deg. 31 min. to find
out the diſtance of the Sun's Place from the North Pole of the Globe,
and the Right Aſcenſion of the ſaid Place.
.
Example.
Let us ſuppoſe the Sun to appear to an Eye placed upon the Suc-
face of the Earth in 27 deg. 33 min. 57 ſec. of Taurus, and lec ic
be required to find the Right Afcenfio. 1 of that point, and the diſtance
that the Sun appears from that point in the Heavens that is oppoſite
to the Northern Pole of the Earth...
The.
:
576
Of the Solution of Aſtronomical Problems
The Stereographic Solution.
1. Having drawn the Ecliptic
R
% TW, the Equino&ial Colure
Per. and at Right-angles to it
and thro the Center e, the Solſti-
tial Colure e V, ſet off the Se-
P
mi-tangent of 23 deg. 29 min. the
conſtant Inclinations of the Axises
of the Equator and Ecliptic from
the point e the Repreſentative of
the Pole of the Ecliptic in the Sol-
ftitial Colure, from e towards *
N
to p, this will give the Place of the
78
North Pole in this Proje&ion.
2. Becauſe the Sun appears 57 deg. 33 min. 57 ſec. from the next
Equinoctial point Aries, ſet off the Chord of 57 deg. 33 min. 57 ſec.
in the Ecliptic from 1 towards s to , this will give the Place of
the Sun.
3. Thro the point and the Pole of the Equator P, draw the
Circle of Right Aſcenſion OPN, by Cafe the ad of Problem the iſt,
of Sestion the ad of Part the 4th, then will po repreſent the appa-
rent diſtance of the Sun from the North Pole, which may be mea-
ſured by Caſe the 3d of Prob. the 7th, of Sext. the ad of Part the 4th,
and the Angle “Po the Right Aſcenſion from the Solſtitial point
O, which may be meaſured by Caſe the 3d of Problem, the 10th of
the ſame Section, and to pronounce their juſt Quantity by Calculation.
In the Spherical Triangle P SO, Right-angled at 5, are given
ps the fnclination of the Earths Axis to the Plane of the Ecliptic,
and the diſtance of the Sun from the next Solftitial point %,
whence, firſt to find Po the apparent diſtance of the Sun from the
North Pole, it will be by the firſt Caſe of Right-angled Spherical Iria
anglesy as R:cs,p::cs, %:cs,P O. That is
As the Radius
To the Co-fine of the Inclination of the Axis 66.31.00-9.6004090
E
quinoctial point Aries 57.33.57
9.9263467
To the Co-line of its diſtance from the North Pole} 9.5269557
70.20:49
10.0000000
1
2. To
by the Copernican Syſtem
577
2
To find the Angle Ops, the Right Afcenfion from the Solfti
al point , it will be by the ad Caſe of Right-angled Spherical Tri
angles,
As S. PSR :: t, % 0 :t, <% PO; that is,
As the Sine of the Inclination of the Axis 66,31.00 9.9624527
10.0000000
To the Radius
So is the Tangent of the Sun's Longitude from the Sol-?
ſticiai point Caneer 32-26.03
9.8030856
To the Tangent of the Right Aſcenfion from the
Solſtitial point Cancer 34 42.563:
9.8406329
Which taken from a Quadrant, will leave 55 deg: 17 min. 03
ſec ả, for the Right Aſcenſion from the Equino&ial point of Aries, or
by the ſame Caſe, thesRight Aſcenſion from the Equinoctial point
sries may be found at once, by ſaying
As R:ct, P0::ct, $ 0:ct,<SPO.
Again,
If about the Pole point p, the great Circle qr, (which will be
the Repreſentative of the Equator) be deſcribed and produced, till
it meet the Circle of Right Afcenfion NP in R, in the Spherica
Triangle rOR Right-angled at R, are given, the Angle OR
the conſtant Inclination of the Planes of the Ecliptic or the Equator;
equal to 23 deg. 29 min. the Arch of the Ecliptic ro, or the Hy
pothenuſe, equal to the Sun's Longitude from the Equino&ial point
Aries, whence by the Laws of Spherical Triangles may oŘ the
Complement of the Sun's diſtance from the North Pole be de-
termined, and the Arch of the Equator r R che Right Aſcenſion of
the point o from the Equino&ial point; and after this manner may
the Right Aſcenſion of the Sun, and his diſtance from either Pole
be determined in any other point of the Ecliptic
.
By reaſon of the immenſe diſtance of the Fixed Stars, which is
ſo great, that if two Lines be extended from the extream points of
the Diameter of the great Oțb, to the neareſt Fixed Star, they will
in a manner coincide, and form no ſenſible. Angle : the whole Circle
in the Circumference of which the Earth is carried about, is but as a
Phyſical point with regard to the Orb, in which the Fixed Stars
ſeem to be placed, and whereas to an Eye placed in the Sun, the
Earth does really appear to move round the Sun in that Circle which
we call the Ecliptic, amongſt the Fixed Stars once in a Year to the
fame
son
ol
.
T
S
KU
me
de
on
C
C
}
It i
Ꮪ S
T
ST
.
1
20
œ
N
X
T
S
Herr
A' A...
de 1.4-.
41. "I'm
4. SEVILA
Thi
0
between 576.577.
Plate 6
578
of the Solution of Aſtronomical Problems
fame Eye placed in the Earth, the Sun according to the known Laws
of Optics, will ſeem to move annually in the ſame Orb round the
Earth, and to deſcribe the ſame Circle amongſt the Fixed Stars ;
and conſequently may be conceived at any time to be in that point
of the Ecliptic in the Earths Disk that is oppofite to the Sun, or
where the Line conne&ting the Centers of the Sun and Earth cuts the
Superficies of the Earth, which therefore may not improperly
be called the Place of the Sun in the Ecliptic ; thus in the former
Figure the point in the Ecliptic row, which incompafies
the Circle e pso , the Repreſentative of the Earths Disk thro
avhich the Sun would appear to an Eye placed in the Center of the
Earth ate, is called the Sun's Place in the Ecliptic ; inaſmuch as
it anſwers to, and correſponds with the ſame point in the great E-
cliptic in the Heavens, in which the Sun would appear to a Spe&a-
tor Place upon the Earth ; and conſequently it we conceive the
Ecliptic, the Equator, the Colures,&c. as a fixed Rete cloſely
inveſting the Earth, while ſhe turns round within them, or which is
the ſame thing, if we ſuppoſe an Artificial Sphere, furniſhed with
the several great Circles, contrived and invented by the Ašti ono-
mers,
for the Diviſion of the Heavens into the ſeveral Parts, necella-
ry for the Solution of the uſual Phenomena ; and at the ſame time the
Éarth to be ſo placed within it, as to be concentrical to the Sphere it
felf, and to revolve upon the common Axis of the Sphere, inaſmuch as
in whatſoever part of the Orb the happens to be, ſhe is ſtill in a Phy-
fical Senſe ſuppoſed to be in the very Center of the Heavens, the Gir-
cles of our imaginary Sphere will become coincident with the great
imaginary Circles in the Heavens, they are ſuppoſed to repreſent, and
.confequently all Concluſions drawn from their ſeveral mutual Inclina-
tions, Interſections, and the leveral Triangles that they form amongſt
themſelves, will anſwer and correſpond to the Arches or Portions of the
Heavens that they repreſent, whence every Solution drawn from theſe
Premiſes, muſt exactly accord and agree with the eſtabliſhed Rules
and Order of Nature: theſe things rightly underſtood, will contri-
bute very much to the forming of a juft Idea in the Mind (of the young
Beginners) of the firſt Rudiments of this Science, and ſhew them by
what Steps Aſtronomers proceed to attain their Ends, and open a
ready Way to the perfe& Underſtanding of whatſoever ſhall be of
fered upon this Subje&t. This being premi ed let us proceed in
.:
Prob.
.
by the Copernican Syſtem
579
Prob. II.
::
To inveſtigate the time when the Sun will appear to Riſe and Set,
and at what diſtance from any of the four CardinalPoints of the Ho-
rizon at London, which is placed at the diſtance of 38 deg. 28 min.
from the North Pole, on the 7th May 1720; when according to the
firſt Problem, the Sun will be found to appear at the diſtance of
70 deg 20 min, 49 lec. ;, from the ſame Pole. The Sun is ſaid to
riſe when the Vertex of any Place, or rather (as the Earth revolves
upon her Axis) when the Place it ſelf juſt arrives at the Horizon
of the Disk, paſſing out of the obſcure into the illuminated
part
of
the Disk, and to ſet when the Vertex of the ſame Place (by which
we are hereafter to underſtand the Place it ſelf) arrives at the
oppo-
ſite Semi-circle of 'the Horizon of the Disk paſſing out of the illu-
minated part of the Disk into the obſcure again. Wherefore,
1. Having drawn the Ecliptic
qis V, the Equinoctial Colore
pes, and at Right-angles to it
and thro the Center, the Solſti-,
tial Colure se V, ſet off the half
Tangent of 23 dg. 29 min. from H
e in the Solſtitial Colure to p, then
will p repreſent the North Pole.
2. At the diſtance of 38 deg. 28
min. the Complement of the Lati-
tude of London, and about the point
Pas a Pole, deſcribe the ſmall Cir-
cle Z t, by Caſe the 3d of problem
ms
the 4h, of Sextion the ad of Part
the 4th, this will repreſent the Path
of the Vertex of London.
3. Becauſe the diſtance of the Sun from the North Pole, is
equal to 70 deg. 20 min. 49 ſec. if about the ſame point P, at the
diſtance of 70 deg. 20 min.49 ſec. another ſmall Circle be deſcribed by
the ſame Caſe, it will cut the Ecliptic in the Place of the Sun in
this Projection.
4. Thro the point and the Center e, draw the Diameter O'e N,
this will reprelene the Circle of Longitude paling thro the Sun and
Ffff *
if
વી
N
19
580 of the Solution of Aſtronomical Problems
it at Right-ingles to it, and thro the Center e, the Diameter
He O be drawn, it will be che Reprefentative of the Horizon of
the Disk in this Projection,
As the Earth revolves about her own Axis from Wifi to Eaſt,
when the Verrex of London arrives at us where the path interſects
the Horizon of the Disk, the Sun appears to riſe, and continues vi-
lible, till the ſame point of the Path anivesar the point. z, in the
oppoſite part of the Disk, where the Sun appears to ſet ; and
the Sun never ſees the Vertex till it arrives again at u, whence it
comes to paſs in the preſent Cafe, that ul2 is the diurnal part,
and um z the nocturnal part of the Path, and to find the value of
each of theſe, or the Quantity of Time anſwering to either of
them,
5. Thro the Place of the Sun, and P the North Pole, draw
the great Circle PON, by Caſe the ad of Problem the it, of Sec-
tion the ad of Part the sale, this will repreſent the proper Meridian
of the Place, and the point / where it interſeets the Paths of the
Vertex, will be the Place of the Sun at the Noon of the fame Day.
Wherefore,
6. If thro the points y and Z, where the Path of the Vertex cuts
the Horizon of the Disk, and the Place of the Sun, iwo Arches
of great Circles he drawn, by Cafe the 3d of Prob. she iſt of Stereo-
graphic Problems; and thro the Pole P and the ſame points Z and
u the Hour Circies P 1 and P Z be drawn by the fame Rules, the
Angle O P 4 equal to O P 2, will new bow many Hours he ri.
ſes before, or ſets after Noon, as will the Angles ou P equal to
OZP, thew how far he riles from the Meridian towards the Eaſt.
or how far he fets from the fame Meridian towards the Weſt : and for
the fame Reaſon, the Angle u P d equal to & Pd, will chew how long
time the Sun rifes after, or ſets before Mid-night ; as will tle An-
gle Pudor Pzd, ſhew how far the Sun riſes from the Eaſt or ſets
from the Wett, towards the North or South ; each of which may
be meatured by Caſe the 3d of Stereographic Problems ; but to de
termine the juſt Quantity by Calculation,
In the Great Triangle Pou, equal to Poare given, PO
the diftance of the Sun from the North Pole, Pu the Semi-di-
ameter of the Path, equal to the Complement of the Latitude of
the Place, and the Side O w equal to a Quadrant, whence the An-
gles Pu the time from Noon, acd # P the Azimuth from the
Meridian may be eaſily inveſtigated by the 11th Caſe of Right-angled
Sple
i
1
by the Copernican Syſtem.
581
Spherical Triangles : Or in the leſſer Triangle Pud equal to P z d,
are given, pu the Semidiameter of the Path equal to the Complement
of the Latitude of the Place, Pd equal to the Complement of the
Sun's diſtance from the Pole, and the Angle at e Right, whence
the Angle u Pe the time the Sun riſes after, or fets before Midnight,
and the Angle Pue the diſtance that he riſes or Sets from the Ė. or
W. point may be readily found, by the 3d Caſe of Right-angled Sphe-
rical Triangles,
As the Solution by the little Triangle Pud is much more ſimple
and eaſy than by the great TriangleP O e, it ought to have the Pre-
ference, wherefore to find the Angle d P u it will be,
As the t, Pu:t, P d: :R:cs,<d P4. That is,
As the Co-tangent of the Latitude 51.32.00
9.9000865
To the Co-tangent of the Sun's diſtance from the Pole?
9.5528217
70.20.49
So is the Radius
10.0000000
Pole}
3
To the Co fine of the Semi-nocturnal Arch 63.17.1749.6527352
1
Which being reduced into time, will give 04h. 13 m 09 f. for
the time the Sun ſets before, or riſes after Midnight.
2. To find the Angle Pud it will be, As S. Pu:R::S.Pd:
S. <Pud. That is,
As the Co-fine of the Latitude 51.32 00
-9.7938317
10.0000000
To the Radius-
So is the Co-line of the Sun's diſtance from the Pole
70.20.49}
9.5267557
Pole}
To the Sine of the dift. of the points of rifing or ſetting} 9.7329240
:
So that in the preſent Cafe the Sun riſes at 4 hot 13 min. og ſec.
32 deg. 43 min. 42 ſec. to the Northward of the Eaſt point, and
ſets at 7 hor. 46 min. 51 ſec. 32 deg. 43 min. 42 ſec., to the North-
ward of the Weſt.
Prob.
Ffff 2*
582
Of the Solution of Aſtronomical Problems
Prob. III.
The ſame things being ſuppoſed given as before, viz. the diſtance
of the Path of the Vertex from the Pole, and the Sun's diſtance from
the ſame point, to find what will be the Sun's Altitude, when he
ſhall be removed at the diſtance of Six Hours from the Meridian, and
upon what Azimuth Circle he will appear.
Example.
Let it be required to find at London, in the Latitude of sr deg.
32 min. oo ſec. when the Sun is removed at the diſtance of
70 deg.
20 min. 49 ſec. from the North Pole, what will be his Altitude
when he ſhall appear upon the Six a Clock Hour Circle, and what
Angle the Vertical Circle pafling thro the Sun that time will form
with the Meridian. When the path of the Vertex arrives at thatCircle
that cuts the proper Meridian at Right-angles, and palles thro the Pole
P, the Sun then appears to the Obſerver to be removed at the diſ-
tance of Şix Hours from the Meridian. Wherefore,
1. Having drawn the Ecliptic
* T W, the Equino&ial Colure
er, and at Right-angles to it
the Solſtitial Colure seve, ſet off
the half Tangent of the diſtance
of the two Poles of the Equator
and Ecliptic from a to P, as in the
former Cafe.
3
N
2. About the point P at the dil-
tance of the Sun from the Pole 70
deg. 20 min. 49 ſec. , deſcribe
a ſmallCircle by the IſtStereographic
Problem to cut the Ecliptic in o
the Place of the Sun.
3. Thro the point o and the Pole, draw the Hour Circle OPN,
this will repreſent the proper Meridian, and if thro the Pole point
By and at Right-angles to this, Meridian, the great Circle t Psabe
drawn, by the oth Stereographic Problem, it will be the Repreſenta-
time of the Six a Clock Haur Circle in this Proje&ion.
4. IF
+
by the Copernican Syſtem
1
583
4. If about the Pole P at the diſtance of 38 deg. 28 min. the ſmall
Circle ts be deſcribed, it will cut the Six a Clock Hour Circle be-
fore deſcribed in the points sandt, at either of which Places when
the point repreſenting London arrives, it will ſee the Sun reinoved at
the diſtance of Six Hours from the Meridian. Wherefore,
5. If through the point and the points sand t, the two Verti-
cal Circles Ot N and os Nibe drawn, we ſhall have the Arches
Os and @t for the diſtance of the Sun from the Verzex, or Com-
plement of his Altitude at each of thoſe times, and the Angles P's
or Pit o for the Azimuth of the Sun from the Meridian at the
ſame time, each of which may be meaſured by the known Laws of
Stereographic Projection, and to find each of them by Calculation,
In the SphericalTriangle © P s or · Pr Right-angled at P are gi-
ven in each, Ps equal to Pt the Complement of the Latitude of
the Place, O P common, equal to the diſtance of the Sun from
the Pole, whence to find out firſt, Os the diſtance of the Sun front
the Zenith, it will be by the ift Caſe of Right-angled, &c
As the R:cs, P::CS, PN:cs, s; that is,
As the Radius
10.000000
To the Co fine of the Sun's diſtance from the Pole
70.20.491
So is the Sine of the Latitude 51 32 00
ole} 9.5267557
9.8937452
To the Sine of the Height at that time 15.16.043 9.4205009
2. To find the Angle P. 0, it will be by the 2d Caſe of Right
angled, &c.
· As S. Pu:R::t,PO: t,<P 40. That is,
As the Co-line of the Latitude 51.32.00
9.7938317
To the Radius-
10.0000000
So is the Tangent of the Sun's diſtance from the Pole?
70.20,495
To the Tang of the Azim. from the Meridian 77.28 29 9.6533466
ole} 10-4471783
Vlience:
?
584 Of the Solution of Aſtronomical Problems
Whence it appears that ar Six a Clock in the Morning, the Sun
will appear 77 deg. 28 min. 29 ſec. to the Eaſtward of the South,
and 12 deg. 31 min. 31 ſec. to the Southward of the Weſt, at Six
a Clock in the Afternoon, at each of which times he will be diſtant
from the Vertex 74 deg. 43 min. 56 ſec.
Proba IV. -
The diſtances of the Sun and the Vertex of any Place from the
ſame Pole being given, to find at what time of the Day the Sun will
appear due Eaſt or due Weſt, and what will be his diſtance from
the Vertex at that time.
I.
wo, and
Example
In the Latitude of si deg. 32 min. oo fec. North, when the dif-
tance of the Sun from the North Pole ſhall be found to be 70 deg.
20 min. 49 ſec. $', let it be required to find at what time of the Day
the Sun will appear due Eaſt or due Weſt, and what will be his di-
fance from the Zenith at that time.
The Solution Stereographically.
1. Having drawn the Ecliptic
e 4 or v the Equino&ial and
0
Solftitial Colures V, SV,
found the point P, the Place of the
Pole as in the former Problem, about
the Pole point P, draw two ſmall
P
Circles, one at the diſtance of 38
op deg. 28 min. to repreſent the Path
of the Vertex of London, the other
at the diſtance of 70 deg. 20 min.
49 lec. , the diſtance of the Sun
N
from the Pole, in order to find 0
To
the Place of the Sun in the Ecliptic,
as in the two fornier Problems.
2. Thro the point draw the Diameter o N, and at Right-
angles to it the Diameter He0. Now inaſmuch as when any Place
ſuppcfe London, arrives in ſuch a part of the Path as that the Me-
ridian fhall cut the Vertical Circle at that time, paſſing thro the Sun
at
by the Copernican Syſtem
585
ar Right-angles, the Sun will then appear due E. or W. to the Ob-
ſerver thus cituated,wherefore if about the point P at the diſt. of sid.
32 min. oo lec. a ſinall Circle be deſcribed, it will cut the Horizon
of the Disk in the point the Pole of the VerticalCircle to be drawn,
and if thro this point G and the Pole P an Hour Circle be drawn,
it will cut the Vertical Circle before drawn in the point Z, the Place
of the Zenith at that time, and conſequently Z O will be the Zenith
diſtance, as will the Angle ZP @ bc the Hour from Noon when this
Appearance will happen, each of which may be meaſured by the
known Laws of Stereographic Projection, and to determine their juſt
Quantities by Calculation ; in the Spherical Triangles oZP Right-
angled at Z are given, GP the diſtance of the Sun from the Pole,
pż the Semi-diameter of the Path, whence to find o Z the diſta
ance of the Sun from the Vertex az chat time, it will be by the 14th
Caſe of Right-angled Spherical Triangles,
As cs, ZP:R: :CS, PO:cs, Z O. That is,
As the Sine of the Latitude 51.32
9.8937452
To the Radius
So is the Co-fine of the Sun's diſtance froin the Pole
70 20.49)
10.0000000
9.5267557
To the Co-line of the Zenith-diſtance 64.33.401
9.6330105
2. To find the Angle O P Z the time from Noon, it will be by
the ad Cafe of Right-angled Spherical Triangles,
Asct, ZP:CC, PO::R:cs, OPZ. That is,
As the Tangent of the Latitude 51.32
10 0999135
To the Co-tangent of the Sun's diſtance &c. 70.20.49; -95528257
So is the Radius
10.0000000
To the Co-fine of the Angle O PZ 73.31.31
94529082
Equal in time to 04 hor. 54 min. 06 ſec. whence it is manifeft,
that the Sun will appear due Eaſt at 7 hor. 3 min. 54 ſec. in the
Morning, and due Weſt at 04 hor. 54 min. 06 ſec in the After-
noon ; at each of which times he will be removed at the Diſtance of
64 deg 33 min 40 ſec, from the Vertex.
Prob
586
of the Solution of Aſtronomical Problems
Prob. V.
Let us now ſuppoſe the Earth to be in Libra, viewing the Sun in
the oppoſite point Aries, in this poſition the Sun will appear in the
Equator, or at aQuadrantal diſtance from the Pole, and lec it be requi-
red to find how far the Sun will appear from theZenith of London, at
half an Hour paſt Three in the Afternoon, and what will be the Angle
which the Vertical Circle at that time paſſing thro the Sun forins
with the Meridian.
?
The Stereographic Solution.
m
1. Having drawn the Ecliptic
room Vý, the two Colures meer
and toe w, and the Path of the
Vertex as in the former Problem,
thro the Pole P and the Sun in
the firſt point of Aries at r, draw
the great Circle PE, this will
repreſent the proper Meridian ; and
if thro the fame point P the Hour
Circle P % be drawn, forming an
N
79
Angle of 52 deg 30 min. equal to
3 h. 30 m. in time, with the proper
Meridian Pr, by the 9th Ste-
reographic Problem, where this in-
terſects the Path of the Vertex as in Z, it will give the Place of
London at that time. Wherefore,
If thro O the Sun and this point Z, a Vertical Circle as Erzo
be drawn, the Arch 20 the diſtance of the Sun from the Vertex,
and the Angle P 2 O the Angle that the Vertical Circle at that time
paſſing thro the Sun forms with the Meridian, may be meaſured by
the known Laws of Stereographic Projection, and to find the ſame by
Calculation, in the Spherical Triangle Pr z Right-angled ar y are
given, P Z the Semidiameter of the Path, and the Angle Z Pr the
Complement of the Hour from Noon; whence to find firſt, Zr
the Sun's Altitude, or the Complement of the Zenith-diſtance, it
will be by the 5th Caſe of Right angled Spherical Triangles,
As
by the Copernican Syſtem
587
As R:S. PZ:: S.<Z Pr:S. Zr. That is,
As the Radius
10.0000000
To the Co-fine of the Latitude 51.32.00
9.7938317
So is the Co-line of the Hour from Noon 52.30.00-9.7844471
To the Co-line of the Zenith-diſtance 67.44.513 9.5782788
2. To find the Angle PZx the Azimuth, it will be, by the 3d
Caſe of Right angled Spherical Triangles,
As R:cs, 2P: :t,<Z Prict, P2r. That is,
As the Radius
10.0000000
To the Sine of the Latitude 51.32'00
9.8937452
So is the Co-tangent of the Hour from Noon 52-30-9,8849805
To the Co-tangent of the Azim. at that time 59.00:10 -9.7787257
Whence it appears that at half an Hour paſt Three, the Sun ap-
pears South 59 deg oo min. 10 ſec. Weſt, at the diſtance of 67 deg.
44 min, si fec. from the Zenith.
#
.
Problem VI.
.
TheEarth being in any point of her Orb, let it be required to find
at what diſtance from the Vertex, and upon what Azimuth the Sun
will appear at any given time, and at any given Place:
Example
Let us now ſuppoſe the Earth to be in 27 deg. 33 min.
57
ſec.
of Scorpio, when the Sun according to the firſt Problem, will be teen
at the diſtance of 70 deg. 20 min. 49 ſec. from the Pole, and ler
it be required to find at what diſtance the Sun will appear from the
Vertex of London, at half an Hour paſt Eight in the Morning, and
what will be the Angle that the Vertical Circle at that time paſſing
thro the Sun, forms with the Meridian.
Gggg*
The
588
Of the Solution of Aſtronomical Problems
2
A
The Solution Stereographically.
1. Having drawn the Ecliptic
or v the Equino&ial Co.
lure mer, the Solſtirial Colore
Sew, and the Path of the Vertex
H H
about the point P, after the manner
fhewn in the former Problem, thro the
z
ſame point P draw the HourCircle
ရာ
PZ, forming an Angle Z P, with
the
proper Meridian OPN, drawn
thro the Sun 0, and the Pole P,
of 52 deg. 30 min. equal to the
Hour from Noon reduced into time,
no
where this great Circle interleats
the Path of the Vertex as in the
point Z, it gives the Place of the
Zenith of London at that time ; wherefore if thro this point Z and
the point », a Vertical or Azimuth 'Circle as o Z be drawn, the
Arch of it o. 2, intercepted between 0 and 2, will be equal to
the diſtance of the Vertex of London from the Sun at that time, and
the Angle O Z P will be the Azimuth at that time, each of which
may be meaſured by the Rules laid down in Section the 2d, of Part
the stb, and to determine their Values by Calculation,
In the Spherical Triangle POZ we have given, Po the Sun's
diſtance from the North Pole, PZ the diſtance of the Vertex of
London from the fame Pole, and the Angle OP Zithe Hour from
Noon, whence to find iſt, ſuppole O ZP, the Azimuth from the
Meridian, it will be by the 4th Axiom of Spherical Trigonometry de-
monſtrated in Page 133d of Volume the 2d,
Sum
Sun
the
To the Sine of half their Difference 15 56.245
9.4387535
So is the Co-tang of half the Hour from Noon 26.15 10.3970250
To the Tangent of a fourth Arch 34.24-193 9.8355970
And
by the Copernican Syſtem
589
And again,
As the Co-fine of half the Sum, &c. 54.24.245
-9.7649423
To the Co-fina of half the difference 15.56.245 -9.9829715
So is the Co-tangent of half the Hour 26.15.00 – 10 3070250
To the Tangent of a ſeventh Arch 73.22.461 -10.5250542
Now if to the 7th Arch 73 deg. 22 min. 46 ſec. }, be added the
4th Arch 34 deg. 24 min. 19 fec. , the Sum 107 deg. 47 min. 06
fec. will be the Azimuth counted from the Northern part of the Me-
ridian; but being taken from 180 deg. oo min. oo ſec. will leave
72 deg. 12 min. 54 ſec. for the Azimuth from the Southern part of
the Meridian. Again,
If from the 7th Arch 73 deg. 22 min. 46 ſec. š, be taken the 4th
Arch 34 deg. 24 min. 19 ſec. Ź, the Remainder 38 deg. 58 inin.
27 ſec. will be the Angle formed at the Sun by the Vertical and
Horary Circle, which Angle is uſually called the Angle of Pofition,
whence to find the diſtance of the Sun from the Vertex it will be, by
the 2d Caſe of Oblique-angled Spherical Triangles,
As the S. <P ZO:S.<ZPO: :S, PO: $.2@. That is,
As the Sine of the Azimuth 72.12.54
9,9787325
To the Sine of the Hour from Noon 52.32.00 -9.8994667
So is the S. of the Sun's diſt. from the Pole 70.20-49* 9.9739342
To the Sine of his diſtance from the Vertex 5 1 41131-9.8946684
Whence it is manifeft, that at half an Hour paſt Eight in the Morn-
ing, the Sun appears at the diſtance si deg. 41 min 13 ſec. { from
the Vertex, when his Azimuth is South 72 deg 12 min. 54
ſec.
Eaſt.
But if inſtead of the Hour of the Day, the diſtance of the Sun from
the Vertex had been given, that is, in
Caſe II. Of Prob. VI.
Suppoſe it were required to find at what Hour in the Afternoon,
and on what Azimuth the Sun will appear at London on the 7th of
May 1720 ; (when the Earth views the Sun in 27 deg: 33 min. 57
ſec.
Gggg 2
0
0
fame Pole, and Ž O si deg. 41 min. 13 fec. , the diſtance of the
590 Of the Solution of Aſtronomical Problems
ſec. of Taurus) at which time the Sun is diſtant from the North Pole
70 deg. 20 min. 49 ſec. 4, and when the diſtance of the Sun from the
Vertex ſhall be found by Obſervation to be si deg. 41 min. 13 .ec .
The Stereographic Solution.
1: Having drawn the Ecliptic
rs in W, the two Colures re
and sew, the Path of the Vertex
Zlú, the proper Meridian OPN,
H
according to the former Directions
P
about the point @ as a Pole, de-
ſcribe a ſmall Circle at the diſtance
op of 70 deg. 20min. 49 ſec. to interſect
the Weſtern part of the Path of the
Vertex in Z the Place of the Ze-
nith at that time, wherefore if
N
thro this point Z and the Pole P,
an Hour Circle as ZP be drawn,
we ſhall have formed the Angle
2 P O equal to the Hour from Noon, and the Angle P 2 O equal
to the Azimuth from the Meridian, each of which may be meaſured
by the 10th Stereographic Problem: and to find their Value by. Cal-
culation,
In the Oblique angled Spherical Triangle ZPO we have given,
Z P 38 deg. 28 min, oo fec. the diſtance of the Vertex from the
Pole, R 70 deg. 20 min. 49 ſec.,, the diſtance of the Sun from the
N
Sun from the Vertex at the time of the Obſervation ; whence to find
firſt, the Hour from Noon, it will be by the inth Caſe of Oblique-
angled Spherical Triangles,
+
As the Radius
camper
10.000000
To the S. of the Diſtance of the Path from the Pole 38.28 9.7938317
So is the S. of the Sun's Diſt. from thePole 70:20.494-9.9739342
To a fourth Sine
9.7677659
As
by the Copernican Syſtem
591
97677659
As the fourth Sine
To the Sine of half the Sum of the diſtances of the Sun?
and the Pole from the Vertex, and the Diſtance of $ 9.9936845
the Vertex from the Pole 80.15.01.
So is the Sine of the Difference between the half
Sum and the Diſtance of the Sun from the Ver 9.6795455
tex 28.3347
To a ſeventh Sine
9.9054641
To which if you add the Radius 10.000000, half (the Sum
19-905464!, will be 9.9527320, the Co-fine of 26 deg. 14 min. 59
dec. which being doubled will give 52.29.58 equal in time to 3
hor.
30 min. oo lec: the Hour from Noon, whence to find the Azimuth
from the Meridian, it will be by the ift Cafe. of Oblique-angled Sphe-
rical Triangles,
As S. Z :S. PO : S. <ZPO:S. <PZ. That is,
As the S. of the Sun's Dift from the Vertex 51.41.131 9.8946686
To the $. of the Sun's diſtance from the Pole 70.20.49% 9.9739342
So is the Sine of the Hour from Noon 52.29.589.8994635
To the Sine of the Azimuth from the Merid. 77612.50-9.9787291
So that on the 7th of May 1720, when the diſtance of the Sum
by Obſervation was found to be si deg. 41 min. 13 fec. , the Hour
at that time was 3'hor. zo min. oo ſec. in the Afternoon, at which
time the Sun appeared upon the Weſt South Weſt half Weſt Azimuth
nearly:
The ſmallneſs of the Compaſs allowed me in this place, as well
as the Obligations I lye under, of enlarging upon thisSubject on ano-
ther Occaſion, have obliged me to put an End to this Section, being
at the fame time well aſſured, that he that underſtands what has
been ſaid, and is Maſter of the sth Part of this work, will readily
give an anſwer to any one of the great Variety of Problems that a-
riſe upon this Head,
The Hypothefis of the Earths Motion, upon which theſe So.
lutions are built, as it is undoubtedly the moſt ancient, ſo it is
certainly the rational and moſt agreeable to Reaſon and Senſe,
and
592 Of the Solution of Aſtronomical Problems
and as flich has been receiv'd and approv'd of, by the moſt eminent and
skillful Aſtronomers of all Times, but more eſpecially of the latter ; for
wlio is he that can compare the Sun with our Earth, and find him to
be a Million of times bigger in Magnitude, and knows at the ſame time
that every fixed Star is at leaſt as big as the Sun, and ſome probably
bigger, who that conſiders the immenſe Diſtance of the neareſt fixed
Star, which is ſo great, (as I have already mentioned) that the Circle
incloſed by the great Orb in which the Earth moves (and whoſe
Diameter is equal in Length to Twenty Thouſand Diameters of our
Earth, the Diameter of the Earth being about Four Thouſand Engliſh
Miles) is but as the Center of that Orb in which the neareſtFixed Star
is placed, who I ſay that duely weighs theſe things can conceive, with
what an incredible Degree of Velocity ſuch an almoſt infinite Num-
ber of glorious Bodies, (ſome of which, if not all, have probably
Worlds attending them) muſt move, to be carried about our little Ter-
raqueous Globe in the Space of 24 Hours, to anſwer no other End
or Purpole, as appears at preſent, than to gratifye the Humour and
Fancy of whymſical and prejudiced Perſons ? Or who can imagine
that the Great and Wife Architet who made the Univerſe and all
that therein is contained, ſhould contrive Epicicles or little Circles,
for the Planets to move in, to produce the retrograde Motions
which they appear to have, when whatſoever is to be perform-
ed by them, Imight much more eaſily be effected by one con-
tinual progreſſive Motion in a fimple Line; as was firſt diſco-
vered by the learned Kepler, from his own and the Obſervations
of the famous Tycho Brahe, and has ſince been confirmed by the
concurrent and manifold. Obſervations of the beſt Astronomers as well
as from the Eſtabliſhed Laws of Motion: And as this Hypotheſis is daily
more and more confirmed by the frequent Diſcoveries that are made
in the Heavens, there is little room left to doubt of its Certainty, and
that in Time it will be univerſally received.
As the firſt Impreſſions made upon the Mind of Man are common-
ly the ſtrongeſt and hardeſt to be erazed, fo Perſons that have all a-
long uſed themſelves to confider the Earth as fixed and quieſcent in the
Center of the Univerſe, and the Sun conſtantly to move, are brought
with great Difficulty to think the quite contrary Way : Ard inaf-
much as the Solution of all Problems are done with as great Eaſe by
this true Hypothefis as by the Ptolomaic it ſelf, I have often wondered
with my ſelf, conſidering the Advantage of having our firſt No.
tions
by the Copernican Syſtem
593
tions rightly ſettled, eſpecially ſince almoſt all Mankind agree
in the Motion of the Earth, that we are ſtill obliged accord-
ing to Cuftom, to Write and Inſtruct according to the Ptolo-
maic Hypotheſis, when if People whoſe Buſineſs it is to teach others,
would but once agree among themſelves, it would ſoon univerſally
ſpread, and Perſons thus Inftru&ed would be ſaved the trouble of
unlearning what they have once taken fome Pains to obtain.
***************************************
Section XXIII.
Of the Uſe of the Globes.
A
Globe in the Language of the Geometers, is a round Solid
Body, formed by the Roration of a Semicircle about its Di-
ameter, remaining fixed until it return to the Place again from whence
it began to move, by which means it comes to paſs that that point
which was before in the middle of the Diameter, and was the Center
of the generating Circle, becomes now the Center of the Qlobe it
felf, as being equally diſtant from every point in the generated Sur-
face.
This Artiſicial Spherical Body from the exact Similitude that it
has to our Ball of Earth, as has been proved from manifold Obſer-
vations, and particularly from the Obſervations made of Ecliples
has been pitched upon and choſen by Geographers, as the fitceſt and
propereſt Inſtrument to delineate upon its outward Surface, the ex-
ternal Appearance of this our Globe, ſo that the Terreſtrial Globe
is an Artificial Repreſentation of this our Terraqueous Globe, up-
on whole Convex Superficies the form of this our habitable World
is depicted, and all the parts of the Earth and Sea are deſcribed in
their natural Form, Order, Diſtance and Scituation.
And becauſe the Cæleſtial Bodies appear to us as if they were all
placed in the ſame Concave Sphere, Aftronomers have likewiſe made
Uſe of the external Surface of the Globe, to lay down the Stars up-
on, in their juft and due Poſition and Diſtance, and according to their
ſeveral Magnitudes. So that
As'
594
: Of the Uſe of the Globes
As the Terreſtrial Globe is an Artificial Repreſentation of the Sur-
face of this our Terraqueous Globe, as it would appear to us if placed
at a convenient Diſtance without it, and we could either be con-
veyed round it, or view it revolving about its own Axis. So
The Cæleſtial Globe is an Artificial and lively Repreſentation of
the Starry Heaven, ſuppoſing our ſelves placed in the Center and
the Globe to be tranſparent, as it appears to us when placed upon
the Convex Surface of our Earth, and which
conſidering its infinite
ſmallneſs, with regard to the unmeaſureable Diſtance of the Fixed
Stars, may be conſidered as the Center of the Univerſe.
And as the Geographers for the ready Diſtinction of Places, have
divided the Surface of the Earth into ſeveral Parts, ſuch as Europe,
Aha, &c and theſe being divided again into Empires, Kingdoms, &c.
to the 4jtronomers that they may the better diſtinguiſh the Fixed
Stars one from another, have in like manner divided them into ſeveral
Aſteriſms or Conſtellations, each of which contain a Syſtem of ſeveral
Stars that are ſeen in the Heavens near to one another, and have re-
duced theſe Conſtellations (the better to give each Star a particular
Name, and to retain it afterwards in Memory) into the Forms of
certain Animals, ſuch as Bulis, Bears, Lyons, Men, &c. or into the
Images of Things known to us, as of à Crown, a Harp, &c.
which were eſteemed by the Ancients upon Religious and Civil Ac-
counts, and are ſtill retained by the Modern Aftronomers, to avoid
that Confuſion which would neceſſarily follow, in comparing their
Obſervations with thoſe that were formerly made.
And as the Ancient Astronomers for theſe Reaſons, very judicoufly
enclofed almoſt all the Fixed Stars that were known to them, in a-
bout Forty-eight Conſtellations, ſo their Succeſſors with equal Right
when by their diligent and frequent Obfervations, they had greatly
increaſed the Number of Stars in their Catalogue, filled up moſt of
the Caſons with new Conſtellations of their own, and which are ſtill
retained with great Advantage to the Obſervators.
Hence it appears, that Globes are not only the moſt proper Con-
trivances for accounting for all the heavenly Appearances ariſing from
the diurnal and annnual Motion of the Sun, or the Rotation of the
Earth about her own Axis; but the nobleft Inſtrument as well for in-
forming the Mind, and giving it the cleareſt and moſt diſtin& Idea
of the thing propoſed to be done, as for giving of Solutions to all
Problems reſulting from it ; and it is from a Contemplation of theſe
Bodies and their ſeveral Appendages, (which will be explained here-
after)
Of the Uſe of the Globes.
595
after) that Aſtronomers have diſcovered ſo many excellent Methods for
the inore eaſy and exact Solution of the ſeveral Aſtronomical, Geo-
graphical, &c. Propoſitions, whence it muſt be allowed, that to have a
perfect Knowledge of them is one of the ſureſt and beſt Foundations
that can be laid, by any one who would underſtand Aſtronomy, Geo-
graphy, Navigation, and their dependent Sciences.
Now in order to lay down the Places of the Stars upon the Globe
in their due Scituation and Order as well as to transfer the leve-
ral Divifions of the Earth, and to deſcribe on the Terreſtial Globe,
the irregular Terminations of Land and Sea, it was abſolutely ne-
cellary to have ſome fixed or certain Points, and conſequently Lines
to begin to count or mealure from ; and this put Mankind upon
contriving thoſe ſeveral Circles that are deſcribed upon the Surface
of the Globes, which have their Names given to them, ſome
according to the Uſes for which they are deſigned, and others upon
the Account of thoſe Places that they are ſuppoſed to pofleſs; which
Circles the Reader ought to be very well acquainted with, and of
which he will meet with a full Account in the 1ſt Section of the sth
Part, to which I refer him ; and ſhall therefore proceed to give
an Account of ſuch Appurtenances belonging to the Globes, as have
not hitherto been taken Notice of: And,
s. The Horizon (defined in Section the oft of Part the 5th) is
that great broad Circle that divides the Globe into two equal Parts,
termed the upper and lower Hemiſpheres.
This Circle is diſtinguiſhed into two kinds, the one is called the
Senſible or Natural Horizon, and is that Circle that Bourds the ut-
moſt Proſpect of our Sight, when we view the Heavens round from
any Part of the Surface of the Earth orSea, the other is called the Trus
or Mathematical Horizon, to which all Aſtronomical Calculations refer,
and which obtains only in the Mind, and ſuppor s theEye to be placed
in the very Center of the Earth, beholding half of the intire Firmament
at one View ; which is repreſented by the uppermoſt Surface of
that broad Wooden Circle upon the upper Surface of the Frame in
which the Globe is fixed, having two Norches, the one in the North,
the other in the South part of it, for the brazen Meridian hereafter
to be deſcribed, to ſtand in.
And as the Globe is made to ſlide up and down in theſe Norches,
this Circle is of great Uſe in determining the Times of the Riſing
and Setting of the Sun or Stars, and their continuance above the
Horizon ; in ſhewing us the Reaſon of the Increaſe and Decreaſe
Hhh l *
of
596
Of the Uſe of the Globes.
of the Length of the Day and Night, &c. in all Places upon the
Earth by Inſpection. Upon this broad Horizon are deſcribed ſeveral
other ſmall Circles.
The firſt or innermoſt contains the Twelve Signs of the Zodiac,
each Sign being equal to 30 Degrees,
In the ſecond they are diſtinguiſhed by their proper Names, Marks,
or Chara&ers. The third contains the Days and Months of the
Year, diſpoſed of according to the Old Stile or Julian Account, ſo
called from Julius Cæfar the firſt Inventer, with the Times when
the Principal fixed Fcaſts or Faſts are to be celebrated.
The fourth contains the Gregorian Kalendar according to the New
Stile or Foreign Account, inſtituted by Pope Gregory the 13th, who
flouriſhed about the latter End of the 16th Century.
The fifth or ourermoſt Circle contains the points of the Compaſs,
with the Names of the Winds, as they are called by our Seamen.
The next Great Circle is the Meridian, (deſcribed in Section the
ift of Part the 5th) and is repreſented by the brazen Frame or Cir-
cle to which the Globe is fixed, by the two Wires which repreſent
the two Poles of the World, or the two Extremities of the Axis in
which the Globe hangs ; within which it turns dividing the Globe
into two equal Parts called the Eaſtern and Weſtern Hemiſpheres,
and from its being made of Brafs it is uſually termed the Brazen Me-
ridian. This Circle is divided into four 90 Degrees anſwering to
360 Degrees, the intire Circumference of every Circle, two of which
commence at the Equator and increale towards each Pole, in order
to ſhew the Declination of the Sun or Stars on the Cæleſtial Globe,
and the Latitudes of Places on the Terreſtrial Globe; and the other
two 90 Degrees commence at either Pole and increaſe toward the
Equator for the more eaſy and ready adjuſting of the Globe to the
Latitude of any particular Place, for as the Globe hangs in the Meri-
dian, and this is madeto ſlide eaſily thro the Norches made in the N.
and S. points of the Horizon, it is very eaſy to elevate or depreſs either
Pole of the Globe, ſo as that it ſhall ſtand at any particularElevation.
And as the Globe is made to turn within this Circle upon the
two Extremities of the Axis, called the two Poles, this brazen Me-
ridian may be made to repreſent the Meridian of any Place upon
the Terreſtrial Globe, or a Circle of Right Afcenfion, to any point
wha:ſoever upon the Cæleſtial Globe, and is of great Uſe in the
Solution of all Problems relating to either Globe; as will appear in
the Sequel of this Diſcourſe.
The
termined
Of the Uſe of the Globes.
597
j
The next great Circle is the Equino&ial, as it is called on the
Cæleſtial Globe, or the Equator on the Terreſtrial (deſcribed in Seat:
the iſt of Part the 5th) and divides the Globe into two equalParis,
called the Northern and Southern Hemiſpheres ; this Circle is of
Uſe in determining the Right Aſcenſion of the Sun or Stars on the
Coeleſtial Globe, and the Longitudes of Places on the Terreſtrial
and is that Circle from whence the Declination of the Sun or Stars
and the Latitude of Places take their Commencement.
The next great Circle is the Ecliptic; divided into Twelve Signs,
and each Sign into 30 Degrees (as has been explained in the ift Seet.
of Part the 5th) dividing the Globe into two equal Parts, called
the Northern and Southern Hemiſpheres, with regard to the Lati-
tudes of the fixed Stars and Planets.
The Sun in his annual Motion according to Prolomy, paſſes thro
the Ecliptic, and if we add to it a broadSpace of about 8 Degrees on
each ſide, we ſhall have the Zodiac ; in which are the Twelve Afte-
riſms, the moſt of which having the likeneſs of ſome living Creature,
was the Occation of giving it this Name; and inaſmuch as the great-
eſt Latitude of any of the Planers never amounts to 8 Degrees, the
Motion of the Moon as well as of the reſt of the Planets, will always
be performed in this Space.
The Quadrant of Altitude, is a narrow thin Plate of pliable Braſs,
exa&ly anſwering to one fourth Part of the Meridian, and divided
into go deg. oo min. having a Notch, Nur and Screw at one End,
to faſten it to the Zenith in the Meridian, and being thus fixed, and
turning round upon a ſmall pevit, it ſupplies the place of an infinite
Number of Vertical Circles, and is very uſeful in determining the Al-
titudes and Azimuths of the Coleftial Bodies, and in finding the
Times, &c.
The Hour Circle is a flat Ring of Braſs, fo contrived that it
may
be taken off and fixed about either Pole of the Globe, and when it
is ſo faſtened to the brazen Meridian, the Pole becomes its Center,
and to that End of the Axis there is fixed an Index, that turns round
as the Globe it felf is turned round, and points out upon the Ho-
rary Circle the Hour given or required, for this Horary Circle being
the Repreſentative of the Equator, which is carried about in
one Day on its upper Surface are inſcribed the 24 Hours of the
Natural Day, at equal Diſtances from one another; the 12th
Hour next towards the Zenith repreſenting 12 at Noon, and the o-
ther neareſt to the Nadir repreſenting : 2 at Night ; and the Hours
Hhhh 2*
On
598
Of the Uſe of the Globes
on the Eaſtern ſide repreſenting the Morning Hours, and thoſe on the
Weſtern-lide repreſenting the Afternoon Hours; each Hour anſwering
to, or correſponding with 15 degrees of the great Equinoctial, and is
placed there for no other Reaſon, but to ſave the trouble of reducing
the Degrees of the Equino&ial into time; and the contrary,
The Semicircle of Poſition, which is rarely or cver affixed to
the Globe, is a narrow thin Plate of Braſs exactly anſwering to
one half of the Horizon, divided into 180 Degrees, and uſually af-
fixed to the North and South Ends of the Horizon or Meridian, and
is of Uſe in marking out the Cufps of the Twelve Houſes as they are
uſually called, in meaſuring the Diſtances betwecn any two Places
upon the Sirface of the Earth, or any two Stars upon the Cæleſtial
Giobe, and may ſerve for a double Quadrant of Altitude, &c. And
laſtly, if a Mariners Compaſs duly touched with a Loadſtone, be fixed
u on the Pediinent or Wooden Frame which contains the Globe, or
in ſome part of the broad Wooden Horizon ; ſo that the true Meridian
of he Compaſs may lie exactly parallel to the Plane of the Brazen
Meridian, the Globes are prepared for any Uſe to which they inay
be applied to; and ready for the Solutions of all Problems that can be
propoſed relating to them.
And inaſmuch as Problems of this Kind are in a manner innumera-
ble, I ſhall ſele& out ſuch only, as the narrowneſs of the Compaſs
to which I am confined will allow me, and which are more imme.
diately applicable to my preſent Deſign.
As the Degrees of Exactneſs to which Queſtions may be anſwer-
ed, is in Proportion to the largeneſs and goodneſs of the Globes,
and the Care taken in the Operation, I ſhall give the Anſwers to
Minutes, and leave the Practitioner to try how near he can approach
to them.
Problem 1.
To find the Latitude of any given Place upon the Earth.
Inaſmuch as the Latitude of any Place, is equal to the neareſt Di-
ftance of that place from the Equator, meaſured by an Arch of the
Meridian paſſing thro the given Place, and intercepted between it
and the Equator.
Turn the Globe about, till the given Plače lye exactly under the
Eaſtern ſide of the Brazen Meridian, then will the Degrees in the
Meridian direaly over it, ſhew the Latitude; which is Northerly
if
Of the Uſe of the Globes.
592
if the Place lye in the Northern Hemeſphere, or on the Northern
fide of the Equator, but South, if the place lye in the contrary He-
miſphere, or on the Southern-líde. Or,
Take the neareſt Diſtance of the given Place from the Equator
with a pair of Compaſſes, and apply it to the Equator, or to any
graduated Circle, and the Degrees of that Circle intercepted between
the points of the Compalles will give its Latitude ; thus the .
Latitude of Londone will be found to be si deg. 32 min. North,
of Rome. 41 deg. 51 min. North, of the Lizard so deg. oo min.
North, of Barbadoes 13 deg. 30 min. North, and of St. Helena 16
deg. 03 min. South.
And inaſmuch as all Places chat lye at the ſame Diſtance from the
Equator, on either Side of the Equator, are ſcituated in the faine
Degree of Latitude, if any given Place be brought to the Meridian,,
and after that the Globe be turned about upon its Axis, all Places
that paſs under the ſame Degree of the Meridian with the gia
ven Place, will have the ſame Degree of Latitude, and will have
the Length of their Days and Nights, and the time of the Riſing:
and Setting of the Sun, the ſame at all tiines of the Year.
Prob. II.
To find the Difference of Latitude between any two given Places...
The Difference of Latitude between any twoPlaces, is equal to an
Arch of the Meridian intercepted between the two Places if they are
fcituated under the ſame Meridian, or the Diſtance between the Pa-
rallels paſſing thro the two Places, if they lye under different Meri-
dians. Wherefore,
Bring each of the Places propoſed, ſucceſſively to the. Brazen Me-
ridian, and obſerve 'where they interſect it, count the Number of
Degrees of the Meridian contained between the two Interſections,
and it will be the Difference of Latitude required, Or,
If the Places propoſed are both on the ſame ſide of the Equator,
having firſt found their Latitudes, fubftra& che lefler from the greate-
er; but if they are placed on contrary Sides of the Equator, add the
lelſer to the greater, and the Difference in the first Care and the Sum
in the latter Cate, will give the Difference of Latitude required to:
be found.
Thus the Difference of Latitude between London and Rome will
be found to be 9 deg. 41 min. between the Lizard and Barbadoes
deg. 30 min. and between Barbadoes and St. Helena 29 deg.: 331 min.
Probs
30
600
Of the Uſe of the Globes.
Prob. III.
To find the Longitude of any given Place upon the Earth.
The Longitude of any Place is an Arch of the Equator, intercep.
ted between the firſt Meridian, and the Meridian paſſing thro the
Place propoſed. Therefore,
Bring the given Place to the Meridian, and the Degree of the
Equator cut off by the Meridian, will give the Longitude ofthe Place
propoſed.
Hence if the Longitude be reckoned from the Meridian of London,
on Eaſtwardly, till it end at the firſt Meridian from whence it began
to be counted, we ſhall find the Longitude of the Lizard to be 254
deg 46 min. of Barbadoes to be 301 deg. 48 min. and of St Helena
to be 353 deg. 32 min
And becauſe all Places that have the ſame Degree of Longitude,
lye under the ſame Meridian, it follows, that if any Place be brought
to the Meridian, all thoſe Places that Iye under the ſameSemicircle of
the Meridian, will have the ſame Longitude ; and all thoſe People
who inhabit under that part of it that is contained or intercepted be-
tween the Pole and the Horizon, will have their Noon or the Sun
upon their Meridian, at the ſame point of abſolute Time.
As the fixing of the firſt Meridian from whence the Longitude of
Places are reckoned is arbritary, and according to the Humour of the
Geographer: Ptolomy having placed it at one Degree to the Weſtward of
the Fortunate or Canary Iſlands, after him, by ſome it was made to paſs
thro St. Nicholas, one of the Cape de Verd Iſlands ; by others thro St.
Fago, by others thro Corvo, one of the Weſtern Iſlands ; by the Dutch
it was made to paſs thro Tenerif"; by the French thro the middle of
Faro, the Weſtermoſt of the Canary Iſlands; and at preſent by every
Geographer and Hydrographer thro the Capital City of his Country, or
Principal Headland, and by our Seamen in their Reckoning from the
Land they laſt ſaw ; hence it is that in comparing of the Longitudes of
Places taken from different Maps or Globes, they will ſeldom or e-
ver agree, tho the Places are never ſo true laid down ; but the Dit-
ferences of Longicude if they are truly ſcituared upon the Globe or
Map, will always anſwer one to another; and this being the Prin-
cipal Thing required in all Calculations, it is of no great Moment
where the firſt Meridian is placed, which is the Realon thar Man-
kind are ſo indifferent about it, tho if they could once agree upon
Of the Uſe of the Globes.
601
a fixed Place, it would undoubredly ſave young Beginners, and Per-
ſons that are not fo well acquainted with theſe things as they ought
to be, a great deal of Trouble; and prevent them from running into
a great many Inconveniencies and Abſurdities, as they do now for
want of being better informed, and for whoſe Sakes I judge it neceſſa-
ry to give this ſhort Account.
Prob. IV.
To find the Difference of Longitude between any two Places gi-
ven upon the Earth.
Bring each of the given Places ſucceſſively to the Meridian, and
obſerve where the Meridian cuts the Equator each time, count the
Degrees of the Equator intercepted between the two Places, and
you have the Difference of Longitude required. Or,
,
Having the Longitude of each of the given Places, ſubſtract the
Jeffer from the greater, and the Remainder will be the Difference of
Longitude, if the Longitude be counted on Eaſtwardly till it end where
it firit began, but if it be counted on Eaſtwardly and Weſtwardly till
it end in the oppoſite Meridian, then add or ſubftra& the Longitudes
of each of the given Places, according as they fall on the ſame or on
contrary ſides of the Meridian; and their Sum or Difference will be
the Difference of Longitude.
Thus the Difference of Longitude between London and Rome will
be found to be 12 deg, 55 min. between the Lizard and Barbadoes
48 deg. so min. and between Barbadoes and St. Helena se deg. 44
min. doc.
By the reverſe of the Method for finding the Latitude and Lon-
gitude of any Place upon the Globe, may any Place be laid down
upon the Globè, from its Latitude and Longitude firſt given, by
bringing the Degree of the Equator anſwering to the Longitude of
the given Place under the Meridian, and right under the Degree of
the Meridian anſwering to the given Latitude, will be the Place re-
quired.
And after the ſame manner may any other place be laid down, and
the Surface of the Earth or Water be deſcribed.
Prob. V,
Let it be required to find the Angle of Poſition between any two
given Places upon the Earth
By
:
602
Of the Uſe of the Globes
By the Angle of Pofition is meant an Angle formed by the Me-
ridian of one Place, and a great Circle palling thro the iwo given
Places, which is meafured by an Arch of a great Circle deſcribed a-
bout the Place given as a Pole, at which the Angle is required.
Wherefore,
Having elevated the North Pole of the Globe, if the Place at
which the Angle of Poſition be required be in North Latitude, but
the South Pole if che Place propoſed be in South Latitude, to the
Latitude of the Place firſt given, bring that Place under the Meri-
dian, over which affix the Quadrant of Altitude, and move it about
till the graduated Edge juſt touches the other place, then will the
Arch of the Horizon (in this caſe) Numbred from the North or
South points of the Horizon, ſhew the Angle of Poſition at the Place
propoſed.
Thus the Angle of Poſition at the Lizard, between the Lizard
and Barbadoes will be found to be S. 68 deg: 57 min. Weſt, but the
Angle of Poſition between the ſame places at Barbadoes will be North
38 deg. 05 min. Eaſt; after the ſame manner the Angle of Poſition
at Barbadoes, between Barbadoes and St. Helena, will be found to be
South 61 deg. 36 min. Eaſt, but the Angle of Poſition at St. Hele-
ua, between it and Barbrdoes, will be North 77 deg. 04 min. Welt,
whence it is maniteſt, that neither of theſe is the true Rhomb AiD:
or Point of the Compaſs, leadirg from one place to the other, or
according to which if a Ship ſhape her Courſe in failing from one
Place the will be carried to the other.
For inaiinuch as the Rumb-lines form cqual Angles with every
Meridian thro which they paſs, conſtantly approaching the Pole
but never faling into it, and therefore becoming Aſymptotes to it,
it is impoſſible for any one of theſe to coincide or becoine a Circle,
except the North and South Line which is the Meridian it ſelf, and
the Eaſt and Weſt Line under the Equator. And
Herce we ſee the Reaſon of that Geographical Paradox, that if
Bermudus (for Example) lies due Weſt of the Lizard (as it nearly
does) for that very Reaſon the Lizard will nor lye due Eaſt from
Beimudus ; for becauſe the Eaſt and Weſt Lines, and conſequently
all the other intermediate Rumb-lines between it and the Meridian
are dependant upon the Meridian or North and South Lines, and
always form equal Angles with it; and fince all the Meridians
(to which the Necdie always conforms it ſelf, conſtantly lying in the
ſame Plane with it excluding the Conſideration of the Variations,
which
+
Of the uſe of the Globes.
603
moeten
as the Lizard was one
which in the preſent Caſe may be totally negle&ed) meet it the Poles
and form Angles with each other in thoſe points, the Eaſt and Weſt
Lines which form Right-angles with theſe, will meet and interfeét
each other allo, when continued upon the Surface of the Globe ; in-
aſmuch as they at that time become great Circles in tit given points,
and conſequently that Place which appears due Eaſt in one Meridian
from a given Place in another Meridian, will be fárther diſtant from
the next Pole than the Place firſt given ; and conſequently if we re-
move to that place in the ſecond Meridian, that appeared dove Eaſt
from us when ſcituated under the firſt Meridian, the Place under the
firſt Meridian being nearer to the Pole than the ſecond Place, cannot
appear due Weſt, but conſiderably nearer to the Northward; and
ſo much the more, by how much the Diſtance be wett the twoPla-
ces is greater; as will evidently appear to any one who will bot be
ar the Pains to draw two Lines to repreſent ewo Meridian Lines,
to interſect each other in a given Point, and to draw two other Lines
at Right-angles to them, at the ſame Diſtance from the interſection-
ary Pole whence it is that if we fail from one place according
to the Direction of the Eaſt and Weſt Line of the Compaſs, as
ſuppoſe from the Lizard, we ſhall not arrive at Bermudus, which
appears due Weſt from the Lizard, but at another Place, as ſuppoſe
Pengwin on the coaſt of Newfoundland, as far diſtant from the Pole
we ſhall fail on, till having made one intire Revolution, we ſhall ar-
rive at the fame Place we ſet ſail from ; always keeping at the ſame
Diſtance from the Pole, and deſcribing that Path which is repreſent-
ed upon the Surface of the Globe, by the Parallel of Latitude pafi-
ſing thro the Lizard or Place firſt given.
But it the Ship fail from the Lizard upon any other Rumb, ſhe will
never arrive at the Lizard again, but will approach nearer and nearer
to the North Pole, if the Courſe be Northerly; but to the South
Pole, it the Courſe be Southerly; and never fall into it, according to
the Diređion of the Compaſs, and in order to determine the Place
of the Ship at any time, and how much ſhe has altered her Latitude
and Longitude
1. Turn the Globe about, till the given Rumb cut the Meridian
in the Latitude of the given Place
2. From that Interfe&ion in the given Rumb, fet off upon the gi-
ven Rumb the Miles of Diſtance Run, taken off from the Equator.
Iiii
*
8.
Bring
604
Of the Uſe of the Globes.
3. Bring this laſt Place to the Meridian, and the Degree cut off
by it will ſhew the Latitude the Ship is in, as will the Arch of the
Equator intercepted between the two points where the Meridians cut
the Equator in the two different Stations of the Ship, gives the Diffe-
rence of Longitude the Ship has made, whence the true Place of the
Ship.upon the Globe, may eaſily be found.
Thus a Ship in failing from the Lizard 27. deg 30 min. equal to
550 Leagues or 1650 Miles upon the South South Weſt half Wett
Rumb, will be found to be in the Latitude of 25 deg. 55 min. N.
and to have altered her Longitude 1004 Miles, or 334} Leagues,
equal to 26 deg. 44 min. and after the ſame manner may the Place
of the Ship be found from any other Courſe and Diſtance, and this
being well underſtood, it will be eaſy to reſolve any other Nautical
Problem that can be propoſed by the Globe it felf.
Prob. VI.
To find the Diſtance between any two Places upon the Earth.
The neareft Diſtance of any two Flaces is equal to the Arch of a
great Circle, paſſing thro the two Places and intercepted between
them ; wherefore, take the Diſtance between the points of the
Compaſſes and apply it to the Equator, or to any other great gradu-
ated Circle, and the Number of Degrees centained or intercepted
between them, is equal to the Diſtance required.: Or,
If the graduated Edge of a Quadrant of Altitude be applied to the
two Places, it will give the Number of Degrees of Diſtance between
them. Or,
Having re&ified the Globe according to the Dire&ions given in the
Jaft Prob. for finding the Angle of Poſition, obſerve the Degree in the
Quadrant of Altitude cut by the Place over which it is laid, this fub-
ftracted from 90 Degrees, will leave the Diſtance, which being re-
duced into Miles, by allowing 60 Miles, &c. to a Degree, will
give the Diſtance in Geographical or Nautical Miles
Thus the Diſtance between London and Rome will be found to be
13 deg so min. equal to 790 Miles, the Diſtance between the Li-
zard and Barbadoes 56 deg. 16 min. equal to 3376 Miles, and be-
tween Barbadoes and St. Helena 59 deg. 04 min. equal to 3544
Miles And,
Inaſmuch as the Standard Miles of all Countrys differ more or leſs,
according to the particular Laws of that Country, Geographers and
Navi-
Of the Uſe of the Globes.
605
Navigators have agreed upon a Standard Mile, for to pronounce the
Diſtances between Places by, which is equal co só part of the Degree
of a great Circle, or about 69* Engliſh Miles; and therefore when
the Reader meets with the Diſtance between two Places, expreſſed in
Miles in Books. of Geography, &c, he muſt form a Conception of
them after this manner
Prob. VII,
The Hour of the Day or Night at one place being given, to find
the correſponding Hour at another Place. Or,
Becauſe the Difference of the Times is ever equal to the Difference
of Longitude between the two Places, having found the Difference
of Longitude between the two Places, reduce it into Time, by
allowing is Degrees to an Hour. Then,
If the place where the Hour required lie to the Eaſtward of the
Place where the Hour is given, add the Difference of Longitude re-
duced into Tiine, but if it lye to the Weſtward, ſubſtrač it from
the Hour given, and the Sum or Remainder will be the Hour re-
quired.
Or having turned the Globe about, till the Place at which the
Hour is given lye, under the Brazen Mcridian, ſet the Index of the
Hour Circle at the given Hour, then turn the Globe about till the
Place at which the Hour is required lye under the ſame Meridian,
then will the Hour Index point out the Hour at the Place required..
Thus when it is 12 of the Clock at London, it will be found to be
12 hor. simin 40 ſec. at Rome, at the Lizard it will be but it hor.
39 min. 04 ſec. at Barbadoes it will be but 08 hor. 07 min. 12 ſec.;
and at St. Helena it will be ri hor 34 min. 08 ſec.
On the contrary, the Difference at the Times of any two Pla-
ces being known or given, the Difference of Longicude between the
two Places is given alſo.
1
Prob. VIII.
Given the Day of the Month, required the Sun's Place in the E-
cliptic.
Having found the Day of the Month in the Kalendar upon the
Horizon, right againſt ic you wil find the Sign the Sun is in, and the
Degree of that Sign.
Iiii 2
Thus
606
Of the uſe of the Globes.
Thus on the 1ſt of May the Sun will be found to be in 21 deg. 51
min. of Iaurus, on the ilth of Auguſt he will be found to be in 29
deg. 27 min. of Leo, on the 13th of November in'oz deg. 30 min. of
Sagittary, and on the oth of February in or deg. 09 min. of Piſces :
Theſe Places anſwer to the reſpective Days in the Year 1728, as
the Reader will ſee if he conſult the 2d Volume, and the Reaſon of
my giving them in Minutes as I ſhall all the following Anſwers, is
not becauſe it can be ſuppoſed that any one can determine them lo
nice by the largeſt Globes, for to Halfs and Quarters of a Degree is
fufficient, but that the Practitioner may ſee how near he can ap-
proach the Truth
On the contrary the Sun's Place being given, the correſpondent
Day of the Month may be found, by finding the given Place upon
the Wooden Horizon, and ſeeing what Day of the Month ſtands a-
gainſt it, for that will be the Day of the Month required.
Prob. IX
Given the Sun's Place to find his Declinacion.
Becauſe the Declination is an Arch of the Meridian contained be-
tween the Sun and the Equator, which is called Northern Declina-
tion if the Sun be on the Northern ſide of the Equator, but South
Declination if the Sun be on the Southern ſide of the Equator.
Having found the Sun's Place in the Ecliptic, bring it to the Eaſt-
ern ſide of the Brazen Meridian, and the Degree in the Meridian
dire&ly over the Sun's Place, will be his Declination.
Thus when the Sun is in 21 deg. 51 min. of Taurus, his Decli-
nation will be found to be 18 deg. 16 min. North, when he is in
29 deg. 27 min. of Leo it will be 11 deg. 41 min. North, when he
is in z deg. 30 min. of Sagittary it will be ao deg. 42 min. South,
and when he is in or deg. og min. of Piſces it will be found to be
18 deg. os min. South.
On the contrary by knowing the Quantity of the Declination,
which way it is, and whether it be increaſing or decreaſing, may
the Place of the Sun be found, by turning the Globe till ſome point
of the Ecliptic cuts the given Dectination, for that point will thew
the Place of the Sun.
Prob.
Of the uſe of the Globes.
607
:
Prob. X.
:
Given the Sun's Place to find his Right Aſcenſion.
Inaſmuch as the Right Aſcenſion, is that point of the Equator
that Riles or Sets with the Sun, in a Right Poſition of the Sphere,
or ſuch a Poſition where the Poles lye in the Horizon, and the Equi-
noctial cuts it at Right-angles, and conſequently muſt be the ſame
Degree that tranſits the Meridian with the Sun.
Bring the Suns Place in the Ecliptic to the Meridian, and the De-
gree of the Equino&ial cut by the Meridian at that time, will give
the Right Aſcenſion.
Thus the Sun being in 21 deg. si min. of I aurus, his Right Af-
cenſion will be found to be 49 deg. 26 min. when he is 29 deg. 37
min. of Leo it will be 151 deg. 35 min. when he is in 2 deg. 30
min. of Sagittary it will be 246 deg, 26 min. and when he is in i deg.
9
min of Piſces it will be found to be 333 deg. 12 min.
On the contrary, by bringing the Sun's Right Afcenfion under the
Brazen Meridian, and counting the Declination of the Sun from the
Equator either Northward or Southward, according as the Caſe re-
quires, may the Place of the Sun in the Ecliptic be found at any time,
from their Right Aſcenſion and Declination firſt given.
Prob. XI.
Given the Day of the Month or the Sun's Place in the Ecliptic,,
together with the Latitude of the Place, to find the Sun's Oblique.
Aſcenſion and Defcenfion.
Becauſe the point of the Equinoctial that aſcends with the Sun in:
an Oblique Poŝtion of the Sphere, or ſuch a Pofition where the E-
quinoctial cuts the Horizon at unequal Angles, and which happens
to all the Inhabitants, except fuch as are under the Poles or the E-
quator, is called the Oblique Aſcenſion, as the point that deſcends
with the Sun in the ſame Poſition of the Sphere is called the Oblique
Defcenfon - Therefore,
Having elevated the Pole according to the Latitude of the Place;,
that is, the North Pole if the Place be in North Latitude, but the
South Pole if the place be in South Latitude; which is called re&ti-
fying the Globe.
Bring the Place of the Eclipric (anfwering to the Sun's Place) to
the Eaſtern ſide of the Horizon, the Degree of the Equinoctial cut
by
608
w
Of the Uſe of the Globes
Taurus, his Amplitude will be found to be 30 deg. 6 min North,
by the Horizon at that time, will give the Oblique Afcenfion, and
it the Globe be turned about till the ſame Degree of the Ecliptic in
which the Sun is, be cut by the Weſtern ſide of the Horizon, the
Degree of the Equino&ial cue by. the Horizon at that time, will
give his Oblique Deſcenfion.
Thus when the Sun is in 21 deg '5.1 min. of Taurus, his Oblique
Aſcenfion at London will be found to be 24 deg. 53 min. and his o-
blique Defcenfion 73 deg. 59 min when he is in 29 deg. 27 min. of
Leo, his Oblique Aſcenſion is 136 deg. 30 min. and his Oblique
Deſcenfion 166 deg. 40 min. when he is in 2 deg. 30-min of Sazit-
tary, his Oblique Aſcenſion is 268 deg go min. and his Oblique
Deſcenſion 2 2 deg. 02 min. and when he is in or deg. og min. of
Piſtes, his Oblique Aſcenſion will be 347-deg. 28-min, and his O-
blique Deſcenſion 318 deg, -56.min.
Prob. XII
Given the Sun's Place in the Ecliptic, or the Day of the Month
and the Latitude of the Place, to find the Sun's Amplitude.
-Inaſmuch as the Sun's Amplitude is an Arch of the Horizon, in-
tercepted between the place where the Sun Riſes and the Eaſt point
of the Horizon, or between the point where he Sers and the Weft
point of the Horizon.
Having re&ifyed the Globe according to the Directions given in
the laſt Problem, bring the Sun's Place to the Eaſtern or Weſtern ſide
of ihe Horizon, and the Number of Degrees (counted in the Hori-
zon) between the place where the Sun Riſes and the Eaſt point of
the Horizon, or between the Weſt point and the point where the
Sun Sets, will give the Amplitude, which is always the ſame way
with the Declination, as is manifeſt by Inſpection, that is, if
the Sun be on the North fide of the Equino&ial hè Riſes or Sets to
the Northward of the Eaſt and Weft points of the Horizon, but if
he be on the Southern ſide of the Equino&ial he Riſes and Sets
to the Southward of the Eaſt and Weſt points of the Horizon.
Thus on the iſt of May when the Sun is in 21 deg. 51 min. of
on the Hih of Auguſt when he is in 29deg. 27min. of Lev, his Ampli-
tude will be 1.9.deg, oo min. North, on the 13th of November, when
he is in 2 deg. 30 min. of Sagittary, it will be 34 deg 38 minSouth,
and
of the Uſe of the Globes.
609
and on the 9th of February,' when he is in or deg og min. of Piſces,
it will be found to be 18 deg. oi min. South
Prob. XIII.
Given the Sun's Place and the Day of the Month; to find the
Meridian Altitude, or the Meridional Zenith-diſtance. !
Becauſe the Meridional Altitude is the greateſt Altitude or Height
when he is under the Meridian, as his Meridian Zenith-diſtance is
his leaſt Zenich-diſtance, or his Diſtance from the Zenith when he
is upon the Meridian.
• Having rectified the Globe, bring the Sun's Place to the Meridi-
an, and count the Number of Degrees between the Place of the Sun
and the Horizon, and it will give the Meridional Altitude, as will
the Number of Degrees between the Sun and the Zenith, give hisi
Meridional Zenith diſtance, both of which in one Sum muſt make a.
Quadrant or 90 Degrees.
But if the Declination of the Sun be given, then the Meridional
Altitude inay be found, by adding the Declination to the Comple,
ment of the Latitude if they are both North or both South, but if
one be North and the other South, then by ſubſtrading the Decli-
nation from the Complement of the Latitude, and the Sum in the
firſt Caſe, and the Difference in the latter, will be the San's Me...
ridional Altitude
In like manner may the Sun's Meridional Zenith diſtance' bence
found, by ſubftracting the Declination from the Latitude of the
Place, if the Latitude and Declination are both the ſame way, but
by adding the Declination to the Latitude if they are contrary the:
one to the other, and the Diference in the former Cafe and the Sum
in the latter, will give the Meridional Zenith-diſtance.
Thus on the 1st of May, when the Sun is 21 deg. 51 min. of
Taurus, his Meridional Altitude will be found to be so deg. 44 min.
his Meridional Zenith-diſtance 33 deg. 16 min. on the 11th of Au-
guſt when he is in 29 deg. 27. min. of Leo, his Meridional Altitude :
will be so deg. 09 min, and his Meridional Zenith diſtance 39 deg:
si mi on the 13th of November, when he is in 2 deg. 30 min. of:
Sagittary, his Meridional Altitude will be 17 deg. 46 min. and his.
Meridional Zenith-diſtance 72 deg. 14 min. and on the 9th of Febru-
ary, when he is in or deg og min. of Pifces, his Meridional Alci-
tude.
3
610
Of the Uſe of the Globes
tude will be found to be 27. deg. 23 min. and his Meridional Zenith
diſtance 62 deg. 37 min.
Prob. XIV.
Given the Latitude of the Place, the Day of the Month, or the
Place of the Sun in the Ecliptic, to find his Afcenfional Difference.
The Aſcenſional Difference is always equal to the Difference be-
tween the Right and Oblique Aſcenſion or Defcenfion... Wherefore
The Globe being re&ified, bring the Sun's Place to the Meridian
and obſerve the point of the Equator then culminating, or lying un-
der the Meridian, then turn the Globe about, till the Place of the
Sun in the Ecliptic cut the Eaſt or Weſt ſide of the Horizon, and ob-
fèrve what Degree of the Equator is cut off by the Horizon, at that
rime count the Number of Degrees in the Equator berween this point
and the point laſt found, and it will give the Aſcentional Difference.
Thus on the iſt of May, when the Sun is in 21 deg. só min. of
Taurus, his Aſcenſional Difference will be found to be 24 deg. 33
min. on the 11th of Auguſt, when he is in 29 deg. 27 min. of Leo,
it will be 15 deg. 08 min. on the 13th of November, when he is
in 2 deg. 30 min. of Sagittary, it will be 28 deg. 24 min. and on
the oth of February, when he is in i deg, og min. of Piſces, it will
be 14 deg. 16 min.
The Right and Oblique Aſcenſion or Defcenfion being known,
the Aſcenſional Difference may be found, by taking the Difference
between the Right and Oblique Aſcenſion or Deſcenſion, or by ta-
king half the Difference between the Oblique Aſcenſion and De-
fcenſion.
Problem
XV.
Given the Latitude of the Place, the Day of the Month, or the
Place of the Sun in the Ecliptic, to find at what time the Sun will
Riſe and Set, and the Length of the Day and Night:
The Degree of the Equinoctial that is under the Meridian in any
Scituation of the Globe, is called the Right Afcenfion of the Mid-
Heaven at that time.
Now
2
The uſe of the Globes
611
3
Now ſince the Hour of the Sun's Riſing is ever equal to half the
time of his continuance below the Horizon, and as this is ever e-
qual to, or meaſured by the Arch of the Equator that tranſits the
Meridian during that Space of time. Therefore,
Bring the point in the Ecliptic (oppoſite to the Sun's Place) to the
Meridian, and obſerve the Right Aſcenſion of the Mid-Heaven at
that time, then turn the Globe about, till the point in the Ecliptic
anfwering to the Sun's Place lyes under the Eaſtern-fide of the Ho-
rizon, and obſerve the Right Afcenfion of the Mid-Heaven at that
time.
Subſtract the latter Right Aſcenſion from the former, or ſubſtract
the leffer from the greater, and if the Remainder be more than 180
Degrees, fubftra&t it from 360 Degrees, and convert the Difference
into Time, by allowing's Degrees to an Hour, C. and it will
give the time of the Sun's Riſing; this therefore being doubled, will
give the Length of the Night, and being ſubſtracted from 12 Hours
will give the time of the Sun's Setting; which being added to its
felf, will give the Length of the Day: Again
Inalmuch as the Hour of the Sun's Setting is ever equal to half
the time of his Continuance above the Horizon, and as this is mea-
ſured by an Arch of the Equnoctial intercepted between the points
of the Equator that culminates with him, and that tranſits the Me-
ridian at the time of his Setting.
Bring the Sun's Place to the Weſtern-ſide of the Horizon, and
ſee what Degree of the Equinoctial is then under the Meridian
Subſtract this from the Right Aſcenſion of the Sun, or the Degree
of the Equator that culminates together with him, (and the Re-
mainder converted into time) will give the Hour of the Sun Setting;
which being doubled will give the Length of the Day, and ſubſtrac-
ted from Twelve Hours will give the Time or Sun Riſing, and laſtly
this being doubled, will give che Length of the Night.
Or,
If the Right Aſcenſion of the Mid-Heaven at the time of the Sun's
Riſing, be ſubſtracted from the Right Aſcenſion of the Mid-Heaven
at the time of his Setting, the Remainder converted into time will
give the Length of the Day; and on the contrary, if the Right Af-
cenſion of the Mid-Heaven at the time when the Sun arrives at the
Weſtern-ſide of the Horizon, be ſubftra&ed from the Right Aſcenſion
of the Sun, when he juſt afcends above the Horizon on the Eaſtern-
ſide, the Remainder converted into time will thew the Length of
Kkk k *
Thus
the Night-
612
of the Uſe of the Globes.
Thus on the ilt of the May when the Sun is in 21 deg. si min.
of Taurus, he will be found to riſe at 4 hor. 22 min. to ſet at 7 hor.
38 min. the Length of the Day will be equal to 15 hor. 16
min, and the Length of the Night 8 hor. 44 min. on the 11th of
Auguſt, when he is in 29 deg. 27 of Leo, he will riſe at 5 hor. and
det at 7 hor. oo min. the Length of the Day will be equal to
14 hor. oo min. and the Length of the Night ro hor. oo min. on the
13th of November, when he is in 2 deg, 30 min. of Sagittary, he will
riſe at 7 hor. 53 min. , and fet at 4 hor. 06 min. , the Length
of the Day will be equal to 8 hor. 13 min. and the Length of the
Night is bor: 47 min. and on the 9th of February, when he is
in i deg. 09 min. of Piſces, he will riſe at 6 hor. 57 min. and ſet
at os hor. 03 min. the Length of the Day will be io hor. 6 min.
and the Length of the Night 13 hor. 54 min.
As the Aſcenfional Difference found by the laſt Problem, ſhews
how much the Semi-Day is longer or ſhorter than Six Hours, ſo it is
ever equal to the Quantity of time that the Sun riſes and ſets before
or after the Hour of Six ; this therefore converted into time and added
to Six Hours, will give the time of the Sun's ſetting, the Semi-
Length of the Day when the Latitude of the Place and Declination
are both North or both South; but the time of the Sun's rifing the
Semi-Length of the Night when they are contrary ; but being ſub-
ſtracted from Six Hours, will give the time of the Sun's riſing when
they are both the ſame way, but the time of the Sun's ſetting when
they are contrary. Or,
Having re&ified the Globe, bring the Sun's Place to the Meridian,
and ſet the Index of the Horary Circle to Twelve at Noon, then bring
the Sun's Place to the Eaſtern-ſide of the Horizon, and the Index will
point out in the Horary Circle the time of the Sun's riſing; and the
Globe being turned about till the Sun's Place arrives at the Weſtern-
ſide of the Horizon, the fame Index will point out in the Horary
Circle the time of the Sun's ſetting. Again,
The Place of the Sun being brought to the Eaſtern-ſide of the Ho-
rizon, and the Index placed at Twelve, if the the Globe be curned
about till the Sun's Place arrive at the Weſtern-fide of the Horizon,
the Index will point out upon the Horary Circle, how mapy Hours
the Day is long; and the Globe being thus ſcituated, if the ſame
Index be placed at Twelve at Night or Noon, and the Globe be
then turned about till the Sun's Place arrive ar the Eaſtern-ſide of the
Horizon, the fame Index will point out upon the Horary Circle the
Length
Of the uſe of the Globes.
613
:
Length of the Night, or how long the Sun has continued below the
Horizon. When the Declination of the Sun becomes equal to the
Complement of the Latitude of any Place, and both are on the ſame
ſide of the Equator, the Sun never deſcends below the Horizon, but
paffes the Meridian on the North or South points of the Horizon,
according as the Place is ſcituated either on the North or South ſide
of the Equator; after which he conſtantly continues above the Ho-
rizon, till having pafled thro the greateſt Declination, he returns to
that point in the Ecliptic where the Declination becomes the ſame
as before: Let it therefore be required in
Prob. XVI.
To find what Number of Days the Sun doth conſtantly ſhine at any
Place within the Artic Circle, at what Day it begins, and at what time
it ends, as alſo the time of his Continuance below the Horizon at the
oppoſite tine of the Year, or the Length of the longeſt Night, and ar
what time it begins and ends,
Having elevated the North Pole of the Globe according to the
Latitude of the given Place, turn the Globe about till ſome point in
the Ecliptic in the firſt Quadrant of the Ecliptic, interſect the Meri-
dian in the North point of the Horizon, and right againſt that point
of the Ecliptic in the Horizon, ſtands the Day of the Month when
the longeſt Day begins
And if the Globe be turned about, till ſome point in the ſecond
Quadrant of the Ecliptic interfe&t the Meridian in the ſame point of
the Horizon, it will thew the Place of the Sun when the longeſt Day
ends, whence the Day of the Month may be eaſily found. Again,
If the Globe "be turned about, till ſome point of the Ecliptic in
the third Quadrant of the Ecliptic interfect the Meridian in the
South point of the Horizon, it will ſhew the Place of the Sun when
the longeſt Night begins : And laſtly, if the Globe be again made
to revolve about its Axis till ſome point in the fourth Quadrant of the
Ecliptic interſect the Meridian in the ſame point of the Horizon, it
will ſhew the Place of the Sun when the longeſt Night ends.
The Beginning of the longeſt or forreſt Day being known, it
will be eaſy to find the End, for by counting the Number of Days be-
tween the Beginning of the longeſt or ſhorteſt Day, and the Day
when the ſucceeding Solſtice happens, and counting that time from
the Solftitial Day, you will have the time when it Ends.
K kkk 2
Thus
614
Of the Uſe of the Globes
Thus at the North Cape, which is ſcituated in the Northern Latitude
of 71° 35' the Sun will not Ser, but will paſs the Meridian in the Nor.
thern point of the Horizon, and after that time will continue above
the Horizon till the 19th of July following, during the Space of 77
natural Days; after which he will continue to Riſe and Ser till the
4th of November following, when he will but juſt touch the Horizon
in the Southermoſt point, and will not aſcend, but .conſtantly con-
tinue below the Horizon till the 16th Day of January following;
when he will juſt appear to Riſe upon the Meridian, after he has
continued below the Horizon for the Space of 73 natural Days;
which is the Length of the longeſt Night.
So that the longeſt Day is longer than the longeſt Night by the
Space of Four natural Days, which Inequality ariſes from the Ex-
centricity of the Earths Orb.
After the ſame manner may the Length of the longeſt and ſhorteſt
Day be found in any other Place within the Artic Circle, and mu-
tatis mutandis, may the Length of the longeft and fhorteſt Day be
found, in any place within the Antartic Circle.
Problem XVII.
The I atitude of the Place, the Day of the Month; or the Sun's
Flace in the Ecliptic being given, to find the Beginning and End
of Morning and Evening Twilight.
It has been found by Obſervation that the Solar Rays are viſible
till the Sun has deſcended 18 Degrees below the Horizon, at which
time the total Darkneſs begins. Wherefore,
Having re&tified the Globe to the Latitude of the Place, and fixed
the Quadrant of Aliitude to the Zenith of the Place, or at the di-
ftance of the Complement of the Latirude of the Place from the
Pole, cauſe the oppoſite point to the Sun's Place in the Ecliptic to
aſcend, till it cue 18 Degrees above the Weſtern fide of the Hori.
zon, on the Quadrant of Altitude, from the Right Afcenfion of the
Mid Heaven at that time, ſubſtract the Right Aſcenſion of the Sun,
and the Remainder converted into Time will ſhew how long the
Twilight begins and ends, before and after Noon; this therefore
will give the End of the Evening Twilight ; and being fubftracted
from 12 Hours, will give the Hour when the Morning Twilight be-
gins ; and if the Globe he turned about, till the ſame point in the
Ecliptic juſt touch the Weſtern edge of the Horizon, the Difference
bes
Of the uſe of the Globes,
615
between the Right Aſcenſion of the Mid-heaven at each Scituation
of the Globe, will ſhew the Duration of Twilight. Or,
Having rectified the Globe as before, and let the Index of the
horary Circle to 12, turn the Globe about, till the point oppoſite
to the Sun's Place be gotten 18 Degrees above the Weſtern lide of
the Horizon, then will the Index point out upon the horary Circle,
the beginning of Morning Twilight : And if the Globe he ſtill turn-
ed about its Axis, till the ſame oppoſite point to the Sun's Place
cut the Weſtern Limb of the Horizon, the Difference between the
Hour pointed out before, and in the preſent Scituation, will ſhew
the Duration of Twilight: and if the Globe be ftill turned about,
till the point in the Ecliptic oppoſite to the Sun, be deſcended 18
Degrees below the Ealtern-fide of the Horizon, the Index will point
out upon the Hour Circle the End of the Evening Twilight in the E-
vening.
Thus upon the oft of May, when the Sun is in 21 deg. 51 min.
of Taurus, the Begining of the Morning Twilight will be at i hor.
21 min. and the End of the Evening Twilight at 10 hor. 39 min,
after it has conrinued 3 hor. oi min. on the ilth of Auguſt,
when the Sun is in 29 deg. 27 min. of Leo, it will begin at 2 hor..
39 min. in the Morning and end at 9 hor, 21 min. in the Evening,
and continue 2 hor. 21 min. On the 13th of November it will be
gin at 5 bor. . 47 min. in the Morning, and continue 2 hor. 7 min.
and end at 6 hor. 13 min. in the Evening: And on the gth of Fe-
bruary it will begin ac 05 hor. 03 min. in the Morning, end at
6 hor. 57 inin, in the Evening ; and continue for che Space of i bor.
57 min.
N. B. That the Duration of Twilight is from its Beginning in
the Morning to the time of Sun Riſing, and from the Setting
of the Sun in the Evening till the End of Evening Twilight;
and that when the Declination of the Sun becomes equal to
the Complement of the Latitude leffened by the Depreſſion,
the total Darkneſs ceaſes, and the Twilight continues from Sun
Set to Sun Riſe.
Prob. XVIII,
Given the Latitude of the Place, the Day of the Month, or Place
of the Sun in the Ecliptic, together with the Hour of the Day, to
find the Altitude of the Sun, and his Azimuth ar that time.
Having
616
Of the Uſe of the Globes
Evden
Having re&tified the Globe and fixed the Quadrant of Altitude
to the Zenith;
Convert the Time from Noon into Degrees and Minutes, and add
it to the Sun's Right Aſcenſion if the Time given be in the After-
noon, but ſubſtract it from the Right Aſcenſion if the time given be
in the Forenoon; or convert the Time reckoned from the preceding
Noon, and add it to the Right Aſcenſion of the Sun at the time
of the preceding Noon, and you will have the Right Aſcenſion of
the Mid-heaven at the time propoſed.
Bring the Degree of the Equino&tial anſwering to the Right Al-
cenfion of the Mid heaven under the Meridian, and lay the Quadrant
(t Altitude over the Sun's Place in the Ecliptic, then will the Arch
of the Quadrant of Altitude intercepted between the Sun's Place
and the Horizon, give the Altitude; as will the Arch of the Hori-
zon contained between that point where the Quadrant of Altitude
cuts it, and the Meridian give the Sun's Azimuih. Or,
Having re&tified the Globe and Quadrant of Altitude as before,
bring the Sun's Place to the Meridian, and fix the Index againſt 12
in the horary Circle, then turn the Globe about till theIndex point our
the given Hour in the horary Circle, and the Quadrant of Altitude
heing laid over the Sun's Place, will ſhew his Altitude and Azimuth
as before.
Thus on the 1ſt of May, when the Sun is in 21 deg. Si min. of
Taurus, ac 30 min. paft Eight in the Morning, his Altitude will be
found t
to be 37 deg. 14 min. and his Azimuth South 71 deg. 8 min.
Altirude at 9 in theMcrning will be found to be 36 deg. 6 min. and
his Azimnth South 58 deg. 59 min. Eaſt, on the 13th of Novem-
ber when he is in 2 deg. 30 min. of Sagittary, at half an Hout after
Two, his Altitude will be so deg: 39min. and his Azimuth S. 35 deg.
25 min. W. and on the oth of February, when he is in i deg. 9 min.
of Piſces, at 2 in the Afternoon his Altitude will be 22 deg. 12
min. and his Azimuth $. 32 deg. oo min. W.
Prob. XIX,
Given the Latitude of the Place, the Day of the Month, or the
Sun's Place in the Ecliptic, together with his Altitude, to find the
Hour of the Day and Azimuth of the Sun.
Having rectified the Globe and the Quadrant of Altitude, as in the
former Problem,
Turn
Of the uſe of the Globes.
617
Turn the Globe about, till the Sun's Place in tlie Ecliptic cut the
given Altitude, on the Quadrant of Altitude, in the Eaftern or Weſt-
ern Semicircle; according as the Caſe requires, and note the Right
Afcenfion of the Mid-heaven at that time : Now if from the Right
Aſcenſion of the Mid-heaven at that time be taken away the Right
Aſcenſion of the Sun, the Remainder converted into Time, will give
the Hour from the preceding Noon. Or,
Having rectified the Globe as before, bring the Sun's Place to
the Meridian, and ſet the Hour Index at 12, and turn the Globe an
bout fill the Sun's Place cut the given Altitude on the Quadrant of
Altitude either in the Eaſtern or Western Semicircle, according as
the Caſe requires ;" then will the Index point out on the horary Cir-
cle the Hour at the time of the Obſervation,
And in either of theſe caſes, the Arch of the Horizon contained
or intercepted between the point where the Quadrant of Altituds
cuts the Horizon, and the North or South points of the ſame, will.
thew the Sun's Azimuth at that time.
Thus on the i of May, when the Sun is in 21 deg. 51 min. of.
Taurus at 40 deg. oo min. of Altitude in the Eaſtern Semicircle,
it will be found to be 8 h. 49 m. and his Azimuth at that time will
be $. 66 deg. 33min. E. cn the uth of Auguſt when he has 35 degrees
of Altitude in the Weſtern Semicircle, theHour will be 3 hor. 8 min.
and the Azimuth South 61 deg. 8 min. Weft on the 13th of Novem-
ber when he is in 2 deg. 30 min. of Sagitrary, at 12 degrees of
Altitude in the Eaſtern Semicircle, the Hour will be 9 hor. 46 min,
and the Azimuth S. 31 deg. 58 min. E, and on the gih of February
when he is in or deg. og min. of Pifces, at 20 degrees of Altitude
in the Weſtern Semicircle, the Hour will be found to be 2 hor.
24 min. and the Azimuth S. 38 deg. 07 min. W.
Prob, XX.
Given the Day of the Month and Hour of the Day, to find
1. In what Places of the Earth the Sun is Vertical, or in the Ze
pith at that time,
2. In what Places of the Earth the Sun is, then Riſing,
3. In what Places of the Earth the Sun is then Setting.
4. In whit Places of the Earth he is then culminating, or on the
Meridian.
5. In what Places of the Earth it is then Midnight.
6. In what Places of the Earch the Twilights are then Beginning
7. If
618
Of the Uſe of the Globes.
7. In what Places of the Earth the Twilights are then Ending.
8. What Zone of the Earth at that time enjoys nothing but
Twilight.
9. What is the Height of the Sun in any part of the illuminated
Hemiſphere. And
10. What is his Depreſſion at any Place in the obſcure Hemiſphere
of the Earth.
Having found the Sun's Place and his correſpondent Declination,
elevate the North Pole of the Globe, if the Sun have North Decli-
nation, but the South Pole if the Sun have South Declination, a-
bove the Horizon, till the Arch of the Meridian contained between
the Pole and the Horizon be equal to the Declination of the Sun at
that time.
Convert the time from the preceding Noon into Degrees and Mi.
nutes, and if the Hour given be in the Morning, add it to the Lor-
gitude of the given Place, but if the time given be in the Afternoon,
ſubſtraat it from the given time, and bring that Degree of the E-
quator under the Meridian. Or,
Having brought the given Place under the Meridian, ſet the Index
at the given Hour, and turn the Globe about till the Index point at
Twelve at Noon upon the horary Circle. Then will
1. The Sun be Vertical to that point of the Earth that is then
under that Degree of the Meridian which correſponds with the Sun's
Declination, and conſequently in the Nadir, to the oppoſite point
in the lower Hemiſphere.
2. All thoſe Places that are in the Weſtern Semicircle of the Ho-
rizon, will have the Sun Riſing.
3. All thoſe people that dwell in thoſe Places of the Earth that
lye in the Eaſtern Semicircle, will ſee the Sun Setting.
4. To all Places that lye under the upper Semicircle of the Meridi-
an, theSun will be upon the Meridian and conſequently it will be their
high Noon.
5. And all thoſe People who dwell under the lower Semicircle of
the Meridian, will have it Midnight.
6. In all thoſe Places that are depreſſed 18 Degrees below the Weſt-
ern Semicircle of the Horizon, the Twilight is juſt beginning in the
Morning. And,
7. In all thoſe Places that are depreſſed 18 Degrees below the
Eaſtern Semicircle of the Horizon, the Twilight is there Ending in
the Evening, and the total Darkneſs beginning.
8. And
tes
The Ule of the Globes
619
8. And in all that Zone of Earth that is contained between the
Horizon and a ſmall Circle drawn parallel to it, and beneath it at
the diſtance of 18 Degrees, it will be Twilight.
9. It a Quadrant of Altitude be fixed in that point where the
Sun is Vertical, and laid over any Place, the Arch of that Quadrant
contained between the Place and the Horizon, will ſhew the height
of the Sun at that Place.
10. If the fame Arch of the Quadrant be continued below the
Horizon, ſo as to paſs over any Place in the obſcure Hemiſphere of
the Globe, the Portion of it contained between the Place and the
Horizon, will ſhew the Depreſſion of the Sun in that place.
From this Poſition of the Globe other uſeful Concluſions might
be drawn, but theſe ſhall fuffice at this time.
$
Prob. XXI.
To find the Latitude and Longitude of a fixed Star.
Inaſmuch as the Latitude of a fixed Star or Planer, is the neareſt
Diſtance of that Star from the Ecliptic, meaſured by an Arch of a
Circle of Longitude paſſing thro the Star, and intercepted between
that Star and the Ecliptic it ſelf, and the Longitude the Arch of the
Ecliptic contained between that point of the Ecliptic where the
Circle of Longitude cuts it, and the firſt point of Aries: therefore,
Having elevated the North Pole of the Globe if the Star be in
the Northern Hemiſphere of the Ecliptic, but if the Scar be in the
Southern Hemiſphere the South Pole, 66 deg. 29 min. the Comple-
ment of the greateſt Obliquity, above the Horizon, turn the Globe
about till the Solſtitial Colure lye under the Brazen Meridian, then
fix the Quadrant of Altitude to the Meridian in that point which is
dire&tly over the Pole of the Ecliptic, and which is fcituated 23 deg.
29 min. from the Pole is felf, and laying it over the Center of the
Star, tie Degree of the. Quadrant or Altitude cut off by the star,
will ſhew irs Latitude; which is North if the Star be on the North
fide of the Ecliptic, but South if the Star lye on the South ſide of
the Ecliptic; and the point of the Eclipric cut by the Quadrant of
Altitude, will ſhew the Place of the Star in the Ecliptic; as will
the Arch of the Ecliptic it ſelf, contained between that point and
the firſt point of Aries ſhew the Longitude of the fame Srar,
:: -Thus Aldebaran or the Southern Eye of the Bull will be found
to be in Gemini os deg. 37 min. having os deg. 30 min. South De-
L111 *
cil.
620
Of the Uſe of the Globes
clination, as will Caftor the Head of the Northern Twin, be found
to be in Cancer 16 deg. 25 min. having 10 deg. 04 min. North La.
titude; allo the Bright Foot of Orion called Regel, in Gemini 13
deg. oo min. having 31 deg. 10 min. South Latitude, and Syrius,
or the Great Dog Star, in io deg. 19 min. of Cancer having 38 deg.
32 min. South Larirude.
By the reverſe of this Method, may a Star belaid down from its
Latitude and Longitade, viz. by preparing the Globe as before, and
by laying the Quadrant of Altitude over the poirt in the Eclipric
anſwering to the Stars Place, and counting the Lalitude of the Star
from the Eclipric in the Quadrant of Altitude, either Northward or
Southward according as the Care requires, and the point upon the
Surface of the Globe, directly under that Degree of the Quadrant of
Altitude, will be the Place of the Star upon the Globe; and after
this manner may all the Stars or Planers themſ lves he laid down
upon the Globe, their Latitudes and Longitudes being known.
Prob. XXII,
To find the Right Afcenfion, Declination, Meridional Alcitude,
or Zenith Diſtance, Amplitude, Oblique Aſcenſion, Oblique Deſcen-
fion, and Aſcenſional Difference, of any fixed Star.
Now becauſe the Right Aſcenſion, Declination, Meridional Al-
titude, Zenith-diſtance, Oblique Deſcenfion, Oblique Afcenfion, Ain-
plitude, and Aſcenſional Difference of a Star is the ſame with that
of the Sun, fuppofing the Sun to be in the Place of the Star, or the
Star to be in the Place of the Sun; they are found upon the Globe, in
the ſame manner as they are found for the Sun in the 9th, 101h, rith,
12th, 13th, and 141h preceding Problems, by fi:ft re&tifying the
Globe to the Latitude, and bringing the Center of the Star to the
Meridian, to obtain the Declination, Right Aſcenſion, Meridional
Alcitude, or Meridional Zénith-distance, and to the Eaſt and Weſt-
fides of the Horizon, to obtain the Oblique Aſcenſion, Oblique Di-
ſcenfion, and Aſcenſional Difference.
Thus the ſeveral Requiſites for the Stars Aldebaran, Caftor, Re-
gel and Syrius, will be found as follows.
For
-
Of the uſe of the Globes,
621
165 03
109 16
For
Aldebaran Cuftor Kegel
Syrius
{ight Afcenfior-
75 21 98 16
declination
15 55N;:32 27 N 18 335 16.215
Meridional Altitude 54.23 7055 29 55 22.07
Meridional Zenith-diſtancel 35 37 19 os
60 05067 53
Oblique Aſcenſion
56 06
64.27 76 36
Amplirude
26 09N.' 59 36N. 13 505 26 54S
Osl que Defcenfion
86 c6 162 26
86 15.119 56
AfcenGon Difference
21 03
53 10
21 40
44 00
ID 54
Prob. XXIII.
The Place of the Sun in the Ecliptic, or the Day of the Month be-
ing giien, to find the time of any Stars coming to the Meridian.
As the Heavens appear to make an intire Revolution in the Spice
of 24 Hours, and conſequently the 24th Part of the Circumference
out the Equinoctial or 's Degrees, muſt tranſit the Meridian in one
Hour : Hence it is that the Arch of the Equinoctial intercepred be-
tween th: Circles of Right Aſcenſion paſling thro the Sun, (the
Governour and Director of the Solar Day, from whoſe arrival
at the Meridian of any place, the Aſtronomical Day cakes its Com-
mencement) and any fixed Star reduced into Time, ſhews how
long that Star will tranſit the Meridian after the Sun ; and conſe-
quenily the Time of the Day at which it will Culminate. Therea
fore,
Having brought ihe Sun and the Star ſucceſſively to the Meridian
of the Globe, obſerve the Number of Degrees of the Equa or that
tranfits the Meridian between them; or count the Number of Degrees
intercepred between the Points of the Equinoctial that culminates
with the Sun or Star; for this Arch reduced into Time, will give
the Hour, &c. of the Stars Culmination. Or,
Having brought the Sun's Place to the Meridian, fet the Index of
the horary Circle to Twelve at Noon, then turn the Globe rill the
Srar ir ſelf arrives under the ſame Meridian, then will the Index
point out upon the horary Circle the time of the Star's tranſicing
the Meridian.
Thus upon
the 241h of Auguſt, the Star Aldebaran will paſs
the Meridian at 17 hor. 28 min. and on the ist of January, the Siar
Caftor will Culminate at 11 hor. 41 min. on the 2d of September
Lill 2*
the
$
622
of the uſe of the Globes
the Bright Foot of Orion tranfits the Meridian ar 17 hor: 37 min. ),
and on the 25th of December, the Great Dog St ar Syrius, will appear
upon the Meridian at 11 hor 29 min.
The Quantity of the Sun and Star's Declination being found, the
time of the Stars Tranſit may be found by ſubftra&ing the Right
Aſcenſion of the Sun from the Right Aſcenſion of the Star, and re-
ducing the Remainder into Time; and after the ſame manner may
the time of the Culmination of any of the Planets be found.
Problem XXIV.
'
Given the Latitude of the Place, the Day of the Month, or the
Sun's Place in the Ecliptic, to find the time of a Stars Riſing and
Setting
Inalmuch as the Hour of the Stars Riſing is equal to the Arch of
the Equinoctial, that tranfits the Meridian between the time of the
Sun's paſſing the Meridian, and the time that the given Star aſcends
on the Eaſtern-ſide of the Horizon ; as the Arch of the Equator
that paſſes over the Meridian, between the cime the Sun paſles, and
the ſame Star arrives ar the Weſtern-ſide of the Horizon, is equal to
the time of the Stars Setting. Therefore,
If the Difference between the Right Aſcenſion of the Sun and the
Right Aſcenſion of the Mid-heaven at the time of the Stars Riſing
and Setting be converted into Time, we ſhall have the time of the
Stars Riſing and Setting. Or,
Having re&ifyed the Globe, and brought the Sun's Place to the
Meridian, and ſee the Index to Twelve at Noon, upon the horary
Circle, if the Globe be turned about till the Star juſt begins to aſ-
cend on the Eaſtern-ſide of the Horizon, the Index will point out
upon the horary Circle, the time of the Stars Riſing: And if the
Globe be ſtill turned about, till the ſame Star arrives at the Weſt-
ern-ſide of the Horizon, the Index will point out upon the horary
Circle, the Hour of the Stars Setting; and afrer the ſame manner
may the time that the Moon or any Planer will Riſe or Set be de-
termined, provided its Place upon the Globe be known. Or, ,
The Aſcenſional Difference of a Star being known, if it be converted
into Time, and added to, or fubftracted from Six Hours, according as
the Declination of the Star and Latitude of the Place, are the ſame or
contrary one to another, we ſhall have half the time of the StarsCon-
tinuance in our Hemiſphere; this therefore being added and ſubftrac-
ted
Of the uſe of the Globes.
623
ted to and from the time of the Stars Southing, will give the time
of the Stars Riſing and Setting.
Thus on the 24th of Auguſt, Aldebaran will be found to Riſe at
London at io hor. 04 min. and to ſet at 24 hor: 53 min. or 53 min.
after Twelve the next Day: On the 1ſt of January at the ſame
Place, the Star Caftor will Riſe at hor. 09 min. and Set at 21 h.
14 min. On the 2d of September the Star Regel will Riſe at the ſame
Place at 12 hor. 21 min. and Set at 22 hor. 54 min. And on the
25th of December, the Dog Star Syrius will Riſe at 06 hor. 35 min.
and Set at 16 hor. oz min.
Prob, XXV.
Given the Latitude of the Place, the Day of the Month, or Place
of the Sun in the Ecliptic, and the Hour of the Night, to find the
Altitude and Azimuth of the Star.
Having rectified the Globe and the Quadrant of Altitude for the
Latitude of the Place, reduce the Time from Noon into Degrees and
Minutes, and add it to the Right Aſcenſion of the Sun, the Sum
will be the Right Aſcenſion of the Mid-heaven at the time propo-
fed : Turn the Globe about till that Degree of the Equino&ial an-
fwering to it lye under the Meridian
#
Or,
Having brought the Sun's Place to the Meridian, and ſet the In-
dex of the Hour Circle to Twelve at Noon, turn the Globe about
till the Index point out the given Hour upon the horary Circle.
Then bring the Quadrant of Altitude over the Center of the Star,
and the Arch of the Quadrant contained between the Star and the
Horizon will give the Altitude ; as will the Arch of the Horizon
contained between the Meridian and the point in the Horizon where
the Quadrant of Altitude cuts it, give the Azimuth,
Thus the Altitude of Aldebaran on the 24th of Auguſt at 14 hor.
30 min. at London will be found to be 39 deg. si min, at which
time the Azimuth will be S. 61 deg. 33 E. alſo the Height of Caſtor
on the ift of January at 9 hor. 30 min. will be 59 deg. 26 min. and
the Azimuth at the ſame time will be South 64 deg. 10 min. Eaſt.
In like manner the Height of Regel at 19 hor. 00 min. on the
ad of September will be 27 deg: 20 min. and the Azimuth S 23 deg.
12 min. W. and on the 25th of December at 14 hor. oo min. the Alti-
tude of Syrius will be 14 deg 34 min. and the Azimuth S. 37 deg.
22 min. W.
Probe
624
Of the Uſe of the Globes.
Prob. XXVI.
Given the Latitude of the Place, the Day of the Month or Place
of the Sun in the Ecliptic, and the Height of any known Star, to
find the Hour of the Night and Azimuth of the Star.
Having elevared the Globe according to the Latitude of the
Place, and fixed the Quadrant of Altitude to the Meridian at the
Zenitḥ, turn the Globe about till the Center of the Star cut the gi-
ven Altitude on the Quadrant of Altitude, in the Eaſtern or Weſtern
Semicircle, according as the Caſe requires, note the Degree under
the Meridian at that time: Now if from the Right Aſcenſion
of the Mid-heaven at that time, be taken the Right Alcenſion of the
Sun, the Rcmainder converted into Time, will fhew the Hour of
the Night Or,
Having brought the Sun's Place to the Meridian, and ſet the In-
dex of the Hour Circie to Twelve, and turn the Globe about till
the Star cut the Quadrant of Altitude in the given Altitude, then will
the Index point out the Hour of the Night upon the horary Circle,
and the Arch of the Horizon in either Caſe contained between thé
Meridian and that point in the Horizon where the Quadrant of
Altitude cuts it, will fhew the Azimuth.
Thus at London on the 24th of Auguſt, when the Star Alaebaran has
Isdeg. 40min. of Altitude in the Eaſtern Semicircle, it will be found
to be 11 hor, 49 min. when its Azimuth will be South 83 deg 59
min. Eait; the Star Castor on the ift of January, at co deg. oo min.
of Altitude in the Eaſtern Semicircle, will have its Azimuth South
63. deg os min. Eaſt, when the Hour will be a hor. 34 min. on
the ad of September, when the bright Foot of Orion has 24 deg. 45
min. of Altitude in the Weſtern Semicircle, the Hour will be found
to be 19 deg 35 min and its Azimuth S. 32 deg. 30 min. W; and
on the 25th of December, when the Dog Star Syrius has is deg 30
min, of Altitude in the Eaſtern Semicircle, the Hour will be found
to be og hor. 08 inin, and the Azimuth of the Star at that time
South 35 deg. 04 min. Eaſt.
Prob.
XXVII.
Given the Latitude of the place, the Day of the Month, or
Place of the Sun in the Ecliptic, and the Hour of the Night, to find
what
Of the Uſe of the Globes.
625
what Stars are then Riſing and Sercing, what Stars are culminating,
and the Height and Azimuth of any Star above the Horizon, c.
Having elevated the Globe to the Latitude of the Place, and fix
ed the Quadrant of Altitude in the Zenith, reduce the time from
Noon into Degrees and Minutes, and add it to the Right Aſcenſion
of the Sun for that Day, and the Sum will be the Right Aſcenſion
of the Mid-heaven at the time propoſed ; and turn the Globe about
till the Degree upon the Equinoctial correſponding to this Number,
lye under the Meridian. Or,
Having brought the Sun's Place to the Meridian, ſet the Index to
Twelve upon the Hour Circle, and turn the Globe about till the In-
dex point out the given Hour upon the horary Circle, then will
1. All thoſe Stars that lye on the Eaſtern-lide of the Horizon be
Riſing
2. All thoſe Stars that lye in the Weſtern Limb of the Horizon),
be Setting
3. All thoſe Stars that lye under the Brazen Meridian be culmi-
4. If the Quadrant of Altitude be laid over the Center of any
Star, the Degree cut off by the Star in the Quadrant, will give the
Altitude of that Star at that time; as will the Arch of the Horizon
contained between the Meridian and the point where the Quadrant
cuts it, be the Azimuth, c.
nating. And
Prob. XXVIII.
To determine all thoſe Places upon the Earth, where an Ecliple
of the Moon, or of any of the Satellites of Jupiter is viſible.
Having found the Declination of the Sun, elevate that Pole of the.
Globe which is moſt remote from the Sun, till its Height is equal to
the Sun's Declination
Convert the time of the Beginning of the Eclipſe from Noon, into
deg. and min. and if the Hour given be between Noon and Midnight,
fubftract it from the Longitude of the given Place to which it is compu-
ted, but if it happen between Midnight and the ſucceeding Noon, add
it to the Longitude of the given Place, and bring the point in the E-
quino&ial oppoſite to this, to the Meridian, then a Line drawn by the
Eaſtern Edge of the Horizon, will paſs thro all thoſe Places where
the Moon appears to begin to be Eclipſed at her Serting; and it from
the Degree of the Equator then under the Meridian, be fubdiacted
nie
626 .
of the Ufe of the Globes
the Duration of the Eclipſe reduced into Time, and the Globe He
turned about till thatDegree under the Equincatial come to the Me-
ridian, a Line drawn by the Weſtern-edge of the Horizon will paſs
thro all thote Places where the Eclipſe will end at the Time of the
Riſing of the Moon ; and conſequently within all that Tra&t of
the Earths Superficies will the Eclipſe be viſible again.
To find all that Space of Earth and Warer, where an Eclipſe of
one of the Satellites of Jupiter will be viſible.
Having found that Place upon the Earth in which the Sun is Ver-
tical at the time of the Eclipſe by the 2016 Problem, elevare the
Pole tliat is neareſt to the Sun, till its Height be equal to the Sun's
Declination at that time, and bring the Place over which the Sun
is Vertical, under the Meridian, then if Fupirer be in Confequence of
the Sun, a Line drawn on the Globe on the Eaſtern ſide of the Horizon :
will paſs over all thoſe Places where the Sun is Setting at that time,
But it Jupiter be in Antecedence of the Sun, draw the Line on the
Weſtern-ſide of the Horizon, and it will ſhew all thoſe Places where
the Sun is tben Rifing.
If Jupiter be in Conſequence of the Sun, add the Difference be-
tween the Right Aſcenſion of the Sun and Fupiter to the Longitude
of that place where the Sun is Vertical at the given time, and bring
that Degree of the Equator under the Meridian and elevate the
North Pole, if Jupiter be on the North-fide of the Equator, but if
he be on the Southern-fide elevate the South Pole, till it be equal
to the Declination of Jupiter, in this pofition of the Glohe draw
a Line along the Eaſtern-ſide of the Horizon, the Space between
this Line and the Line that determined the place where the Sun was
Setting, will comprehend all thoſe Places of the Earth where Ju-
piter will be viſible, from the Setting of the Sun to the Setting of
Fupiter.
But if Jupiter be in Antecedence of the Sun, fubftract the Diffe-
rence between the Right Aſcenſion of the Sun and Jupiter, from the
Longitude of that place where the Sun is Vertical at the time of the
Eclipſe, and bring the Degree of the Equator anſwering to the Re
mainder under the Meridian, and the Globe being elevated as before
taught, draw a Line by the Weſtern Limb of the Horizon, and
the Space contained between this Line and the Line of the Sun's
Riſing before drawn, will comprehend all thoſe Places on the Earth
where the Eclipſe is viſible, between the times of the Riſing of the
Sun and Fupiter:
Theſe
The Elements of Chronology.
627
Theſe Problems being well underhood, there is ſcarce any Pro.
blem that can be propoſed, bur the skilful and diligent Reader will
be able to give a Solurion to it himſelf; and as theſe Problems that
I have relolved are the moſt general and uſeful thár can be, ſo I have
given each of them a general Anſwer, and ſhewn how each Problem
may be Solved independently of the other, from the moſt fimple Da-
ta, and that by the help of the Equator which gives Solutions to all
Queſtions, to a much greater Degree of Exactneſs than the horary
Circle can do, by how much the Radius of the Equator is greater
than the Radius of the horary Circle, the common Method made
Uſe of in all Books that treat of the Ule of the Globes, which be-
ing calculated for the Uſe of ſuch People only as do not underttand
the four Fundamental Rules of Arithmetic, ought to be laid quite
aſide by People that are Mafters of them ; tho for Conformities fake,
and for the Benefit of ſuch Perſons as are not capable of adding, c.
I have ſhewn how the Problems themſelves may be relolved by the
Hour Circle and Index only.
How the ſeveral Triangles are formed upon the Surface of the
Gisbe, by the mutual Interſection of the ſeveral great Circles made
Uſe of in theſe Solutions, whence the Length of the ſeveral Parts
of them may be Calculated by the Trigonometrical Laws, will be
very obvious to any one who underſtands what has been faid in the
preceeding Sellion but one, and in the 2d Settion of Part the 5th, I
Íhall therefore paſs it over, and leave it as an Exerciſe to the young
Beginners.
Spa
Section XXIV.
2
Of the Diviſion of Time and its Parts; of the Cycles,
Epoch's, Era's, Periods, Moveable Feaſts, &c.
YHronology as to its Etymolgy,denotes noother than a Diſcourſe
concerning Time, and is an Are that reaches us how to diftin-
guith rightly the Times, in which memorables Things have hap-
pened; this gives a Form to Hiſtory (which is a Narration of
paft Tranſactions) and Hiſtory furniſhes this with Matter; for as
Mmmm
Hif.
628
The Elements of Chronology
Hiftory deſcribes the Things done, fo Chronology aſſigns them their
proper Places in the Order and Series of Time, and is made up of
Two Parts.
The firſt explains the Nature and Parts of Time.
The ſecond treats upon certain Characteriſtics or Indices, by
which Times are diſtinguiſhed.
Abſolute True and Mathematical Time, conſidered in it ſelf, or
in its own Nature, without Relation to any external Being, flows
equally or uniformly, and is otherwiſe called Duration ; to diſtinguiſh
it from Relative, Apparent, and Vulgar Time, which is a fentible
external Meaſure of Duration by Morion, and this is what Aftrono.
mers, Chronologers, &c. mean by the Word Time : So that by
Time we are to underſtand a certain Part of Duration meaſured out
to us by the fimple and uniform Motion of ſome Body, that al-
ways goes on at the ſame Rate, ſuch as is the Motion of the Calef-
tial Bodies, but chiefly the Sun and Moon; which have not only
been agreed upon by the common conſent of Mankind in all Ages,
to meaſure Time by, but ſeemed deſigned for this purpoſe, a
mong their manifold Uſes, by the Wiſe Creator of the Univerſe
himielf.
Time thus meaſured out to us, is diſtinguilhed into Years, Months,
Weeks, Days, Hours, and Scruples.
A Scruple is a very ſmall Part of Time, and is either Hörary or
Diurnal,
A Horary Scruple which is alſo called a Prime, is the One Six-
tieth Part of an Hour, it is called a Moment or Minute, and is by
Aſtronomers divided into Sixty Parts called Seconds of a Minute, or
fimply Seconds, and each of theſe again into 60 Equal Parts more
called Thirds of a Minute, °C.
A Diurnal Scruple is the One Sixtiech Part of a Day, and is divi-
ded into Seconds, Thirds, Fourths, &c.
An Hour is a certain determinate Part of the Day, and is either
equal or unequal.
An Equal Hour is the Twenty-fourth Part of a Natural Day, or
of that Space of Time that flows while the Sun goes from any Me-
ridian, till it arrives at the ſame Meridian again.
An Unequal Hour is the Twenty-fourth Part of an Artificial Day,
or of that Space of Time that flows while the Sun is moving from
the Eaſtern Limb of the Horizon to the Weſtern; or the time that
he continues above the Horizon; as is the Twenty-fourth Part of
the
The Elements of Chronology
629
the Artificial Night, or time the Suuntinues below the Horizon,
called a Nocturnal Hour; theſe Hours are likewiſe called Temporary
Hours. becauſe at different Seaſons of the Year they are of different
Lengths : For a Diurnal Hour in the Summer is longer than one in
the Winter, and a Nocturnal Hour is ſhorter ; but in the Equinoc-
tial Day, the Hours in the Day and Night are equal to each other,
and to a Man berween the two Extreams; whence it comes to paſs
that equai Hours are ſometimes called Equino&tial Hours.
Theſe Haus begin at the Riſing and Sening of the Sun, and were
of Uſe among he Jews and Romans, and at this Day among the Turks;
they are alſo called Planetary Hours, becauſe in every Hour the Ar-
cients luppoſed one of the Seven Planets to preſide over the World,
which they took by turns; that the firſt Hour after Sun Riſing on
Sunday, tell to the Sun, the next to Venus, the third to Mercury,
and therelt in Order to the Moon, Saturn, Jupiter and Marsi By
this means on the firit Hour of the next Day the Moon preſided, and
01 dat Account gave the Name to the Day which we call Monday,
and foon; whence it comes to paſs that every other Day of the
Weck took its Name from that Planet that was ſuppoſed to 'go-
vern the first Hour of it.
The Hebrews, and many of the Eaſtern Pecple, as well as the
Romans, abour Three Hundred Years after the Foundation of Rome,
divided their Artificial Day into Four equil Parts, the Firſt of which
began at the Riſing of the Sun, and this Firſt Part continued Three
Hours, and was called the Firſt Hour ; and at the End of it a Signal
was given to the People to let them know that the Firſt Hour was
pilt, and that the Second was begun, which was called the Second
Hour; which alſo continued Three Hours, and then another Sig-
nal was given to inform them that the Second was paſſed and that
the l'hud was begun; and that it was then the middle of the Day,
which was called the Sixth Hour, and this Part continued alſo
Three Hours; which being ended the Fourth and Laſt Part of the
Day begun, which was called the Ninth Hour, becauſe this was
the Ninth Part of the Day, which began at Sun Riſe and ended at
the Serting of the Sun, and this Part of the Day they called the
Veſpers.
And the Church of Rome makes uſe of theſe Hours at this Time
for the performing of Divine Service in, and which has been in
Uſe ever lince the Year 543. The Night alſo was divided into
Four equal Parts, which were called Watches, each Part contain-
Mmmm 2 *
ing
:
630
The Elements of Chronology
S
ing Three unequal Hours; Firſt Watch began at the Setting of
the Sun and continued Three Hours; the Second continued till Mid.
night when the Third begin ; and the Laſt continued till Sun Rife.
And inaſmuch as the Sun in one diurnal Revolution paſſes hy Four
remarkable Points, viz. two in the Horizon, which are the points
of Riſing and Serting, and two in the Meridian, which are Mid-
day and Mid-nighr; tunce it comes to paſs that different People be-
gin their Day at different Times, the Babylonians, the Aſyrians,
with many orker Eaſtern Nations, as well as the Inhabitants at
Majorca, Minorca, and Nuremberg, began their Day at Sun Riſing
The Fews, the Athenians, formerly the Greeks, the Italians, the
Scicilians, the Bohemians, the Auſtrians, the Polanders, do at this
Time begin their Day at the Setting of the Sun, and the Hour af-
ter the Sun is Ser they call the Firſt Hour and count on till they
come to the Twenty-fourth.
The Egyptians with the Ancient Romans began their Day ar Mid-
night,which was followed and made Uſe of by Hipparchus in his A-
stronomical Obſervations, and is ſtill retained in Britain, France, Spain,
and in moſt Places of Europe, and by the Church in the Celebration
of her Feafts, tho the Arabs and the modern Aſtronomers take the
Commencement from Noon, or when the Sun is upon the Meridian,
A Week is a ſucceſſion of Seven Natural Days, to which the
Gentiles gave the Names of the Seven Planets, to the Firſt Day they
gave the Name of the Sun, to the Second the Name of the Moon,
to the Third the Name of Mars, to the Fourth the Name of Mer-
cury, to the Fifth the Name of Jupiter, to the Sixth the Name of
Venus, and to the Seventh the Name of Saturn. But the Hebrews
called every Day of the Week by the Name of Sabbaths, and becauſe
the Day of the Sabbath was their Principal Day, (as our Sunday is
to us) the Day after the Sabbath they called the Firſt Day of the
Sabbath, which is our Sunday; the following Day and which is our
Monday, they called the Second Day of the Sabbath, and ſo on till
the Seventh Day which was the Sabbath it felf, and the Day before
they called the Preparation for the Sabbath ; but ſince the Com. .
mencement of Chriſtianity the Names of theſe Days have been chan-
ged into that of Feriæ, which ſignifyes Feafts Days, becauſe all the
Days of the Week are Feaſt Days for the Chriſtians, unleſs other
ways fet a-lide; and the Firſt Day they called Feria Sunday, or the
Lords Day, in Commemoration of the Reſurrection, the Second
Day or Monday, they called the Second Feria ; the Third Day or
Tueſ:
1
The Elements of Chronology 6:31
S
,
Tueſday, they called the Third Feria, 3 c. till they come to the 705
Day which they called the Sabbath Dry, which fignifies a Day of
Reít, but che People in General at this Day uſe the ſame Names
that were given them by the Gentiles.
A Month is a certain Syſtem of Days confifting of ſomething
more or leſs than Thirty Days, and is either Aſtronomical or Civil,
the Aſtronomical which are likewiſe called Natural, are thoſe that
are governed by the Courſe of the Sun and Moon, and are either So-
lar or Lunar.
A Solar Month is properly that Space of Time that the Sun em-
ploys to run thro any sign of the Zodiac, which is the Twelfth Part
of the whole Eclipric; and of theſe there are two Sorts, the Mean
Solar Month, which is the Time that the Sun requires to move thro
the Twelfth Part of the Zodiac, according to mean Motion, and
which conſiſts of 30 days to hor. 29 min. 6 ſec. 18 thirds so fourths,
and the true Solar Month which is the time that the Sun requires
to move through one of the Signs by his true Motion, and as there
Times are unequal to the Months themſelves are alſo unequal; be-
ing greatelt when the Sun is in the Apogaum, which happens when
the Sun is in about 08 deg. of Cancer, and thorteſt when he
is in his perigeum, which happens whenhe is in about 08
Capricorn.
The Lunar Months are thoſe which are meaſured by the Motion
of the Moon, and of the te there are Three Sorts, the Periodical,
the Synodical, and that of Illumination.
The Periodical is that Space of Time that the Moon employs to
run thro the Ecliptic, or till he returns to the ſame place from
whence he began to move, which according to mean Motion is per-
formed in 27 days 7 hor, 43 min. 7 ſec. but according to his true.
Motion in an Hour more or leſs.
The Synodical Month, is that Space of Time that elapſes between
one Conjunction of the Sun and Moon and the next following, and
is that which agrees with the mean Motion of the Sun and Moon,
and is equal to 29 days 12 hor. 44 min. 3 ſec. 11 tbirds, and is
the Meaſure of all the Lunar Months ; but the True Synodical Monib
is that which is made by the true Conjunction of the Sun and Moon
and its Difference from the Means is Tometimes greater and ſomea
times leſſer than 14 Hours.
deg. of
---
The
632 The Elements of Chronology
The Month of Illumination or Apparition, is the Interval of Time
between the Day that the Moon begins to appear after the Con-
junction, to the Day that ſhe dilappears ; and this Month ordina-
rily conſiſts of Twenty-eight Days, but not exactly, for ſometimes
ſhe is ſeen ſooner or later, according as ſhe is to the Northward or
Southward, or according as her Motion is ſwitter or ſlower, or ac-
cording to the Signs aſcending or deſcending in which ſhe is.
The Civil or Political Months are different from each of theſe, and
couliſts of a certain Number of Days, fewer or more, according to
the Laws and Cuſtoms of the Country in which they are kept.
The Egyptian Months conſiſted of Thirty Days, and at the End of
each Year they added Five Days, which they called Epagomenos
Dies, and the Perfi.ın Months conſiſted of Thirty Days as the E-
syptians did, and they added Five Days at the End of every Year,
which they called a Common Year, and Six Days in the Intercalary
Tears, and theſe Days they called Muſteraka,
A Year is a Syſtem of Months, and is either Aſtronomical or Civil.
The Sydereal Tear is a Space of Time that flows while the Sun
is inoving from a Fixed Star or Planet, till it return to, or over-
take the fame Fixed Star or Planet again ; whence it comes to paſs
that there are Two Sorts of Sydere al Tears, the one fixed the other
moveable ; the fixed Sydereai Year, is that Space of Time that elapſes
betwixt the Conjunction of the Sun and any fixed Star, and the
next Conjun&ion of the Sun with the fame Star, which is always
of the ſame Length, and contains 365 days 6 hor. 9 niin. 14 fec.
The Moveable Sydereal Tear, is that which ariſes from the Con-
jandion of the Sun with any of the Planets, and is contained between
that and the next Conjun&ion ; this Year is unequal, being different
according to the different Planets; this Year with Saturn conſiſts of
378 days 2 hor, 12 min. 13 ſec with Jupiter it is made up of 398
days 2 i hor. 19 min. 9 ſec. with Mars it made up of 779 days 22
hor. 22 min. 40 ſec. Mean Time.
The Tropical Year, is a Space of Time that flows while the Sun
is inoving from any one of the Tropical or Equino&ial points, or
from any fixed point in the Ecliptic, till he returns to the famë point
again; this Year is likewiſe called the Natural Tear, and feems as
if it were deſigned by the God of Nature for to meaſure Time by,
and conſiſts of 365 days 5 hor. 48 min. 57 ſec.
But becauſe while the Sun is moving thro the Ecliptic, the Equi-
nođial points themſelves are found to recede at the Rate of so Se-
conds
+
The Elements of Choronology
633
conds; hence it is that the Sun requires 365 days 6 hor. 9 min. 14
fec. to return to the ſame point in the Heavens that he poſſeſſed the
Year before ; and this is called the Periodical Solar Tear, and theſe
are what are called Solar Tears.
A Lunar Year is a certain Syſtem of Months, and is either Com-
mon or Emboliſmic. A Common Lunar Year conſiſts of Twelve Sy-
nodic Lunations, but the Emboliſmic of Thirteen, and each of theſe
Years are either Mean or True, the Mean Lunar Year, which is
that which governs the Year, and is what people make Uſe of, con-
ſiſts of 354 days 8 hor. 48 min 38 ſec. and the Mean Emboliſmic Tear
of 383 days 21 hor. 32 min. 41 ſec. the Irue Lunar Year differs
from the Mean Lunar Year, and is ſometimes greater and ſometimes
lefſer ; but this is not regarded in common Uſe.
Now inaſınuch as the Common I.unar Tear which conſiſts of Twelve
Synodick Months, or of 12 Lunations, confifts but of 354 days 8 hor
48 min. 38 fec. whereas the Solar Year conſiſts of 365 days s hor.
48 min. 57 ſec, it is plain that the Solar Year exceeds the Lunar Tear
by to days 21 hor. oo min. 19 ſec. and conſequently in the Space
of about Thirty-three Years, the Beginning of the Lunar Year will
have moved thro all the variety of Seaſons, whence it is called the
Moveable Lunar Year; and this Form of the Year is at this Time.
obſerved by the Turks and Arabs. And,
Becauſe that in the Space of Three Years, this Difference between
the Solar and Lunar Tear amounted to 32 days 15 hor. oo min 57 ſec.
therefore to keep the Lunar Months in the ſame Seaſons and Time of
the Solar Tear, as near as poſſible, they added a whole Month every
Third Year to the Lunar Year, and made it conſiſt of Thirteen Months,
by which Means the Beginning of the Lunar Year was kept nearly
in the fame Seaſon, and this Year they called the Emboliſmic Tear,
and the Month it ſelf the Embolimaan or Intercalary Month, and theſe
were called the Fixed Lunar Tears, and were obſerved by the Greeks
and Romans till the Time of Julius Cæfar.
The Egyptians made Uſe of the Solar Tears, and made it conliſt
of 365 Days juſt, and this being leſs than the True Solar Tear by fix
Hours nearly, hence it comes to paſs tha: by looſing a Day in every
Four Years, in Four times 365 Years, that is in 1400 Years, (which
was called the Great Canicular Tear or Sothiacal Period) the Begin-
ing of the Year had moved thro all the Seaſons, this therefore was
called the Moveable Solar Tear; and is the ſame with the Year of Na-
bonaffar, of great Eſteem among the ancient Aftronomers, made Ure
of
634
The Elements of Chronology
of by Prolomy, Copernicus and others and by the ancient Perſians, for
the Space of 477 Years ; till by the Command of their Emperor
Abb. Arſalam, about the Year 1079, they began to add a Day at the
End of every Fourth or Fifth Year, and in 648 Years to include
236677 Days, ſo that their Mean Year conſiſted of 365 days 5 hor-
48 min 53 ſec
But Julius Cafar lorg before this time, finding the Incon-
venicncies that aroſe from this Method of Computation, and con-
fidering that it was neceſſary the Civil Tear ſhould always com-
mence on the ſame Day, and at the Beginning of that Day, which
it would not do if the Six Hours were added to every Year; and
being at that time Pontifex Maximus, did by the Advice and
Alliſtance of Sfiogenes the Mathematician, ſet about correcting the
Year, and ordered that every Fourth Year ſhould have an Intercalary
Day, which therefore ſhould conſiſt of 366 Days, and that this ad-
ded Day ſhould be put in the Month of February, and becauſe in
the Common Year the 24th of February in the Roman Way of Rec-
koning, was the Sixth of the Kalends of March, he ordered that for
that Year there ſhould be Two 6ths, or that the 6th of the Kalends
of March ſhould be twice Reckoned ; upon which Account the Year
was called Billextile, is made Uſe of by us at this very Day, and is
called the Leap Year.
But becauſe the true Length of the Year conſiſts but of 365 days
5 hor. 49 min. nearly, hence it comes to paſs that after this Way of
Reckoning, at the End of every Four Years the Civil Tear will be-
gin 44 Minutes looner than it did before, and conſequently by this
Way of counting, in 131 Years it will anticipate one whole Day;
ſo that if at any time the Equinoctial happened upon any one Day of
the Year, it will happen after the Term of 131 Years a Day foon-
er, and this was the Reaſon why Pope Gregory the 13th of that
Name, in the Year 1582, being willing to celebrate Eaſter according
to the Original Inſtitution, and to keep up to the Letter of the Order
of the Nicene Council, which was held in the Year 325, ſet himſelf
about reforming the Year, and finding that between the Time of the
Nicene Council and his Time, that the Equinox had anticipated Ten
whole Days, ordered that theſe Ten Days ſhould be taken out of the
Kalendar that year, and that the 11th Day of March ſhould be recko-
ned as the 21ſt, and to prevent the Seaſons of the Year from going
backwards as they did before, he ordained that every Hundredth
Year which in the Julian Form was to be a Biffextile, Aould be a
7
com.
The Elements of Chronology
635
common Year, and conſiſt only of 365 Days; but becauſe that
was too much, every Four Hundredth Year was to remain Biſextile ;
This new Form of the Year being Eſtabliſhed by the Authority of
Pope Gregory himſelf, is called by the Name of the Gregorian Tear;
and is received in all thoſe Countries where the Pope's Authority is
acknowledged, and in the Year 1700, it was received in ſeveral Pla-
ces where the Reformed Religion is profeflia.
Beſides the Political Diviſion of the Month into Weeks, the Ro-
mans according to the Inſtitution of Romulus, the Firſt Founder of
their State, divided their Months after a peculiar manner, and that
was this, the Firſt Day of every Month they called the Kalends ; be-
cauſe on that Day the Prieſt, whoſe Buſineſs it was to obſerve the
Moons, according to which they reckoned their Month, gave No-
tice on theſe Days to the Preſident over the Sacrifices, and he called
the People together, and declared unto them how they were to rec-
kon the remaining Days of the Month, and which were the proper
Days ſet a-ſide for religious Uſes ; and the Day immediately prece.
ding this they called the Pridie Kalende ; that immediately preceding
this they called the Third of the Kalends ; the next immediately go-
ing before they called the the Fourth of the Kalends, ác. till they
came to the ides of the preceding Month; the Day immediately
ſucceeding the Kalends was called the 4th or the 6th of the Nones,
according as the Months it happened in, for in the Four full Months of
Romulus, viz. March, May, July and O&tober ; the 7th Day from thence
was called the Nones, and in the reſt of the Months which were called
Hollow Mon:hs, the 5th from that Day was the Nones ; and the 8th
Day from the Nones, which was the 15th Day in the full Months and
the 13th in the reſt, was called the Ides ; and theſe as well the Kalends
were all Numbred in an inverted Order ; and the next Day before
the Nones or Ides was called Pridie Nona or Pridie Idus ; and that
which immediately preceded, was called the 2d Day, &c. according
to the following Lines.
Prima dies menfis cujuſq eft dieta Kalenda,
Sex Nonas Majus, Oétober, Julius et Mars,
Quatuor at reliqui : dabit Idus quilibet Oto
Inde dies reliquos omnes dic elle Kalendas.
Quas retro Numerans dices a menſe ſequente.
And this way of Reckoning was followed by all the Latin Church-
es and is made Uſe of by the Church of England at this Day, tho
for Civil Uſes we as well as the reſt of Europe, reckon our Days ac-
cording to the Place they obtain in the Month
Nnnn
From
636
The Elements of Chronology
From this Eutom of calling the Firſt Day of every Month the
Kalends of thas Moneh, we have been led to call thoſe Tables Ka-
lendars, in which are ſet down all the Days of the Year in a regular
Diſpoſition according to their Months, each. Day of every week be
ing diſtinguiſhed from another by one of the Firſt Seven Letters of
the Alphabet, viz, A, B, C, D, E, F, G, the firſt of which A
is placed againſt the iſt of January, the ſecond B is placed againſt
the za Day of January, the third Letter C againſt the 3d Day of
January, and ſo on to the ſeventh Letter G which is placed againſt
the 7th Day of January; after which the Letter A is placed againſt
the 8th, the Letter B againſt the 9th, the Letter C against the roth,
&c. to the End of the Year; whence it comes to paſs that whatſoe-
ver Letter liappens to be placed againſt any Day of any Week, the
fame Letter will point out that Day throughout the whole Year
Thus, if the 1ſt Day of January, againſt which the Letrer A is pla-
ced be a Sunday, all the Days in the Kalendar that have an A put to
them will be Sundays; in like manner if the 3d of January againſt
which the Letter C is placed, happen to be a Sunday, the Letter
will point out all the Sundays throughout that Year; and this Let-
ter whichſoever it is, is called the Dominical or Sunday: Letter for
that Year; and for the fame Reaſon whatever Letter is joyned
to the firſt Monday. in January, the ſame as often asic is repeated in
the Kalendar, ſhews all the Mondays in that Year.
But becauſe the common Year conſiſts of 365 Days, which is e-
qual io 52 times 7 and one over, hence it is that the Letter that
ſtands againſt the firſt Day of the Year, will ſtand againſt the laſt
Day; and if the Year begin on a Sunday it will end on a Sunday,
and the firſt Day of the next Year will be a Monday; and the Sunday
will fall on the 7th Day of January, to which is annexed the Letter
G, which therefore will be the Sunday Letter for that Year; and ſince
the Year began on a Monday it will alſo end: on a Monday, and the ſuc-
ceeding Year will begin on a Tueſday, and the ift Sunday of the Year
will fall on the 6th of January, againſt which Day is put the Letter
F, which is the Sunday Letter for that Year; and for the fame Realon
the Sunday Letter for the next Year will be E, for the Year after that
D, and thus the Sunday Letter will continually change, in a ſucceſſive
but retrograde Order, till it has gone thro all the Series of Letters,
after which it begins again of a new,
If the Year conſiſted of 365 Days exa&ly, after a Period of Se-
Fen Years, the fame Day of each Month would fall on the fame Day
of
The Elements of Chronology
637
of the Week, but becauſe of.the Intercalary Day, every fourth Year
conſiſts of 366 Days, which is equal to Seven Wecks and Two
Days over ; hence it is that if that ¥ear begin with a Sunday it will
end on a Monday, and the next Year will begin on a Tueſday, and
the firſt Sunday of that year will fall on the oth of January, to
which is annexed the Letter F, which will be the Dominical Letter
for the Year following the Leap-Tear, whoſe Dominical Létter was A;
and inaſmuch as the Leap-Tear returns every 4th Year, the Series of
Dominical Letters ſucceeding each other is interrupted, and does not
return in Order till after Four times Seven or Twenty-eight Years:
And hence ariſeth the Cycle of Twenty-eight Years, which is the So-
lar Cycle, which being compleated, the Days of the Month return
in the ſame Order to the ſame Day of the Week.
And becauſe in every 4th Year the Intercalary Day is placed be-
tween the ažå and 24th of February, making thereby two 240hs of
February, hence it is that the 24th and 25th of February according to
our way of counting, are eſteemed as one and the fame Day, and
have the ſame Letter annexed to them, whence the Order of the Sunday
Letter at that time is interrupted, and the ſucceeding Letter takes
Place ; thus if the Sunday Letter be A, the 24th of February will fall
on a Tueſday and the 25th on a Wedneſday; and inaſmuch as both
theſe Days are marked in the Kalendar with the Letter C, the follow-
ing Day which is Thurſday will be marked with the Letter D, the
next after that which is Friday with the Letter E, the Saturday with
the Letter F, and the ſucceeding Sunday which before was marked
with the Letter A will now have the Letter G annexed to it;
which Letter will continue to point out all the Sundays, and conſe-
quently will be the Dominical Letter the remaining part of the Year;
and heuce.it is that che Leap Tear has two Dominical Letters belong-
ing to it, the firſt of which ſerves from the beginning of the Year till
the 24th or 25th of February, and then the ſecond takes place and
ferves for all the reft of the Year.
As the firſt Year of the Solar Cycle was piaced in a Leap Year, hence
the Dominical Letters anſwering to it are G and F, whence the Sun-
day Letter for the ſecond Year of the Cycle will be E, for the third
D, for the fourth C, and becauſe the fifth Year of the Cycle is a
Biſextile, the Dominical Letters belonging to it will be B and A,
whence the Dominical Letter for the 6th Year will be G, and for the
reſt of the Years of the Gyde as in the following Table
.
Nnnn 2*
Тbe
638
The Elements of Chronology
The TABLE.
IG F 51B A 9 D C|131F E 171A G 211C B125 E D
E
6 G
B 14 D 18 F
IO
2
22 A
261 C
Il
B
c
A | |
F
II A
7
3/ D
151 C
E 23
1 l
19
G
27) B
41 C
8
E
G
I 2
161 B
20 D
D 241 F
28 A
Hence the Year of the Cycle being known, the Dominical Letter
is eaſily had,
And if the Number of Years elapſed between the Commence-
ment of any Cycle and any other given Time be divided by 19, the
Quotient will ſhew the Number of Cycles paft, and the Remair, der
the Current Year of the Cycie.
If the Cycle had commenced with the Chriſtian Era, that is if
the Letter F had been placed in the Ecclefiaftical Kalendar againſt
the il Day of January, then to have found the Dominical Letter
we had had no more to do but to have divided the Year by 28,
and the Remainder would have given us the Dominical Letter for
that Year ; but becauſe the Firft Year of Chriſt happened on the 101b
Year of the Cycle, as it had been before ſettled; therefore
To find the Year of the Cycle of the Sun for any Year of the Chris
Stian Era.
To the given Year we muſt add 9 and divide that Sum by 28,
then will the Quotient Mew the Number of Cycles that have revol.
ved ſince the firft Year before Chriſt, and the Remainder if there be
any, will ſhew the current Year of the Cycle ; but if there be no
Remainder it ſhews that that Year is the laſt, or the 28th Year of
the current Solar Cycle: So that if it be required to find what
Year of the Solar Cycle the Year 1723 is, and what is the Dominical
Letter.
To the given Year 1723 I add 9, and divide the Sum 1732 by 28,
the Quotient 61 ſhews me that there has elapfed fince the Year be
fore Chriſt 61 Cycles, and the Remainder 24 ſhews me that it is
the 24th Year of the current Cycle, whence the Dominical Letter
will be found to be F.
But the Dominical Letter may be directly found without the
help of the Cyclear Table.
It
The Elements of Chronology.
639
It has been already ſhewn that if the Year conſiſted of 365 Days
exaêtly, after a period of 7 Years the Dominical Letters would return
in the ſame Order, and each Day of the Month would fall on the
ſame Day of the Week, and conſequently if the Number of Years
poſt from any fixed Time were divided by 7, the Remainder would
ihew the Dominical Letter from the Letter in the fixed Year, in a
retrograde Order.
But inaſmuch as in every 41b Year there is an additional Day,
therefore to find the Number of Periods of the Dominical Letter
for any fixed certain Nurnber of Years we muſt add to the Num-
ber of Years patt, (one fourth part of the fam) reje&ting the Re:
mainder if any there be, fince the additional Year is made at the
End of the fourth Year, and divide the Sumn by 7, then the Quotient
will ſhew the Number of Periods, and the Remainder the Surplus
Thus for Example, luppoſe it were required to find the Number of
Periods of the Dominical Letter in 1723 Years, to the Number 1723
I add its 4th 430, rejecting the Remainder3, the Sum 2153 I divide
by 7, and the Quotient 307 (hews me that there has been ſo many
Revolutions, and the Remainder 4 the Surplus.
Now if the Solar Cycle had been made to commence at the ſame
Time with the Chriftian Era, then inaſmuch as there is 1723 Yeais
paft ſince the Beginning of the fame Era, the Surplus 4 would have
given us the Dominical Letter in a retrograde Order' from A the
firſt, and conſequently the Letter would have been C; but because
the Cbriftian Era commenced in the job Year of the Solar Cycle
when the Dominical Letter was B, therefore if from © (excluſive)
the Dominical Letter for the preceding Year we count 4 the Surplus
in a retrograde Order, viz. B, A, G, to F or from C, its Comple-
ment to 7, in a direct Order, as D, E, F, we ſhall have F for the
Dominical Letter for the Year 1723 propoſed; wherefore to find
the Dominical Letter for any other Year ſince Chrift.
To the Year add its fourth, and divide the Sum by 7, if nothing
remain, the Dominical Letter is C, buc if there be any Remainder
count. it backwards from C, or its Difference from 7 forwards from
the ſame Letter, and you will have the Dominical Letter; but becauſe
the Letter C is the fourth in the retrograde Order from G, the last of
the Dominical Letters, therefore if to the Year you add its 4th and
4, and divide that Sum by 7, if nothing remain the Dominical Let-
ter is G, but if there be any Remainder it (hews the Letter in a reg
trograde Order from 6, and being fubftra&ted from 7 will give the
Qra.
640 The Elements of Chronology
Order or Place of the Letter from the Seat of A; which Rule is ex.
preſs’d by Street in his memorial Verſes, thus ;
Divide the rear, its Fourth and Fouý by Seven,
Whai's left fubtract from Seven, the Letter'sgiven.
Reckonning for i A, for 2 B, for 3 C, &C.
So that if by this Rule it were required to find the Dominical
Letter for the Year 1723.
To 1723, add its 415 430, and 4, the Sum 2157, being divided
bb 7 will give 308 for the Quotient, and i for the Remainder, which
yeing fubftracted from 7 w.ll leave 6 for the Seat of the Dominical
Letter, from A incluſive, and conſequently the Letter will be F.
But becauſe the Dominical Letter in the ſtcond Year of the Chri-
ſtian Era was A, if from the Year we fubftract 1, and to the Re.
mainder add its 4th, and divide the Sum by 7, the Remainder if
any there be, taken from 9, will give the Dominical Letter, count-
ing A for 1, B for 2, c. but if there be no Remainder A witl be
the Dominical Letter.
Thus if from 1723 I take 1, it will give 1722, and if to this I add
its 4th 430, the Sum 2152 divided by 7, will leave 3 for a Re-
inainder, which taken from 9 will leave 6 for the Alphabetical Or.
der of the Letter which therefore must be F.
The Dominical Letter being known it will be eaſy to find upon what
Day of the Week that Year begins, for if the Dominical Letter be
A the Year begins on a Sunday, if it be B it begins the Day before
Sunday or on a Saturday, if it be Con a Friday, &c, and conſequent-
ly in the preſent Cafe when the Dominical Letter is Fit must begin
on a Tueſday.
Or if the Remainder after the Diviſion of the Year its 41b and 4
be reckonned from Monday, it will give the ſame Day; thus in
the former Example, where the Remainder was 1, it ſhews that the
Year begins the next fucceeding Day or Tueſday, and ſo for the
relt; and by proceeding after this manner we may eaſily find upon
what Day of the Week the firſt Day of every Month will happen,
and conſequently upon what Day in the Week any Day in the Year
will fall.
As the Periodical returns of the Sun and Moon are found to be
conſtantly the ſame, the Moon moving with near thirteen times the
velocity of the Sun ; hence it müft come to paſs that after a cer-
tain
i
The Elements of Chronology 641
tain Number of Revolutions they muſt neceffarily meet in the famo
point of the Heavens, and this was firſt pronounced by Meton tho
Athenian, to happen once in Nineteen Years juſt, and is therefore
called the Meronic Cycle, and was of great Uſe among the Ancients,
for adjuſting of the times of the New and Full Moons.
For inaſmuch as according to this Period the New and Full
Moons happen on the fame Day, and at the ſame time of that Day,
after a Space of 19 Years, if the time of the New and Full Moons
be had for any Term of 19 Years, the times of the New and Full
Moons for any Year paſt or to come is given alſo.
And therefore at the Time of the Council of Nice; when they were
about ſettling the Time for the Celebration of Eaſter, this Cycle was
had in ſo great Eſteem that they founded their Eccleſiaſtical Compu-
tation upon it, and finding by Obſervation that in the firſt Year of the
Cycle the New Moons happened upon the 23d of January, the 21ſt
of February, the 23d of March, &c. againſt thoſe Days in the first
Column of the Kalendar they put the Number 1, and finding that
in the 2d Year of the Cycle, the New Moons happened upon the
1:2th of January, the 10th of February, the 12th of March, &c. a-
gainſt theſe Days in the Kalendar they writ the Number 2, and dies
gainſt the 1st of January, the 3d of February, the ift of March, &c.
on which Days they obferved the New Moons to be celebrated, in
the 3d Year of the Cycle they annexed the Number 3; and after
the ſame manner to every Day in the Year during the whole Cycle
on which they found the New Moon to happen, they inſcribed the
Number anſwering to the current Year of the Cycle; whence it
comes to paſs chat by knowing in what Year of the Cycle any Year
happened, the Days on which the New Moons happened during
that whole Year, were had by Inſpection, for by finding the Year of
the Lunar Cycle in the firſt Column of every Month in the Kalen-
dar, againſt that Number we ſhall have the Day of that Month on
which the Change of the Moon happens, and for this excellent Uſe
of theſe Numbers they were wont to be ſet in Golden Letters in the
Kalendar, and from them the Year of the Cycle for any Year was
called the Golden Number of that Year, but in our Liturgy it is cal-
led the Prime.
And in order to find the Year of the Cycle or the Golden Number,
divide the Year of the Julian Period (which will be explained here-
atrer) by 19, and the Remainder will thew you what Year of the
Cycle the Current Year is, but becauſe the firſt Year of the Chriſtian
Era happened in the ſecond Year of the Lunar Cycle ; therefore if
to
642
The Elements of Chronology
to the Year from Chriſt be added 1, and that Sum be divided by 19,
the Quotient will ſhew how many Cycles have revolved ſince the
Cominencement of che Chriſtian Era, and the Remainder will hew
what Year.of the Cycle the given Year is; hut it nothing remain
then the given Year is the latt Year of the Cycle, and the Prime or
Golden Number is 19.
And her.ce we are taught a ready way of finding on what Day of
aly Month, in any Year the New Moon will happen, for by finding
the Prime or Golden Number for that Year, and ſearching for that
Number in the firft Column of the given Monthin the Kalendar,
Fight againſt will ltand the Day of the Month required.
As ſuppoſe it were required to find on what Day of March in
the Year 1723, the New Moon will be celebrated.
To the given Year 1723 I add 1, and divide the Sum 1724 by
19, the Remainder 14 Thews me that it is the 1415 Year of the Lu-
nar Cycle, and that therefore the Prime or Golden Number is 14.
Entering therefore the firſt Column of the Month of March in the
Kalendar, with the Number 14, I find it to ſtand againſt the 30th
Day, which ſhews that upon that Day the Moon changes: After the
1eme minner by ſeeking for the ſame Number 14 in the firſt Co-
luinn of the Months of January, February, April
, &c. I find that ac-
cording to this Computation the New Moons happen the 30th of
January, the 28th of February, the 28th of April, &c. and hence
the Day of the Month being known the Age of the Moon on that
Day was known alſo.
Hence we are taught a very ready wayof finding in what Month, and
on what Day of the Month Eaſter Day falls on in any Year, according
to the Decree of the Nicene Council, which was firſt that Eaſter
Day ſhould be celebrated after the Vernal Equinox, which then hap-
pened upon the 214 Day of March, ficondly that it ſhould be kept
after the 14th Day of that Moon, which ſhould happen after the 21ſt
of March, and thirdly that the Sunday immediately following that
141h Day ſhou'd be Eaſter Sunday; ſo that if it were required to
find in what Month and on what Day of the Month Efter Day
would happen in the Year 1723 of the Chriſtian Era.
I ſearch for the Golden Number 14 in the firſt Column of the
Month of March in the Kalendar, and find that it ſtands againſt
the 30th Day of the Month, counting therefore from thence 14 Days
inclufive, I find that the Pafchal Full Moon happened upon the 12th
Day of April, againſt which ſtands the Letter D, and becauſe the
Sun
The Elements of Chronology
643
ܪ
day Letter for the fame Year is F, it ſhews that the next Day or
the 14th of April is Eaſter Day. Again.
Suppoſe it were required to find on what Day of the Year Eaſter
Day falls in the Year 1731, when the Golden Number is 3 and
the Dominical Letter C.
Entering the Kalendar in the Month of March, I find the Num-
bor 3 to ſtand againſt the 1ſt Day, from which reckonning 14 Days
I find that the next Full Moon happens on the 14th Day of the
fime Month, which becauſe it precedes the 21ſt Day, I ſearch
again in the fame Month, and find that the Number 3 ftands againſt
the 3 ft Day of the ſame Month, which is therefore the Day of the
Commencement of the Paſchal New Moon, from whence counting
14 Days, I find that the Paſchal Full Moon happens on the 13th
Day of April, againſt which ſtands the Letter E, but becaufe the
Dominical Letter is C, therefore Eaſter Day falls upon the 18th Day
of the fame Month; and this Rule is ſtill kept by our Church,
which ordains that Eafter Day is always the firit Sunday after the
firſt Full Moon which happens next after the 21ſt Day of March ;
and if the Full Moon happens upon a Sunday, Eaſter Day is the
Sunday after : where it is to be underſtood,
1. That in Leap-year ioftead of the 21ſt of March we muſt uſe
March the 20th.
2. That the Full Moon meant is the 14th Day of the Moon, aco
cording to the Kalender in the Common Prayer Book, counting that
Day for the firſt of the Moon which hath the Golden Number of
the Year collateral to it in the firſt Column of the ſaid Kalendar.
And
3. That theſe words (next after the 21ſt of March) are meant
incluſively, as if it had been ſaid (next after the commencement of
March the 21ſt) ſo that if the Full Moon happens on March the 21ſt,
the ſame muſt be the Paſchal Full Moon; whence it comes to paſs that
whereas the Pafchal Full Moon can never happen before the 21ſt of
March and after the 18th of April incluſively, Eaſter Day can never
happen ſooner than the 22d of March, nor later than the 25th of
April inclufively.
But becauſe the Lunar Synodical Month conſiſts but of 29 days
12 hor. 44 min. 03 ſec. therefore 235 Lunations, which are ce
Number of Lunations that happen in 19 Julian Tears, are performed
in 6939 days 16 hor. 31 min. 45 ſec. whereas 19 Fulian Tears are
equal to 6939 days and 18 hours, ſo that 235 Lonations are lefs
0000
than
644
The Elements of Chronology
than 19 Fulian Tears by s hor. 28 min. 15 ſec. and conſequently, the
New Moons afier 19 Julian Tears will not return to the lane Hour
of the Day, but will buppen i h. 28 m. 15 1. ſooner which in the
Spice of 3 10 Years will amount to one intire Day, whence it comes
to paſs that in the Space of 310 Years, the New Moons will anti-
cipite one Day; and conſequently fince the time of the Council
of Nice which was in the Year 325, that is during the Space of
1398 Years they have anticipated above tour Days and an haif, ſo
that if in the Year 325 when the Kalendar was firſt composed, the
New Moon happened upon the ift of March, which hy the
Church Kalendar ir appears to have done, in the Year 1723 it will
happen be: ween the 241h and 25th of February, and in the firſt
Year of the Chriſtian Era it was celebrated upon the ad of Mirch.
A d inaſmuch as the Compilers of the Kalendar did not ſo much
conform themſelves to the Aſtronomical Methods of computing the
New Moons as they did to the Cuſtom of the Jews, who follow-
ed the vulgar way of Reckoning, and counted the Age of the Moon
not from the time of its Conjun&tion with the Sun, but from the
firſt Day of its Appearance, which could never happen till the next
Day after the Change, hence the times ſet down in the Kalendar
are later than the tree times by 5 Days and an half at preſent,
and will continue to fall later a Day in 310 Years, ſo that in the
Year 1875 they will happen 6 Days later than the true and
Aftronomical New Moon, and are therefore for Diſtinction fake
called ihe Ecclefiaftical New Moons.
Wherefore inalmuch as the times of the New Moons puined out
by the Kalendar, were not the times when the New Moons bap-
pered, but the times when the Moon firſt appeard, which was the
firſt Day after theChange, or when ſhe was one Day old ; therefore if
from the time of the Ecclefiaftical New Moon be taken 5 Days and
an half, it will ſhew the time of the Aſtronomical New Moon ac
this time, thus the time of the Aſtronomical New Moon in March
1723, will be found by reckonings Days and an half backwards from
the 30th Day of March, which is the time of the Ecclefiaftical New
Moon in the Kalendar, to happen between the 241h and 2515 Day,
which agrees as near as ſuch a manner of computing can well
do, for the true Aſtronomical New Moon happens on the 241h Day
at 20 hor. 48 min. nearly.
'Twas in Conformity to the Cuſtom of the Jews, who were
commanded by God himſelt (as we may read in the 12th of Exodus
and
The Elements of Chronology
645
and at the 6th Verfe, and again in the 23d. of Leviticus and at the sth
Verſe) to celebrate the Paſſover in the firft Month, and on the 1415
Day of that Month in the Even, that our Primitive Fathers ordered
that the 14th Day of the Moon from the Kalender New Moon,
which imniediately follows after the 21st of March, ar which time the
Vernal Equinox happened apon that Day ſhould be decmed the Paſchal
Full Moon, and that the Sunday after (becauſe our Saviour Roſe
upch
the Day after the Jewiſh Papover) ſhould be Eaſter Day, and
'tis upon this Account that our Rubric has appointed it upon the
firſt Sunday after the firſt Full Moon (becauſe the 14th Day of the
Jewiſh as well as the Ecclefiaftical Moon, was the 15th Day of the
Aſtronomical Moon, and conſequently the Day of the Full Moon it
ſelf) immediately following the 2117 Day of March, upon which
Day at that time the Vernal Equinox happened ; wnence it appears
that the true time of celebrating Easter, according to the Origi-
nal Inſtitution of the Feaſt of the Pallover, as well as according to
the intent of the Council of Nice, was to be the firſt Sunday af-
ter the firſt Full Moon immediately following the Vernal Equinox, or
when the Sun entered into the firſt point of Aries ; and this was the
Principal View (that is the Celebration of Eaſter according to the
intent and meaning of the Nicene Council) that Pope Gregory had,
when he reformed the Kalendar, tho he mified his Aim, and is
what the Proteſtant Body in Germany, happily acconipliſhed in the
Year 1700, when they conformed themſelves to the Gregorian Year,
and 'tis for want of this Reformaliorrin England, that in this preſent
Year 1723, we celebrate Enſter on the 14th of April, when in reali-
ty, and according to the Original Inſtitution, it ought to be kept on
the 17th of March Old Stile, upon which Day it is obſerved in all
the Chriſtian Churches where the New Stile is received.
It has been already ſhewn (in Page the 633d) that the Solar Year
exceeds the Lunar Year by 10 days 2 1 hor. oo min. 19 ſec. or is
Days nearly, theſe therefore are called the Intercalary Days or Epaits,
and are of very great Uſe in adjuſting the Length of the Lunar Year
to the Solar, and in finding how long time has elapſed from the pre-
ceding Conjunction in any Lunation, at any time propoſed, which
is called the Age of the Moon at that time.
For inaſınuch as the Lunar Year is leſs than the Solar Year by II
Days in round Numbers, if the Sun and Moon happen to be in Conjun-
aion on the laſtDay of any Year, at the End of the next Year the Moon
will be paſt the Conjunction by It Days nearly; at the End of Two
00002
Years
646
The Elements of Chronology
و
Years it will want 22 Days; but becauſe at the End of Three Years
it amounts to 33 Days, it fhews that in that Year there has been
13 Conjun&ions or New Moons and Three Days over ; whence
it appears that at the End of the fourth Year, the
Moon will be paſt
the Conjunction 14 Days; at the End of the 5th Year 25 Days, &c.
conſtantly increaſing by. Eleven Days per Annum, till atter the End.
of Nineteen Years it becomes the ſame as it was Nineteen Years be-
fore ; and the fame Series of Epacts return, and the Day of the Lu-
nation is the ſame as formerly.
Hence it appears that the Epact for any Year being known, the
Epait for any Year paſt or to come may be found readily, for if the
Lunar Cycle happen to commence from that Year in which the Con-
junction of the Luminaries, or the time of the New Moons hap-
pened on the laſt Day, or on the 31ſt Day of December, of the prece-
ding Year, then the Epa£t for the firſt Year of the Cycle will be 11,
for the ſecond Year 22, for the third Year 3, for the fourth Year 14,
for the fifth Year 25; 6c. conſtantly increaſing by 11; hence it
is manifeft, that to find the Epait for any Year of the Cycle, we
have no more to do but to multiply the Year of the Cycle (which is
called the Golden Number for the Current Year of Chriſt) by 11, and
divide the Product by 30, and the Quotient if there be any, will
fhew how many Embolimean or Intercalary Months has happened ſince
the firſt ear of the Cycle, and the Remainder the Epait for that
Year, or Number of Days between the laſt Day of the former Year,
and the immediate preceding Conjun&ion or New Moon: Thus it
it be required to find the Epact for the given Year 1723, or Number
of Days elapſed between the 31ſt Day of December 1722, and the
Day of the immediate preceding New Moon, b:cauſe the given Year
1723 is the 14th Year of the Cycle, that is the Golden Number for
the 1723d Year of the Chriſtirn Era, I multiply the Golden Number
14 by 11, and becauſe the Product 154 exceeds 30, I divide it by
30, and the Quotient s, Thews me that there has been 5 Intercalary
Lunar Months ſince the Commencement of the Cycle which was in
the Year 1709, and the Remainder 4, ſhews me that on the 31ſt
Day of December 1722, the Moon was 4 Days paſt the Conjun-
&ion, and that therefore the Epact for the preſent Year 1723 is 4.
And hence we are taught how to find on what Day of any Month
the New. Noon will happen.
For inaſmuch as the Synodical Month or time between any two
immediate Conjun&ions of the Luminaries is equal to 29 days 12
min.
The Elements of Chronology 647
min, or 29 Days and an half, and that the Epal for any Yearſhews
us how many Days are elapſed between the laſt Day of the prece-
ding Year and the next preceding Conjun&tion to that Day, if from
29 Days and a half be ſubftrated the Epait for any Year, che Re-
mainder will ſhew us on what Day in January the next New Moon
will happen ; and becauſe the Month of January confifts of 3.1
Days, which is more than the Synodical Month by One Day and an
halt, if to the Epa& for the Year, we add 1 Day and a half, and
fubftract that from 29 Days and an half, we ſhall have the Day
of the New Moon in February, and becauſe in common Years the
Months of January and February do make up juſt two Synodical
Months or 59 Days, the Epall for the Year will ſhew how many
Days it is paft the New Moon on the laſt Day of Eebruary, and be
ing therefore taken from 29 Days and a half, will give the Day
of the New Moon in March ; and thus by increaſing the Epal con-
tinually, by the Exceſs of the Month above a Synodical Month, and
lubftra&ting that Sum from the Length of the Synodical Month, we
ſhall have the Day of the Change in any Month, and by adding
14. Days and three quarters to the time of the New Moon, we ſhati
have the Day when the Full Moon will be celebrated, but to
to avoid Fractions, in the common Method of Calculation they al.
low 30 Days to the Synodical Month, and for February they allow
2, for March 1, for April 2, for May, 3, for June 4, for Fulys,
for August 6, for September 8, for October 8, for November 10, and
December 10, and makes the Epad to take place on the iſt of Jane
U iry; and by proceeding after this manner we ſhall find the New
Moon in January 1723, to happen on the 26th Day in the Month
o February in the fame Year, on the 241h Day in the Month of
of March, on the 25th in the Month of April, on the 24th in
May, on the 23d in June, on the 22d in July, on the 214 in Aug
gust, on the acth in September, on the 18th in O&tober, on the 1815
in November, and on the 16th of December, which ſeldom differ
above half a Day from the true Conjun&tion, and ſometimes agree
pretty well with it, and are therefore exact enough for common
Uſe: And if to each of theſe times thus found, we add is Daysaxi
we ſhall have the times of the ſucceeding Full Moons.
And inaſmuch as the Epad for the Year as has been before taken
Norice of, thews us how many Days has happened between the latt:
Day of the preceding Year and the immediate preceding New Moon,
and is therefore equal to the Age of the Moon, at that time, if it
be
$
648 The Elements of Chronology
be required to find the Age of the Moon any Day in January we
hare no more to do but to add the Day of the Month to the Epact,
and the Sum if it be leſs than 30, will ſhew the Moons Age ; but if
the Sum be more than 30 the Exceſs above 30 will give the Age of
the Moon ; thus in the Year : 723, where the Epact is 4, the Moon
will be found to be 24 Days old on the 20th of January, but on the
28th of the fame Month it will be found to be 2 Days old, ſince 4
and 28 makes 32, which exceeds the Synodical Month by 2, and
if to the ſame Epact we add 2 for the Month of February, í for the
Month of March, 2 for April, &c. we ſhall have the Age of the
Moon at the Beginning of every Month, and if to Thar Sum there-
fore be added the Day of the Month when the Age is required, if
the Sewn be leſs than 30 it will be the Moons Age, but if it more
than 30, then the Exceſs above 30 will give the Age required; as is
abundantly manifeſt from what has been ſaid
But becauſe the New Moons do not return at the End of Nine-
tecu Years, exa&ly at the ſame time of the Day, but i hor. and 28
min nearly ſooner than they did before, and that of Conſequence in
310 Years they will anticipate one whole Day ; hence it comes to
paſs that the Annual Epact it ſelfvaries, and that in the Space of
310 Days it will be increaſed one Day, ſo that if I was to find
the Age of the Moon on any Day of the Year 325, to which the
Ecclefiaftical Kalendar is fitted, when according to the common Me-
thod of Complitation the Epact is 3, inſtead of ſaying add .to the
Epact the Day of the Month, and for January o, for February 2,
for March 1, for April 2, 6c. we muſt ſay, add to the Epact the
Day of the Month, and for January ſubſtract 4, (becauſe of the An-
ticipation which is four Days ſince that time) for February 2, for
March 3, for April 2, for May 1, for June o, and for July add 1,
c. and if we were to find the Age of the Moon in the firſt Year of
Chriſt, which was 324 Years before that, we muſt ſay, ſubſtract 5
for January, 3 for February, &c or which is the ſame thing we
mult find the proper Epact for that Year by ſubſtracting 4 from the
Epaxt found after the common Method, which reduced it to 28
and 5 from 22 the Epact, for the firſt Year Current of Chriſt, which
will give us 17 for the Epaxt for that year, and then we muſt pro-
ceed as before. In like manner, if we were to find the Age of the
Moon in any Month of the Year 200, when the Epat according
to the common Method of Computation is 26, we muſt either add
I to the Epact found, which will give us 27 for the true Epact for
that
1
The Elements of Chronology.
649
t
ܪ
that Year, and then proceed according to the General Rule ; or elſe
it we will make Uſe of the Epact 26, we muſt ſay, initead of ad-
ding for January o, for February 2, for March 1, &c we muſt ſay,
for January add 1, for February 3, for March 2, &c and the ſame
Law muſt be obſerved at all other different times: thus the Age of
the Moon on the firſt of February in che Year 13, will be found to
be 2 Days, on the firſt of February 1723 ir will be found to be 7
Days, and on the firſt of February 1951 it will be found to be Ś
Days;
in each of which Years, the Epact according to the common
Method of Computation will be 4.
By the help of the Epacts thus determined, may the time of the
Pafchal Full-moon be found, and from thence and the Dominical Letter
the Day on which Eaſter happens.
For inaſrnuch as in the preſent Age, the Epact is equal to the
Age of the Moon, on the laſt Day of February ſave one, if we take
the Epact from 28, the Remainder will fhew us on what Day in
March the Astronomical New-moon will happen, and if to this we add
15, the Sum will give us the Day of the Aſtronomical Full-moon; and
becauſe the Aſtronomical Full-moon is 4 Days ſooner than the Eccle-
fiaftical Full-moon, if to the laſt Sum we add 4, we ſhall have the
Day of the Eccleſiaſtical Full-moon, which is uſually called Eaſter Li-
mit; that is, if from (the Sum of 28, 15, and 4 which is equal to)
47, we take the Epact for the Year, the Renainder will give us the
Day of the Ecclefiaftical Full-moon or Eaſter Limit, reckored from the
If March incluſive : thus in the Year 1723, when the Epaxt is 4, Eaſter
Limit will be equal to 47 leſs 4, equal to 43 ; ſo that the Ecclefiafti-
cal Full-moon falls on the 12th of April, and becauſe the Sunday
next ſucceeding the Limit is Easter Day, and the Dominical Letter
ſtanding againſt the firſt of March in the Kalendar is D, if we count
the Number of Letters between D and the Dominical Letter for
the given Year in a direct Order, it will ſhew us how many Days
after the Paſchal-Full-moon Eaſter Day happens, this being therefore
added to the Day of the Paſchal Full-moon, will fhew us on what
Day of the Month Eaſter Day falls; thus becauſe in the Year 1723,
the Dominical Letter is F, which is the ſecond Letter in the Alpha-
betical Order from E exclufive, if we add 2 Days to Eaſter Limit
which is 43, the Sum 45 reckoned from the firſt of March incluſive,
or if we count 2 Days from the 12th of April excluſive, which is
the Day of the Paſchal Full-moon, it will give us the 14th Day of
April for Eaſter Day; but inaſmuch as the Lecter D which ſtands a-
gainſt
650
The Elements of Chronology
gainſt the firſt of March in the Kalendar, is the fourth Letter in
the Alphabet ; if to the Dominical I.etter for the given Year be ad-
ded 4, if the Sum be leſs than 7, it ſhews us on which Day of the
Month the firſt Sunday in March falls; if the Sum be greater than 7
it ſhews us on what Day of the Month the ſecond Sunday happens,
this therefore taken from Eaſter Limit, ſhews us how many Days
it is from that Sunday to the Paſchal Full-moon, this Remainder there.
fore taken from the next greateſt Number of Sevens, will ſhew us
the Number of Days between the Paſchal Full-moon and the fuc-
ceeding Sunday, which therefore being added to Eaſter Limit, will
fhew us in how many Days from the firſt of March incluſive, and
confequently on what Day of the Month Eaſter Day falls. Thus,
In the former Example, where the Dominical Letter is F, if to
the Number 6, which anſwers to the Seat of this Letter in the Al-
phabet, be added 4, the Sum 10, ſhews us that the 2d Sunday
in March happens on the 10th Day; this therefore taken from 43
which is Eaſter Limit, will give 33 for the Number of Days between
the 2d Sunda; and the Paſchal Full moon, this laſt Remainder 33
therefore taken from 35 the next greater Number of Sevens, will
leave 2 for the Number of Days between the Pafchal Full moon and
Easter Day, tie ſame as was before determined; whence ic follows,
that Eaſter Day will happen on the 14th of April.
But when the Epact is 28 or 29, if it be taken from 28, it will
give us the New-moon in February, which therefore cannot in this
Caſe be the Paſchal New-moon ; fo that we muſt add 30 Days to
the Time thus found, or which is the ſame thing, inſtead of taking the
Epact from 47, we muſt take it from 47 plus 30 or 77, and the Re-
mainder will give us the true time of the Pafchal Full-moon or Enfter
Limit, reckoned from the firſt of March incluſive, whence the Day
on which Eafter falls may be determined by the former Rules, and
the ſame Law will hold good for any Time paſt or to come.
Thus ar the Time of the Council of Nice which was held in the
Year 325, when the Epact was 3 and the Dominical Letter C, if
from 47 be taken 3, the Remainder 44 ſhew'd that the Ecclefiaftical,
which at that Time was the Aſtronomical Full moon, happened on the
13th of April, when chc Dominical Letter being C, Eaſter Day was
the 5th Day following, or the 18th Day of the ſame Month : For
inaſmuch as the New-moon happened in this Month upon the ift
and 31ſt Day of the Month, the latter of which was the Paſchal New-
moon, and ſince this muſt be 44 Days from the iſt of March inclu-
five
A
The Elements of Chronology 651
five, and the Epic at that time being 3, the Fix'd Number for
finding Eaſter Limit by the Epiêt must at this time be 47, (qual ro
the Sum of 44 and 3, and the fame will hold good throughout all
the Cycles of Epact.
But if inſtead of theſe Fix'd and Sert!ed Epacts, we ſhould make
Uſe of the true Epacts or Ages of the Moon upon the laſt Days of
December and February; then the Numbers 47 and 77 muſt be in-
creaſed by Unity for every 310 Years after the 16th Century, and
diminiſhed by the ſame Qnantity for every Three Centurys back.
ward from the ſame time : Thus in the Year 325, if I make uſe
of the true Epact or Age of the Moon upon the laſt Day of February,
which is 29, then inſtead of the Number 47 and 77 I muſt uſe the
Fixed Number 43 and 73, and the ſame Limits will be produced.
And after this manner may the Time when Eaſter falls (upon
which all the moveable Feaſts and Terms depend) according to the
Supputatior of the Church of England be found, for any Time paſt
or to come independently, without the help of Tables or Kalen-
dar, by Addition and Subtraction only.
From the Cycles of the Sun and Moon (already explained) mula
tiplied into one another, arifes a third Period of 532 Years (for
28x19=532) which is called the Vi&torian or Dionyſean Period,
after the Completion of which Period, not only the New and Full
Moons return to the ſame Days of the Month, but alſo the Days of
the Month return to the fame Days of the Week, and therefore the
Dominical Letter and the moveable Feaits return again in the fame
Order; whence this Cycle is called the great Pafchal Cycle. And
To find the Year of the Dionyſean Period for any Year of Chriſt,
to the Current Year add the Number 457, the Year of that Cy-
cle when the Chriſtian Era commenced, and divide the Sum by
532, the Remainder after the Diviſion is the Year of the Period;
thus the present Year 1723 will be found to be the sad of the Vic-
torian Period.
Beſides the Cycles of the Sun and Moon, there is another Cycle
which the Romans made uſe of, confiſting of 15 Years, which is
called the Cycle of Indi&tion, having no connection with the Coe-
leftial Motion, and was inſtituted as fome fuppofe (cho the true
Reaſon does not appear) for the ſake of a certain Tribute, and was
by the Command of Juſtinian the Emperour, inſerted in their
Publick Inſtruments, and is made Uſe of at this Day by the Popes
of Rome in their Bulls Diplomas, &c. and commenc'd with them
Pppp
20
652
The Elements of Chronology
at the Kalends of Fanyary, but the Romans made it to commence
at different Times of the year,
And inaimnch as the firſt Year of the Chriſtian Era happened in
the fourth Year of this Cycle, therefore to find what Year of the
Cycle correſponds with any Year cf Chriſt, to the given Year add
3 and divide by 15, the Remainder will thew what Year of the Cy.
cle it is ; thus the Year 1723 will be found to be the ift Year of
the Cycle.
From the Multiplication of theſe three Cycles, viz, the Lunar
of 19 Years, the Solar of 28 Years, and the Indi&tion of is Years,
ariſes the grear Fulian Period conſiſting of 7980 Years ; this Period
is ſuppoſed to have its Beginning 764 Years before the Creation, whei!
all the 3 Cycles begin together, and is not yet complealed; and there-
fore it comprehends all other periods, Cycles and Epocha's:. There
is but one Year in the whole Period which has the fame Numbers
for the three Cycles of which it is made up, and therefore if the
Hiſtorians, had remarked in their Annals the Cycles of each Year,
there had been no Diſpute about the Times of any Action.
Th. Year before Chrift happened in 4713 Year of the Julian Period,
therefore to find what Year of the Fulian Period any Current Year
anſwers to; to the Current Year of Chriſt add 4713, the Sum will
be the Year of the Fulian Period; thús The Year 1723 will be
found to be the 5+361b Year of the Julian Period.
Theſe are the chief and moſt remarkable Periods, but there were
ſeveral others made Uſe of, ſuch as was the Year of the Jubilee a-
inong the Jews, which conſiſted of Seven Sabbattical Years, and re-
turned in the Space of 49 or 50 Years. The Olympiads among the
Grecians, which conſiſted of Four Years, and the great Platonic Year,
ſo celebrated among the Hiſtorians, which contained an intire Rea
volution of the Stars to the ſame point in the Heavens from whence
they ſet out,
As in the Heavens there are certain Fixed Points from which A-
Stronomers begin their Computations, fo alſo there are certain Points
of Time from whence Hiſtorians begin to Reckon, and theſe points
or Roots of: Time are called Epocha's or Era's, from which we ge-
nerally count our Years and Time, and in which all memorable AC-
tions are diſpoſed and recorded according to the Series of Years which
tollowed from that Root, the moſt Ancient is that of the Creation
of the World,3950 Years before Chriſt; the next is that of the Deluge.
2956 Years before.Chrift; and the third is that of the Olympiads,
which commenced in Greece 776 Years before. Chriſt, and in the
7
3938
The Elements of Chronology. 653
3938 Year of the Julian Period when the Solar Cycle was 18, the
Lunar Cycle 5, and the Roman Indiction 8.
The next and moſt remarkable Epocha, is that of the building of
Rome, in the Spring, about the End of the Third Year of the Sixth
Olympiade 753 Years before Christ, in the 396 1$7 Year of the Ju-
lian Period, the Cycle of the Sun being 13, the Cycle of the Moon
9, and the Indiction 1.
The next and moſt celebrated Era was that of Nabonafar King
of Babylon, ſo famous among the Aſtronomers, being made Uſe of
by Ptolemy, Albategnus, Alphonſus, Copernicus, and others, as a very
proper Era for computing the Mocions of the Stars from; it began
according to Ptolomy, on the 4th of the Kalends of March, being Wed-
neſday, 747 Years before Christ, in the 3967th Year of the Ju-
lian Period; the Firſt Year of the Eighth Olympiade, and in the
Sixteenth Year of the building of Rome, when the Solar Cycle was
19, the Lunar Cycle 15, and the Indiction 6.
Next after this we have the Epocha of Alexander the Great, 424
Egyptian Years after the Nabonaſarean Era began, for Alexander
died at Babylon in the 33d Year of his Age, in the Firſt Year of the
114th Olympiade the 3d of the Month Deſij, (according to ſome
Hiſturians) but according to others on the 23d or 27th ; thoſe who
determine the Day of his Death to the Julian Kalendar, make it
to happen on the 20th of May; others on the 9th or 23d of June,
and ſome on the 25th of July: but the Astronomers who have com-
puted their Times from thence, ſuch as Albategnus, and others make
it to happen on the 12th of November at Noon being Sunday, on
the Firſt Day of the Egyptian Month Thoth, 324 Year's Current be-
fore Chriſt, in the 43 90th Year of the Julian Period, 279 Years
before the Julian Epocha began, and 424 Egyptian Years compleat,
after the Commencement of the Era of Nabonalar.
The Epocha of the Syrians and Chaldeans began in the Reign of
Seleucus Nicator, who ſucceeded Alexander the Great, and reigned
in Syria and part of Africa about Twelve Years after that of the
Death of Alexander the Great.
The Julian Epocha which was approved of by Julius Cafar him-
felf, took its Commencement after the Year of Confufion upon the
iſt of January, which then happened on a Friday; this Prince ob-
ſerving that the Years inſtituted by Numa Pompilius the 2d King of
Rome, who ſucceeded Romulus the firſt Founder of the State, con-
filted but of Twelve Lunar Months, which had been made Uſe
Pppp 2 *
of
.
654
The Elements of Chronology
of till that Time by the Romans ; did no ways agree with the Pe-
riodical return of the Sun, the ſeeming fitteſt Motion to govern the
Year by, ordained in the 4th Year of his Confulſhip, 708 Years af-
ter the building of Rome, in the 731ſt Olympic Ycar, 45 Years before
the Birth of Chriſt, that for the future, the Year fhould conſiſt of
365 Days 6 Hours, which Year was ever after made Uſe of by the
Romans, and continued down to the preſent time, and is called the
Julian lear.
The Spanib Era began in the Reign of Auguſtus, in the 7th Year
of the Julian Era, 38 Years before Chrift, 715 Years after the build-
ing of Rome, and the 738th Year of the Olympiads; the Occaſion
is ſaid to be this, the Romans having divided the Empire, Spain fell
to Auguſtus's Share, who immediately took Poffefſion of it; and to
render this Day memorable, they took this time for their Root or E-
pocha, and began after that time to reckon Ab Exordio Regni Augufti,
which words being abriged, and in proceſs of time the Initial Letters
being made Uſe of in the room of the words themſelves, gave Birth
(as ſome affirm) to the Word ERA, which is made Uſe of to ſignify
a point of Time, from whence Hiſtorians began to reckon.
The next in order of Time which is moſt famous and beſt known
to us, is that which commences from the Incarnation of our Lord and
Saviour Jesus CHRIST, and in Honour of him is called the Chri-
ftian Era ; it began at the Noon of the 3 ſt of December immedi-
ately following his Birth, being Saturday, in the 4714th Year of the
Julian Period, the 753d Years after the building of Rome, sand in the
747th Year of Nabonafar, and in the 324th Year after the Death of
Alexander the Great, when the Solar Cycle was 10, the Lunar Cy-
cle 2, and the Roman Indi&tion 4
And altho the Chriſtians in General begin their Epocha from the
laſt Day of December, yet the Engliſh and Iriſh have an Epocha a
whole Year poſterior to this, which they commonly uſe in all Public
and Ecclefiaftical Affairs, for they do not begin their Year with the int
of January, but with the 25th of March ; and hence it comes.to paſs
that the Engliſh reckon from the Feaſt of the Annunciation 1723, that
there are compleated 1722 Years, but from the Birth of our Lord
to the Feaſt of the Nativity 1722, they number only 1721 Years e-
lapſed, whereas all the reſt of the Chriſtian World count 1722 Years,
but for the greateſt Part of the Year the Engliſh deſign it by the farne
Number that the reſt of the Christian World does; but from the
Beginning of the Year to the 25th of March, they write one Year
ters.
There
of the Nature and Conſtruction of Logarithms: 655
There are other Era's or Epocha's of leſs Uſe, ſuch as that of
Dioclefian the Roman Emperor, which began on the 21st of A.
pril, the Day on which the Foundation of Rome was laid, in the
Year of Chriſt 284, and the 4997th of the Julian Period; it is alſo cal-
led the Epocha of the Ethiopians, Abiſins, and that of the Martyrs,
becauſe of the great Perſecution that the Chriſtians ſuffered in bis
Reign. Alſo the Epocha of the Turks which they call their Hegira,
which commenced from the Flight of Mahomet, from the City of
Mecha to Jebril, on the 16th of July being Friday, in the 622d Year
of Chriſt Current, which was the time that Mahomet began to preach
and ſpread bis falſe Do&rine: As alſo that of the Perſians which is
called Tezdegiid from a King of that Name, who died the 16th
of June being Tueſday, in the Year of our Lord 1632.
The Principal End of this section being to explain the Kalendar,
and ſhew the Reaſon of the Rules commonly delivered for computing
the Golden Number, Epact, Age, and Time of the New and Full-
moon., &c. I judge this little ſhort Account of the ſeveral Periods
and Epocha's made Uſe of by. Afti onomers and Hiſtorians would not
be unacceptable to the curious Reader.
La la la
Section. XXV.
Of the Nature and Conſtruction of Logarithms.
Ogarithms juſtly eleemed one of the greateſt and moſt uſeful
Diſcoveries that the laſt Century, has produced, were firſt in-
venced by John Neper, Baron of Marchiſton in Scotland, and by him
firſt publiſhed at Edenburgh in the Year 1614, and ſoon after by him-
ſelf and Mr. Briggs (then Savilian Profeſſor of Geometry in the Uni-
verſity of Oxford) reduced into a better Form and publiſhed at Lói--
don in the Year 1624, in the ſame Form and in the ſame Nanner ás
they are uſed at this Day.
And notwithſtanding that theſe Tables were computed with the
greateſt Care and are ſufficiently exa& for any Uſes to which Loga-
rithms can be applied, yet their frequent and extenſive Uſe, and which
reaches almoſt to every uſeful Branch of the Mathematical Sciencesy
has.
656 of the Nature and Conſtruction of Logarithms
has encouraged the greateſt Geometricians that have lived ever ſince,
to inquire more particularly into the Nature and Property of theſe
Admirable Numbers, and in which they have ſo well ſucceeded,
that by their frequent Diſcoveries they have rendered their Con-
ſtruction much more eaſy and facile than by thoſe operoſe Methods
that were at firſt made Úſe of, by their noble Inventor.
And inaſınuch as they bear ſo grear a Share in this work, it is
abſolutely neceſſary that I ſhould give ſome Account of them, and
ſhew the Manner how they are made, as well for the Satisfa&ion of
thoſe who are deſirous of being thorough Maſters of ſo noble and bene-
ficial a Branch of Science, aş to gratify thoſe whofe Curioſity may lead
them either to try the Tables already made, or carry them on to great-
er Degrees of Exactneſs, in ſuch Caſes where it ſhall be found re-
quiſite ; which I ſhall endeavour to do with all the plaineſs and per-
fpicuity that the Nature of the Subject will allow of.
Logarithms may be conſidered as, and were uſually defined to be,
a Series or rank of Numbers in an Arithmetical Progreſſion, fitted
or adapted to a Series of Numbers in a Geometrical Progreſſion;
that is,
If to a Rank of Continual Proportionals in a Geometric Pro-
greſſion from 1, Suppoſe
1, 2, 4, 8, 16, 32, 64, 128, 60
be accomodated'a Rank of Numbers continually proportional in
an Arithmetic Progreſſion beginning from o. Suppoſe
0, 1, 2, 3, 4, 5, 6, 7, c.
Inaſmuch as for every Multiplication or Diviſion of the Terms
in the Geometrical Progreſſion, there is an anſwerable Addition or
Subfraction of the correſpondent Terms in the Arithmerical Pro-
greſſion; that is, As 2 the ſecond Term in the Geometric Series,
multiplied by 4 the third Term produces 8, the fourth Term in the
fame Series, ſo i the fecond Term in the Arithmetic Series added to
2 the third Term produces 3, the fourth Term in the fame Series ;
and again, as 16 the fifth Term in the Geometric Series, divided by
2 the ſecond Term produces 8, the fourth Term in the ſame Series,
ſo if from 4 the fifth Term in the Arithmetic Series, be taken 1 the
ſecond Term, there will remain 3 the fourth Term in the fame Se-
ries; or univerſally, if inſtead of 2, the ſecond Term in the Geo-
metric Series we ſubſtitute x, and in the room of 1 the ſecond Term
in the Arithmetical Series, we ſubſtitute a, that is, if to a Series of
Geometrical mean Proportionals, as
I, XX,
Of the Nature and Conſtruktion of Logarithms. 657
I, *, xx, x, xt, xos
x", *', %7, &C.
be accommodated a series of Arithmetical mean Proportionals, as
0, 2, 20, 30, 4a, 50, 60, 70, &c.
Inaſmuch as the Multiplication of any cwo Terms in the Geome-
tric Series, as x and x2 produces the Term.x", which anſwers or corre-
fponds to the Term 3a in the Arithmetic Series, the Sum of the Terms
a and 2 a in the fame Serics, and the Diviſion of any two Terms in
the firſt Series, as x? by *, produces **, which correſponds with
the Term 2 a in the ſecond Series, the Difference between the cor-
reſpondent Terms 3. a and a in the fame Series; and inafinuch as
this is the Ellential Property of the Logarithms, therefore the Terms
in the latter Series are the Logarithms of their correſpondent Term's
in the firſt Series; and becauſe the Terms a and x may fand for any
Numbers taken at Pleaſure, therefore Logarithms may be of as ma-
ny different Sorts as there can be allumed different Values of the
Quantities x and a.
If the Quantity a in the ſecond Series be put equal to t, then the
ſevcral Terms of the ſecond Series, viz.
0, 1, 2, 3, 4, 5, 6, 7, Oc.
become the Indices or Exponents of the ſeveral refpe&tive Terms or
Powers in the firſt Progreſſional Series of Geometrical Proportionals
whoſe firſt Term is i and ſecond Term x, inaſmuch as they point
out the Places or Diſtances that each Term obtains from Unity; for
Example, as the Numbers in the ſecond Series ſtands againſt the
Quantity xs in the firſt Series, it points out and ſhiews that the Quan-
tity as is in the fifth Place from Unity; in like manner as the Num-
ber 6 in the ſecond Series ſtands againſt x' in the firſt Series, it ſhiews
that the Quantity x' is in the ſixth Place from Unity; or which is
the ſame thing, is removed at fix times the Diſtance from Unity,
that the Quantity * is that immediately follows: And hence
it is that Logarithms are ſometimes defined to be the Exponents
or Indices of the Powers of the reſpective Numbers to which they
belong
In the progreſſional Series of Geometric Proportionals; if between
the Terms i and x be inſerted a mean Proportional which is Vx,
its Index will be , inaſmuch as its Diſtance from Unity will be
but one half of the Diſtance of x from Unity, and conſequently the
Root of x will be expreſſed by }: In like manner if between x and
mes we inſert a mean Proportional, its Index will be i and 1 or }, in-
aſmuch.
.
*
--
I
3
I
I I
658 Of the Nature and Conſtruction of Logarithms
aſmuch as its Diſtance from Unity is once and a half the Diſtance
of x from the ſame Place of Unity.
Again, if between 1 and * be inſerted two mean Proportionals;
the firſt of theſe will be the Cube Root of x or v'x, and its Index
will be }, inaſmuch as its Diſtance from the Units place is but
of the Diſtance of x from the lame Place of Unity; and conſequent-
ly vix is expreſſed by #39 whence it follows, thar the Index of U.
nity is o, inalinuch as Unity cannot be removed at any Diſtance
froin its ſelf.
And the fame Series of Geometrical Proportionals may be con-
tinued on the contrary Side of Unity, or towards the Letr Hand,
and which therefore will decreaſe in the ſame manner or in the fame
Ratio as the Terms placed on the Right Hand increaſes.
For the Terms contrast , I, *, *, *, **, *5,&C.
are all in the ſame Geometrical Progreſſion continued, and have all
the ſame common Ratio ; wherefore inaſmuch as the Diſtance of
* from Unity towards Right Hand is Poſitive, or ti, the Di-
ſtance of from Unity towards the Left Hand, which is c-
qual to the former is Negative or --1, which is therefore the
the Index of and conſequently for may be written
Again, inaſmuch as x2 is on the Right Side of Unity, and Pofi-
tive, and its Index +2, the Index of fin which is as far diſtant
from Unity on the contrary Side, or towards the Left Hind
will be Negative, and -2, and conſequently
is the ſame with
: In like manner is the ſame with
33
203?
X
I
X
I
I
*
2
3
I
73, and
fame with Jat; whence it comes to paſs that theſe Negative Indices
ſhew that the Terms to which they belong, are as far diftant from
Unity towards the Left Hand, as the Terms whoſe Indices are the
fame and Poſitive, are removed from Unity on the contrary Side
or towards the Right Hand.
Again,
of the Nature and Conſtruction of Logarithms. 659
8
I
I
4
Again, as between 1 and x, in the Series increaſing from or on the
Right Hand of Unity, be inſerted a mean Proporcional as Vx, and
between this mean Proportional and Unity be interced another mean
Proportional as Vx, and between this laſt meanProportional and Unity
be inſerted another mean Proportional as v *, C..whoſe reſpeđive
Indices will be ', 4, , 6c. ſo between Unity and the next term
in the Series decreaſing from Unity, or on the Left-hand of U-
nity, may be inſerted a mean Proportional as.
s to whoſeCorreſpon-
dent Index will be- ; and again between this mean Proportional
and Unity, may another mean Proportional as be inſerted, whoſe
Vx
correſpondent Index will be-, and between this and Unity may
another mean Proportional as : be inſerted; whoſe Index will be
VX
I, c. and as this may be done between any other two Terms
of the Progreſſion, it follows; that between any two Terms may
an infinite Number of mcan Proportionals be inſerted, whoſe re-
ſpective Indices will become the Logarithms of the reſpective
Terms to which they belong: And as theſe mean Proportion-
als the greater they are in Number, between the two Terms of a-
ay Ratio, the nearer do they approach to a Ratio of Equality,
with their correſpondent Indices or Exponents.
Hence we are let into the Reaſon of the ancient way of making
Logarithms, and which was made Uſe of by their firſt Inventors, and
thar was, to extract the Square Root out of the Square Root, c.
of any Number, in order to find out a Series of continual mean Geo-
metrical Proportionals; till the Number of Cyphers contained be-
tween Unity and the firſt ſignificant Figure of the Root, was equal
to the Number of Places that the intended Logarithm ſhould confiſt
of, and at the ſame time to find out a like Number of correſpondent
Arithmetical Means, which would be the Logarithms of the Geoma-
trical Means reſpe&ively.
And how operoſe this Method is, may be eaſily gueſſed at, by
any one who has been at the Pains to extract the Roor of a large
Number; and may be gathered from this, that tú find a Logarithm
to Seven places only, requires at leaſt 27 Extractions of Root out of
*
*
Q999
Roor,
660 of the Nature and Conſtruction of Logarithms
Porno
2
1
2
X
Roor, the Square conſiſting of 16 Figures at leaſt; and that if in a-
ny one of the Operations an Error happens to creep in, the whole
work muſt be repeated
In the adjacent Figure, if the Line ſtanding againſt the Num-
ber i be made to repreſent Unity, then the double of that Linc will
repreſent the Number 2, and its triple will repreſent the Number 3 ;
its one half will repreſent
the fractional Number
and its one third part will
repreſent the Frađion }
c. and conſequentiy if the
Line ſtanding againſt x' be
ſuppoſed to repreſent the
Value of x, the Line ſtand-
ing againſt x?, which we
ſuppoſe here to be a third
3
Proportional to the Line re-
preſenting Unity, and to
the Line x will repreſent
the Quantity *? in the for-
mer Series of Geometrical
6
Proportionals: In like man-
ner the Line ſtanding againſt
x?, which is ſuppoſed to be
7
a fourth Proportional Geo-
metrical, to the Lines re-
preſenting Unity, x and xạ, will ſtand for, and repreſent x>, in the
former Geometrical Series ; and conſequently the ſeveral Lines which
ftand againſt **, *', *°, 47, 6c. which are ſuppoſed to be a
Geometrical Scale of Progreſſional Lines; infinite in Number, will
repreſent the feveral Terms **, *, *, *, &c. in the geometric
Series of Progreffional Numbers, beginning from Unity and increaſing
towards the Right-hand; and it the Lines which ſtand againt
-", -2, 6c, on the contrary Sides of the Line reprefenting U-
nity, be ſuppoſed to diminiſh in the ſame Ratio as the Lines x';
xoc, increaſe on the contrary Side, we ſhall have a Geometri-
cal Scale of Progreſſional Lines infinite in Number ; which will be
equal in Value and Analogous, to the ſeveral Terms of the infinite
Progreſional Series of Quantities following, which are here fuppoſed
to repreſent an infinite Series of Numbers.
Óc
5
X
of the Nature and Conſtruction of Logarithms 661
*
$
&c. x-², x , 1, ****, **, **, **, *", x', &c.
For as the Terms 1, *, *', are in a continued Ratio of 1 to
x, and as the Indices affixed to each proportional Term pronoun-
ce the Place or Diſtance that each Term obtains from Unity, lo
the Magnitudes of the ſeveral Lines 1, *, *?, &c. have the ſame
Ratio of I to *; and the Indices of the Powers of the ſeveral
Lines, ſhew the Order that each Line is placed in, or obtains from
the Line repreſenting Unity; for as the Term $3 is in the third
place from Unity, in the Series of Geometrical Proportionals, ſo
the Line repreſenting x?, is the third in Order from the Line re-
preſenting Unity, and is placed at three times the diſtance from
the Line repreſenting Unity, as the Line repreſenting the Quantity
%, which immediately follows,
Again, as between 1 and x in the progreſſional Series of Propor-
tionals, there may be inſerted a Geometrical Proportional as Vx or
1 /
ſo between the Lines repreſenting Unity and the Line x may
be found and inſerted a Geometrical mean Proportional Line, which
will repreſent and be equal in Value to the Term in the Pro-
greſſional Scale of Quantities, which will have the ſame Ratio
to the Line repreſenting Unity, as the Term ** in the progreſſion-
al Series has to Unity it felf; and the fame will hold good if be-
tween any other two Lines there be found a Mean Proportional
.
Again, as between 1 and Vx or x ž in the progreſſional Series of
Proportionals, there may be inſerted a Geometrical mean Propor-
tional as V* or x
Px orx, fo between the Line repreſenting Unity and
12
the Line repreſenting %], may be found and inſerted a Geometri-
cal mean Proportional Line which will repreſent and be equal in
124
Value to the Term x* in the progreſſional Scale of Quantities,
which will have the ſame Ratio to the Line repreſenting Unity,
as the Term x4 in the progreſſional Series has to Unity it ſelf; and
univerſally, as between 1 and xin the progreſſionalScale of Quantities
there may be inſerted an infinite Number of Geometrical mean Pro-
portionals, ſo between the Line repreſenting Unity and the Line x,
may be found and inſerted an infinite Number of Geometrical
mean Proportional Lines, which will be every way Analagous to
Q9992
the
662 of the Nature and Conſtruction of Logarithms.
the infinite Series of proportional Numbers or Quantities between
1 and the Number repreſented by *; and the ſame will hold good
between any other two Terms whatſoever.
In the following Figure therefore upon the Line AG infinitely ex-
tended cath way, let there be taken AB, BC, CD, DE, EF, &c.
towards the Right-hand, alſo A, OP, &c, towards the Left-
3
:
P
O
A b Bc Cd De E F F
G
Ꭹ
R
8
Q
Q
S
р
q
k
K
1
L
1
i
m
M
T
n
n
N
hand, all equal among themſelves, and from the ſeveral points P,
0, A, B, C, D, E, F, &c. let there be drawn the Lines RP, OQ,
А н,
of the Nature and Conſtrution of Logarithms. 663
را -
-2,
AH, BI, CK, DL, EF, FN, &c. equal in Length to the ſeveral
Lines x2,
1, *, **, *, **, *', '', *", &c. in the
Geometrical Scale of Progreſſional Lines infinite in Number, theſe
therefore will repreſent ſo many Terms in the Progreſſional Series
of Proportionals, of which the Line AH reprefents Unity, and
confequently the Lines 4B, AC, AD, AE, AF, &c. -AO-AP,
&c. will expound the Diſtances of the ſeveral Terms in the Pro-
greſſion from Unity, or the Place and Order that each Term in
the Geometrical Scale obtains, according to the Unites Place; thus,
if AD be triple the Diſtance of AB, the Line DL will repreſent the
third Term or x?, in the Geometrical Series of Quantities, of
which B I repreſents the firſt, or x'. In like manner, if AF be
quintuple the Diſtance of AB, the Line FN will repreſent the fifth
Term or x', in the Series of Progreſſional Quantities or Numbers :
Again, inaſmuch as AP is double the Diſtance of A B, but on the
contrary Side of the Line AH repreſenting Unity, PR will repreſent
the ſecond Term in the Geometrical Series or x on the Left-
hand of the Unites Place, O.
Now if the Extremities R, G, H, I, K, L, M N, &c of the
Geometrical Scale of mean Proportionals, be connected together by
Right-lines, the Figure PRNF will be a Poligon, conſiſting of more
or fewer Sides, according as the Terms in the Scale of mean Pro-
portionals are more or fewer in Number.
If the Diſtances AB, BC, CD, DE, EF, &c. are biſe&ed in the
points b, c, d, e, f, &c. and from theſe points be drawn Lines, as
bi, ck, dl, em, fn, &c perpendicular to AG, which will
be ſo many Proportionals between A H, BI, CK, DI, EM,
FN, &c. there will ariſe a new Scale or Series of Propor-
tionals, double in Number to thoſe in the former Scale ; and
conſequently the Difference between the Terms will be leſs, and
the Terins themſelves will approach nearer to a Ratio of Equality
than the Terms in the former Series, and inaſmuch as in this New
Scale of Proportionals the Lines AP, and AB, expound the Diſtan-
ces of the Terms FN, and B I, from Unity ; it is manifeſt, that
as AF is removed ten times as far as Ab, the Term FN will
be the tenth Term of the Series from Unity; of which Ab is
the firſt.
In like manner, as A c is triple the Diſtance of Ab, the Line c K',
will repreſent the third Term in the New Scale of Proportionals;
and as between AH and CK there are three mean Proportionals,
fo
664. Of the Nature and Conſtruction of Logarithms.
ſo between AH and FN, which is three times the Diſtance, there
will be nine mean Proportionals.
Now it the Extremities H,,, I, k, Kl, S, &c. of this new Scale
of Proportionals be connected together by Right-lines, there will
ariſe a new Poligon, which will conſiſt of a greater Number of
Sides; but each Side lefſer in Magnitude.
If we conceive the Diſtances Āb, b B, B c, &c. to be again bi-
fc&ted in the middle, and between every two Terms a new Scale of
mean Proportionals be inſerted, there will ariſe a new Scale of
mean Proportionals, double in Number to thoſe in the former Scale,
from the Place of Unity ; the Difference of which Terms will be
ſtill lefler, and the Terms themſelves will approach ſtill 'nearer to a
Ratio of Equality than the former. And,
If the Extremities of this new Scale of mean Proportionals bc con-
nected together by Right Lines, we ſhall have a new Poligon of
double the Number of Sides to the former ; but each Side will be
ſtill leſs in Magnitude, inaſmuch as the diſtance of the Terms from
each other, is ſtill leſs than before.
Moreover, in this New Scale the diſtances AF and AB, ſtill de-
termine the Order or Place that the Terms FN and BG obtains
from Unity; and if A F be quintuple the diſtance of A B from
Unity, and FB the fourth Term in this new Scale of Proportion-
als, the Line FN will repreſent the twentieth Term in the ſame
Scale.
After the ſame manner, if the diſtance between every two Terms
be continually biſe&ed, and to each of the Points ſo found, there
be applied a Series of mean Proportionals, the Number of Terms
in this new Scale, as alſo the sides of the Poligon will by this means
become infinite ; and each side of the Poligon will be leſs than any
given Right line, and conſequently the Poligon will be changed into
à Curve-lined Figure ; ſince every Curvilinear Figure may be con-
ſidered as a Poligon, compoſed of an infinite Number of Sides, each
of which are infinitely ſmall in Magnitude,
The Curve-line thus deſcribed is called the Logarithmic Curve, in
which it the Right-lines AH, BI, &c. which ſtand at Right-angles
to the Axis AG, reprefent ſo many Terms in a Series of Numbers
in a Geometrical Progreſſion, the Portions of the Axis AB, BC,
&c. between the Place of Unity and any given Term, will fhew
the Place or Order that each Number in the scale of Geometrical
Proportionals obtains from the Place of Unity in the ſame Series :
For
of the Nature and Conſtruction of Logarithms. 665
For Example, if AF be quincuple of AB, and between Unity and
FN, there are jooo mean Proportionals between Unity and B1,
there will be 200 of the lame mean Proportionals; or B I will be
the 200th Term of the Scale of Geometrical Proportionals from U-
nity; and univerſally, whatever be the Number of Terms between
Unity and FN, between Unity and BI, there will always be one
fifth of the fame Number of mcan Proportionals; and the ſame will
obtain between any other two Terms in the Progreſſional Series,
This Curve may be concieved to be deſcribed by a two-fold
Motion, the one equable and uniform, the other acceler,
ated or retarded in a given Ratio : For Example, if the Right-
line AH moves uniformly along the Right line' AG, ſo that the
point A deſcribes equal Spaces in equal Times, while the Live it
felf AH increaſes in ſuch manner as to acquire Increments in equal
times, which ſhall be conſtantly proportional to the whole increa-
fing Line ; that is, if while the Line AH is moving into bi, it acquires
the Increment i 0, and in moving from thence into B l in the ſame
Space of time, it is increaſed by a ſimilar part Ip, which ſhall be
to bi as the Increinent io is to AH: In like manner while the
fame Line A Hin moving from BI into the Place of ek in the ſame
time acquires an Increment qk, which ſhall be to Blas 1p to
ib; or as i o to A H: That is, that the Increments acquired in
equal times, may always be proportionnal to the Wholes. Or,
If the Line AH in moving backwards from its place in AH di-
miniſhes in a conſtant Ratio, that is, while it moves over equal
Spaces AO, OP, it is leſſened by equal Decrements H x and gy,
which are proportional to the whole Lines 20, and PR, the Ex-
tremity H of this flowing Line, will deſcribe what is called the
Logarithmic Curve
For becauſe AH:10: :ib: 1p: :1Bkq; it will be by com-
pounding AH:ib (=botoi): ib: : (Bp-+pi=BI: B1:(09+
ck=) e k, &c
From this Conſtru&ion it is manifeft, that all the Terms that are
placed at equal diſtances from each other, are continually propor-
tional; and that of any four Terms as AH, BI, EM, FN, if the
diſtance between the firſt and ſecond Term be equal to the diſtance
between the third and fourth, whatſoever the diſtance be, thoſe Terms
will be proportional. For, becauſe the diſtances AE and e fare
equal, A H will be to the Increment IS, as E M to the Increment
NT; wherefore by compounding, A H will be to B i as E M to
FM
666 of the Nature and Conſtruction of Logarithms
FM; and on the contrary, if any four Terms are proportional, the
uiſtance between the firſt and ſecond Term, will be equal to the dist-
ance between the third and fourth.
The diſtance therefore between any two (Terms, which repre-
ſent ſo many) Numbers in this continued Scale of Geometrical
Proportions, is the Logarithm of the Ratio of thoſe two Numbers
the one to the other; and is meaſured and eſtimated not by the Ratio
it felf, but by the Number of mean Proportionals in the continued
Scale of Geometrical Proportionals contained between the two
Numbers, and expounds the Number of equal Ratiunculæ, of which
the given Ratio is compounded
For if the Diſtance berween any two Numbers be the double
of the diſtance between any other two Numbers, then the Ratio
between the two former Numbers will be the duplicate of the
kario between the two latter Numbers : For Example, if the di-
(tance between the two Terins E M and FN, be double of the dif-
rance of Ab, which is contained or intercepted between the two
Terms A H and bi, then the Ratio of EM FN will be dupli-
care of the Ratio of AH to bi. Let EF be biſected in the point f,
becauſe Ab=Ef=fF, the Ratio of EM to Pn, will be equal to
the Ratio of AH to bi, but the Ratio of EM to FN is duplicate of
the Ratio of EM to fn; wherefore the Ratio of EM to FN will be
duplicate of the Ratio of A H to bi: In like manner, if the dif-
tance CF be triple of the diſtance AB, then the Ratio of CK to FN,
will be triplicare of the Ratio of AH to BI ; for becauſe the di-
Itance between CK and FN is criple of the diſtance between AH and
B I, there will be triple the Number of mean Proportionals between
C K and FN, as there is between AH and BI; but the Ratio of
CK and FN, and of AH to BI, is compounded of all the inter-
mediate equal Ratio's ; wherefore inaſmuch as the Number of e-
qual Ratio’s between CK and F N are treble the Number of
the fame, and equal Ratio's contained between AH and BI, the
Ratio of CK to FN, will be triplicate of the Ratio of AH tó BI;
alſo for the ſame Reaſon, if the diſtance D F be quadruple of the
diſtance Ab, the Ratio of DL to F N, will be quadruplicate of
the Ratio of AH to bi, c.
The
will be
Of the Nature and Conſtruction of Logarithmis 667
The Logarithm therefore of any Number, is the Logarithm of the
Ratio of Unity to the Number it ſelf; or it is the diſtance between
Unity and the ſame Number, in the Geometrical Scale of Propor-
tionals, and is meaſured by the Number of Proporcionals contained
between them: Logarithms theretore expound the Dignity, Place,
or Order that cvery Number obtains from the Units Place in the
continued Series or Scale of Proportionals, infinite in Number;
thus, if between Unity and the Number 10 there be ſuppoſed 10.
000.000 6c mean Proportionals, that is, if the Number 10 be placed
in the 10,000,000th Place from Unity between 1 and 2, there will
be found to be 3010300 of ſuch Proportionals; that is, the Num-
ber 2 will be placed in the 3010300oth Place from Unity, between
1 and 3, there will be found .4771213 of the ſame Proportionals,
and the Number 3 will be found to ſtand in the 4771213th Place
in the Scale of Proportionals infinite in Number, which Numbers
10000000, 3010300, 4771213, are therefore the Logarithms of
10, 2 and 3 ; or more properly the Logarithms of the Ratio's of
thoſe Numbers one to the other.
Again, if between Unity and the Number 10 there be ſuppoſed an
infinite Scale of mean Proportionals whoſe Number is 2.3025851,
Oc. that is, if the Number 1o be placed in the 2.3025851t, c. place
from Unity, in the infinite Scale of Proportionals between 1 and
2, there will be found to be 6931471, c of ſuch Proportionals;
that is, the Number 2 will ſtand in the 693 1471th, &c. Place from
Uniry, between 1 and 3, there will be found to be 5.0986122 of
the fame Proportionals; and the Number 3 will be found to ſtand
in the 1098612 zih Place, in the fame Geometrical Scale of Pro-
portionals, &c. ſo that if the firſt Term of the Series from Unity be
called x, the ſecond Term will be xạ, the third Term will be x?,
c. and if the Number to be luppoled to be the 1000000th Term
in the Series, the Number 2 will the 3010300th Term, and the
Number 3 will be the 4771213th Term in the ſame Series ; and
conſequently ſince shodóóooo=10, *3016300=2, and 34170303
3, &c. and again, if the Number 10 be ſuppoſed the 23025857th
Term in the Series, the Number 2 will be the 69314711 Term,
3
ries, and conſequently in this caſe 10=*23025851, 4093147152,
and x'0986122=3, .
And inaſmuch as the infinite Number of mean Proportionals be-
tweet any two Numbers may be affumed at Pleaſure, hence Loga-
rithms
Rrrr
allir
a
665 of the Nature and Conferaction of Logarithms.
rithms may be of as many different Forms (that is, there may be
as many different Scales of Logarithms) as there can be affumed
different infinite Scales of mean Proportionals 'between any two
Nurnbers. Every Number therefore is ſome certain Power of that
Number which is placed next to Unity, in the infinite Scale of Pro-
portionals between Unity and the given Number, and the Index of
that Power is the Logarithm of the Number:
Logarithms therefore may be of as many Sorts as you can al-
fume different Indices of the Power of that Number whoſe Logaa
rithm is required.
If the Index be affumed equal to 10000000, Oc infinitely, the
Logarithms produced will be thoſe invented by Neper bug
if the Index be afumed equal to 23025851, 6c, che Loga-
rithms produced, will be thoſe that wețe made by Mr. Briggs,
And inaſmuch as the Ratio of Uniry to any Number, is meaſured
by the Number of Raliuncula contained in that Ratio.
Hence we may value Ratio's by the Number of Ratiuncula con-
tained in eacb Ratio, and may conſider them as Quantitates Süi, ge-
mative when the Ratio is increaſing, as of Unity to a greater
Number; but Negative when decreaſing, as of Unity to a leffer
Number.
So that Ratio's are one to another as the Number of like and e.
qual Ratiuncula contained between the two Terms of the Ratio
wherefore, if the Ratio of 1 to 10 contain roo00000,6c equal Ra-
tiuncula, that of its duplicate or of 10 to 100, will contain 20000-
000, OC that of its iriplicate, or of 100 to 1000, will contain
30000000,
6c. and that of its quadruplicate, or of iooo to 10000,
will contain 40000000, 6c of the ſame like and equal Ratiuncula;
lo that the Logarithms of Numbers, or the Values of Ratio's, are
in an Arithmetical Progreſſion; and hence ariſes that vulgar Definition
Arithmetic Progreſſion, fitted or aſſigned to a Rank of Numbers in
a Geometrical Progreſſion.
If the diſtance between Unity and any Number be ſuppoled cô
confift always of the ſame equal Number of mean Proportionals, as
ſuppole 1000000o, Sc. or which is the ſame thing, it we ſhall
take the Ratio of Unity to any Number, to conſiſt always of the
fame infinite Number of Ratiuncula, the Magnitudes of each in this
Cale, will be, as their Number in the former; ſo that it between
Unity
3
N
-n.
22
Alſo nD=SELL
Of the Nature and Conſtruction of Logarithms. 669
Unity and any Number propoſed, there be taken any Infinity of
mean Proportionals, the infinitely little Augment or Decrement of
the firſt of thoſe means from Unity, will be a Ratiuncula ; and fee-
ing this is ever proportionable to the Momentum or Fluxion of the
Ratio of Unity to the ſaid Number ; and inaſmuch as in theſe con-
tinual Proportionals all the Ratiuncula are equal, their Sum or the
whole Racio, will be as the ſaid Momentum or Fluxion is directly:
wherefore, if the Root of the infinite Powers be extracted out of a-
ny Number, the Difference between that Root and Unity, will be,
as the Logarithm of that Number ; ſo that the infinite Root of any
Number being found, the Logarithm of that Number is eaſily had;
how the infinite Root of any Number may be found, comes next
to be ſhewn.
Let pt px be a given Quantity, whoſe n Root is to be extracted
where n ltands for any Power or Number taken at Pleaſure.
Then will p+px=pxitx
3
4 5
Aflume 1+x= 1+Ax+B x +Cx+DxtEx, &c.
n-1
:
3.
4:
Then, nx1 *x* xAx+2 Bxx+ 3 Cxx+4 Dxx+s Exx, Oc:
3 4
And nx1+x A +2Bx+3 C*44Dxts Éx, or.
n.
nxi + x
A 42Bx + 3Cx4Dx: +5 Ex4, Go.
And
-122 ITAx+ Bx + 2xy + Dx+y Ex', 6c
1+x
And,
n+nAx+nBx' +nCx' +cDx“, t..=A+2Bx**3 C14DX34-5Ex*,
&c
+AX-+-2Bx211-3 Cx-t4Dx4,
So And n=A, alſo, nA=2B+A alſo nB=C+28.
And, nA-AZ2B, And, nB-2B=30
And n-1 XA=2B,
And n–2XB=3C,
B-
Antti Stringan XiAEB
And,
*B-C.
Alſo, fic=4D13C
And, nC-3C=4D,
And, 11D-4D=5E,
And, n4-3*C=4D, And, r-4xD=SE,
3
And, XC=D. And,
XD=E.
5
2
+;
- 2
2
3
1
n 4-3
n -
6:9,
Rrrr 2
*
Where-
670 Of the Nature and Conſtruction of Logarithons.
Wherefore, A=n, B=nx = C=nx
2, D=nx
2 >
n-3
n
!
2
x
2
3
n-3 n-3 n-4
Х
Х x
3. 4
nI n-2
xtnx
3.
I
and E-nx
2 X 3 X 4
n
nat
And, 1 tx=i+nx - x 2
11 -2
X
X
xº, pc.
3 4
5.,.
*
2 tnx
X
2
-3
N-I
n
0 2
n 3
Il-11
2
рх
n
n
ng
Х
2
2
n
n2
n
n
•B xt
text
+
2
ره
1
I
1
And, pfpx =pta putnx px tnx
3
n 1
DI
x 3
Put pA, npx= B, nx I p x:=C, nx
3 P
= EC.
n!
4
Then,
p FnAx*
ptpx=
3
5
Ex, c.
Which is the celebrated Binomial Series invented by the great Sir
ſaac Newton, for railing of a given Quantity to any given Power,
or extra&ing any Root out of the ſame Quantity ; inaſmuch as n
ſtands for any Number whether it be a whole Number or a Frac-
tion, wherefore, becauſe when n is finite.
I - 30tann 1--ontrinon3
ontonon
ta
*3.-+
Itx"==+
63
**, &c. when it becomes infinite.
1
I
I
n=1X xx-
tx - X 4+-X', 66.
n
20 3n 4n
nn being infinitely infinite, and conſequently whatſoever is divided
thereby vaniſhing, whence it follows, that ñ multiplied into x-
*x*-+*x-_***, &c. is the Augment of the firſt of the mean Pro-
portionals between Unity and itx, and is therefore the Logarithm
of the Ratio of 1 to itx.
I
But becauſe the infinite Root of 1–x, or 1-X is l_nX-2n
1x* c. therefore multiplied into x++ x2+ {x' +
3n
4n
x ola
n
2 nn
24n4
I
I
it x
5n
n.
I
4x4
1
I
1
X
13
*4 do
11
1/2 2
{ z
b
.
b = Log.
a
5253 &c. or
927&c. will be
Of the Nature and Conſtruction of Logarithms. 671
1*++*', &c. will be the Logarithm of the Ratio of 1 to 1-%, of
the Logarithm of a Number leſs than Unity ; and whereas the in-
finite Index n may be taken at pleaſure, the ſeveral Scales of Lo-
garithms to ſuch Indices will be reciprocally as nı or the ſeveral In-
dices, and if the Index be taken 10000000, &c, the reſpe&ive Lo-
garithms will be ſimply.
-
to X', &c .
3 n
-|-4n
5n
Let a repreſent the leaſt of any two given Numbers, and b the
greateſt, x the Sum of the two Numbers, and d the Difference,
and let us ſuppoſe the Ratio of a to b, to be divided into that of
a to 1%, and of į < to b; that is into the Ratio of a to the A-
rithmetical mean between the two Numbers and the Ratio of the
faid Arithmetical mean to the greater Number 6, then becauſe
=
the Log. of an t- Log.
Log in or becauſe
1/2 z
b b
the Log. of {2+- Log Log. 5, that is the Sum of
XT
the Logarithms of theſe two Ratio's will be the Logarithms of the
Ratio of a to b; and to find the Value of each of theſe Ratio's,
in the Terms of 1 and 1 x we muſt ſay, { 2:2 ::1:1-*)
d
whence x =
and again, as i 2:6: :111tx, whence
b./
d
and ſubftituting
d in the room of x, in each cf
d
d?
the former Series, we ſhall have.
ds
d d:
5253 &c. for the Log
: of the Ratio of a to {z, and 2 m
dº d
+ 나
​&c. for the Logarithm of the Ratio of į z
2d, 2 d3
b; and conſequently the Sum of theſe two Series comx
d
ds
d?
-X 2.X
fo 아
​7 Zbora
the Logarithm of the Ratio of a to b, whoſe Difference is d and
Sum z; whence to find the Logarithm of any Prime Number, we
have this General Rule,
то
+
1 / 2
11
%
2 =
>
d+
I
--
X
/
--
2
N
22
3 2
4277
and-*
2
d?
+
32
3
4
4%
525
to
xoto
323
Z
2 d's
1
d
07
1
+
N
2
32
5 z'
:
67: Of the Nature and Conſtruction of Logarithms
To the the given Number add 1 for a Denominator or Diviſor,
and ſubftra& from the ſame given Number for a Numerator or
Dividend, then of the Vulgar Frađion thence relulting, compoſe
all the odd Powers thereot, theſe will form a Series of Numbers
which being divided by their reſpective Exponents, viz. 1, 3, 5,
7, 9, &c. will produce a Series of Quotes, whoſe Sum will be the
Natural Logarithm of the Number propoſed, and being doubled,
will give the Logarithm of the ſame Number, according to Nepers
form.
This Rule is not only very eaſy to be retained in the Memory,
but very proper for the Practice of making of Logarirhms, which
it performs with very great Expedition, as will appear hereafter ;
and is ſo very plain that it may be taught to Perſons of a mean Cam
pacity.
It was firſt publiſhed by Mr. James Gregory, in a Treatiſe of his
written by himſelf, and princes in the Year 1668, wherein that ex-
cellent Geometrician has fhewn che Analogy between the Loga-
rithms and certain Hyperbolic Spaces, and by ſhewing us how to
(quare the one, he has taught us how to conſtruct the other; and the
Truth of the ſame Law has long ſince that time been demonſtrated,
independent of the Hyperbola ; from a Conſideration of the Pro-
perties of the Logarithms only, and by the help of Sir Iſaac Newton's
General Theorem, (demonſtrated in the former Page) for Excrading
of Roots and raiſing of Powers.
Let it be required to find the Logarithm of 2, the firſt Prime
Number
Becauſe a=1, and b= 2, therefore, d=b-a=2-1=1,
d
da
46=1+2=3 ; wherefore, and =ý:
Hence ;-1X,-|--*,7*txtér, &c. equal to titos
f-1, &c will be the Natural Logarithm of 2, and conſequent-
ly 18i-1-121, Trios, &c. will be the Logarithm of the fame Num-
bor in Neper's Form, whercfore if the ſeveral Fra&ions be ad-
ded together and reduced into an Equivalent Decimal, we fhall
have the Logarithm of the Number given ; but becauſe to mul-
tiply by, and divide by 9, will produce the ſame Efect. If
the firſt FraAion be reduced into a Decimal Fraction, and that
be divided by 9, and each fucceffive Quotient by the fame, we
ſhall have a Series of Quotients, which being divided by the leve-
ral Co-efficients 1, 3, 5, 7, 9, 6c the Sum of theſe laſt Quotients
will give the Logarithm of the Number given ; and in order to.
and za
obtain
of the Nallire and Conſtruction of Logarithmis. 673
obtain the Logarithm true to any Number of Places, it is neceflary
to continue the Fraction to one or two more Places than the in-
tended Number of Places the Logarithm is to conſiſt of ; ſo that
to have the Logarithm true to 10 places, it will be convenient
to find the Value of į in Decimals to it or 12 places.
The Operation is as follows.
I
I
7
1
ja lamamin na mambo mlahat ng na
= 333.333333333=
37.037 037.037 its
4.115.226.337 its :
-457.247.371 its
-50.805.263 its
글
​5.645.029 its
627.225 its :
-69.691 its ,
7.743 irs_.!
860 its
90 irs,
333.333-333-333-4-
12.345.679.012+
823.045.268-
65.321.053
5.645 029+
513.185m
48.248
4646
I
1
: 3
456
45+
.
5
The Nat. Logarithm of 2 is
346.573.590280
2
.
The Neper Logarithm of 2 is
693.147.180.560
42
Hence 346. 573. 590. 280. the Natural Logarithm of 2 being
do bled will give 693. 147. 180. 560. for the Natural Logarithm
of 4; and this again, being doubled, will give 1386. 294. 361.
120, c. for the Natural Logarithm of 16, c. in like manner
1386. 294. 361. 520. the double of 693. 147. 180. 560. the Ne-
per's Logarithm of 2 will give the Neper's Logarithm of and
this again being doubled, will give 2772. 588.722. 2 40. for the
Neper's Logarithm of 16, obc
Again, as the Natural Logarithm of 2 being multiplied by 3,
will produce 1039. 720. 770. 840. for the Natural Logarithm of
8; ſo the Neper's Log. of 2 being trebled, will give 2079. 441.
$41. 68o. for the Logarithm of 8, c.
I
Again,
674 of the Nature and Conſtruction of Logarithms.
Again, it the Logarithm of 8 be multiplied by the Logarithm of
1., we hall have the Logarithm of 10 ; and to find the Loga-
rithm of 1 ) we have given, a=1, b=14=, whence z=*, and
d.
d=s, and conſequently š, and more
d
-
LIIII1.111.111
-1.371.742.112 its
16 935.087 its
of
209 075 its
2.581 its
JI1.III III.111
457.247.371
3.387.017
TT
3
29.868
287
-3
32 its i
The Natural Logarrithm fit
I11.571.775.657
657
The Neper's Logarithm 1= 223 143.551.314
As 1039. 720.770. 840. the Natural Logarithm of 8 being
added to 11. 571. 775.657. will give 1151.292. 546. 497. for
the Natural Logarithm of 10 ; ſo 2079. 441. 541. 680. the Ne-
per's Logarithm of 8 being added to 223. 143. 551.314. the New
per Logarithm of 1 i will give 2302.585.092. 994. for the Neper's
Logarithm 10.
As theſe Logarithms which are called Neper's Logarithms, were
thoſe that were firſt publiſhed by Neper the Inventor himſelf in the
Year 1614 ; ſo theſe that are called the Natural Logarithms were
thoſe that were published in the Year 1619. by John Spidel, then a
Profeffor of Mathematics in London, to which they fitted Tables
of Artificial Sines, Tangents, and Secants; but as theſe were found
not ſo proper for the Buſineſs as could be withed for, the Inventor
himſelf, with the Aſſiſtance of Mr. Henry Briggs, alter'd the Form,
and made the Logarithm of 10 to be 1,0000000000, &c. and not
2,30258509299, &c. that is, he made the Number 10 the
10000000000 th, c. Term in the Series, whence the logarithm
of 100 will be 20000000000, 6c that is the Number 100, will
be placed in the 200000090ooth place in the Series, and the Loga-
rithm of 1000 will be the 30000000000, c. that is, the Num-
ber 1000 will be the 3000000000oth Term in the Series,
c. whence the Logarithm of all Numbers between 1 and so wil!
begin from o, that is they will have c, for the firſt Term towards
the
.
of the Nature and Conſtruction of Logarithms. 075
the Left Hand, inaſmuch as they are each of them leſs than the Lo-
garithm of 10, which has Unity in the firſt place ; in like manner
the Logarithms of all Numbers between 10 and 100 begin from
Unity, inaſmuch as they are all greater than 1,00000000, G. and
leſs than 2,0000000000, &c. for the ſame Reaſon the Logarithms
of all Numbers between 100 and 1000 begin from 2, ſince they are
all greater than thc Logarithm of 100, which is fixed at 2, and leſs
than the Logarithm of 1000, which is made equal to 3, after the
ſame manner the Logarithm of all Numbers between roooard 10000
will begin from 3, and the Logarislims of all Numbers betweco 10000
and 100000 will begin from 4; that is, they wiil have a 4 for the
firſt Figure towards the Lelt Hand, c.
The firſt Figure of every
Logarithm towards the i cft Hand, which is ſeparated from the reſt
by a Point is called the Index or Charaâerillic of that Logarithm,
inaſmuch as it ſhews or points out the higheſt or remoteſt Place of
that Number from the place of Unity in the infinite Scale of Pro-
portionals towards the Left Hand. Thus if the Index of the Lo-
garithm b: 1, it ſhews that its higheſt place towards the Left Hand
is the Tenth Place from Unity. If the Index be 2, it ſhews that its
higheſt place towards the Left Hand from the place of Unity is the
Hundredth place; and, as all the Logarithms that have 1 for heir
Index will be found between the roth and looth place from Unity
in the Order of Numbers ; ſo all the Logarithms which have 2 for
their Index, will be found between the rooth and 100oth place, in
the order of Numbers, c. whence the Index or Chara&eriſtick of
any Logarithm is always leſs by one than is the Number of Figures,
which correſpond or anſwer to the Given Logarithm.
Now inaſmuch as all Scales of Logarithms are made up of fimi-
lar Quantities, it will be eaſy from any Scale of Logarithms whatloe-
Ver to form another Scale of Logarithms in any given Racio, and
conſequently to reduce any Table of Logarithis into arother of
any given Form, by ſaying as any one Logarithm in the given
Form is to its Correſpondent Logarithm in the New Form ; ſo is
any other Logarithm in the given Form to its Correſpondent Lo-
garithm in the required Form ; and hence we are taught how to
reduce either Veter's or the Natural Logarithms into the form of
Briggs; and the con fary
For as 2,302, 585, 092, 994, the Neper Logarit'im of 10 to
1.0000000000, the Brigg's Logarithm of 10; fo is any other Loga-
rithm in Neper's Form to the Corre pondent Tabular Logarithm in
Brigg's Form; and inaſmuch as the ż firſt Numbers conſtantly re-
Ssss
main
%6 Of the Nature and Conſtruction of Logarithms.
given.
main the ſame; if the Neper's Logarithms of any one Number
be divided by 2.302,585 oC. or multiplied by 434,294,481,903,
the Ratio of 1,000,000; &i. to 2,302,585, c. as is found by divi-
ding 1,000,000, c. by 2302,58, &c. the Quotient in the firſt Cafe
and the Product in the laſt Cafe will give the Correſpondent Brigg's
Logarithm, and the contráry.
And again as 1,151, 292,546, 497 the Natural Logarithm of 10
is to 1,000,000,000,000 the Brigg's Logarithm of 10; ſo as any
other Natural Logarithm to the Correſpondent Logarithm in Brigg's
Form; wherefore if the Natural Logarithm of any one Number be
divided by 1,151, 292, 546, &c. or multiplied by (the Quotient of
1,000,000, c. divided by 1,151,89, c. equal to) 868,588,963,
806. the Quotient in the firſt Caſe, and the Product in the latter
Caſe will give the Brigg's Logarithm Correſpondent to the Number
Hence by dividing the Neper's Logarithm of 2, which has been
found to be 693,147,180,56, by 2,302,585,092,9, or by multiplying
the ſame Logarithm of Neper, viz. 693,147,180,56 by 434,294,
481, 903, the Quotient in the firſt Caſe and the Product in the lat-
ter Caſe will give 301,029,995,66, for the Brigg's Logarithm of 2.
Again, by dividing 346,573,590,28, the Natural Logarithm of
2 by 1151,292,546,497, Øc or by multiplying the fame Natural
Logarithm by 868,588,963,806, we ſhall have in either Caſe 305,
029,995,66 for the Brigg's Logarithm of 2, as was before found.
d
If inſtead of finding a Decimal Fraction equal to
we divide
the Reciprocal Ratio of the the Neper or Natural Logarithm to
the Brigg's by the reciprocal Ratio of d to 2, and divide that Quo-
tient by the Square of z, and each ſucceſſive Quotient by the ſame
Square of 2, we ſhall have a Series of Quotients, which being divi-
ded reſpectively by the ſeveral Co-efficients 1, 3, 5, 7, 9, c. will
produce a New Series of Quotients whoſe Sum will give the Brigg's
Logarithm of the Number propoſed. If we make uſe of the Reci-
procal Ratio of the Natural Logarithm to the Brigg's; but the
half Sum if we uſe the Reciprocal Ratio of the Nepers to the Brigg's
Logarithm.
Let it therefore be required to find the Brigg's Logarithm of 2 to
10 places, which is as far as the largeſt Tables now publiſhed extend,
from the Reciprocal Ratio of the Natural to the Brigg's Logarithm.
Put therefore R=868,588,963,806, and
Becauſe a=1, b=2, d=1, and 2=3; therefore
Rx+x=;xšxist1x2,567, 6o=L, 2.
289
Z
Of the Nature and Conſtruction of Logarithms. 677
I
5
on ki o moje
289.529.654605
32.169.961.62 its
3.574-440:18 its
397.360.02 its ;
44.128.89 its
4.903.21 its i
-544.80 its
60.53 its
6.72 its
T?
289.5 29.654.60
10.723.320.54
-714.888.04
56.737.14
4.903.21
445.75
41.91
4.03
40
4
I
jarja
74 its
The Briggs Logarithm of 2 is
301 029.995.66
Z
R
2 + x 1
32
Again, if it be required to find the Brigg's Logarithm of 3.
d
Becauſe a=1,b=3, d=2, and z=4; therefore,
=. And,
R
R
tx &c. will be the Logarithm of 3 ; where-
8
fore, if half of R be divided by 4, and each Quotient ſucceſſively
by the fame Number, and thoſe again by the Indices of the odd
Powers, the Sum of theſe laft Quotients will be the Brigg's Loga-
rithm of 3; but inaſmuch as this Series converges very ſlow,
the ſame Logarithm of 3 may be found much quicker, by finding
the Difference between the Logarithm of 2 already known, and
the Logarithm of 3 required.
Put therefore 8=2, b=3, then will z=5, and d=1,7= , and
d?
the common Multiplicator equal to 1', wherefore if R be di-
vided by 5, and that Quote by. 25, s. and thele ſeveral Quotes
3, , 7, Quotes
3.
d
2
2
garithm of 2, "will give the Logarithelaf Ruotes added to the Lo-
2
The
SS SS 2
7
678 of the Nature and Conſtručtion of Logarithms
The Operation is as follows,
; -868.588.963 80=R
asam 173.717 792.765
25-6.948.711.71 its
277.948 47 its
11.117.94 its
444.73 its
17.79 its
2 ई
3
5
173.717.792.76
2.316.237-24
55.589.69
1.588.28
4941
-1.62
5
I
9
3
-71 its
I
I
13
Diff. between the Log. of 2 & 3=
176.091.25905
Wherefore if to the Logarithm of 2, equal 301:029.995.66, be
added 176.019.259,055, the Sum 477.121.254.71, will be the Lo-
garithm of the given Number 3. Again, becauſe 2x2=4. There-
fore.
To find the Logarithm of 4.
Add 301.029.995 66, the Logarithm of 2, to its ſelf; or multi-
ply the ſame Logarithm of 2 equal to 301.029.995.66 by 2, the
Produa 602.059.991.37, will be the Logarithm of 4
And again, becauſe 2x5=10.
To find the Logarithm of 5.
From the Logarithm of 10
y.c00.000 000.00
Take the Logarithm of 2
3.01.029.995.66
There remains the Logarithm of s
698 970.004:34
And becauſe 2x3=6. Therefore,
To find the Logarithm of 6.
To the Logarithm of 3
Add the Logarithm of 2
477.12 1.254.72
301.029.995 66
The Sum will be the Logarithm of 6 -
778.151.250.38
The Logarithm of 6 being known, the Logarithm of 7 will be
readily had by the General Theorem.
For
Of the Nature and Conſtruction of Logarithms.
679
d
2²
For putting 4-6, 6-7, we ſhall have z=13 and d=1, and
=
=ıss, whence xR will be equal to 66.814.535.68, and conſe-
quently,
176 66.814.535.68= 66.814,535 68
-395.352.28 its
131.784.09
467,87
1.98
1
ju
3
177
1m
2.339.30 its
13 84 its
9
t's
8
The Diff of the Log. of 6 & 7=
66-946 789.63
Wherefore,
To the Logarithm of 6
Add the Difference found
778 151.250.38
66 946 789.63
The Sum will be the Logarithm of 7
845 098 040:01
And becauſe 23=8, if 3.301.029.995.66 be trebled, the Product
0.903.089.986.99, will be the Logarithm of 8.
Again, becauſe, 3x3=9=3?, therefore 2 L3=L9'; wherefore,
To find the Logarithm of 9
Multiply the Logarithm of 3 equal to
477.12 1.254.719
By the Number 3
2
The Product will be the Logarithm of 9 954.2424509:438
From the Logarithm of 10 equal to 1.000 000 000.00, may the Lo-
garithm of 11 be eaſily found, by the General Theorem
For putting a =10, b=11, and <=21, and d equal to 1, and
dd
d
=
จาก and
XR=41.361.379.23, and conſequently,
71 * 41.361.37923
41 361,379 23,
**1
31.263 32
141
42.53,
z z
3
93.789.97 its
212.67 its
49 its
I
7
The Diff between the Log of 10 & 11
41392.685.15
This therefore added to 1.000.000 000.00 the Logarithm of ro,
will give 1.041392-685.15, for the Logarithm of 11.
And
680 of the Nature and Conſtruction of Logarithms.
And again, beacauſe 6x2=12, if to the Logarithm of 6 equal
to 778.151.250.38, be added the Logarithm of 2 equal to 301.029
.995.66, 'the Sum 1.079.181.246.04, will be the Logarithm of 12.
Whence,
To find the Logarithm of 13 the next Prime Number. Putting
-d
a=12, b=13, z=25, d=1, we ſhall have
XR=34.743.558
-552; and conſequently,
34.743.55855
34 743.558.55
55.589.69 its
18.529 90
88.94 its
1
625
[
625
ohnt enim
6 2 5
17.79
1
63
14 its
The Diff between the Log, of 12 & 13
34.762.106.26
of 13:
Which being therefore added to the Logarithm of 12 equal to
1.079.181.246.04, will give 1.113.943.352.30, for the Logarithm
And again, becauſe 7x2=14 Therefore,
If to the Logarithm of 7
0.845.098.040.01
Be added the Logarithim of 2-
-0.301.029.995.66
The Sum will be the Logarithm of 14
---1,146.128.035.67
Again, becauſe 5x3=15. Therefore,
If to the Logarithm of s
0.698.970.004.33
Be added the Logarithm of 3
-0.477.121.254.72
The Sum will be the Logarithm of 15
1.176.091.259.05
Again, becauſe 4X4=4'=16. Therefore,
If the Logarithm of 4 equal to
-0.602.059.991.32
2.
Be multiplied by 2
The Product will be the Logarithm of 16-1:204.119.982.64
2
Hence the Logarithm of 17 may be eaſily computed : For
put-
d
ting n=16, b=17, %=33, and d=1, we fhall have XR = 26.
3 20,877.69; and confequently,
ܝܐ
26
of the Nature and Conſtruétion of Logarithint. 681
175 26.320.877.69 26.320.87769
24.169.77 its
22.19 its
8.056.52
3
7099
I099
1
5
4.44
The Diff
' betw, the Log. of 16 & 17
26.328.938.72
17
Which therefore added to 1.204,119.982-69 the Logarithm of
16 before found, will give 1.230 448.921.3, for the Logarithm of
Again, becauſe 2x9=18.
If to the Logarithm of 9 equal to -
0'954 242.509 44
De added the logarithm of 2 equal to 0:301 029.995.66
The Sum will be the Logarithm of 18 equal to — 1.255.272.505.10
Whence to find the Logarithm of 19, the next Prime Number,
d?
d
put a=18, b=19, then will z=37, and d=1, and = 1367, and
XR=23 475.377.400. Whence,
23 475 377.40
23.475.377.40
17 147.83 its
5.715.94
2.51
2
z
T
1379
er mwl
T
3
I
1
'13
12-52 its
The Dif.betw. the Log of r8 & 19
23.481.095.85
This therefore added to 1.255.272.505.10 the Logarithm of 18,
will give 1.278.753.600.95, for the Logarithm of 19; and becauſe
2X10=4x5=20, therefore,
If to the Logarithm of 10
1.000.000.000 20
Be added the Logarithm of 2
301.029.995.66
The Sum will be the Logarithm of 20
1.301.029.995.16
Or,
If to the Logarithm of 4 equal to
Be added the Logarithm of s equal to -
0.602.059.991.33
0.698.970.004:33
The Sum will be the Logarithm of 20
1.301.029.995.66
Hence
682 of the Nature and Conſtruction of Logarithms
Hence the Logarithm of 1 being oc0,000,000, &c.
The Logarithm of 3
is 0,301,029,995,66
3
0,477,12 1,254,72
0,602,059,991,33
0,698,970,004,34
6
0,778,151,250,38
7
0,845,098,040,01
8
0,903,089,986,99
9
0,954,242,509,43
1,000,000,000,00
1,941,392,685,16
. 1,079,181,246,05
13
1,113,943,352,3 1
1,146,118,035,68
15
1,176,091,259,06
16
1,204,119,982,66
17
1,230,448,921,38
18
1,255,272,525,10
19
1,278,753,600,95
1,301,029,999,56
o con ant
10
II
1.2
JA
20
After the ſame manner may the whole Table be conſtructed.
But if the Logarithm be computed to Seven Places only, which
is as far as the Tables commonly uſed do extend ; which are ſufficient-
ly exa& for all Uſes, except in very extraordinary Caſes, two Thirds
of the Trouble will be ſaved,
The Excellency of this Method will better appear, by comparing
the ſeveral preceding Computations of the Prime Numbers toge-
ther; by which it is obvious, that as the Prime Number it ſelf in-
creaſes, the Operation decreaſes; and the fewer Steps are requiſite,
till at laſt the firſt Step will fuffice to find the Difference to any Num-
ber of Places requifite.
Thus in the common Tables which extend but to Seven Places
in Decimals, when the Difference d becomes the one Two Hun-
d
dredth Part of the Sum 2, the firſt ſtep-R, will give the Diffe.
rence between the Logarithms true to the ſame Number of Places
2
For
of the Nature and Conſtruction of Logarithms. 683
For if 868.588.9 be divided by 201, the Sum of the Numbers 100
and 101, the Quotient 0043213, will give the Difference between
the Log. of 100 and 101 true to 7 Places, which therefore being addd
to 2.0000009, will give 2.0043213 for the Logarithm of 101 ; fo
that the Logarithms of the Prime Numbers under 100 being found,
the Logarithm of all the Prime Numbers above 100 may be found
by one ſingle Diviſion, and the Logarithms of all the intermediate
Numbers by the Addition and Subſtraction of those already found.
As the Addition of Logarithms anſwers to the Muiciplication of
the Numbers they repreſent, and the ſubſtracting one Logarithm from
another, produces the Logarithm of the Quotient reſulting from
the Diviſion of the one of the correſponding Numbers by the other,
ſo when we are about to find the Logarithm of any Prime Nun:ber,
if the Numerator of the Frađion reſulting (by adding and ſubſtract-
ing of 1, to and from the given Number; be more than 1, (inaſ-
much as the Series formed from it deverges but very ſlowly) the
Logarithm of ſuch a Number is more readily inade by the Loga-
rithms of its two Compoſers or Factors, whoſe Fac or Product is
equal to the given Number, wherein one of the Factors is ſuch a
Number whoſe Logarithm is already known, the other ſuch a Num-
ber as that by Addition and Subſtraction of 1, there may reſult
ſuch a Vulgar Fra&tion as ſhall have i for its Numerator ; by which
means the ſeveral Powers thereof may be formed with greater Eale,
and conſequently all the Operations more expedited and facilitated
than when the Numerator is a Number greater than Unity ; and by
how much the greater is the Denominator than Unity, by ſo much
the ſwifter does the Series converge.
And in the Choice of theſe Factors conſiſts chiefly the Expeditious
and Eaſy Conſtruation of a Table of Logarithms.
Thus if we are about to make the Logarithm of 3 by two Num-
bers whoſe Fac is 3, the two Numbers moſt convenient will be 2
and I i, for the Logarithm of 2 being known, and í being added
to, and ſubſtracted from 1i, will give * ; the Square of which be-
ing zs, will give us 2,5 for a continual Divilor, which will cauſe
the Fraction to converge inuch ſwifter than by dividing by +, the
Denominator of the Fraction reſulting from the Addition and Sub-
ftra&ion of 1, to and from the given Number 3:
Again, if we are about to make the Logarithin of 7, the next Prime
Number, it is much eaſier performed by conſidering the Number 7
as made up,or produced by the Multiplication of o by it, (which
Logarith. of 6) being a Compound of the Logarithms of 2 and 3,
Titt
ſup-
ز
684 Of the Nature and Conſtruction of Logarithms.
w
fuppoſed to be salready found) whence the Derfominator of the re-
ſulting Fraction will be 13, and its Square 169 for a common Di.
viſor,) then by the Additionof 1, to or from the given Number 7,
which will produce &, the Square of which is is for a common
Multiplicator.
In like manner the Logarithms of the firſt three Prime Numbers
2, 3 and 5, are much more eaſily formed by their component
Factors I, I, I, than by the Numbers themſelves; as any
one may ſee who will but be at the Pains to compute them.
As the Logarithms cfi, 10, 100, 1000, 10000, c. in Nepers
Form, are made equal to 0, 1, 2, 3, 4, 6c.
Hence it is that the Logarithms of all Numbers which increaſe
or decreaſe in a Ten-fold Proportion, differ from each other only
in their Characteriſtics or Indices, the Fractional Number remain-
ing conſtantly the ſame : And hence it is, that the Fractional
Number .30102.99956.6 which is the Logarithm of 2, when the
Index is a Cypher, becomes the Logarithm of 20, when the Index is
made equal to 1. the Logarithm of 200, when the Index is put equal to
2, and the Logarithin of 2000, when the Index is put equal to 3, &c.
and the ſame Logarithm of .30102.99956.6 which is the Logarithm
of 2 when it has nothing for its Index, becomes the Logarithm of
Tö, when the Index is made - the Logarithm of tão, when the
Index is --2 the Logarithm of toot, when the Index is made ---3,
Gr. the Frađional Number remaining ſtill the ſame ; for becauſe r is
to 2, as ro to 20; as 100 to 200; as 1000 to 2000;c the diſtance
between the Numbers 1 and 2, 10 and 20, 100 and 200, 1000
and 2000, c. are equal to each other, and the Number of Termis
in the infinite Scale of Proportionals between the Number 1 and 2, 10
and 20, 100 and 200, 1000 and 2000, that is the Logarithm of that
diſtance will be the ſame; and ſince the Logarithm of 1 is 5.000.000.
000 0, of 10 is 1.000.000.000.0, of 100 is 2.000.000.000 o, and the
Logarithm of 2 is 30102.99956.6 the Logarithm of 20 will be
1.30102 99956.6, of 200 2.30102299956.6, of 2000 3,3010299956,
and of 20000 4:30 102.999566 &c. wherefore, inaſmuch as the
Numbers 37251, 3725.1, 372.51, 37.251, 3.7251-37251,
037251, .100.37251, are concinual Propoportionals, viz. in the
Ratio of 10 to 1, the Number of Terms between each in the in-
Sinite Scale of Proportions will be equal : And inaſmuch as the
Logarithm of 37251 is 4 57113.79358, the Logarithm of the other
Numbers will be 3-57113.79358 2.57113.79358; 157113.79358,
0.57113.79358,-1.57113.79358, -2.57113.79358, -3.57113.
79358,0
This
,
of the Nature and Conſtruction of Logarithms. 685
This Property of having the ſame Number repreſent or become
the Logarithm of an infinite Series of Numbers by increaſing or dimi-
niſhing the Characteriſtic by Unicy, is peculiar only to the Brigg's
Form, and gives it a Preference beyond all other Kinds of Loga-
ritlims hitherto known, not only for the ready finding of the Lo-
garithm to any given Number, and the contrary; but for finding
Logarithms to Numbers and Numbers to Logarithms which exceed
the Bounds of the Table.
It has been already thewn, if z ſtand for the Sum of any two
d
Numbers a and b, and d for their Difference, that R, will give
the Difference of the Logarithms between a and b true to 7 Places
when the Number is greater than 100; and the ſame will hold
good to 10 Places when the Number is greater than 1000. Let it
therefore be required to find the Logarithm of 37 271 256, from the
next leſs Tabular Logarithm to the given Number, by the ſame Law.
inaſmuch as the next leſs Tabular Numer to the given Numter
is 37271000, if 868.588 963,8 equal to R, be divided by 291181,
the Ratio of 74542256, the sum of the given Numbers
37271256 and 37271000 to their Difference 256, the Quotient
29829 added to 57137.10453, the Frađional part of the Logarithm
anſwering to 37271000, will give -5713740282 for the 'ractional
part of the Logariihm required ; to which prefixing 7 for the Index
becauſe the given Number conſiſts of 8 Figures, we ſhall have
7.5713740282 for the true Logarithm of the given Number
37271256.
But inaſmuch as while the Numbers themſelves increaſe, the Dif-
ference of their correſpondenc Logarithms conſtantly decreaſe, and
approach nearer and nearer to an Equality of Ratio with the Num-
bers themſelves; if the given Number conſiſts but of a few Figures
more than the Tabular Numbers, the proportional Logarithmic
Augment or Decrement may be found, from the common Difference
between the next greater and lefler Tabular Numbers, to the given
Number, by the common Rule of Proportion, by ſaying;
As Unity (with as many Cyphers as the Number of Figures in
the given Number, exceeds the next leſs Tabular Number)
Is to the Difference between the Tabular Logarithm of the two
next Numbers, above and below the given Number :
So is the Exceſs of the given Number above the next leſs Tabu-
lár Number:
Tttt 2*
To
686 Of the Nature and Conſtruction of Logarithms.
To a fourth Proportional; which being added to the Tabular
Logarithm of the next leſſer Number to the given Number, will
give the Logarithm of the given Number.
Thus in the foriner Example, to find the Log. of 37271256, from
the Logarithm of the next greater and lefler Tabular Numbers,
it will be,
As 1000 tO 116521 the Difference between 57138.26974 the
Fractional Part of 3727200, the next greater Tabular Namber to
the given Number, and 573710453 the Fractional Part of the next
leſler Number, to the given Number; fo is 256 the Exceſs of the
given Number above the next leſs Tabular Number, to 29829 the
proportional Augment; which therefore added to 5713710453,
the Fractional Part of the Logarithm of the next leſs Tabular Num-
ber will give 5713740282, for the Fractional Part of the Number
to which prefixing 7 for the Index, we ſhall have 7.5713740282 for
the Logarithm of the Number 37271256, the ſame as was found
before,
Hence the Natural Sine, Tangent, or Secant of any Arch being
given, we are taught how to find the Artificial Sine, Tangent,
or Secant of the ſame Arch, viz. by entering the Logarithmic
Table, and finding out the Tabular Logarithm of the given
Number, expreſſing the Artificial Sine, Tangent or Secant of the
given Arch; for thatNumber will be the Logarithmic Sine, Tangent
or Secant of the Arch propoled : Thus the Natural Sine of 80 deg.
being 9848,0775.30 ſuch parts as the Radius is 1.0000 0000.00
the correſpondent Logarithmic Sine will be found to be 9.9933.
514589, the Radius of the Artificial Sine being 10.0000 0000.00.
In like manner the Logarithmic Secant of 10 degrees will be
found to be 10.0066485411, the Natural Secant being 1.0154.267
and the Logarithmic Tangent of so Degrees, will be found to be
10.076 1.864698, the Natural Tangent being 11917536.
After the ſame manner may the Logarithmic Siue, Tangent, or
ecant of any Arch be found, and conſequently the whole Table of
Artificial Sines. Tangents and Secants may be readily formed; the
Tables of Logarithms and Tables of Natural Sines, Tangepts, and
Si cants, beirg firſt given.
But the Logarithmic Sine, Tangent or Secant, of any Arch may
be directly found, independent of the Tables of Logarithms; for
S-1
by putting S for the Sine of any Arch, and
9
St.
we ſhall have
1
Of the Nature and Conſtru&tion of Logarithms. 687
1.
N
72
6
8
IO
Ioney
25 2
a
15
14 15
xatiqi t'; q*+*9', &c. for the Logarithmic Sine of the
fame Arch ; and if in the room of S in the former Series, we put
t for the Tangent, or C for the Secant of any Arch, the ſame Series
will produce the Logarithmic Tangent or Secant of the given Arch.
But if inſtead of Š the Right-line in the former Series, we put a
for the Length of the Arch it felf, then will xn-a-vat
a a
&c. be the Logarithinic Co-ſine
of the ſame Arch in Brigg's Form, the Radius being aſſumed equal
to 10.000 000.0, and n equal to 1.30585, Oc. Thus,
Suppoſe it were required to find the Logarithmic Sine of 8. De-
grees, from the ſame Series
Becauſe by Se&t. the ad of Part the 2d, where the Radius is af-
ſumed equal to Unity, the Arch of the Semicircle will be 3 1415.
9265.358, if this be divided by 18, the Quotient 1745.3292.519
will be the Length of the Arch of 10 Degrees in the ſame Parts:
Put this therefore equal to ,
Then will {a? be equal to
0152-3087.10
And , at equal to
7732.65
And aequal to
62.81
And são a' equal to
58
And conſequently, ta’tréat+45 as tajži na = 0152.0883.14
ܘ ܕ ܪ ܐ
This therefore taken from n=23025.8509.29, will leave 2.2872•
7626.15, which being multiplied by 4342.9448.190, will give
9933.5145.89 for the Fractional Part of the Logarithmic Sine of 80
Degrees, to which therefore prefixing 9 for the Index, becauſe the
Radius is put equal to 10, we ſhall have 9.9933514589 for the
Brigg's Logarithmic Sine of 80 Degrees; and after the ſame manner
may the Logarithmic Sine of any other Aich be found, independant
of the Logarithmic Tables, and the whole Table of Sines con-..
ſtructed.
The Logarithmic Sines being thus obtained, the Logarithmic
Tangents and Secants may be found, by the ſimple Addition and
Subſtradion of the Logariihmic Sines, to or from each other.
For becauſe the Co-line of any Arch is to its Right-fine, as the -
Radius is to the Tangent of the ſame Arch.
If
688 of the Nature and Conſtruction of Logarithms
If from theSum of the Logarithms of the Radius and Right-line of
an Arch be taken the Logarithmic Co-fine, the Remainder will be
the Logarithmic Tar.gent of the ſame Arch: Thus if it were re-
quired to find the Logarithmic Tangent of 80 Degrees, from the
Logarithmic Sine and Co-fine of the ſame Arch.
To the Logarithmic Sme of 8o deg.
9.99335.14589
Add the Logarithmic Radius
IO 00000,00000
From this Sum-
Take the Logarithmic Co-fine of 8o deg.
19.99335.14589
9.23967.02300
The Remain will be the Log. Tangent of 80 deg. 10.75368.52289
Again, becauſe the Radius is a mean Proportional between
the Co-fine and Secant of the ſame Arch, if from twice the Radius
be taken the Logarithmic Co-fine of an Arch, the Remainder will
be the Logarithmic Secant of the ſame Arch: Thus, if it were
required to find the Logarithmic Secant of 10 Degrees, from the
Logarithmic Co-ſine of the ſame Arch
From twice the Logarithmic Radxus-
20 20000.00000
Take the Logarithmic Co ſine of 10 deg.
-9.99335.14589
Remains the Logarithmic Secant of so deg.
- 10 00664.85411
Again, inaſmuch as the Radius is a mean Proportional between
the Logarithmic Sine and the Logarithmic Co-ſecant of the fame
Arch.
If from twice the Logarithmic Radius
20 00000,00000
Be taken the Logarithmic Sine of ro deg.
9.23967.02300
There will remain theLogarithmic Secantof 80 deg. 10-76032.97700
So that the Logarithmic Sine and Co-fine of any Arch being
known, the Logarithmic Tangent, Secant, and Co-ſecant of the
ſame Areh is readily found.
But the Logarithmic Secant it ſelf of any Arch may be dire&tly
found, without the help of the Logarithmic Sine.
For if we put a for the length of the given Arch, then
+ ka't11m trsa', ci will be the Logarithmic Secant of the
fame
1
XR
of the Nature and Conſtruction of Logarithms. 689
ſame Arch : Thus if it were required to find the Logarithmic Se-
cant of so Degrees, from the length of the Arch of 10 Degrees e-
equal to 1745,3293,52, &c. before found,
To į a' equal to
0152.3087.10
Add at equal to
7732.65
Alſos a equal to
62.81
And 51: aequal ro
-58
I
The Sum equal to
0153.0883.14
Being added to n equal to 2 3025 8509 29, will give 2.3178.
9392.43, which being multiplied by 4342.9448.1903, and adding
the proper Index, will give 10.0066.485411, for the Logarithmic
Secant of 10 Degrees.
The Logarithinic Secant being thus obtained, the Logarithmic
Sines and Tangents themſelves may be found, by the Addition and
Subſtraction of the Logarithmic Secants, to and from each other,
according to tlae former Rules.
xatárt
}&
equal to the Logarithmic Tangent of 45 Degrees to the given
Arch a in Brigg's Term. If it be required to find the Logarith-
mic Tangent of 50 Degrees, put a =1745.3292.52, &c. the length
of the Arch of 10 Degrees, the Radius being 1,000000, &c. Then,
Toa the length of the Arch of 10 deg.
1745.3292.42
Add al egaal to
8.8609.52
Allo 4a' equal to
674.80
And 74 aequal to
-5.97
And 77fc a? equal to -
-6
The Sum
1754.2582.97
Being multiplied by 4342:9448.19, will give 0761.8646 98, to
which prefixing 10 for the Index we ſhall have 10.07618.64698 for
the Brigg's Logarithmic Tangent of so Degrees
Again, becauſe the Radius is a mean Proportional between the
Tangent and Co-tangent of an Arch.
If from twice the Radius be taken the Logarithmic Tangent of
an Arch, the Remainder will be the Logarithmic Tangent of the
Complement of that Arch to a Quadrant:
Sup-
690 Of the Nature and Com Praption of Logarithms
Suppoſe it were required to foc Logarithmic Targent of
40
Degrees, from the Logarithmic te gent of 50 Degrees first given
From twice the Logarithinic Rasius
Take the Logarithmic Tangent of 50 Degrees-10 07618.64098
The Remainder will be the Log. Tang. of 40 Deg: 9.92381.35302
20.00000.00000
Hence the Logarithmic Tangent being made for either half of
the Quadrant, the reſt may be found by Subftra&ion.
The Logarithmic Tangents being thus obrained, the Logarith-
mic Sine and Secant may be readily found by Addition and Sub-
ſtraction only, by the Rules delivered in Page 688.
And hence we are taught various Methods for computing the
Logarithmic Sines, Tangenes and Secarts of any Arch, and conſe-
quently a ready and expeditious Way of Conſtructing the Tables
themſelves.
It has been already ſhewn, that Xx—***+*—*%4, 6.
I
7
I
N
I
•N
N
L' +
equal to 1-tox. I will be the Logarithm of the Ratio of 1 to 1+x,
where n repreſents any infinite Index, and x any Number taken
at Pleaſure. Now if I be put for the Logarithm it felf, then
11 / 을
​itxho will be equal to L, conſequently, itx will be equal to
I+1, and itx equal to Łti and becauſe Ltr =it nit
2 n-n ni +3ni tan n4613-trinon
L? +
L*, &c. by
24
the General Theorem, in Page the 669th, when n is finite, when
n becomes infinite L+I will be equal to itnL+in’L+?n?L?
+ n°14 tiça n'L' +10nºl•+574om"",=1+x. In like
*************4+*3 bx', &c. e-
22
qual to 1-1q is the Logarithm of the Ratio 1 to 1-X, 1-14
2
6
•N
I
manner, becauſe
N
2
n
n
will be equal to L, conſequently i-*=1-1, and Ix=1- L,
where-
of the Nature and Conſtruction of Logarithms. 691
n
+
f
2
I
Or,
et
wherefore by the ſame General Theorem, 1-x=1-1, will be e-
qual to 1-nl to n? L'—*n'Z' trin'24-36 n'l'ttt néL'-
soon? L', &c.
Whence I
+nl+fm?***n'I.' + ts mit L
n' L', &c. which is a General Theorem for finding the Number
from the Logarithm given, of any Form whatſoever.
+
And becauſe n=1000000,6c. in Nepers Form i +2+
iĽ+IL’t zil+ tris L', &c. will give the Number anſwe-
ring to any Logarithm in Nepers Form.
Hence any one Term of the Ratio whereof L is the logarithm,
being given, the other Term will be obtained readily; for putting
a for the lefler of the two Terms, and b for the greater, axl -
+L. L' +424 FEL', &c. will give the greater ; and bx i
-L+ {L'-;L+TL', &c. will give a the lefler, in Ne-
per's Form.
axiton Ltın? L’tn 13 + sänt 1++:L', &c.=b: And,
axi-nLt į n’L'- n L +24n+ L+ - 1L", &c=a, in any
Form
Let d ſtand for the Difference between the given Logarithm and
the next neareſt Tabular Logarithm; then will axitd. tid +
*d +24 d'+its d', &c.=N.
Or,
bx,1-d.Lid-d +2nd+mikod', &c.=N, the Number requi-
red in Nepers Form.
AXIn d+ d+3 aº d+s4=* 4*+s nº dº, &c=N.
And,
bxr-nd + Ž na d—ind tzant d' thons d', &c, =N,
the correſpondent Number in Logarithms of any Species
Let it therefore be required to find the Number anlwering to
7,5713770282 in Brigg's Form.
Becauſe 7.5713710453 (the neareſt tabular Logar. to this) is leſs
than the given Number, put 37271000 the Number anlwering to
it, equal to a, and the Difference between the two Logarithms
0000029827 equal to d, then will n equal to 2.30258, &c, multipli-
ed by d produce o000068683 ; which being increaſed by s, and mul-
tiplied by 37271000, the next neareſt Number in the Table, will
give 37271255.988 for the Number anſwering to the given Loga-
rithm : Or, if oooo068683 be multiplyed by a, equal to 37271000,
Uuuu *
the
7
Or,
ol
692 of the Nature and Conſtrution of Logarithms.
the Product 255.998 increaſed by a, equal to 37271000 will
give 37271:55,988 for the Number required; and after the ſame
inanner inay the Number anſwering to any given Logarithm be
found
Hence and from the General Theorem, may the Number anſwering
to any Logarithm be found, and conſequently an Anti Logarithmic
Carion be conſtructed, ſhewing the Natural Numbers anſwering
to every Logarithm, fet down in a Natural Order from 1 to 100 000,
whence the Number anſwering to any Logarithm might be
fo: nd with the fame Eaſe as we find the Logarithm for any Num-
ber in the Logarithmic Canon, which would render the Logarith-
mic Tables compleat, and ſich a Canon was confirućt :d ma-
ny Years ago, as Dr. Wallis informs us, but loſt for want of due
Encouragement to Prin: it.
But the Number anſwering to any given Logarithm may be found
much more eaſily, and ſufficiently exact for common Uſes, by the
common Rules of Proportion.
For having found out in the Logarithmic Canon, the rext leſs
Logarithm to the given Logarithm, fubftract it from the next greater
Tabular Logarithm, and call the Remainder the Tabular Difference.
Then ſay,
As the Tabular Difference,
To an Uaire, with as many Cyphers as there are Places wanting
So is the Exceſs of the given Logarithm above the next lefs Ta-
bular Logarithim,
To a fourth Proportional Number.
Or,
So is the Defect of the given Logarithm, to the next greater Tabu-
iar Logarithin,
To a tourth Proportional:
Which fourth Proportional being added in the firit Cafe to the
Tabular Number, ſtanding againſt the next lefs Tabular Logarithm ;
to the given Logarithm but in the ſecond Caſe ſubſtracted from the
Tabular Number ſtanding againſt the next greater Tabular Loga-
rithm, to the given Logarithm, will give the abſolute Number an-
ſwering to the given Logarithm.
Thus if it were required to find the abſolute Number anſwering
to 7.5713740282, a given Logarithm, as in the former Example.
Searching the Tabular Canon, I find the next leſs Tabular Lo-
garithm to che given Logarithm to be 7.5713710453, and the ab-
ſolute Number anſwering to it to be 37271.000; Likewiſe in the
fame
;
of the Nature and Conſtruction of Logarithms. 693
fame Canon I find the next greater Logarithm to the given Loga-
rithm to be 7.5713826974, and the abſolute Number anſwering to
it to be 37272000; then I ſay, as 116521, the Difference betwixo
te two Tabular Logarithms, to 1000; lo is 29829, the Exceſs
of the given Logarithm above the next leſs Tabular Logarithm, to
255.99, the fourth Proportional; which being therefore added to
37271000, the Number anſwering to the next lefs Tabular Loga-
richin, will give 37271255-99, or 37271256, for the abſolute Num-
ber anſwering to the given Logarithm ; or ſo is 86692, the Exceſs
of the next greater Tabular Logarithin above the given Logarithm,
to 744, a fourth Proportional; which being ſubtracted from
37272000, the Number anſwering to the next greater Tabular Lo-
garithm, will give 37271256, for the Number anſwering to the gi-
ven Logarithm agreeing with the former Aníwers.
From the Principles here laid down, leveral other Series might
be drawn, and conſequently ſeveral other Methods inight be thew'ı
for the coſtructing of Logarithms, and the contrary; but as the
chict End of this section was only to explain the Nature of them,
and ſhew the Manner how they may be made, I fall leave the far-
ther Proſecution of them for this time, being well aſſured that there
is norning here delivered, but may be ciearly underſtood by any
Perſon who is but moderately skill'd in the Fundamental Rules of
coinmon Algebra.
As my chief View in this Section, as well as throughout the whole
Work, has been to deliver every thing in as plain a manner as can
be, and ſo as that it may be underſtood by the meaneſt Capacity ;
'cis for this Reaſon that I have explained the Nature of Logarithms
by the lielp of the Logarithmic Curve ; wherein the Logarithms or
Sum of the Ratio's are expounded by Right-lines, and the Rario's
themſelves by the Magnitudes of the Lines compared with each other.
For altho' the Subject of Logarithuns be a Thing purely Arith-
metical, wherein nothing beſides the Nature and Properties of
Numbers are to be conſidered, yet this Way of repreſenting them
by Lines, as it is a Thing allowed of, and practiſed by the greateſt
Geometricians, ſo it certainly fixes a more juſt, diſtinct, and laſting
Idea in the Mind than can be done from the Conſideration of abſtra-
Eted Numbers only ; wherein the forming a true Conception of the
Nature of Ratio's and Ratiunculæ, is a Thing nor ſo eaſily attain-
ed by Beginners; and of theſe as well as of other parts of purc
Mathematicks, that Perſons have formed to themſelves falſe Notions,
U uu u 2 *
and
694 Of the Nature and Conſtruction of Logarithms,
and their Minds have not rightly been informed, there need: no
greater Proof than the ſeveral Miſtakes and Blunders that have
been committed by Perſons, in Things purely Mathematical, and
which would admit of the ſtrictest Demonſtration.
Logarithms are not only very uſeful in facilicating Trigonometrical
Calculations, for which they were principally invented; but in the
Solution of all Arithmetical Queſtions wherein Multiplication and
Diviſion is concerned, and are capable of giving Solutions to jome
Problems that are ſo operoſe by the common Way,that it is next to an
Impoſſibility to give an Anſwer; for inaſmuch as Multiplication is
performed by the Addition of the correſpondent Logarithms, and
Diviſion by ſubſtracting of one Logarithm from the other, ſo Powers
are raiſed by inultiplying the Logarithm of any Number by the
Ir.dex of the Power, and Roots are extracted by dividing the Lo-
garithm of the Number, by the Index of the given Power, and
this it performs to ſufficient degrees of Exactneſs for almoſt all
Uſes, let the Powers be never ſo high ; which is impoſible to be
done by the common Methods of Operation; and of this I ſhall
give an Inſtance, and that is in the purchaſing of Annuities.
If a be put for any Annuity, p for the preſent Value, x the A-
mount of one Pound for one Year at any Rate, and n for the Num-
ber of Years. Then,
Becauſe x :1: :2:
the preſent Value at the iſt Years End.
*
And, x:1:
:
the preſent Value at the 2dYearsEnd,
* *
And, * :1::
the preſent Value at the 3d YearsEnd
***
a
a.
+
+
will
&c. to
3
X
د کی
Then +
+
X
be the preſent Value equal to p.
72
But as
:
or as x : 1:: pa
: 0
n
X
go
Therefore, -
no X-R.
And, p x-p=-
2.
And
Of the Nature and Conſtruction of Logarithms. 695
And p
n
XI
3
i
X2
Let it therefore be required to find the preſent Value of an An-
nuity of sol. per Ann. to continue for Ninety Years, at the
Rate of 5 per cent. per Ann. here a =Son=90, and x =1.05. Now
in order to obtain the Anſwer, we inuſt find the Ninetieth Power
of x, or of 1.05 ; that is, we muſt mulcipiy 1.05 Ninety times into
its ſelf, and how operoſe that is by the Common Way any one may
jiunge ; but by the Logarithms it is done with the greateſt Eaſe ;
for if .c211893 the Logarithm of 1.05 be multiplied by 90,
the Product 19070370 will be the Logarithm of the Nintich
Power of 1.05, which being therefore ſubſtracted from 1.6989706,
the Logarithm of a equal to so, will leave 9.7918330, the Lo-
garithm of .619203, which being ſubfiracted from so, and divi.
ded by x-15.05, will give 987.61594 equal to 9871. 12 s. 3 d.
for the Value of the Annuity fufficiently exact. And as from the
gencral Equation -
px - may Theorems be deduced for
finding the Value, Anruity, Rate or Time, and as the principal Dif-
ficulty in applying thele Theorems to Pra&ice confifts in raiſing the
Powers of x in the Theorems, and as theſe are performed with
the greatet Eaſe by the Help of thele Artificial Numbers, it ſhews
of what greac Benefit the Invention of theſe Numbers are to
· Mankind.
I ſhall give one Inſtance more of the great Uſe of Logarithms
in Arithmetical Calculations, and that is in the Caſe of Sefa an
Indian, as it is relared by Dr. Wallis in his Opus Arithmeticum from
Alſephad (an Arabic Writer) in his Commentaries upon Tograius's
Verſes, who having firſt found out the Game of Chere, and ſhew'd
it to his Prince Sheram, the King who was highly pleaſed with it,
bid him ask what he would for the Reward of his Invention;
whereupon he asked that for the firſt liicle Square of the Chelſe-board
he might have one Grain of Wheat given him; for the ſecond, two,
and ſo on; doubling continually according to the Number of
the Squares of the Chelle-board, which was ſixty four. And when
the King, who intended to give a very noble Reward was much
diſpleaſed that he had asked ſo trifling an one, Sella declared that he
would
!
696 of the Nature and Conſtruction of Logarithms
would be contented with this finall one The Reward he had fixed
upon was ordered to be given him, but the King was quickly aſto-
niſhed, when he found that this would riſe to ſo vaſt a Quantity
that the whole Earth it ſelf could nor furniſh our ſo much Wheat;
and how great the Number of theſe Grains is, may be found by
doubling One continually 63 times; ſo that we may get the Num-
ber that ftands in the laſt Place, and then one time more, tor to
have theSum of all, ſince the double of the laſt Term, leſs by one, is
the Sum of all the Terms : Now thiswill be nore expeditioully done
by the help of Logarithms and accurately cnough too for this purpoſe.
For if to the Logarithm of 1 (which is o) we add the Logarithm
of 2, (which is 3010300) multiplied by 64, the Product will be
19,2659200, and the abſolute Number agreeing to this will be
great
er than 18446,00000,00000,00000, and leſs than 18447,00000,
00000,00000.
And laſtly, as all Operations in Natural Numbers that are
performed by Multiplication and Divilion, are performed by the Ad-
dition and Subſtraction of theſe Artificial Numbers, he that knows
how to manage the one cannot be ignorant how to apply the o-
ther, in all Arithmetical Caſes whatſoever.
But before I conclude this Sestion, it may not be amiſs to give
fome Account of the Manner how the Lines upon the Gunter Scale
are formed.
The Line of Numbers is no other than a Logarithmic Scale of
Proportionals, wherein the Diſtance between each Diviſion is
equal to the Number of mean Proportionals contained between the
two terins, in ſuch Parts as the Diſtance between 1 and 10 is
1000, Gr. Wherefore,
If the Diſtance between 1 and 10 upon the Scale be made equal
to 10000, 6c. cqual Parts. And.954, the Logarithm of 9 of the
fame Parts be ſet of from 1 to 9, it will give the Diviſion ſtanding a
gainſt the Number 9. In like manner, if 903, c. 845, c. 778,&c.
which are the Logarithins of 8, 7, and 6, &c. of the ſame equalParts
be ſet off from 1 to 8, to 7, to 6, &c. they will give the Diviſions
anſwering to the Number 8, 7, 6, c. upon the Scale.
Again, If 995,991, 986, 6c of the ſame equal Parts, which
are the Logarithims of 99, 98, 97, &c. be ſet of from 1, &c to-
wards 10, they will give the ſeveral Diviſions upon the Scale that
anſwer to theſe Logarithms, and after the ſame manner may the
whole Line be disided.
Again,
Of the Nature and Conſtruction of Logarithms. 697
Again, If Numbers anlwering to the Natural Signs and Tan-
gents of any Arch, in ſuch Parts as the Radius is 10000, oc be
found out upon the Line of Numbers right againſt them in the ſame
Scale will ſtand the reſpective Diviſions anſwering to the rulocatie
Arches, for becauſe the Narural Siae of 30 Degrecs is 5000, cc.
right againſt 80, upon the Line of Numbers ſtands the Sie 9: 30
degs. upon the line of Sines; and becauſe the Natural Tannent
of zo dets. is 577 of the ſame Parts right againſt 577 in the Line
of Numbers ſtands thc Tangent of 30 deg, upon the Line of Tan-
gents, or which is the ſame thing, If the Diſtance between 50
and 100 upon the Line of Numbers be ſet of in the Line of Sines
from go towards the Left-hand it will give the Point anſwering to
the Sine o zo deg. lipon the Scale; alſo if the Diſtance between
57,7 and 100 upon the Line of Numbers be ſet off in the Line of
Tangenes from 45 towards the Left-hand, it will give :le Point
upon the scale anſwering to the Tangene of 30 deg. and after the
faine manner may the whole Line of Sir's Tangents and Verſed
Sines be divided
The Lines being thus conſtructed all Problems relating to Arith-
meric, Trigonometry, and their dependent Sciences may be reſolved
by the Extent of the Compailles only; and as all Queftions whatſo-
ever are reduceable into Proportions, the general Rule is this, to
extend the Compaſes from the firſt term to the ſecond, and the
fame Extent of the Compaſſes will reach from the third term to the
fourth, which fourth Term muſt be ſo contrived as to be the thing
required, which a little Practice will render eaſy.
The End of the firſt VOLUME,
Jul 12 1920
.
1
!
-
:
*
,
1
1
i
.
.
1
3
:
(
,
1
WONINIWASANNONS
UNIVERSITY OF MICHIGAN
3 9015 06725 2174
C
the most
.