This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal. The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission.
Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford.
EEBO-TCP is a partnership between the Universities of Michigan and Oxford and the publisher ProQuest to create accurately transcribed and encoded texts based on the image sets published by ProQuest via their Early English Books Online (EEBO) database (http://eebo.chadwyck.com). The general aim of EEBO-TCP is to encode one copy (usually the first edition) of every monographic English-language title published between 1473 and 1700 available in EEBO.
EEBO-TCP aimed to produce large quantities of textual data within the usual project restraints of time and funding, and therefore chose to create diplomatic transcriptions (as opposed to critical editions) with light-touch, mainly structural encoding based on the Text Encoding Initiative (http://www.tei-c.org).
The EEBO-TCP project was divided into two phases. The 25,363 texts created during Phase 1 of the project have been released into the public domain as of 1 January 2015. Anyone can now take and use these texts for their own purposes, but we respectfully request that due credit and attribution is given to their original source.
Users should be aware of the process of creating the TCP texts, and therefore of any assumptions that can be made about the data.
Text selection was based on the New Cambridge Bibliography of English Literature (NCBEL). If an author (or for an anonymous work, the title) appears in NCBEL, then their works are eligible for inclusion. Selection was intended to range over a wide variety of subject areas, to reflect the true nature of the print record of the period. In general, first editions of a works in English were prioritized, although there are a number of works in other languages, notably Latin and Welsh, included and sometimes a second or later edition of a work was chosen if there was a compelling reason to do so.
Image sets were sent to external keying companies for transcription and basic encoding. Quality assurance was then carried out by editorial teams in Oxford and Michigan. 5% (or 5 pages, whichever is the greater) of each text was proofread for accuracy and those which did not meet QA standards were returned to the keyers to be redone. After proofreading, the encoding was enhanced and/or corrected and characters marked as illegible were corrected where possible up to a limit of 100 instances per text. Any remaining illegibles were encoded as <gap>s. Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor.
The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines.
Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements).
Keying and markup guidelines are available at the Text Creation Partnership web site.
Imprimatur;
COSMOGRAPHIA, OR A VIEW OF THE Terrestrial and Coelestial GLOBES, IN A Brief Explanation OF THE PRINCIPLES Of plain and solid GEOMETRY, Applied to Surveying and Gauging of CASK.
The Doctrine of the Primum Mobile.
With an Account of the Juilan & Gregorian
Calendars, and the Computation of the Places of
of the Sun, Moon, and Fixed Stars, from such De
To which is added an Introduction unto GEOGRAPHY.
By John Newton, D. D.
London, Printed for Thomas Passinger, at the Three Bibles on London-Bridge 1679.
HE that adventures upon any
thing contrary to the General
received practice, what ever his
own courage and resolutions
are, had need to be supported, not only by
the most Wise and Honourable, but also
and have and do declare,
that the multitude of Schools for the learning
of the Latine and Greek Tongues, are destructive
both to our youth and the Commonwealth;
and if the Opinion of Sir Francis Bacon in
his Advice to King Iames concerning Sutton's
Hospital, be not sufficient to warrant
my Assertion, I could heartily wish that
no such Evidence could have been produmateria rerum novarum; whether
this be an essential or an accidental Effect,
London, there are
but few Places in this Nation, where a man
can put his Son, to be well instructed in Arithmetick,
Geometry, Musick and Astronomy;
and even that Famous City was without a
Publick School for Mathematical LearnArithmetick, Geometry, Musick and Astronomy;
before the Latine and Greek
MY Design in publishing these InGeometry and Astronomy,
is so well known by all
the Epistles, to my other TreatiGrammar, Arithmetick, Rhetorick,
and Logick, that I think it needless to tell
thee here, that it is my Opinion, that all the
Arts should be taught our Children in the
English Tongue, before they begin to learn
the Greek or Latin Grammar, by which means
many thousands of Children would be fitted
for all Trades, enabled to earn their own Li
Christ's Church London; I am
not now without hopes, to see the same effected
in many other Places in this Kingdom; and
to this purpose I have to my Introductions to
the other Arts, added these also to Geometry
and Astronomy; which I call by the
name of Cosmographia; and this I have
divided into four Parts; in the first I have
briefly laid down the first Principles belonging
to the three kinds of Magnitude or continued
Quantity, Lines, Planes and Solids; which
ought in some measure to be known, before
we enter upon Astronomy, and this part I
call an Introduction unto Geometry:
The second and third Parts treat of Astronomy;
the first of which sheweth the DoPrimum Mobile, that is, the
Declination, Right Ascension, and Oblique
Ascensions of the Sun and Stars, and such
other Problems, as do depend upon the Do
The second Part of Astronomy, treateth
of the motion of the Sun, Moon and Fixed
Stars; in order whereunto, I have first givJulian and Gregorian Calendar, and of both
from the true; for it must be acknowledged
Anno 1674. upon the 19
day of April, so the Almanacks for that Year,
as well as the General Table set before the
Book of Common Prayer; but by the Rule in
the said Book of Common Prayer given, the
Feast of Easter should have been upon the
twelfth of April, for Easter-Day must always
be the first Sunday after the first Full
Moon, which happeneth next after the one
and twentieth day of March, and if the
Full Moon happen upon a Sunday, Easter-Day
is the Sunday after; Now in the Year
1674. the 19 of April being Friday was Full
Moon, therefore by this Rule, Easter-Day
should be the twelfth, and by the Table and the
Common Almanacks April the tenth; but
this learned man must know, that the mistake
is in himself, and not in the Rule or Table
set down in the Book of Common Prayer; for
if he please to look into the Calendar, he will
find that the Golden Number Three, (which
was the Golden Number for that Year) is
March, and
therefore according to the supposed motion of
the Moon, that Day was New Moon; and then
the Full Moon will fall upon the fourteenth
day of April, and not upon the tenth, and so
by consequence the Sunday following the first
Full Moon after the 21 day of March was
the nineteenth of April and not the twelfth.
And thus the Rule and the Table in the
Book of Common Prayer for finding the Feast
of Easter are reconciled; and when AuthoriNicene Council it was appointed, that
the Feast of Easter should be observed upon
the Sunday following the first Full Moon
after the Vernal Equinox, which then indeed
was the 21 of March; but now the tenth,
and in the Year 1674. Wednesday the 11 of
March was Full Moon, and therefore by this
Rule, Easter-Day should have been upon
March the fifteenth, whereas according to
the Rules we go by, it was not till April the
nineteenth.
Primum Mobile, by reason of the gene
What I have done in this particular, as
it was for mine own satisfaction, so I am
apt to believe, that it will be pleasing to many
others; and although I shall leave every one
to abound in his own sense, yet I cannot think
Street's Astronomia Carolina,
and the several Books written in English
by Mr. Wing.
The fourth Part of this Treatise is an InGeography, in which I have
given general Directions, for the understandAstronomy;
the place where, and this depends upon
Geography; and the Person by whom any
memorable Act was done, and this must be
had from the Historical narration thereof;
and he that reads History without some knowAstronomy and Geography will
find himself at a loss, and be able to give but
a lame account of what he reads; but after
the learning of these Arts of Grammar, (I
mean so much thereof, as tends to the unArithmetick, Geometry and Astronomy;
a Child may proceed profitably to Rhetorick
and Logick, the reading of History, and the
learning of the Tongues; and sure there is
Musick in the Worship and Service of God
be not Argument enough to allow that a place
among the Arts, let that poor end of De
The great Obstruction in this Work, is
the general Ignorance of Teachers, who beSed
Cynthius aurem vellit; Therefore I will
forbear and leave what I have written, to be
perused and censured as thou shall think fit.
Of the Definition and Division of Geometry.
GEometry is a Science explaining the kinds and properties of continued quantity or magnitude.
2. There are three Kinds or Spe
3. A Line is a Magnitude consisting only of length without either breadth or thickness.
4. In a Line two things are to be considered, the Terms or Limits, and the several Kinds.
5. The term or limit of a Line is a Point.
6. A Point is an indivisible Sign in Magni
7. The kinds of Lines are two, Right and Ob
AB lieth streight
and equally between the Points A and B. Fig. 1.
9. An Oblique Line is that which doth not lie equally between its Points, but goeth up and down sometimes on the one side and sometimes on the other. And this is either simple or various.
10. A simple Oblique Line, is that which is ex
11. Thus are Lines to be considered in them
12. In respect of their distances, they may be either equally distant, or unequally.
13. Lines equally distant are two or more,
which by an equal space are distant from one ano
14. Lines unequally distant, are such as do
more or less incline to one another, and these be
15. Concurring Lines are either perpendicu
16. A Perpendicular Line, is a Right Line
falling directly upon another Right Line, not deAB in Fig. 1.
17. A Perpendicular Line is twofold, to wit, ei
18. A line exactly Perpendicular, may be
drawn in the same manner, as any Right Line
Fig. 1.
For Example; Let CD be the Right Line given,
and let it be required, to bisect this Line, and to erect
a Perpendicular in the middle thereof. 1. Then setC, draw the
Arches E and F. 2. Setting one Foot of your ComD, draw the Arches G and H, and from the
Intersections of these Arches draw the Right Line
KL, so shall the Right Line KL be Perpendicular to
the Right-Line CD, and the Right Line CD also diA.
19. A Line Perpendicular to any other Point than the middle is twofold: for it is either drawn from some Point given in the Line; or from some Point given without the Line.
20. From a Point given in the Line, at Perpendicular
may thus be drawn. In Fig. 2. Let
the given Line be CD, and let it be required to draw
a Perpendicular Line to the Point C, your Compasses
being opened to any reasonable distance, set one Foot in
the Point C, and the other in any place on either side
the Line CD, suppose at A, then describe the Arch
ECF, this done draw the Line EA, and where that Line
being extended shall cut the Arch ECF, a Right Line
drawn from C to that Intersection shall be PerpendiC in the Line CD, as was required.
21. From a Point given without the Line, a
Perpendicular may be drawn in this manner.
CD, and let it be reF without the Line. From the Point F
draw a streight Line to some part of the Line CD at
pleasure, as FE, which being bisected, the Point of BiA, if therefore at the distance of AF,
you draw the Arch ECF, the Right Line CF shall be
Perpendicular to the Line CD, as was required.
22. Hitherto concerning a Perpendicular Line.
A Right Line not Perpendicular, is a Right Line
falling indirectly upon another Right Line, in
23. Lines unequally distant, and at last con
24. An Angle therefore is nothing else, then the place, where two Lines do meet or touch one another, and the two Lines which constitute the Angle, are in Geometry called the sides of the Angle.
25. Every Angle is either Heterogeneous, or Homogeneous:
that is called an Hetorogeneous Angle,
which is made by the meeting of one Right Line,
and another that is Oblique and Crooked; and
that is called an Homogeneous Angle, which is
made by the meeting of two Lines of the same
kind, that is, of two Right Lines, or of two
curved or Circular Lines.
26. An Homogeneous Angle made of two curved
or Circular Lines, is to be considered in Geome
27. Right lined Angles are either Right or Oblique.
Thus
in Fig. 1. the Line AK is Perpendicular to the Line
CD, and the Angles KAC and KAD, are right
and equal to one another.
29. An Oblique Angle is that, whose sides are not Perpendicular to one another.
30. An Oblique Angle is either acute or ob
31. An Acute Angle is that which is less than a Right.
32. An Obtuse Angle, is that which is greater
than a Right. Thus in Fig. 1. The Angle BAC
is an Acute Angle because less than the Right Angle
CAK. And the Angle BAD is an Obtuse Angle
being greater than the Right Angle DAK.
The Geometrical Propositions concerning
Lines and Angles are very many, but these follow
To divide a Right Line given into any Number of equal Parts.
Let it be required to divide the Right Line AB
into five equal Parts. From the extream Points of
the given Line A and B, let there be drawn two
Parallel Lines, then from the Point A at any diB set
off the like Parts in the Line BC, and let them be
AB into 5 equal Parts, as was required.
Two Right Lines being given, to find a Mean pro
Let the two Right Lines given be DB and
CB, which let be made into one Line as CD,
which being besected the Point of bisection is A,
from which as from a Centre describe the Arch
CED, and from the Point B erect the PerpendiBE, so shall BE, be the Mean proportional
required; for, BC. BE∷BE. BD.
Three Right Lines being given, to find a fourth proportional.
Let the three given Lines be AB. BC. and
AD. Fig. 5. to which a fourth proportional is
required: draw AE at any Acute Angle, to the
Line AD in the Point A; and make DE paralBC, so shall AE be the fourth proportioAB. BC∷AD. AE.
Vpon a Right Line given, to make a right-lined. Angle, equal to an Angle given.
Let it be required upon the Line CD in Fig. 6.
DAE in
Fig. 5. From the Point A as a Center, at any exBG,
between the sides of the Angle given, and with
the same extent describe the Arch HL from
the Point D, and then make HL equal to BG,
then draw the Line DL, so shall the Angle CDL
be equal to the Angle DAE given, as was re
Of Figures in the general, more particularly of a Circle and the affections thereof.
HItherto we have spoken of the first kind of
Magnitude, that is, of Lines, as they are
considered of themselves, or amongst them
2. The second kind of Magnitude is that
which is made of Lines, that is, a Figure con
3. And in a Superficies there are three things to be considered. 1. The Term or Limit. 2. The middle of the Term. 3. The Thing or Figure made by the Term or Limit.
4. The Term or Limit is that which compre
5. The Term of a Figure is either Simple or various.
6. A Simple Term is that which doth consist
of a Simple Line, and is properly called a Cir
7. A various Term is that which hath bending
or crooked Lines, making Angles, and may there
8. The middle of Term is that which is the
Center of the Figure; for every Figure, whe
9. The Thing or Figure made by the Term
or Limit, is all that Area or space which is included
by the Term or Terms. And here it is to be
observed, that the Term of a Figure is one thing, and
the Figure it self another; for Example, A PeripheArea or space
which is included by the Periphery, for a Periphery is
nothing but a Line, but the Circle is that which is in
10. As the Term of a Figure is either Simple or Various; so the Figure it self is either Simple and Round, or Various and Angular.
11. A Simple Figure is that which is contained by a Simple or Round Line, and is either a Circle or an Ellipsis.
12. A Circle therefore is such a Figure which is made by a Line so drawn into it self, as that it is every where equally distant from the middle or Center.
13. An Ellipsis is an oblong Circle.
14. In a Circle we are to consider the affecti
15. And Right Lines may be applied unto a
Circle, either by drawing them within, or with
16. Right Lines inscribed within a Circle, are either such as do cut the Circle into two equal or unequal Parts, as the Diameter and lesser Chords, or such as do cut the Diameter and lesser Chords into two equal or unequal Parts, as the Right and versed Sines.
17. A Diameter is a Right Line drawn
through the Center from one side of the Circumference
to the other, and divideth the Circle
into two equal Parts, As in Fig. 7. The Right Line
GD drawn through the Center B is the Diameter of
the Circle GEDL dividing the same into the two eGED, and GLD: and this is also called
the greatest Chord or Subtense.
18. A Chord or Subtense is a Right Line inscribed
in a Circle, dividing the same into two
equal or unequal Parts; if it divide the Circle into
two equal Parts, it is the same with the Diameter,
but if it divide the Circle into two unequal Parts it
is less than the Diameter, and is the Chord or Subtense
of an Arch less than a Semi-circle, and also
of an Arch greater than a Semi-circle. As in the
former Figure, the Right Line CAK divideth the
Circle into two unequal Parts, and is the Chord or
Subtense of the Arch CDK, less than a Semi-circle,
and of the Arch CGK greater than a Semi-circle:
and these are the Lines which divide the Circle into
two equal or unequal Parts. And as they divide the
Circle into two equal Parts, so do they also divide one
another; The lesser Chords when they are divided by
19. Sines are right or versed.
20. Right Sines are made by being besected,
by the Diameter, and are twofold, Sinus totus, the
whole Sine or Radius, and this is the one half of
the Diameter, as the Lines BE or BD, and all Lines
drawn from the Center to the Circumference.
21. Sinus simpliter, or the lesser Sines, are the
one half of any Chord less than the Diameter, as
in the former Figure CA or AK, which are the equal
Parts of the Chord CAK, are the Sines of the ArchCD. and DK less than a Quadrant, and also the
Sines of CEG and KLG greater than a Quadrant.
22. Versed Sines are the Segments of the Diaas AD is the versed Sine of CD or
DG and the other Segment AG is the versed Sine of
the Arch CEG or KLG.
23. The Right Lines drawn without the Cir
24. A Tangent is a Right Line touching the
Circle, and drawn perpendicular to the Diame
25. A Secant is a Right Line drawn from the
Center through the Circumference, and extended
to the Tangent. As in the former Figure, the
Right Line DF is the Tangent of the Arch CD,
and the Right Line BF is the Secant of the same
Arch CD.
The Arch of a Circle being given to describe the whole Periphery.
Let ABC be an Arch given, and let the CirA, B, C; open your Compasses to more
than half the distance of A, B, and setting one
Foot in A describe the Arch of a Circle, and the
Compasses remaining at the same distance, setB, describe another Arch so as it
may cut the former in two Points, suppose G,
and H, and draw the Line HG towards that Part
on which you suppose the Center of the Center
of the Circle will fall.
In like manner, opening your Compasses to
more than half your distance of B, C, describe
two other Arches from the Points E and C, cutE and F, then draw the Line
EF till it intersect the former Line HG, so shall
the Point of Intersection be the Center of the
Circumference or Circle required, as in Fig.
may be seen.
The Conjugate Diameters of an Ellipsis being given, to draw the Ellipsis.
Let the given Diameter in Fig. 24. be LB and
ED, the greatest Diameter. LB being bisected in
the Point of Bisection, erect the Perpendicular
AD. which let be half of the lesser Diameter
ED, then open your Compasses to the extent of
AB, and setting one Foot in D, with the other
make a mark at M and N in the Diameter BL,
then cutting a thred to the length of BL, fasten
the thred with your Compasses in the Points NM,
and with your Pen in the inside of the thred deBFKL, so shall you describe the
one half of the Ellipsis required, and turning the
Thred on the other side of the Compasses, you
may with your Pen in the like manner describe the
other half of the Ellipsis GBHL.
Of Triangles.
HItherto we have spoken of the most Simple Figure, a Circle. Come we now to those Figures that are Various or Angular.
2. And an Angular Figure is that which doth consist of three or more Angles.
3. An angular Figure consisting of three
Angles, otherwise called a Triangle, is a Super
4. A Triangle may be considered either in re
5. A Triangle in respect of its Sides, is either
Isopleuron, Isosceles, or Scalenum.
6. An Isopleuron Triangle, is that which hath
three equal sides. An Isoscecles hath two equal Sides.
And a Scalenum hath all the three Sides unequal.
7. A Triangle in respect of its Angles is Right or Oblique.
9. An Oblique angled Triangle, is either Acute or Obtuse.
10. An Oblique acute angled Triangle, is that which hath all the three Angles Acute.
11. An Oblique obtuse angled Triangle, is that which hath one Angle Obtuse, and the other two Acute.
Vpon a Right Line given to make an Isopleuron or an Equilateral Triangle.
In Fig. 8. let it be required to make an EquilaAB. Open
your Compasses to the extent of the Line given,
and setting one Foot of your Compasses in A,
make an Arch of a Circle above or beneath the
Line given, then setting one Foot of your ComB, they being full opened to the same
extent, with the other foot draw another Arch
of a Circle crossing the former, and from the InAC
and AB, so shall the Triangle ACB be Equilate
Vpon a Right Line given to make an Isosceles Tri
In Fig. 8. let AB be the Right Line given,
from the Points A and B as from two Centers,
but at a lesser extent of the Compasses than AB;
AB the greatest Side, at a
greater extent; if you would have it to be the
least Side, describe two Arches cutting one anoF, and from the Intersection draw the
Lines AF, and FB, so shall the Triangle AFB
have two equal Sides, as was required.
To make a Scalenum Triangle, or a Triangle, whose three Sides are unequal.
In Fig. 9. let the three unequal Sides be EFG
make AB equal to one of the given Lines, supG, and from A as a Center, at the extent of
E describe the Arch of a Circle; in like manner
from B at the extent of F describe another Arch
intersecting the former, then shall the Right
Lines AC. CB and BA comprehend a Triangle,
whose three sides shall be unequal, as was required.
Of Quadrangular and Multangular Figures.
WE have spoken of Triangles or Figures conQuadrangle, Quinquangle, Sexangle, &c.
2. A Quadrangle is a Figure or Superficies,
which is bounded with four Right Lines.
3. A Quadrangle is either a Parallelogram or a
Trapezium.
4. A Parallelogram is a Quadrangle whose oppo
5. A Parallelogram is either Right angled or
Oblique.
6. A Right angled Parallelogram, is a Quadrangle
whose four Angles are all Right, and is either
Square or Oblong.
7. A Square Parallelogram doth consist of four
equal Lines. The Parts of a Square are, the
Sides of which the Square is made, and the Dia
8. An Oblong is a Right angled Parallelogram,
having two longer and two shorter Sides.
9. An Oblique angled Parallelogram, is that
whose Angles are all Oblique, and is either a
Rhombus or a Rhomboides.
10. A Rhombus is an Oblique angled and equiParallelogram.
11. A Rhomboides is an Oblique angled and
inequilateral Parallelogram.
12. A Trapezium is a Quadrangular Figure
whose Sides are not all parallel; it is either
Right angled or Oblique.
13. A Right angled Trapezium hath two op
14. An Oblique angled Trapezium is a Quadrangle,
but not a Parallelogram, having at least
two Angles Oblique, and none of the Sides pa
15. Thus much concerning Quadrangles or
four sided Figures. Figures consisting of more
than four Angles are almost infinite, but are re
Polygons are such,
as are contained by equal Sides and Angles, as
the Pentagon, Hexagon, and such like.
17. Inordinate or irregular Polygons, are such
as are contained by unequal Sides and Angles.
The construction of these Quadrangular and
Multangular Figures is explained in the Proposi
Vpon a Right Line given to describe a Right an
In Fig. 10. let the given Line be AB, upon
the Point A erect the PerpendicAD equal
to AB if you intend to make a Square, but longD and B at the distance of AB and
AD describe two Arches intersecting one anoED and EB, so shall the Right angled Figure
AE be a Square, if AB and AD be equal, o
To describe a Rhombus or Rhomboides.
In Fig. 11. To the Right Line AB draw the
Line AD at any Acute Angle at pleasure, equal
to AB if you intend a Rhombus, longer or shortRhomboides, then upon your
Compasses to the extent of AD and upon B as
a Center describe an Arch; in like manner, at the
extent of AB upon D as a Center describe anED and EB, so shall AE be the Rhombus
or Rhomboides, as was required.
Vpon a Right Line given to make a Regular Pen
In Fig. 12. Let the given Line be AB, upon
A and B as two Centers describe the Circles
EBGH and CAGK, then open your Compasses
to the extent of BC, and making G the CenHAFK, then draw the
Lines KFE and HFC: so shall AE and BC
be two sides of the Pentagon desired, and opening
your Compasses to the extent of AB, upon E
and C as two Centers describe two Arches interED and DC, so shall the
Figure AB and DE be the Pentagon required.
To make a Regular Pentagon and Decagon in a given Circle.
In Fig. 13. upon the Diameter CAB describe
the Circle CDBL, from the Center AErect the
Perpendicular AD, and let the Semidiameter
AC be bisected, the Point of Bisection is E, set
the distance ED from E to G, and draw the
Line GD, which is the side of a Pentagon, and
AG the side of a Decagon inscribed in the same
Circle.
In a Circle given to describe a Regular Hexagon.
The side of a Hexagon is equal to the Radius of a
Circle, the Radius of a Circle therefore being
six times applied to the Circumference, will give
you six Points, to which Lines being drawn from
Point to Point, will constitute a Regular Hexagon,
as was desired.
In a Circle given to describe a Regular Hepta
The side of a Heptagon is equal to half the
side of a Triangle inscribed in a Circle, having
therefore drawn an Hexagon in a Circle, the
Chord Line subtending two sides of the Hexagon
lying together, is the side of a Triangle inscribHeptagon, as in
Fig. 14. is plainly shewed.
Of Solid Bodies.
HAving spoken of the two first kinds of Magnitude, Lines and Superficies, come we now to the third, a Body or Solid.
2. A Body or Solid is a Magnitude consisting of length, breadth and thickness.
3. A Solid is either regular or irregular.
4. That is called a regular Solid, whose Bases, Sides and Angles are equal and like.
5. And this either Simple or Compound.
6. A simple regular Solid, is that whith doth consist of one only kind of Superficies.
7. And this is either a Sphere or Globe, or a plain Body.
8. A Globe is a Solid included by one round and convex Superficies, in the middle whereof there is a Point, from whence all Lines drawn to the Circumference are equal.
9. A simple plain Solid, is that which doth consist of plain Superficies.
10. A plain Solid is either a Pyramid, a Prism, or a mixt Solid.
11. A Pyramid is a Solid, Figure or Body, contained by several Plains set upon one right lin'd Base, and meeting in one Point.
12. Of all the several sorts of Pyramids,
there is but one that is Regular, to wit a Tetrahedron,
or a Pyramid consisting of four regular
or equilateral Triangles; the form whereof (as
it may be cut in Pastboard) may be conceived by
Figure 15.
Paralellogram.
14. A Prism is either a Pentahedron, a Hexahedron,
or a Polyhedron.
15. A Pentahedron Prism, is a Solid compreFig. 16.
16. An Hexahedron Prism, is a Solid comQuadrangle,
as Fig. 17.
17. An Hexahedron Prism, is distinguished
into a Parallelipipedon and a Trapezium.
18. An Hexahedron Prism called a Trapezium
is a Solid, whose opposites Plains or Sides, are
neither opposite nor equal.
19. A Parallelipipedon is either right angled or
oblique.
20. A right angled Parallelipipedon is an Hexahedron
Prism, comprehended of right angled
Plains or Sides; and it is either a Cube or an Ob
21. A Cube is a right angled Parallelipipedon
comprehended of six equal Plains or Sides.
22. An Oblong Parallelipipedon, is an Hexahedron
Prism, comprehended by unequal Plains or
Sides.
23. An Oblique angled Parallelipipedon, is an
Hexahedron Prism, comprehended of Oblique
Sides.
24. A Polyhedron Prism, is a Solid compreQuincangle, Sexangle, &c.
25. A regular compound or mixt Solid, is such
Octohedron, the Icosahedron,
of both which the Base is a Triangle; and
the Dodecahedron, whose Base is a Quincangle.
26. An Octohedron is a Solid Figure which is
contained by eight equal and equilateral TrianFig. 18.
27. An Icosahedron is a Solid, which is conFig. 19.
28. A Dodecahedron is a Solid, which is conPentagons, equilateral and
equiangled, as in Fig. 20.
29. A regular compound Solid, is such a So
30. A Cone is a Pyramidical Body, whose
Base is a Circle, or it may be called a round Pyramis,
as Fig. 21.
31. A Cylinder is a round Column every
where comprehended by equal Circles, as Fig.
22.
32. Irregular Solids are such, which come
not within these defined varieties, as Ovals, Frustums
of Cones, Pyramids, and such like.
And thus much concerning the description of
the several sorts of continued Quantity, Lines,
Plains and Solids; we will in the next place
consider the wayes and means by which the Di
Of the Measuring of Lines both Right and Circular.
EVery Magnitude must be measured by some
known kind of Measure; as Lines by Lines,
Superficies by Superficies, and Solids by Solids, as
if I were to measure the breadth of a River, or
height of a Turret, this must be done by a Right
Line, which being applied to the breadth or
height desired to be measured, shall shew the
Perches, Feet or Inches, or by some other known
measure the breadth or height desired: but if the
quantity of some Field or Meadow, or any other
Plain be desired, the number of square Perches
must be enquired; and lastly, in measuring of
Solids, we must use the Cube of the measure used,
that we discover the number of those Cubes
that are contained in the Body or Solid to be
measured. First, therefore we will speak of the
several kinds of measure, and the making of such
Instruments, by which the quantity of any Magni
2. Now for the measuring of Lines and Su
3. An Inch is three Barley Corns in length, and is either divided into halves and quarters, which is amongst Artificers most usual, or into ten equal Parts, which is in measuring the most useful way of Division.
4. A Foot containeth twelve Inches in length,
and is commonly so divided; but as for such
things as are to be measured by the Foot, it is far
5. A Yard containeth three Foot, and is com
6. An Ell containeth three Foot nine Inches,
aud is usually divided into halves and quarters,
and needs not be otherwise divided, because we
have no use for this Measure, but in Shop Com
7. A Pole or Perch cotaineth five Yards and
an half, and hath been commonly divided into
Feet and half Feet. Forty Poles in length do
make one Furlong, and eight Furlongs in length
do make an English Mile, and for these kinds of
of lengths, a Chain containing four Pole, divided
by Links of a Foot long, or a Chain of fifty
Foot, or what other length you please, is well
enough, but in the measuring of Land, in which
the number of square Perches is required; the
Chain called Mr. Gunters, being four Pole in
length divided into 100 Links, is not without just
reason reputed the most useful.
8. The making of these several Measures is
not difficult, a Foot may be made, by repeating
an Inch upon a Ruler twelve times, a Yard is
eight Foot, and so of the rest; the Subdivision
of a Foot or Inch into halves and quarters, may
be performed by the seventeenth of the first, and
into ten or any other Parts by the first Proposition
of the first Chapter, and all Scales of equal Parts,
of what scantling you do desire. And this I
9. The next thing to be considered is the mea
10. And every Circle is supposed to be divi
11. Now then a Circle may be divided into
360 Parts in this manner, Having drawn a DiaFig. 7.
GE. ED. DL and LG, are each of them 90
Degrees; and the Radius of a Circle being equal
to the Chord of the sixth Part thereof, that is to
the Chord of 60 Degrees, as in Fig. 14. if you
set the Radius GB from L towards G, and also
from G towards L, the Quadrant GL will be subGM. 30. MH 30 and HL
30, the like may be done in the other Quadrants
also; so will the whole Circle be divided into
twelve Parts, each Part containing 30 Degrees.
And because the side of a Pentagon inscribed in a
Circle is equal to the Chord of 72 Degrees, or
Fig. 13. therefore if
you set the Chord of the first Part of the Circle
given from G to L or L to G, in Fig. 7. you will
have the Chord of 72 Degrees, and the difference
between GP 72 and GH 60 is HP 12, which be
12. A Circle being thus divided into 360 Parts, the Lines of Chords, Sines, Tangents and Secants, are so easily made (if what hath been said of them in the Second Chapter be but considered) that I think it needless to say any more concerning their Construction, but shall rather proceed unto their Use.
13. And the use of these Lines and other Lines of equal Parts we will now shew in circular and right lined Figures; and first in the measuring of a Circle and Circular Figures.
Of the Measuring of a Circle.
THe squaring of a Circle, or the finding of a
Square exactly equal to a Circle given, is that
which many have endeavoured, but none as yet
have attained: Yet Archimedes that Famous MatheArea of
a Circle is equal to a Rectangle made of the Rodius
and half the Circumference: Or thus, The
Area of a Circle is equal to a Rectangle made of
the Diameter and the fourth part of the Circum
2. But Ludolphus Van Culen finds the Circum&c. the Product
is 7.85395 which is the Area of that Circle, and
from these given Numbers, the Area, Circumfe
The Diameter of a Circle being given to find the Circumference.
As 1. to 3.14159: so is the Diameter to the
Circumference. Example. In Fig. 13. Let the
Diameter IB be 13. 25. I say as 1. to 3. 14159.
so IB. 13.25 to 41.626 the Circumference of
that Circle.
The Diameter of a Circle being given to find the Superficial Content.
As 1. to 78539; so is the Square of the DiaExample, Let the Diameter given be as before IB
13.25 the Square thereof is 175.5625 therefore.
As 1. to 78539: so 175.5625 to 137.88 the Superficial Content of that Circle.
The Circumference of a Circle being given, to find the Diameter.
This is but the Converse of the first PropositiExample, Let the given Cir
As 1. to 318308: so 41.626 to 13. 25. the Diameter required.
The Circumference of a Circle being given to find the Superficial Content.
As the Square of the Circumference of a CirExample, As the Square of
3.14159 is to 7853938: so is 1. the Square of
another Circle to 079578 the Superficial Content
required, and so an Unite for the most easie work
As 1. to 0.79578: so is the Square of 41.626 to 137.88 the Superficial Content required.
The Superficial Content of a Circle being given, to find the Diameter.
This is the Converse of the second Propositi
As 7853978. 1∷1. 1. 27324, and there
The Superficial Content of a Circle being given, to find the Circumference.
This is the Converse of the Fourth Propositi
As 079578. 1 :: 1. 12.5664, and therefore if the Superficial Content given be 137.88, to find that Circumference: I say,
As 1. to 12.5664: so is the 137.88 to 1732.7 whose Square Root is 626 the Circumference.
The Diameter of a Circle being given to find the Side of the Square, which may be inscribed within the same Circle.
The Chord or Subtense of the Fourth Part of
a Circle, whose Diameter is an Unite, is 7071067,
and therefore, as 1. to 7071067: so is the DiaExample, let the Diameter given be 13.25 to find
the side of a Square which may be inscribed in that
Circle: I say,
As 1. to 7071067: so is 13.25 to 9.3691 the side required.
The Circumference of a Circle being given, to find the Side of the Square which may be inscribed in the same Circle.
As the Circumference of a Circle whose Dia
As 3.14159 to 7071067: so 1. to 225078.
And therefore the Circumference of a Circle being as before 41.626, to find the side of the Square that may be inscribed: I say,
As 1. to 225078. so is 41.626 to 9.3691 the side inquired.
The Axis of a Sphere or Globe being given, to find the Superficial Content.
As the Square of the Diameter of a Circle,
which is Unity, is to 3.14159 the Superficial
Content, so is the Square of any other Axis
given, to the Superficial Content required. Example,
Let 13.25 be the Diameter given, to find
the Content of such a Globe: I say,
As 1. to 3.14159: so is the Square of 13.25 to 551.54 the Superficial Content required.
To find the Area of an Ellipsis.
As the Square of the Diameter of a Circle, is
to the Superficial Content of that Circle; so is
the Rectangle made of the Conjugate DiameEllipsis, to the Area of that Ellipsis; And
the Diameter of a Circle being one, the Area is
7853975, therefore in Fig. 26. the Diameters
AC8 and BD5 being given, the Area of the
Ellipsis ABCD may thus be found.
As 1. to 7853975: so is the Rectangle AC
in BD40 to 3.1415900, the Area of the Ellipsis
required.
Of the Measuring of Plain Triangles.
HAving shewed the measuring of a Circle,
and Ellipsis, we come now to Right lined
Figures, as the Triangle, Quadrangle, and Mul
2. And the measuring of Plain Triangles is
either in the measuring of the Sides and Angles,
or of their Area and Superficial Content.
3. Plain Triangles in respect of their Sides
and Angles are to be measured by two sorts of
Lines, the one is a Line of equal Parts, and by
that the Sides must be measured, the other is a
Line of Chords, the Construction whereof hath
4. And first to protract or lay down an Angle
to the Quantity or Number of Degrees propoAD in
Figure 5, then open your Compasses to the NumA, with the other describe the
Arch BG, and from the Point A let it be requiB, with the other make
a mark at G, and draw the Line AG, so shall the
Angle BAG contain 36 Degrees, as was re
5. If the Quantity of an Angle were reBAG, open your
Compasses in the Line of Chords to the extent of
60 Degrees, and setting one Foot in A, with the
other draw the Arch BG, then take in your ComBG, and apply that extent
to the Line of Chords, and it will shew the Num
6. In every Plain Triangle, the three Angles
are equal to two right or 180 Degrees, there
8. In a Right Angled Plain Triangle, one of the Acute Angles is the Complement of the other to a Quadrant or 90 Degrees.
9. In Right Angled Plain Triangles, the Side
subtending the Right Angle we call the Hypotenuse,
and the other two Sides the Legs, thus in Fig. 5 ▪
AE is the Hypotenuse, and AD and ED are the
Legs; these things premised, the several cases in
Right Angled and Oblique Angled Plain Tri
In a Right Angled Plain Triangle, the Angles of one Leg being given, to find the Hypotenuse and the other Leg.
In the Right Angled Plain Triangle ADE in
Fig. 5. Let the given Angles be DAE 36, and
DEA 54, and let the given Leg be AD 476;
to find the Hypotenuse AE, and the other Leg
ED.
Draw a Line at pleasure, as AD, and by your
Scale of equal Parts set from A to D 476 the
Quantity of the Leg given, then erect a PerpenD, and upon the Point A
lay down your given Angle DAE 36 by the
fourth hereof, and draw the Line AE till it cut
the Perpendicular DE, then measure the Lines
AE and DE upon your Scale of Equal Parts, so
shall AE 588.3 be the Hypotenuse, and DE 345.8
the other Leg.
The Hypotenuse and Oblique Angles given, to find the Legs.
Let the given Hypotenuse be 588, and one of
the Angles 36 degrees, the other will then be
54 degrees, Draw a Line at pleasure, as AD, and
upon the Point A by the fourthAC, and from your Scale of
equal Parts, set off your Hypotenuse 588 from
A to E, and from the Point E to the Line AD
let fall the Perpendicular ED, then shall AD beED 345.8 the other.
The Hypotenuse and one Leg given to find the An
Let the given Hypotenuse be 588. and the given
Leg 476. Draw a Line at pleasure as AD, upon
which set the given Leg from A to D. 476, and
upon the Point D, erect the Perpendicular DE,
then open your Compasses in the Scale of Equal
Parts to the Extent of your given Hypotenuse 588,
and setting one Foot of that Extent in A, move
the other till it touch the Perpendicular DE, then
and there draw AE, so shall ED be 345.8 the
Leg inquired, and the Angle DAE, will be found
by the Line of Chords to be 36DEA. 54.
The Legs given to find the Hypotenuse, and the Oblique Angles.
Let one of the given Legs be 476, and the oAD to the extent of
476, and upon the Point D, erect the PerpendiDE to the extent of 345.8, and draw the
Line AE, so shall AE be the Hypotenuse 588, and
the Angle DAE will by the Line of Chords be
found to be 36 Degrees, and the Angle DEA 54,
as before.
Hitherto we have spoken of Right angled plain Triangles: the Propositions following concern such as are Oblique.
Two Angles in an Oblique angled plain Triangle, being given, with any one of the three Sides, to find the other two Sides.
In any Oblique angled plain Triangle, let one
of the given Angles be 26.50 and the other 38.
and let the given Side be 632, the Sum of the
two given Angles being deducted from a Semi-circle,
leaveth for the third Angle 115.50 DeBC 632. and upon the
Points B and C protract the given Angles, and
draw the Lines BD and CD, which being meaBD will be
fouBD 431.09,
Two Sides in an Oblique Angled Triangle being
given, with an Angle opposite to one of them, to find the
other Angles and the third Side, if it be known whe
In an Oblique Angled Plain Triangle, let the
given Angle be 38 Degrees, and let the Side adBC in Fig. 25. protract
the given Angle 38 Degrees upon the Point C, and
draw the Line DC, then open your Compasses to
the Extent of the other Side given 431. 1. and
setting one Foot in B, turn the other about till it
touch the Line DC, which will be in two plaD and E; if therefore the
Angle at B be Acute the third Side of the
Triangle will he CE, according therefore to the
Species of that Angle you must draw the Line
BD or BE to compleat the Triangle, and then
you may measure the other Angles, and the third
Side as hath been shewed.
Two Sides of an Oblique Angled Plain Triangle be
Let one of the given Sides be 632, and the oBC, and by help of your Scale of Equal Parts,
set off one of your given Sides from B to C 632.
then upon the Point B protract the given Angle
26. 50. and draw the Line BD, and from B to D,
set off your other given Side 431. 1. and draw
the Line DC, so have you constituted the Triangle
BDC, in which you may measure the Angles and
the third Side, as hath been shewed.
The three Sides of an Oblique Angled Triangle being given, to find the Angles.
Let the length of one of the given Sides be
632, the length of another 431.1, and the length
of the third 312.4, and Draw a Line at pleasure,
as BC in Fig. 25, and by help of your Scale of EB to C. then open your Compasses in the
same Scale to the extent of either of the other
Sides, and setting one Foot of your Compasses in
B, with the other describe an occult Arch, then
extend your Compasses in the same Scale to the
length of the third Side, and setting one Foot in
C with the other describe another Arch cutting
the former, and from the Point of Intersection
draw the Lines BD and DC. to constitute the
Triangle BDC, whose Angles may be measured,
as hath heen shewed.
And thus may all the Cases of Plain Triangles
be resolved by Scale and Compass, he that desires
to resolve them Arithmetically, by my Trigometria
Britannica, or my little Geometrical, TrigoArea of the Segment of a Cir
In a Right Angled Plain Angle the Hypote
Take the Sums and difference of the Hypotenuse
and Leg given, then multiply the Sum by the Dif
Example. In Fig. 5. Let the given Hypotenuse
be AE 588.3, and the given Leg AD 476, and
let DE be the Leg inquired. The Sum of AE
and AD is 1064.3, and their Difference is 112.3,
now then if you multiply 1064.3 by 112.3, the
Product will be 119520.89, whose Square Root
is the Leg DE. 345. 8.
The Legs of a Right Angled Plain Triangle being gived, to find the Area or Superficial Content thereof.
Multiply one Leg by the other, half the
Product shall be the Content. Example, In the
Right angled plain Triangle ADE, let the given
Legs be AD 476, and DE 345, and let the Area
of that Triangle be required, if you multiply
476 by 345 the Product will be 164220, and the
half thereof 82110 is the Area or Superficial Con
The Sides of an Oblique angled plain Triangle be
Add the three Sides together, and from the
half Sum subtract each Side, and note their Dif
Example. In Fig. 9. Let the Sides of the TriABC be AB 20. AC 13, and BC 11 the
Sum of these three Sides is 44, the half Sum is
22, from whence subtracting AB 20, the DifAC 13, the Difference is 9, and lastly, if you
subtract BC 11 from the half Sum 22, the Diffe
Or thus, from the Angle C let fall the PerpenDC, so is the Oblique angled Triangle
ABC, turned into two right, now then if you
measure DC upon your Scale of Equal Parts, the
length thereof will be found to be 6.6, by which
if you multiply the Base AB 20, the Product will
be 132.0, whose half 66, is the Area of the Tri
The Sides of any Oblique angled Quadrangle be
Let the Sides of the Oblique angled QuadranABED in Fig. 11. be given, draw the DiagoAE, and also the Perpendiculars DC and BF,
then measuring AE upon the same Scale by which
the Quadrangular Figure was protracted, suppose
you find the length to be 632, the length of DC
112, and the length of BF 136, if you multiply
AE 632 by the Half of DC 56, the Product
will be 35392 the Area of ACED. In like
manner if you multiply AE 632, by the half of
BF 68, the Product will be 42976 the Area of
ACEB, and the Sum of these two Products is
the Area of ABED as was required.
Or thus, take the Sum of DC 112, and BF
136; the which is 248, and multiply AE 632 by
half that Sum, that is by 124, the Product will be
78368 the Area of the Quadrangular Figure
ABED, as before.
The Sides of a plain irregular multangled Figure being given, to find the Content.
In Fig. 26. Let the Sides of the multangled FiA. B. C. D. E. F. G. H. be given, and let the
Area thereof be required, by Diagonals drawn
from the opposite Angles reduce the Figure given,
AF the Content required: thus
by the Diagonals AG. BE and EC the mulAFGH. AFEB and BEDC, and each of these
are divided into four Right angled Triangles,
whose several Contents may be thus computed.
Let GA 94 be multiplied by half HL 27 more
Half of KF 29, that is by 23, the Product will
be 21, be the Area of AHGF. Secondly, OB
is 11, and FN 13, their half Sum 12, by which
if you multiply AE 132, the Product will be
1584 the Area of AFEB. Thirdly, let Bp be
18 m D 32, the half Sum is 25, by which if you
multiply AEC 125 the Product will be 3125
the Area of BEDC, and the Sum of these
Products is 6871 the Area of the whole irreguABCDEFGH, as was required.
The Number of Degrees in the Sector of a Circle being given, to find the Area thereof.
In Fig. 27. ADEG is the Sector of a Circle, in
which the Arch DEG, is Degrees. 23.50, and by
1. Prop. of Archimed. de Dimensione Circuli, the
length of half the Arch is equal to the Area of
the Sector of the double Arch, there the length
DE or EG is equal to the Area of the Sector
ADEG: and the length or circumference of the
whole Circle whose Diameter is 1 according to
Van Culen, is 3.14159265358979, therefore
the length of one Centesme of a Degree, is.
0. 01745329259. Now then to find the length of
any Number of Degrees and Decimal Parts, you
must multiply the aforesaid length of one CenArea of a Sector containing
twice those Degrees and Parts. Example, the
half of DEG 23.50 is DE or EG 11.75, by
which if you multiply 0.01745329259, the ProDE, and the Area of the Sector ADEG.
The Number of Degrees in the Segment of a Circle being given, to find the Area of the Segment.
In Fig. 27. Let the Area of the Segment
DEGK be required, in which let the Arch
DEG be Degrees 23.50, then is the Area of the
Sector ADEG 2050761879325 by the last aArea of
the Triangle ADG, the remainer will be the Area
of the Segment DEGK. And the Area of the TriADG may thus be found. DK is the Sine of
DE 11.75, which being sought in Gellibrand's DeAK is the Sine of
DH 78.25, or the Cosine of DE. 9790454724,
which being multiplied by the Sine of DE, the ProAG
Radius by half DF the Sine of the double Arch
DEG, the Product will be 19937453445 as beArea of the Sector ADEG 2050761879325, the
remainer will be 57016434875 the Area of the
Segment DEGL, as was desired.
The Diameter of a Circle being cut into any Num
In Fig. 28. The Radius AD is cut into five EEDFL is made by
the Chord Line ELF at Right Angles to AD
in the fourth Equal Part, or at eight tenths thereArea of this Segment
we have given AE Radius, and AL 8, and thereEL will be found to be
606000, the Sine of ED 36.87, by which if you
multiply 0.0174532, the Product is the Area of
the Sector AEDF 64350286, and the Area of
the Triangle AEF is 48, which being deducted
from the Area of the Sector, the Remainer
16350286 is the Area of the Sector EDFL, as
was required. And in this manner was that TaRadius into 100 Equal Parts.
Another way.
In Fig. 28. Let the Radius AD be cut into
10.100 or 1000 Equal Parts, and let the Area of
GK and M. PN. EL, the
double of each Ordinate, will be the Chords of
the several Arches, and the Sum of these Chords
beginning with the least Ordinate, will orderly
give you the Area of the several Segments made
by those Chord Lines, but the Diameter must be
be divided into 100000 Equal Parts, because of
the unequal differences at the beginning of the
Diameter: but taking the Area of the Circle to
be 3. 1415926535, &c. as before, the Area of
the Semicircle will be 1.57079632, from which
if you deduct the Chord GH1999999, the
Chord answering to 999 Parts of the Radius, the
remainer is. 1.56879632 the Area of the SegGDH. And in this manner by a contiArea
of the Segment of the Circle given, was made
that Table shewing the Area of the Segments of
a Circle to the thousandth part of the Radius.
And because a Table shewing the Area of the
Segments of a Circle to the thousandth part of
the Radius, whose whole Area is Unity, is yet
more useful in Common Practice, therefore from
this Table, was that Table also made by this Pro
As the Area of the Circle whose Diameter is.
Unity, to wit 3.14149 is to the Area of any
part of that Diameter, so is Unity the supposed
Area of another Circle, to the like part of that
Diameter. Example, the Area answering to 665
parts of the Radius of a Circlewhose Area is
3.14159 is 0.91354794 therefore,
As 3.14159265 is to 0.91354794: So is Area required; and the Table
being thus computed to the 1000 parts of the
Radius, we have enlarged it by the difference to
the 5000 parts of the Radius, and consequently
to the ten thousandth part of the Diameter: The
use of which Table shall be shewed when we come
to the measuring of Solid Bodies.
Of the Measuring of Heights and Distances.
HAving shewed in the former Chapter, how
all plain Triangles may be measured, not
only in respect of their Sides and Angles, but in
respect of their Area, and the finding of the Area
of all other plain Figures also, that which is
next to be considered, is the practical use of those
Instructions, in the measuring of Board, Glass,
Wainscot, Pavement, and such like, as also the
measuring or surveying of Land; and first we will
shew the measuring of Heights and Distances.
2. And in the measuring of Heights and DiRadius, and the larger
the Quadrant is, the more exactly may the Angles:
be taken, though for ordinary Practice, four or
five Inches Radius will be sufficient.
Let such a Quadrant therefore be divided in
the Limb into 90 Equal Parts or Degrees, and
numbred from the left hand to the right, at every
tenth Degree, in this manner 10. 20. 30. 40. 50.
60. 70. 80. 90. and within the Limb of the QuaRadius or Tangent of 45 Degrees, and
from thence both ways the Tangents of 63.44
Deg. 71.57 Deg. 75.97 Deg.78.70 Deg. 80.54
Deg. that is, 2. 3. 4. 5 and 6 being set also, your
Quadrant will be fitted for the taking of
Heights several ways, as shall be explained in the
Propositions following.
To find the Height of a Tower, Tree, or other Object at one Station.
At any convenient distance from the Foot of
the Object to be measured, as suppose at C in Fig.
30. and there looking through the Sights of your
Quadrant till you espie the top of the Object at
A, observe what Degrees in the Limb are cut by
the Thread, those Degrees from the left Side or
Edge of the Quadrant to the Right, is the QuanACB, which suppose 35 DeBAC 55 Degrees, beB the Foot of the Object,
to your Station at C, which suppose to be 125
Foot. Then as hath been shewed in the 1. Prop.
Chap. 8. draw a Line at pleasure as BC, and by
your Scale of Equal Parts, set off the distance
measured from B to C 125 Foot, and upon the
Point C lay down your Angle taken by observaB, and let it be extended till it cut the
Hypothenusal Line AC, so shall AB measured on
your Scale of Equal Parts, be 87.5 Foot for the
Height of the Object above the Eye; to which
the Height of the Eye from the Ground being
added, their Sum is the Height required.
Another way.
Let AB represent a Tower whose Altitude
you would take, go so far back from it, that
looking through the Sights of your Quadrant,
to the top of the Tower at A the Thread may
cut just 45 Degrees in the Limb, then shall the
distance from the Foot of the Tower, to your
Station, be the Height of the Tower above the
Eye.
Or if you remove your Station nearer and near&c. As if removing my
Station from C to D, the Thread should hang oBD 62
Foot, then will 124 Foot be the Height of the
Tower, above the Eye.
In like manner if you remove your Station
backward till your Thread fall upon one of those
Figures in the Quadrant; between 45 and 90 De
A Third way by a Station at Random.
Take any Station at pleasure suppose at C, and
looking through the Sights of your Quadrant,
observe what Parts of the Quadrant the Thread
falls upon, and then measure the distance be
Example, Suppose I standing at C, that the
Thread hangs upon 36 Degrees, as also upon 72
in the Quadrant which is the Tangent of the said
Arch, and let the measured distance be CB 125
Foot, which being multiplied by 72, the Product
is 9000, from which cutting off his Figures beRadius is supposed to be 100, the Height
inquired will be 90 Foot, he that desires to per
To find an inaccessible Height at two Stations.
Take any Station at pleasure as at D, and there
looking through the Sights of your Quadrant to
the top of the Object, observe what Degrees are
cut by the Thread in the Limb, which admit to be
68 Degrees, then remove backward, till the Anviz. AD.
if the distance between your two Stations were
326 foot, then draw a Line at pleasure as BD,
upon the Point D protract, the Angle ADB 68
Degrees, according to your first Observation, and
from your Line of equal parts set off the HyD to A, and from the
Point A let fall the Perpendicular AB which be
Or working by the Table of Sines and Tan
As the Radius, is to the measured distance or
Hypothenusal Line AD; so is the Sine of the
Angle ADE, to the height AB inquired.
Another more General way, by any two Stations
taken at pleasure.
Admit the first Station to be as before at D,
and the Angle by observation to be 68 Degrees,
and from thence at pleasure I remove to C, where
observing aim I find the Angle at C to be 32 DeBC, and upon
Clay down your last observed Angle 32 Degrees,
and by help of your Scale of Equal Parrs, set off
your measured distance from C to D 150 Foot,
then upon D lay down your Angle of 68 DeAD and AC meet, let fall the
Perpendicular AB, which being measured in your
Scale of Equal Parts, shall be the height of the
Object as before.
1. As the Sine of DAC to the Distance DC.
So the Sine of ACD, to the Side AD.
2. As the Radius, to the Side AD; so the Sine
ADB, to the Perpendicular height AB inquired.
The taking of Distances is much after the same manner, but because there is required either some alteration in the sights of your Quadrant or some other kind of Instrument for the taking of Angles, we will particularly shew, how that may be also done several ways, in the next Chapter.
Of the taking of Distances.
FOr the taking of Distances some make use of
a Semicircle, others of a whole Circle, with
Ruler and Sights rather than a Quadrant, and al
2. A piece of Board or Brass then about twelve or fourteen Inches Diameter, being made Circular like a round Trencher, must be divided into four Quadrants, and each Quadrant divided into 90 Degrees, or the whole Circle into 360, and each Degree into as many other Equal Parts, as the largeness of the Degrees will well permit: let your Circle be numbred both ways to 360, that is from the right hand to the left, and from the left to the right.
4. You must also have a Ruler with Sights fixed
at each end, for making of Observation, either
fixed upon the Center of your Circle, or loose, as
you shall think best; your Instrument being thus
made, any distance whether accessible or inacces
5. When you are in the Field, and see any
Church, Tower, or other Object, whose Distance
from you, you desire to know, choose out some
other Station in the same Field, from whence you
may also see the Object, and measure the distance
between your Stations; then setting your Ruler
upon the Diameter of your Circle, set your InACD in Fig. 30. Then reD, lay the Ruler on
the Diameter thereof, and then turn the whole InC, and
there fix your Instrument, and then upon the CenA; and observe the DeADC, and let the diDC 326
Foot; so have you two Angles and the side be
As the Sine of DAC, is to DC; so is the Sine
of ACD to the Side AD.
Or, as the Sine of DAC, is to the given Side
DC.
So is the Sine of ADC to the Side AC.
6. There is another Instrument called the plain
Table, which is nothing else, but a piece of Board,
in the fashion and bigness of an ordinary sheet of
paper, with a little frame, to fasten a sheet of pa
Having measured the distance between your
two Stations at D and C, draw upon your paper
a Line, on which having set off your distance
place your Instrument at your first Station C, and
laying your Ruler upon the Line so drawn thereD, then laying your
Ruler upon the Point C, turn the same about till
through the Sights you can espy the Object at A,
and there draw a Line by the side of your Ruler,
and remove your Instrument to D, and laying
your Ruler upon the Line DC, turn the InstruC, and then laying your Ruler upon
the Point D, turn the same, till through the Sights
you can espy the Object at A, and by the side of
your Ruler draw a Line, which must be extended
till it meet with the Line AC, so shall the Line
AD being measured upon your Scale of Equal
D,
and the Line AC shall be the distance thereof
from C.
7. And in this manner may the distance of two,
three or more Objects be taken, from any two
Stations from whence the several Objects may be
seen, and that either by the plain Table, or The
How to take the Plot of a Field at one Station, from whence the several Angles may be seen.
ALthough there are several Instruments by which the Plat of a Field may be taken, yet do I think it sufficient to shew the use of these two, the plain Table and Theodolite.
2. In the use of either of which the same chain
which is used in taking of heights and distances,
is not so proper. I rather commend that which is
known by the Name of Gunter's Chain, which is
four Pole divided into 100 Links; being as I con
3. When you are therefore entered the
Field with your Instrument, whether plain Table,
or Theodolite, having chosen out your Station,
let visible Marks be set up in all the Corners
thereof, and then if you use the plain Table, make
a mark upon your paper, representing your Stati
Example. Let Fig. 31. represent a Field whose
Plot is required; your Table being placed with
a sheet of paper thereupon, make a Mark about
the middle of your Table, as at A. apply your
Ruler from this Mark to B and draw the Line AB,
then with your Chain measure the distance thereA to B, and at B make a mark.
Then directing the Sights to C, draw a Line by
the side of your Ruler as before, and measure the
distance AC, which suppose to be 7 Chains and
44 Links, this distance must be taken from your
Scale, and set from A to C upon your paper.
And in this manner you must direct your Sights
from Mark to Mark, until you have drawn the
Lines and set down the distances, between all the
Angles in the Field and your station, which being
done, you must draw the Lines from one Point to
another, till you conclude where you first began,
so will those Lines BC. CD. DE. FG. and
GB, give you the exact Figure of the Field.
4. To do this by the Theodolite, in stead of
B, and observe how many DeB, which stands for your first
Mark, then measure the distance AB as before,
and set that down, in the fourth and fifth Co
5. Having thus taken the Angles and Distances
in the Field, to protract the same on Paper or
EB representing the DiaA, mark a Mark, and opening your ComA as a Center describe a Circle, then lay your
Field book before you seeing that your first ObC are 40.75 which being taken from your Line
of Chords, you must set them from H to I, and
draw the Line AI. the Degrees at D are 96 which
must in like manner be set from I to K, and so
the rest in order.
This done observe by your Field-book the
length of every Line, as the Line AB at your
first Observation was 11 Chains and 36 Links,
which being by your Scale set from A will give
the Point B in the Paper, the second distance beAI will give the Point C, and so
proceeding with the rest, you will have the Points
BCDEF and G, by which draw the Lines BC.
CD. DE; EF. FG and GB, and so at last you
have the Figure of the Field upon your Paper, as
was required.
And what is here done at one station, may be
done at two or more, by measuring one or two
distances from your first station, taking at every
station, the Degrees and distances to as many An
And as for taking the the Plot of a-Field by In
How to take the Plot of a Wood, Park or other Cham
BY these Directions which have been already
given, may the Plot of any Field or Fields be
taken, when the Angles may be seen alone or more
stations within the Field, which though it is the
case of some Grounds, it is not the case of all;
now where observation of the Angles cannot be
observed within, they must be observed without,
and although this may be done by the plain Ta
2. Suppose the Fig. 32. to be a large Wood
whose Plot you desire to take; Having placed
your Instrument at the Angle A, lay your Ruler on
the Diameter thereof, turning the whole InstruK, then fasten it there, and turn your Ruler upB, the Degrees cut by the
Ruler do give the quantity of that Angle BAK,
suppose 125 Degrees, and the Line AB 6 Chains,
45 Links, which you must note in your Field-book,
as was shewed before.
B, and
laying your Ruler upon the Diameter thereof,
turn it about, till through the Sights you can espy
your third mark at C, and there fasten your InA, the
Degrees cut by the Ruler being 106.25 the quanABC, and the Line BC contain
4. Remove your Instrument unto C, and laying
the Ruler on the Diameter thereof, turn the InD, and fixing of it there, turn the Ruler
upon the Center till you see your last station at B,
and observe the Degrees cut thereby, which supCD 6 Chains
65 Links, which must be entered into your Field-book
also, and because the Angle BCD is an in
5. Remove your Instrument unto D, and Iaying
the Ruler on the Diameter, turn the Instrument aE, and there fixing your Instrument, turn your
Ruler backward till you espy the Mark at C, where
the Degrees cut are, suppose 68.0 and the Line
DE 8 Chains and 23 Links.
6. Remove your Instrument unto E, and laying
the Ruler on the Diameter, turn the Instrument
about, till through the Sights you see the Angle
at F, and there fix it, then turn the Ruler backD, where the DeEF 7 Chains and 45 Links.
F, and laying
your Ruler upon the Diameter, turn the InstruG, where fix the same, and turn the Ruler
backward till you see the Angle at E, where the
Degrees cut by the Ruler are 70, and the Line FG
4 Chains 15 Links, which must be set down with
this
8. Remove your Instrument unto G, and layH, where fix the same, and turn the RuF, where
the Degrees cut by the Ruler are 65.25, and the
Line GH 5 Chains 50 Links.
9. Remove your Instrument in like manner to
H and K, and take thereby the Angles and Di
The taking of the inward Angles BCD and
EFG was more for Conformity sake than any
B to D, from E to G, the Length of the
Lines BC. CD. EF and G, would have given by
protraction the Plot of the Field without taking
these Angles by observation; many other com
10. The Angles and Sides of the Field being
thus taken, to lay down the same upon Paper,
Parchment, another Instrument called a Protractor
is convenient, the which is so well known to In
11. Having then this Instrument in a readiness
draw upon your Paper or Parchment upon which
you mean to lay down the Plot of that Field, a
Line at pleasure as AB. Then place the Center
of your Protractor upon the Point A, and beA
was 115 Degrees 00 Parts, turn your Protractor aAK lie directly under the 115
Degree; and then at the beginning of your ProAB, setA to B.
12. Then lay the Center of your Protractor
upon the Point B, and here turn your Protractor
about, till the line AB lie under 106 Degrees
25 Parts, and draw the Line BC, setting off the
Distance 8 Chains, 30 Links from B to C.
13. Then lay the Center of your Protractor
upon the Point C, and turn the same about till the
Line BC lie under 134 Degrees, but remember
CD, setting off 6 Chains, 65 Links
from C to D.
And thus must you do with the rest of the
Sides and Angles, till you come to protract your
last Angle at H, which being laid down accordHK will
cut the Line AK making AK 11 Chains and HK
6 Chains, 50 Links. This work may be also perAK, you may lay the
Center of your Protractor upon the Point K, and
the Diameter upon the Line AK; and because your
Angle at K by observation was 140 Degrees, you
must make a Mark by the Side of your Protractor
at 140 Degrees; and draw the Line KH, setting
off 6 Chains, 50 Links from K to H. And thus
proceeding with the rest of the Lines and Angles,
you shall find the Plot of your Field at last to
close at A, as before it did at K.
The Plot of the Field being taken by any Instrument, how to compute the Content thereof in Acres, Roods, and Perches.
THe measuring of many sided plain Figures
hath been already shewed in the 13 Proposition
of the 8 Chapter, which being but well conside
3. Having then converted your Plot into Tri
Example. Let 258.94726 be the Sum of seve
How to take the Plot of Mountainous and uneven Grounds, and how to find the Content.
VVHen you are to take the Plot of any
Mountainous or uneven piece of
Ground, such as is that in Figure 33, you must
first place your Instrument at A, and direct your
Sights to B, measuring the Line AB, observing
the Angle GAB, as was shewed before, and so
proceed from B to C, and because there is an asC to D, you must measure the true
length thereof with your Chain, and set that
down in your Book, but your Plot must he drawn
according to the length of the Horizontal Line,
which must be taken by computing the Base of a
right angled Plain Triangle, as hath been shewed
before, and so proceed from Angle to Angle until
you have gone round the Field, and having drawn
the Figure thereof upon your Paper, reduce into
Triangles and Trapezias, as ABC. CDE. ACEF
and AFG. then from the Angles B. C. D. F and
G; let fall the Perpendiculars, BK. CN. DL.
FM. and GH. This done you must measure the
Field again from Angle to Angle, setting down
the Distance taken in a straight Line over Hill and
Dale, and so likewise the several Perpendiculars,
which will be much longer than the streight Lines
measured on your Scale, and by these Lines thus
measured with your Chain cast up the Content;
which will be much more than the Horizontal
Content of that Field according to the Plot, but
if it should be otherwise plotted than by the Ho
How to reduce Statute Measure into Customary, and the contrary.
VVHereas an Acre of Ground by Statute
Measure is to contain 160 square
Perches, measured by the Pole or Perch of sixteen
foot and a half: In many places of this Nation,
the Pole or Perch doth by custom contain 18 foot,
in some 20. 24. 28 Foot; it will be therefore re
2. To do this you must consider how many square Feet there is in a Pole according to these several Quantities.
Now then if it were desired to reduce 7 Acres,
3 Roods, 27 Perches, according to Statute Measure,
into Perches of 18 Foot to the Perch; first re
Then say, as 324. to 272. 25. so is 1267 to
1065. 6. that is 1065 Perches, and 6 tenths of a
Perch. But to reduce customary Measure into sta
Of the Measuring of solid Bodies.
HAving shewed how the content of all plains may be computed, we are now come to the measuring of solid Bodies, as Prisms, Pyramids and Spheres, the which shall be explained in the Propositions following.
The base of a Prism or Cylinder being given, to find the solid content.
The base of a Prism is either Triangular, as the
Pentahedron; Quadrangular, as the Hexahedron,
or Multangular, or the Polyhedron Prism, all which
must be computed as hath been shewed, which done
if you multiply the base given by the altitude, the
product shall be the solid content required.
Example. In an Hexahedron Prism, whose base
In like manner the Base of a Cylinder being 45. 6. and the altitude 15. 4. the content will be 702. 24.
And in this manner may Timber be measured
whether round or squared, be the sides of the squa
Example. Let the Diameter of a round piece of Timber be 2. 75 foot. Then, As 1 it to 785397. so is the square of the Diameter 2. 75. to 5.9395 the Superficial content of that Circle.
Or if the circumference had been given 8. 64. then, As 1 is to 079578, so is the square of 8. 64. to 5.9404 the superficial content.
Now then if you multiply this Base 5. 94. by the length, suppose 21 foot, the content will be 124. 74.
If the side of a piece of Timber perfectly square be 1.15 this side being multiplyed by it self, the product will be 1.3225 the superficial content, or content of the Base, which being multiplyed by 21 the length, the content will be 27. 7745.
Or if a piece of Timber were in breadth 1. 15. in depth 1.5 the content of the Base would be 1.725 which being multiplied by 21 the length, the content will be. 36. 225.
The Base and Altitude of a Pyramid or Cone being given, to find the solid content.
Example. In a Pyramid having a QuadranguFig. 22. The side CF 17. CD 9. 5.
the Product is the Base CDEF. 161. 5, which
being multiplyed by 10.5 the third of the AltiAB 31.5 the Product is 1695.75 the conviz. 53. & 3
being multiplied by the whole Altitude AB 31.5
the Product will be the content as before.
2. Example. In Fig. 21. Let there be given
the Diameter of the Cone AB 3. 5. The Base
will be 96. 25. whose Altitude let be CD 16.92 the
third part thereof is 5.64 & 96.25 being multipli
The Axis of a Sphere being given, to find the so
If you multiply the Cube of the Axis given by
523598 the solid content of a Sphere whose Ax
Example. Let the Axis given be
The Basis and Altitude of the Frustum of a Pyra
If the aggregate of both the Bases of the Frustan
and the mean proportional betweFrustum.
Example. In Fig. 22. Let CDEF represent
the greater Base of a Pyramid, whose superficiHGLKO. 85 the mean proportional between
them is. 1. 2775 and the aggregate of these three
numbers is. 4. 0475. Let the given Altitude be 15.
the third part thereof is. 5 by which if you mulFrustum Pyramid.
And to find the content of the Frustum Cone.
I say.
As. 1. ro 78539. so 20.23 to 15. 884397, the content of the Cone required.
But if the Bases of the Frustum Pyramid shall
be square, you may find the content in this man
Multiply each Diameter by it self and by one
another, and the aggregate of these Products, by
the third part of the altitude, the last Product shall
be the content of the Frustum Pyramid.
Example. Let the Diameter of the greater Base be 144, the Diameter of the lesser Base 108, and the altitude 60.
Which being multiplyed by 20 the third part
of the Altitude, the Product 959040 is the conFrustum Pyramid.
And this content being multiplied by .785 39
the content of the Frustum Cone will be .753 .228.
Another way.
Find the content of the whole Pyramid of the
greater and lesser Diameter, the lesser content deFrustum. To find the content of
the whole Pyramid, you must first find their se
As the difference between the Diameters,
Is to the lesser Diameter.
So is the Altitude given, to the Altitude cut off.
Example. The difference between the former Diameter. 144. and 108 is 36, the Altitude 60. now then As 36. 108∷60. 108. the altitude cut off.
Now then if you mnltiply the lesser Base 1 1664 by 60 the third part of 180 the Product 699840 is the content of that Pyramid.
And adding 60 to 180 the Altitude of the greatFrustum Pyramid as before.
And upon these grounds may the content of Taper Timber, whether round or square, and of Brewers Tuns, whether Circular or Elliptical, be computed, as by the following Propositions shall be explained.
The breadth and depth of a Taper piece of Squared Timber, both ends being given together with the length, to find the content.
According to the last Proposition, find the A
The Sum of these 18.6264 being multiplyed by 8 one third of the length, the content will be found to be 149. 0112. Thus by the Table of Logarithms the mean proportional between the two Bases is easily found, and without extracting the square Root, may by natural Arithmetick be found thus.
A 4 2/2 CX A half C multiplyed by B: And C
more half A multiplyed by D being added toge
Being multiplyed by 8 the third of the length, the content will be. 149. 49000. The like may be done for any other.
The Diameters of a piece of Timber being given at the Top and and Bottom, together with the length, to find the content.
The Proposition may be resolved either by the
Squares of the Diameters, or by the Areas of
the Circles answering to the Diameters given, for
which purpose I have here annexed not only a Ta
And therefore putting S = The Sum of the
Tabular numbers answering to the Diameters at
each end.
X = The difference between these Diameters.
L = the length of the Timber, C = The
content.
Then 1 ½ S = ½ - XX. + L. = C.
If you work by the Table of the squares of Numbers. you must multiply the less side of the Equation, by 0.26179 the third part of 0.78539 the Product being multiplyed by the length, will give the content.
But if you work by the Table of the third parts
of the Areas of Circles in full measure, the ta
But because that in measuring of round Tim
Example. Let the circumference of a piece of
Timber be 8325220 looking this Number in the
second column of that Table, I find the next less to
be 8.168140 and thence proceeding in a streight
Line, I find that in the seventh Column the Num
And although by these two Tables all round Timber may be easily measured, yet it being more usual to take the Circumference of a Tree, then the Diameter, I have here added a third Table, shewing the third part of the Areas of Circles answering to any circumference under 10 foot, and that in Natural and Artificial numbers, the use of which Table shall be explained in the Proposition following.
The Circumference of a piece of round Timber at
both ends, with the length being given, to find the con
The Circumference of a Circle being given, the
Area thereof may be found as hath been shewed,
in the 7 Chapter, Proposition 4. and by the first Pro
The Artificial numbers were computed by ad
And by these Natural and Artificial numbers
By the Natural numbers in the same manner as the content was computed, the Diameters being given, and by the Natural and Artificial numbers both, by finding a mean proportional between the two Areas at the top and bottom of the Tree, as by Example shall be explained.
Let the given Diamensions, or Circumferences be At the Bottom 9.95 Their difference is 6.20 At the Top 3.75
Which being multiplyed by the length 24, the content will be 95. 73468.
Mr. Darling in his Carpenters Rule made easie,
doth propound a shorter way, but not so exact,
which is by the Circumference given in the middle
of the piece to find the side of the Square, nameExample.
The circumference at the top and bottom of the Tree being given 9.95 and 3.75 the Sum is 13.70 The half thereof is the mean circumfer. 6.85 Which sought in the Table, the Numbers are.
The Natural number is 1.244657, which being multiplyed by 3 the Product is 3.733971, which multiplyed by the length 24, the content is 89. 615304.
The Diameters of a Brewers Tun at top and bottom
being given with the height thereof, to find the con
In Fig. 29. Let the given Diameter.
At the top be AC 136 BD 128
At the bottom. KG 152 HF 144 Altit. 51 Inches.
The which by the 5 Proposition of this Chap.
may thus be computed. AC 139 + ½ KG 76 = 212
× BD 128 the Product is 27136.
KG 1524 ½ AC 68 = 220 × HF 144 the
Product is 31680. the Sum of these 2 Products is
58816 which being multiplyed by onethird of 51,
that is by 17, and that Product multiplyed by
26179 the third of 78539 will give the content.
Thus the content of a Tun may be found in In
Or thus; divide the former. 26179 by 282 the quotient will be 00092836. by which the content may be found in Ale Gallons in this manner.
The Diameters of a close Cask, at head and bung with the length given, to find the content.
In the resolving of this Proposition, we are to
consider the several forms of Casks, as will as
the kind of the Liquor, with which it is filled,
for one and the same Rule will not find the con
Frustum of a Spheroid, the
middle Frustum of a Parabolical Spindle, the midFrustum of two Parabolick Conoids, or for the
middle Frustum of two Cones abutting upon one
common Base.
And the content of these several Casks may be
found either by equating the Diameters, or by e
And the Area of a Circle to any given Diame(as I think)
may be more readily made in this manner.
The Square of any Diameter in Inches, being
divided by 3.81972 will give the Area of the Cir
But because it is easier to multiply then divide:
Again if you divide 1024 by 1077.161 the quotient will be 9508, or being multiplied by 00092836, the Product will be 9508.
Lastly if you divide 1024 by 882.755, the quo
And in this manner may the Tables be made for
Wine and Beer-measure, but the second differen
Thus the Squares of 1. 2. 3. and 4 Inches are.
1. 4. 9 and 16 by which if you multiply 00113333,
the several Products will be third part of the Area,
of the Circles answering to those Diameters in
Wine-Gallons. Or 00092836 being multiplied by
those Squares, the several Products, will be the
third part of the Areas of the Circles answering
to those Diameters in Ale-Gallons; the which
with their first and second differences are as fol
And by the continual addition of the second
differences to the first, and the first differences to
the products before found, the Table may be con
The construction of the Tables being thus shewed: We will now shew their use in finding the content of any Cask.
Let S = the Sum of the Tabular Numbers anD = their difference X = the difference of the
Diameters themselves. L = the length of the
Vessel, and C = the content thereof.
1. If a Cask be taken for the middle Frustum
of a Spheroid, intercepted between two Planes
parallel, cutting the Axis at right Angles: Then
1 ½ S + ½ D × L = C.
2. If a Cask be taken for the middle Frustum
of a parabolical Spindle, intercepted between
two planes parallel cutting the Axis at right
Angles. Then 1 ½ S + ½ D × L = C.
Frustum
of two Parabolick Conoids, abutting upon one
common Base, intercepted between two Planes
parallel, cutting the Axis at right Angle: Then
1 ½ S: × L = C.
4. If a Cask be taken for the middle Frustum
of two Cones, abutting upon one common bafe,
intercepted between two Planes parallel cutting
the Axis at Right Angles. Then 1 ½ S—⅓ XX.
× L = C.
In all these four Equations, if you work by the
Table of Squares of numbers, you must multi
But if you work by the Tables of the third
parts of the Areas Circle, the Tabular Numbers
being multiplyed by the length only will give the
content required, only in the fourth Equation
instead of half the Square of the Difference of
the Diameters, take half the Tabular Number
answering to that difference, and you shall have
the content required; as by the following Exam
Examples in Wine-measure by the Table of the
Squares of Numbers.
The Diameter of a Vessel
At the Bung being 32 Inches.
At the Head 22 Inches.
The difference of the Diameters 10 Inches.
And the length of the Vessel 44 Inches.
And here for the Singularity of the Example,
I will set the Dimensions of a Cask lately made
in Herefordshire, for that excellent Liquor of Red
streak Cyder, the like whereof either for the
largeness of the Cask, or incomparable goodness
of that kind of Drink, is not to be found in all
England, nay and perhaps not in the World.
The length of the Cask is 104 Inches.
The Diameter at the Bung 92 Inches.
And the Diameter at the Head 74 Inches.
And thus you have the content of this Cask by
four several Ways of Gauging, but that which
doth best agree with the true content, found by
these that filled the same is the second way or that
which takes a Cask to be the middle Frustum of
a Parabolick Spindle, according to which the
content is 2047 Gallons. That is allowing 64
Gallons to the Hogshead. 32 Hogsheads very
near.
If a Cask be not full, to find the quantity of Li
To resolve this Proposition, there must be gi
As the whole Diameter, is to its wet Por
So is the Diameter in the Table. 10.000 to its like Portion, which being sought in the Table of Segments, gives you a Segment, by which if you multiply the whole content of the Cask, the Product is the content of the Liquor remaining in the Cask.
But in the Table of Segments in this Book,
you have the Area, to the equal parts of one half
of the Diameter only, when the Cask therefore
is more then half full, you must make use of the
dry part of the Diameter instead of the wet, so
shall you find what quantity of Liquor is wanting
to fill up the Cask, which being deducted from
the whole content of the Cask; the remainer is
the quantity of Liquor yet remaining, an Exam
1. Example, In a Wine Cask not half full, let
the great Diameter be as before 32 Inches, the
As the whole Diameter 32. is to the wet part
12. so is 10.000 to 3750, which being sought in
the Table, I find, the Area of that Segment to
be. 342518 which being multiplyed by the whole
content of the Cask 126.25, the Product is
43.24289750 and therefore there is remaining in
the Cask 43 & 1/4 ferè.
2. Example. In the same Cask let the wet part of the Diameter be 18 Inches. I say.
As 32.18 :: 10000.5625 whose Complement
to 10000 is 4375 which being sought in the Ta
As the whole Area of the Circle 1000000 is to the whole content of the Cask 126. 25.
So is the Area of the Segment sought. 420630, to the content 53.1044375 which is in this case the content of the Liquor that is wanting, this therefore being deducted from the content of the whole Cask, 136. 25. the part remaining in the Vessel is. 73. 1455625.
Thus may Casks be gauged in whole or in
part, in which a Table of Squares is sometimes
necessary, as being the Foundation, from whom
the other Tables are deduced; such a Table
therefore is here exhibited, for all Numbers un
The Rectangle made of the Sum and Difference of any two Numbers, is equal to the Difference of the Squares of these Numbers.
Example, Let the given Numbers be 36 and 85
And hence the Square of any Number under 10.000 may thus be found, the Squares of all Numbers under 1000 being given.
Example. Let the Square of 5715 be required.
The Square of 571 by the Table is 326041, there
TABLES FOR THE Measuring OF TIMBER, AND THE GAUGING OF CASKS AND Brevvers Tuns.
LONDON,
Printed for Thomas Passinger at the three Bibles on
London-Bridge. 1679.
Area of the Segments of a Circle whose
Diameter is 2.0000 &c. into a Table
shewing the Area of the Segment of any
Circle whose Area is given.
&c. to every Hundredth
Part of the Radius.
A TABLE SHEWING THE AREA OF THE SEGMENTS OF A CIRCLE WHOSE Whole Area is Unity, to the ten Thousandth part of the Diameter.
COSMOGRAPHIA, THE Second Part. OR, THE DOCTRINE OF THE PRIMUM MOBILE.
Of the Primum Mobile.
Of the General Subject of Astronomy.
AStronomy, is a Science concerning the Measure and Motion of the Spheres and Stars.
2. Astronomy hath two parts, the first is Absolute, and the other Comparative.
3. The Absolute part of Astronomy is that which treateth of the Measure and Motion of the Orbs and Stars absolutely without respect to any distinction of Time.
5. The Absolute part of Astronomy treateth of
the Primum Mobile, or Diurnal Motion of all the
Celestial Orbs or Spheres.
6. The Primum Mobile, or Diurnal Motion of
the Heavens, is that Motion, by which the se
7. This first and common Motion of the Hea
8. This Representation or Description of the
Visible World is by Circles, great and small, some
of which are expressed upon, and others are fra
9. The Circles without the Globe are chiefly
two; the Meridian and the Horizon, the one of
Brass, and the other of WoodHorizon and one Meridian in reHorizons and
Meridians.
11. This Brass Meridian is divided into 4 e
12. The Horizon is a great Circle without the Globe, which divides the upper part of Heaven from the lower, so that the one half is always above that Circle, and the other under it.
13. The Poles of this Circle are two, the one
directly over our Heads, and is called the Zenith;
the other is under feet, and is called the Na
14. The Horizon is either Rational or Sensi
15. The Rational Horizon is that, which divi
17. Besides these two great Circles without the
Globe, there are 4 other great Circles drawn up
18. The AEquator is a great Circle drawn upon
the Globe, in the middle between the two Poles
19. The AEquator is the measure of the MotiPrimum Mobile, for 15 Degrees of
this Circle do always arise in an hours time; the
which doth clearly shew, that the whole Heavens
are turned round by equal intervals in the space of
one day or 24 hours.
20. In this Circle the Declinations of the Stars are computed from the mid-Heaven towards the North or South.
Aries, and Libra, and then he
maketh the Days and the Nights equal: His enAries is in March, and is called the
Vernal Equinox; and his entrance into Libra, is
in September, and is called the Autumnal Equi
22. And from one certain point in this Circle, the Longitude of Places upon the Earth are reckoned; and the Latitude of Places are reckoned from this Circle towards the North, or the South Poles.
23. The Zodiack is a great Circle drawn up
24. The Poles of this Circle are as far distant
from the Poles of the World, as the greatest
Obliquity thereof is from the Equinoctial, that
is 23 Degrees, and 31 Minutes or therea
25. This Circle doth differ from all other Cir
26. The Zodiack then in respect of Longi
27. The 12 Signs into which the Zodiack is
divided, have these Names and Characters. Aries
♈. Taurus ♉. Gemini ♊. Cancer ♋. Leo ♌. Virgo
♍. Libra ♎. Scorpio ♏. Sagittarius ♐. Capricorand Pisces ♓.
28. These two Circles of the Equator and Zo
29. The other great Circles described upon the Globe are the Meridians: Where we must not think much to hear of the Meridians again. That of Brass without the Globe is to serve all turns, and the Globe is framed to apply it self thereto. The Meridians upon the Globe, will easily be perceived to be of a new and another use.
30. The Meridians upon the Globe are either the great or the less: Not that the great are any greater than the less, for they have all one and the same center, and equally pass through the Poles of the Earth; But those which are called less, are of less use than that, which is called the great.
Landmark of the whole Sphere, from
whence the Longitude of the Earth, or any part
thereof is accounted. And it is the only Circle
which passing through the Poles is graduated or
divided into Degrees, not the whole Circle but
the half, because the Longitude is to be reckoned
round about the Earth.
32. The lesser Meridians are those black lines;
which you see to pass through the Poles and suc
And to this purpose also upon the North end
of the Globe, without the Brass Meridian, there
is a small Circle of Brass set, and divided into two
equal parts, and each of them into twelve, that
is, twenty four all; to shew the hour of the Day
and Night, in any place where the Day and Night
exceed not 24 hours; for which purpose it hath a
little Brass Pin turning about upon the Pole, and
pointing to the several hours, which is therefore
the Index Horarius, or Hour Index.
33. Having described the great Circles fraRadius
by which they are drawn.
34. These lesser Circles are either the Tro
35. The Tropicks are two small Circles drawn
upon the Globe, one beyond the Equator towards
the North Pole, and the other towards the South,
Shewing the way which the Sun makes in his Di
36. These Tropical Circles do shew the point
of Heaven in which the Sun doth make either the
longest Day, or the Shortest Day in the Year, ac
37. The Polar Circles are two lesser Circles
drawn upon the Globe at the Radius of 23 De
38. These Polar Circles are 66 Degrees and a half distant from the Equator, and 43 Degrees distant from his nearest Tropick. They are called the Arctick and Antarctick Circles.
39. The Arctick Circle is that which is descri
40. The Antarctick Circle is that which is de
Having thus described the Globe or Astronomi
Of the Distinctions and Affections of Sphe
THE uses of the Globe as to practice, are ei
2. And that the nature of Spherical Trian
3. A Spherical Triangle then is a Figure con
In Fig. 1. The streight Line HAR doth reprePR the height of the Pole aPMS a Meridian, and these
three Arches by their intersecting one another
do visibly constitute four Spherical Triangles.
1. PMR. 2. PMH. 3. SHM. 4. SMR.
And every Arch is less than a semicircle, as in the
Triangle PMR, the Arch PR is less than the SePRS, the Arch MR is less than the SeAMR, and the Arch PM is less than the
Semicircle PMS, the like may be shewed in the
other Triangles.
5. Spherical or circular Lines are Parallel or Angular.
6. Parallel Arches or Circles, are such as are
drawn upon the same Center within, without, or
equal to another Arch or Circle. Thus in Fig.
1. The Arches ♋ M ♋ and ♑ O ♑ are though lesser
Circles, parallel to the Equinoctial AE A Q and do
in that Scheme represent the Tropicks of Cancer
and Capricorn. The manner of describing them
or any other Parallel Circle is thus, set off their
distance from the great Circle, to which you are to
draw a parallel with your Compasses, by help of
your Line of Chords, which in this Example is
23 Degrees and a half from AE to ♋, then draw
the Line A ♋, and upon the point ♋ erect a PerPAS extended, is the Center of that Pa
7. A Spherical Angle, is that which is con
8. A Spherical Angle is either Right or Ob
9. A Spherical Right Angle is that which is
conteined, by two Arches of the greatest Circles
in the Superficies of the Sphere cutting one ano
10. An Oblique Spherical Angle, is that
which is conteined by two Arches of the greatest
Circles in the Superficies of the Sphere, not be
11. An Oblique Spherical Angle is Obtuse, or Acute.
12. An Obtuse Spherical Angle, is that which is greater than a Right Angle. An Acute is that which is less than a Right Angle.
13. If two of the greatest Circles of the Sphere
shall pass through one anothers Poles, those two
great Circles shall cut one another at Right An
Fig. 1. The Arch PM doth
intersect the Meridian and Horizon, but not in
the Poles of either, therefore the Angle HPM
upon one side of the intersection of that Arch with
the Meridian, is more than a Right Angle; And
the Angle MPR upon the other side of the InterPMH
upon the one side of the intersection of the Arch
PM with the Horizon HR, is greater than a
right Angle; and the Angle RMP upon the o
15. A Spherical Angle is measured by the Arch
of a great Circle described from the Angular
point between the sides of the Angle, those sides
being continued unto Quadrants. Thus the Arch
of the Equator TQ in Fig. 1. is the measure
of the Angle MPR, or TPQ, the sides PT and
PQ being Quadrants.
And the measure thereof in the Projection may
thus be found: lay a Ruler from P to T, and it will
cut the Primitive Circle in V; and the Arch VQ
being taken in your Compasses and applyed to
your Line of Chords, will give the Quantity of
the Angle propounded.
16. The Complement of a Spherical Arch or
Angle, is so much as it wanteth of a Quadrant,
if the Arch or Angle given be less than a Qua
17. An Arch of a great Circle cutting the
Arch of another great Circle, shall intersect one
another at Right Angles, or make two Angles;
which being taken together, shall be equal unto
two Right. Thus in Fig. 1. The Axis PAS or
Equinoctial Colure doth cut the Equator AE A Q
at Right Angles; but the Meridian PMS doth
cut the Horizon HMR at Oblique Angles, maPMR less than a Right, and the
Angle SMR more than a Right, and both to
18. From these general Definitions proper to
Spherical Lines or Arches, the general Affecti
19. And the whole Body of the Sphere or Globe, in respect of the Horizon, is looked upon by the Earths Inhabitants, either in a Parallel, a Right, or an Oblique Sphere.
20. A Parallel Sphere is, when one of the Poles
of the World is elevated above the Horizon to
the Zenith, the other depressed as low as the Na
21. A Right Sphear is, when both the Poles
of the World do lie in the Horizon, and the E
22. An Oblique Sphere is, when the Axis of
the World (being neither Direct nor Parallel to
the Horizon) is inclined obliquely towards both
sides of the Horizon, as in Fig. 1. Whence it co
And in this Position of Sphere, the Days are someFig. 1. For when the Sun riseth at M, the Line M
♋ above the Horizon is the Semidiurnal Arch of
the longest day. When he riseth at C, the Arch C
♑ above the Horizon, is the Semidiurnal Arch of
the shortest Day: And when he riseth at A, the
Days and Nights are of equal Length, the SemiAAE, being equal to the SeminoAQ.
Of the kind and parts of Spherical Trian
HAving shewed what a Spherical Triangle
is, and of what Circles it is composed, with
the general Affections of such Lines: I will now
shew how many several sorts of Triangles there
are, of what Circular parts they do consist, and
such Affections proper to them as will render the
so
2. Spherical Triangles are either Right or Ob
3. A Right Angled Spherical Triangle, is that which hath one or more Right Angles.
4. A Spherical Triangle which hath three
Right Angles, hath always his three sides QuaFig. 1. The Spherical Triangle
AZR, the Angles ZRA, RAZ and AZR are
right Angles, and the three sides AZ, ZR and
AR are Quadrants also.
Fig. 1. The sides of the Spherical TrianTPQ, namely TP and PQ are Quadrants,
and the Angles opposite to these sides, to wit, PTQ
and TQP are Quadrants also, and the third
Angle TQ is the measure of the third Angle
TPQ. But the Right Angled Triangle which
hath one Right and two Acute Angles, is that
which cometh most commonly to be resol
6. The Legs of a right Angled Spherical
Triangle are of the same Affection with their opZAR Fig. 1.
The side ZA is a Quadrant, and the Angle at A is
right, because Z is the Pole of the Arch AR and
ZA is perpendicular thereunto. And in the TriRAAE the side RZAE being more then
a Quadrant the Angle RAAE is more then a
Quadrant also, being more then the Right AnRAZ. And in the right Angled Spherical
Triangle APR the side PR being less then a
Quadrant, the Angle PAR is less then a Quadrant
also, being less then the right Angle RAZ.
7. An Oblique angled Spherical Triangle is either acute or obtuse.
8. An Acute angled Spherical Triangle hath
all his Angles Acute, and each Side less then a
Quadrant; As in the Triangles, ZFP. Fig. 2.
The Angles at Z and P are acute, as appeareth by
inspection; and the Angle at F is acute also beCD = EM is less then
a Quadrant.
viz. Acute
and mixt.
10. The Sides of a Spherical Triangle may be
turned into Angles, and the Angles into Sides;
The Complement of the greatest Side or greatest
Angle to a Semicircle being taken in each converZFP into sides in Fig. 3. EAE is
the measure of the Angle at P, and AD in the
Triangle ADC equal thereunto, AC is the
Complement of FZP to a Semicircle, and KM the
the Measure of the Angle at F is equal to DC, and
so the Sides of the Spherical Triangle ADC are
equal to the Angles of the Spherical Triangle
FZP, making the side AC equal to the CompleZ to a Semicircle.
11. In Right Angled Spherical Triangles the
Sides intending the Right Angle we call the Legs;
The Side subtending it the Hypotenuse.
12. In every Spherical Triangle besides the
Area or space contained, there are six parts. viz.
Three Sides and three Angles and of these six there
must be always three given to find the rest, but in
right Angled Spherical Triangles there are but
five of the six parts parts which come into questi
AEZB as may be seen in Fig. 1. In
which the outward Circle HZR doth represent
the Brass Meridian, AEAQ the Equator, and ZC
the Quadrant of altitude.
14. An Oblique Angled Spherical Triangle
may be represented upon the Globe in this manZFP in Fig. 3. In which
ZP is numbred upon the Brass Meridian from S
the Pole of the World to Z the Zenith, ZF the
Azimuth Circle represents the Quadrant of AltiPF the great Meridian upon the Globe
intersecting the Quadrant of Altitude at F.
15. A Right or Oblique Angled Spherical
Triangle being thus delineated upon the Globe,
there needs no further instructions, as to the mea
16. The Circles in the first Figure are the MeHZRN in Fig. 1. Within which all the other
are projected. The Horizon, AEquator, AEquiHAR reAEAQ the Equator, PAS
the Equinoctial Colure and ZAN the East and
West Azimuths, in the drawing of these there
is no difficulty, PMS is a Meridian, and ZCN
an Azimuth Circle, for the drawing of which
there are three points given and the Centers of
the Meridians do always fall in the Equinoctial exAT set from
A to D, or the Secant of the Complement set
from A to D will give the Center of the MeridiPTS. The other two Circles in the 1. Fig.
AE to ♋ and AE to ♑, if you draw a Line from
A to ♋ and another perpendicular thereunto from
♋ it will cut the Axis SAP extended in the CenAE ♋ set from ♋ to the
Axis extended will give the Center as before, and
thus may all other Parallels be described.
17. In the second and third figures, the two
extream points given in the Meridians are not eF and the CenA draw the Diameter TAS, and cross the
same at Right Angles with the Diameter BAG, a
Ruler laid from G to F will cut the primitive CirL, make EL = BL a Ruler laid from G to E
will cut the Diameter SAT in V the Center of
the Circle BDG. Which Circle doth cut the
Diameter HAR in the Pole of ZF, and the DiAEAQ in D in the Pole of PFX, and
a Ruler laid from Z to C will cut the Primitive
Circle in Y, and making Y O equal to Y a Ruler
laid from Z to O will cut the Diameter HAR,
extended in the Center of the Circle ZF.
18. Having drawn the Circle ZFI, in Fig. 13.
The Circle PEX, or any other passing through
the point F, may easily be described. Draw
AEQ at right Angles to PX, a Ruler laid from
G unto (e) will cut the Primitive Circle in (m)
make mn = Bn, a Ruler laid from G to n
TFS in p make Fq =
Fp so shall FQ be the Radius, and the Center
of the Circle PFX as was desired.
19. The preceeding directions are sufficient
for the projecting of several Circles of the Globe
before described upon the Plane of the Meridian,
and the parts of those Circles so described may
thus be measured. In Fig. 1. HZ = CZ = AZ
90 Degrees. Whence it followeth, that the
Quadrant CZ is divided into Degrees from its
Pole M, by the Degrees of the Quadrant HZ,
that is a Ruler laid from M to any part of the
Quadrant HZ will cut as many Degrees in CZ as
it doth in the Quadrant HZ, and thus the Arch
CF = HK the Arch CB = HL, and the Arch
BF = LK.
20. That which is next to be considered is the
projecting or laying down the Angles of a TrianA in the Triangle PAR; or such as are conPMR. Fig. 1. Or such as are conteined by
two circular Lines, as the Angles FZP or ZFP
in Fig. 3. The projecting or measuring the first
sort of these Angles, needs no direction.
21. To project an Angle conteined by a streight
and a circular line as the Angle AEBZ in Fig. 1.
Do thus, lay a Ruler from N to C, and it will cut
the Primitive Circle in K make ZX = HK, a RuN to X will cut the Diameter HAR
in the point M the Pole of the Circle ZCN,
a Ruler laid from M to B the Angular point proI, make
NY = HL a Ruler laid from N to Y will cut
the Circle ZCN in W, a Ruler laid from B to W
will cut the Primitive Circle in A, make AQ equal
to the Angle propounded, and draw the DiameBAQ, then is the Angle AEBZ or NBQ =
NQ as was required.
22. If the Angle had been projected and the
measure required, a Ruler laid from M to B
would give L and making NY = HL a Ruler
laid from M to Y would give W, from B to W
would give A, and AQ would be the measure of
the Angle propounded.
23. To project an Angle conteined by two
circular lines, one of them being an Arch of the
Primitive Circle, as the Angle AEZB, Fig. 1.
Do thus, set off the quantity of the Angle given
from H to G, a Ruler laid from Z to G will cut the
Diameter HAR in the point C, so may you draw
the Circle ZCN and the Angle HZC will be eHG = HC as was required.
24. If the Angle had been projected and the
measure required, a Ruler laid from Z to C would
cut the Primitive Circle in G and HG would be
the measure of the Angle propounded.
25. To project an Angle conteined by two obZFP in
Fig. 3. You must first find the Pole of one of the
two Circles conteining the Angle propounded,
suppose ZBI, a Ruler laid from C the Pole thereF, the Angular point propounded, will cut
the Primitive Circle in a make ab equal to the
Angle propounded, a Ruler laid from F to b will
cut the Diameter AEAQ in D the Pole of the
Circle PEX, a Ruler laid from G to e will cut
the Primitive Circle in m, make mn = Bm
G to n will cut the Diameter TAS
in p, make Aq = Ap so shall Fp be the RadiPFX and the AnZFP = ab, as was propounded.
26. If the Angle had been projected and the
measure required; through the point F draw the
Diameter TFS and the Diameter BAG at right
Angles thereunto, a Ruler laid from G to F will
cut the Primitive Circle in K, and making KE =
BK a Line drawn from G to E will cut the DiaTAS in the Center of the Circle GDB
cutting the Diameter HAR in C the Pole of the
Circle ZBI, and the Diameter AEAQ in D,
the Pole of the Circle PEX and a Ruler laid from
F to C and D will cut the Primitive Circle in a
and b the measure of the Angle required.
Or a Ruler laid from F to K and M will cut the
Primitive Circle in Deg. the measure of the Angle
propounded as before.
Or thus a Ruler laid from C and D to F will
cut the Primitive Circle in ae and h set 90 Degrees
from e and h to f and l a Ruler laid from C to f will
cut ZBI in M and a Ruler laid from D to l will
cut PEX in K. This done a Ruler laid from F
to K and M will cut the Primitive Circle in g and
d the measure of the Angle as before.
And in Fig. 2. The quantity of the Angle ZEP
may thus be found. A Ruler laid from C the Pole
of the Circle ZFI to F the angular point will cut
the Primitive Circle in a, set off a Quadrant from
a to b, a Ruler laid from C to b will cut the
Circle ZFI in the point M. In like manner a
Ruler laid from D the Pole of the Circle PEX,
will cut the Primitive Circle in D, set off a QuaA to h, a Ruler laid from D to P will
PFX in K: Lastly a Ruler laid
from F to K, and M will cut the Primitive Circle
in NS the measure of the Angle KFM or ZFP,
as was propounded.
27. Having shewed how a right or oblique An
28. The Spherical Triangles usually representHypotenusas.
The first is called the Ecliptical Triangle, whose
Hypotenusa is an Arch of the Ecliptick, the Legs
thereof are Arches of the AEquator and MeridiADF, in Fig. 1. In which the five Cir
1. The Hypotenuse or Arch of the Ecliptick
AF.
2. The Leg or Arch of the AEquator, AD.
3. The Leg or Arch of the Meridian DF.
4. The Oblique Angle of the Equator with
the Ecliptick and the Suns greatest Declination
DAF.
5. The Oblique Angle of the Ecliptick and
Meridian, or the Angle of the Suns position
AFD.
The two next I call Meridional, because the
Hypotenusas in them both, are Arches of a MeriMPR in Fig. 1. In which the five Circular parts
are;
1. The Hypotenusa or Arch of a Meridian
PM.
2. The Leg or Arch of the Horizon MR, the
Suns Azimuth North.
3. The Leg or Arch of the Brass Meridian,
representing the height of the Pole PR.
4. The Oblique Angle of the Meridian upon
the Globe; with the Brass Meridian, or Angle
of the Hour from Midnight. P.
5. The Oblique Angle of the Suns Meridian
with the Horizon, or the Complement of the
Suns Angle of Position PMR.
The other Right Angled Meridional Triangle
is noted with the Letters AEG in Fig. 1. In which
the 5 Circular parts are.
1. The Hypotenusa or present Declination of
the Sun, AE.
2. The Leg or Suns Amplitude at the hour of
six, AG.
3. The other Leg or Suns height at the same
time EG.
4. The Angle of the Meridian with the HoEAG.
5. The Angle of the Meridian with the AziAEG.
The fourth Right Angled Spherical Triangle,
I call an Azimuth Triangle, because the Hypotenusa
doth cut the Horizon in the East and West
Azimuths, as is represented by the Triangle
ADV. in Fig. 1. In which the 5 Circular parts
are,
Hypotenusa, or Arch of the Sun or Stars
Altitude AV.
2. The Leg or Declination of the Sun or Star,
DV.
3. The other Leg, or Right Ascension of the
Sun or Star, AD.
4. The Oblique Angle or Angle of the Poles
elevation, DAV.
5. The other Oblique Angle or Angle of the
Sun or Stars Position, DVA.
The fifth and last Right Angled Spherical TriHypotenusa thereof is an
Arch of the Horizon, and is represented by the
Triangle AMT in Fig. 1. In which the 5 Circular
parts are;
1. The Hypotenusa and Arch of the Horizon,
or Amplitude of the Sun at his rising or setting,
AM.
2. The Leg conteining the Sun or Stars DecliTM.
3. The other Leg or Ascensional difference
AT, that is, the difference between DT the
Right Ascension and DA the Oblique Angle.
4. The Oblique Angle of the Horizon and ETAM.
5. The other Oblique Angle, or Angle of
the Horizon and Meridian AMT.
The Oblique Angled Spherical Triangles uFZP in
Fig. 1. Whose Circular parts are;
1. The Complement of the Poles elevation
ZP.
FP.
3. The Complement of the Suns Altitude or
Almicantar FZ.
4. The Suns Azimuth or Distance from the
North FZP.
5. The hour of the day or distance of the Sun
from Noon ZPF.
6. The Angle of the Suns Position ZFP.
The second Oblique Angled Spherical TrianFZP in
Fig. 1. Whose parts are.
1. The Complement of Latitude as before
ZP.
2. The distance between the two places at Z
and F or side FZ.
3. The Complement of the Latitude of the
place at F or side FP.
4. The difference of Longitude between the
two places at Z and F or the Angle FBZ.
5. The point of the compass leading from Z
to F or Angle FZP.
6. The point of the Compass leading from F
to Z, or Angle ZFP.
The third Oblique Angled Spherical TrianFSP in Fig. 4. In which the Cir
1. The distance between the Pole of the
SP.
2. The Complement of the Stars Declination,
FP.
3. The Complement of the Stars North LaFS.
4. The Angle of the Stars Right Ascension
FPS.
5. The Complement of the Stars Longitude
FSP.
6. The Angle of the Stars Position SFP.
29. And thus at length I have performed, what
was proposed in the 15 of this Chapter, that is,
I have shewed how the several Circles of the
Globe, may be projected upon the Plane of the
Meridian, the several useful Triangles that are
described by such projection, with such Astrono
In the Right Angled Spherical Triangle
AEBZ in Fig. 1. The measure of the Angle
AEZB is reckoned in the Horizon from H to C
but to lay down or measure the Angle AEBZ
the readiest way is to describe the Triangle again,
making AEZ = AEB and AEB = AEZ, so
will the Angle AEBZ stand where the Angle
AEZB is, and may be measured in the Horizon
as the other was.
FZP in Fig. 1. The Angles at Z and P are
easily measured or laid down upon the Globe, but
to perform the same with the Angle ZFP, you
may represent it at the Pole or Zenith and find the
measure in the Equator or Horizon.
30. And now having, as I hope, sufficiently prePrimum Mobile, by
shewing how the Heavens and the Earth are repreNepiers Catholick Proposition for the solu
Of the solution of Spherical Triangles.
IN Spherical Triangles there are 28 Varieties or
Cases, 16 in Rectangular, and 12 in Oblique,
whereof all the Rectangular and ten of the Obli
1. Axiom. In all Right Angled Spherical TriHypotenusas are proportional to
the Sines of their Perpendicular.
2. Axiom. In all right Angled Spherical Tri
That all the Cases of a Right Angled Spherical
Triangle may be resolved by these two Axioms,
the several parts of the Spherical Triangle propoHypotenusa, into Bases and Perpendiculars and the
contrary. By which means the proportions as to
the parts of the Triangle given, are sometimes
changed into Co-sines instead of Sines, and into CoNepier observing; those parts of the Right Anviz. The Hypotenuse and the two AMPR in Fig. 1. And these five he calleth the Cir
2. Now if you reckon five Circulat parts in a
Triangle, one of them must needs be in the midMPR if you
make the Leg PR the middle part, then the oMR and the Angle Comp. P. Are the
extreams conjunct, the Hyp. Comp. MP and Comp.
M, are the extreams disjunct, and so of the rest,
as in the following Table.
3. These things premised, the Ld. Nepier as a
consectory from the two preceeding Axioms
hath composed this Catholick and Universal Pro
The Rectangle made of the Sine of the middle part and
Radius is equal to the Rectangle made of the Tan
Therefore if the middle part be sought, the
Radius must be in the first place; if either of
the extremes, the other extreme must be in the
first place.
Only note that if the middle part, or either
of the extremes propounded be noted with its
That these directions may be the better con
By these 10 Rectangles may the 16 Cases of a Right angled Spherical Triangle be resolved, and some of them twice over; for although there are but 16 varieties in all Right angled Spherical Triangles, yet 30 Astronomical Problems may be resolved by one Triangle, as by the following Examples shall more clearly appear.
The Legs given, to find the Angles.
IN the Right angled Spherical Triangle
MPR. The given Legs are MR and RP.
The Angles at M and P are required.
By the first of the 10 equal Rectangles
s PR x Rad.=t MR x ct P. in which P is
fought, therefore putting MR in the first place.
The proportion is. t MR. x Rad.∷s PR.
ct P.
And by the third equal Rectangle. t PR.
Rad.∷s MR. ct M.
The Legs given, to find the Hypotenuse.
In the Right angled Spherical Triangle MPR.
The given Legs are MR and PR. The HypoMP is required.
By the eighth of the 10 Rectangles cos. MP x
Rad.=cos. PR x cos. MR in which MP the
middle part is sought, therefore Radius must be
put in the first place, and then the proportion is.
Rad: cos. PR∷ cos. M. R. cos. MP.
A Leg with an Angle opposite thereunto being given, to find the other Leg.
In the Right angled Spherical Triangle MPR,
let there be given. The Leg MR. The Angle
P. The Leg PR inquired.
By the first of the 10 Rectangles. Rad. tMR
∷cot. P. Sine PR. or The Leg PR and the
Angle M given, to MR.
By the 3 of the 10 Rectangles. Rad. tPR∷ct
M. Sine MR.
A Leg with an Angle conterminate therewith being given, to find the other Leg.
In the Right angled Spherical Triangle, MPR,
The given Leg is MR, with the Angle M. The
Leg PR is required.
By the 3 Rectangle. cot. M. Rad∷Sine
MR. tPR.
The given Leg RP, and Angle P. The Leg
MR is required.
By the 1. Rectangle. ctP. Rad∷sine RP.
tang. MR.
A Leg and an Angle conterminate therewith being given, to find the Hypotenuse.
In the Right angled Spherical Triangle MPR,
let there be given,
MR and the Angle M
PR and the Angle P
MP.
By the 5. Rectangle, t MR. Rad∷cos. M. ct MP.
By the 9. Rectangle. t PR. Rad.∷cos. P. ct MP.
The Hypotenuse and a Leg given, to find the con
In the Right angled Spherical Triangle MPR,
let there be given,
MP,
By the 5. Rectangle, Rad. ct MP∷t MR. cos. M.
By the 9. Rectangle, Rad. ct MR∷t PR. cos. P.
The Hypotenuse and one Angle given, to find the other Angle.
In the Right angled Spherical Triangle MPR,
let there be given,
MP
By the 7. Rectangle, cot. M. Rad∷cos. MP. cot. P.
By the 7. Rectangle cot. P. Rad∷cos. MP. cot. M.
The Oblique A
In the Right angled Spherical Triangle MPR,
let there be given The Angles at P and M, To
find the Hypotenuse PM.
By the 7. Rectangle. Rad. ct P∷cot. M.
cos. MP.
The Hypotenuse and an Angle given, to find the Leg conterminate with the given Angle.
In the Right angled Spherical Triangle MPR,
let there be given,
PM
By the 9. Rectangle, ct PM. Rad∷cos. P. t PR.
By the 5. Rectangle, ct PM. Rad∷cos. M. tMR.
The Hypotenuse and an Angle given, to find the Leg opposite to the given Angle.
In the Right angled Spherical Triangle
MPR, let there be given,
PM
By the 2. Rectangle, Rad. s MP∷s M. Sine PR.
By the 4. Rectangle, Rad. s MP∷s P. Sine MR ▪
A Leg and an Angle opposite thereunto being given, to find the Hypotenuse.
In the Right angled Spherical Triangle MPR,
let there be given,
PM.
By the 2. Rectangle, s M. Rad∷s PR. s MP.
By the 4. Rectangle, s P. Rad∷s MR. s PM.
The Hypotenuse and a Leg given, to find the Angle opposite to the given Leg.
In the Right angled Spherical Triangle PMR,
MP and the Leg MR are given,
the Angle at P is required.
By the fourth Rectangle Sine MP to, Rad∷s
MR. s P.
The Hypotenuse MP and Leg PR given, the
Angle M is required.
By the second Rectangle. sMP. Rad∷s
PR. s M.
The Angle and Leg conterminate with it being given, to find the other Angle.
In the Right angled Spherical Triangle PMR,
let there be given,
By the tenth Rectangle, Rad. cs MR∷s M. cs P.
By the sixth Rectangle, Rad. s P∷cs PR. cs M.
An Angle and a Leg opposite thereunto being given, to find the other Angle.
In the Right angled Spherical Triangle MPR,
let there be given,
By the 10. Rectangle, cs MR. Rad∷cs P. csM.
By the 6. Rectangle, cs PR. Rad∷cs M. sP.
The Oblique Angles given, to find a Leg.
In the Right angled Spherical Triangle MPR,
let there be given, the Angles at M and P, to
find the Legs MR and PR.
By the 10. Rectangle, sM. Rad∷cs P. cs MR.
By the 6. Rectangle, s P. Rad∷cs M. cs PR.
The Hypotenuse and one Leg given, to find the other Leg.
In the Right angled Spherical Triangle MPR,
let there be given,
MP and the Leg
Thus I have given you the Proportions by
which the 16 Cases of a Right angled Spherical
Triangle may be resolved, In which there are
contained 30 Astronomical Problems. Two in
every Case except the Second and the Eighth.
In both which Cases there are but two Problems.
And thus I have done with Right angled Spheri
4. In Oblique angled Spherical Triangles
5. If the Angles at the Base be both acute
or both obtuse, the Perpendicular shall fall with
6. However the Perpendicular falleth, it must
be always opposite to a known Angle, for your
better direction, take this General Rule.
From the end of a Side given, being adjacent to an
Angle given, let fall the Perpendicular.
As in the Triangle FPS in Fig. 4. If there
were given the Side F S and the Angle at S, the
Perpendicular by this Rule must fall from P upon
the Side S P extended, if need require.
But if there were given the Side P S and the
Angle at S, the Perpendicular must fall from F
upon the Side F S.
7. To divide an Oblique angled Spherical
Triangle into two Right, by letting fall a Per
FPS in Fig. 4. let it be required to let fall
a Perpendicular from P upon the Side FS. SupP to stand in the Zenith, where
the Arch FS shall cut the Zodiack, which in
this Figure is at K, make a mark, and from this
Point of Intersection of the Circle upon which
the Perpendicular is to fall with the Zodiack,
reckon 90 Degrees, which suppose to be at P; a
thin Plate of Brass with a Nut at one end thereof,
whereby to fasten it to the Meridian, as you do
the Quadrant of Altitude, being graduated as
that is, but of a larger extent (for that a QuaP and turned about till it cut the Point L in the
Zodiack, will describe upon the Globe the Arch
of a great Circle PEL, intersecting the Side
F S at Right Angles in the Point E, because the
Point L in the Zodiack is the Pole of the Circle
SFK, now all great Circles which passing
through the Point L, shall intersect the Circle
SKG, shall intersect it at Right Angles; by the
13. of the 2. Chapter.
9. And hence to divide an Oblique angled
Spherical Triangle into two Right by projectiFPS, the Pole of
the Circle SFK is L, therefore the Circle
BLP shall cut the Arch FS at Right Angles
in the Point E. And because the Point M is the
Pole of the Circle BFP, therefore the Circle
GMS shall cut the Circle BFP at Right AnD, the Side F P being extend
Two Sides with an Angle opposite to one of them be
In the Oblique angled Spherical Triangle
F P S, in Fig. 4. the Sides and Angles given and
required will admit of six Varieties; all which
may be resolved by the Catholick Proposition,
at two operations, but those two may be re
Two Sides with an Angle appo
In this Case there are six Varieties, all which may be resolved by the Catholick Proposition, according to the Table following.
Two Sides and an Angle opposite to one of them being given, to find the third side.
The Varieties in this Case, with their resolu
Two Angles with a Side opposite to one of them being given, to find the Side opposite unto the other.
The Varieties in this Case, with their Resolu
Two Angles and a side opposite to one of them being given, to find the Side between them.
The Varieties and Proportions, are as follow
Two Angles and a Side opposite to one of them being given, to find the third Angle.
The Varieties and Proportions are as follow
Two Sides and their contained Angle being given, to find either of the other Angles.
The Varieties and Proportions are as follow
Two Sides and their contained Angle being given, to find the third Side.
The Varieties and Proportions are as follow
Two Angles and their contained Side being given, to find one of the other Sides.
Two Angles and the Side between them being given, to find the third Angle.
The Varieties and Proportions are as follow
The three Sides being given, to find an Angle.
This Case may be resolved by the Catholick
Proposition also, according to the direction of
the Lord Nepier, as I have shewed at large in the
Second Book of my Trigonometria Britannica, Chap.
2. but may as I conceive be more conveniently
solved, by this Proposition following.
As the Rectangle of the Square of the Sides containing the Angle inquired;
Is to the Square of Radius: So is the Rectan
To the Square of the Sine of half the Angle inquired.
In this Case there are three Varieties, as in the
Triangle FZP. Fig. 3.
The three Angles given, to find a Side.
This is the Converse of the last, and to be reACD
will be equal to the Angles of the Triangle FZP
n Fig. 3.
And thus the Sides of the Triangle ZPF are
equal to the Angles of the Triangle ACD.
The Complement of the greatest Side PF to a
Semicircle being taken for the greatest Angle
ADC.
And in this Case therefore, as in the preceding,
there are three Varieties which make up sixty Pro
Of such Spherical Problems as are of most
General Use in the Doctrine of the Pri
The greatest Declination of the Sun being given, to find the Declination of any Point of the Ecliptick.
THe Declination of the Sun or other Star, is
his or their distance from the Equator, and
as they decline from thence either Northward or
Southward; so is their Declination reckoned
North or South.
2. The Sun's greatest Declination, which in
this and many other Problems is supposed to be
given, with the Distance of the Tropicks, EleEquator, and Latitude of the Place,
may thus be found.
Take with a Quadrant, the Sun's greatest and
least Meridian Altitudes, on the longest and
shortest days of the year, which suppose at London
to be as followeth.
Now then in the Right angled Spherical TriADF in Fig. 1. there being given.
1. The Angle of the Sun's greatest DeclinaDAF. 23. 525.
2. The Sun's supposed distance from ♈ to ♎
AF. 60 deg.
The Sun's present Declination DF may be
found, by the 10 Case of Right angled Spherical
Triangles.
As the Radius
The Sun's groatest Declination, with his Distance from the next AEquinoctial Point being given, to find his Right Ascension.
In the Right angled Spherical Triangle ADF
in Fig. 1. Having the Angle of the Sun's greatDAF. 23. 525. And his supHypotenusa AF.
60. The Right Ascension of the Sun, or Arch
of the AEquator, AD may be found, by the ninth
Case of Right angled Spherical Triangles,
To find the Declination of a Planet or Fixed Star with Latitude.
In the Oblique angled Spherical Triangle FPS
in Fig. 4. we have given, 1. PS = AE ♋ the
greatest Declination of the Ecliptick, 2. The
Side FS the Complement of the Stars Latitude
from the Ecliptick at K. 3. The Angle PSF
the Complement of the Stars Longitude. To
find PF the Complement of Declination. By
the eighth Case of Oblique angled Spherical
Triangles, the Proportions are.
Whose Complement, is FT. 25. 99. the De
To find the Right Ascension of a Planet, or other Star with Latitude.
The Declination being found by the last ProPFS in Fig. 4. All the Sides with the
Angle FSP 20 deg. or the Complement of the
Stars Longitude. Hence to find FPS by the
first Case of Oblique angled Spherical Triangles,
I say
Whose Complement 67. 72. is the Right Asc. of a Star II. 10. North Lat. 4.
The Poles Elevation, Sun's greatest Declination and Meridian Altitude being given, to find his true place in the Zodiack.
If the Meridian Altitude of the Sun be less
than the height of the AEquator, deduct the MeAEquator,
the Remainer is the Sun's Declination toAEquator, deduct the height of the AEquator
from the Meridian Altitude, what remaineth, is
the Sun's Declination towards the North Pole,
in these Northern Parts of the World: the con
Then in the Right angled Spherical Triangle
ADF in Fig. 1. we have given the Angle FAD
the Sun's greatest Declination.
The Leg DF the Sun's present Declination,
To find AF the Sun's distance from the next E
Therefore by the Case of Right angled Spherical Triangles.
The Poles Elevation and Sun's Declination being given, to find his Amplitude.
The Amplitude of the Sun's rising or setting
is an Arch of the Horizon intercepted betwixt
the AEquator and the place of the Sun's rising or
setting; and it is either Northward or SouthAEquator toAEquator
which is towards the South Pole: That we may
then find the Sun's Amplitude or Distance from
the East or West Point, at the time of his rising
or setting. In the Right angled Spherical TriATM, in Fig. 2. let there be given the
Angle TAM. 38. 47. the Complement of the
Poles Elevation; and TM. 23. 15. the Sun's
present Declination: To find AM the Sun's
Amplitude.
By the eleventh Case of Right angled Spheri
To find the Ascensional Difference.
The Ascensional Difference is nothing else,
but the Difference between the Ascension of any
Point of the Ecliptick in a Right Sphere, and
the Ascension of the same Point in an Oblique
Sphere; As in Fig. 1. AT is the Ascensional difDA the Sun's Ascension in a
Right Sphere, and DT the Sun's Ascension in an
Oblique Sphere. Now then in the Right angled
Spherical Triangle AMT, we have given.
The Angle MAT. 38. 47. the Complement
of the Poles Elevation. And MT. 23. 15. To
find AT the Ascensional difference.
Having the Right Ascension and Ascensional Diffe
In Fig. 1. DT represents the Right Ascension,
AT the Ascensional Difference. DA the Oblique
Ascension which is found by deducting the AscenAT. from the Right Ascension
DT. according to the Direction following.
To find the time of the Sun's rising and setting, with the length of the Day and Night.
The Ascensional Difference of the Sun being
added to the Semidiurnal Arch in a Right Sphere,
that is, to 90 Degrees in the Northern Signs, or
substracted from it in the Southern, their Sum or
Difference will be the Semidiurnal Arch, which
As when the Sun is in 0 deg. ♊. his Ascensio
The double whereof is 235.22 the Diurnal Arch, which being converted into time makes 15 hours 41 minutes: for the length of the Day, whose Complement to 24; is 8 hours 19 minutes the length of the Night; the half whereof is 4 hours 9 minutes 30 Seconds the time of the Sun's rising.
The Poles Elevation and the Sun's Declination being given, to find his Altitude at any time assigned.
In this Problem there are three Varieties.
1. When the Sun is in the AEquator., that is, in
the beginning of ♈ and ♎ in which case suppoB, 60 degrees or four hours
distant from the Meridian, then in the Right
angled Spherical Triangle BZ AE, in Fig. 1. we
have given, AE Z, 51. 53. the Poles Elevation,
and B AE 60 degrees, to find BZ.
Therefore by the 2 Case of Right angled Sphe
Whose Complement BC. 18. 12. is the ☉ Al
The second Variety is when the Sun is in the
Northern Signs, that is, in ♈. ♉. ♊. ♋. ♌. ♍. in
which Case supposing the Sun to be at F in Fig. II
Then in the Oblique angled Spherical Triangle
FZP, we have given. 1. PZ 38. 47 the ComFP. 67. 97
the Complement of Declination. 3. ZPF. 45
the Distance of the ☉ from the Meridian, To
find FZ.
Therefore by the eighth Case of Oblique angled Spherical Triangles.
Whose Complement FC. 44. 55 is the ☉ Alti
The third Variety is when the Sun is in the
Southern Signs as in ♎. ♏. ♐. ♑. ♒. ♓. And in
this Case supposing the ☉ to be ♐ 10 degrees, and
his Declination South Db 22. 03. and his DiZ bP in
Fig. 1. we have given Z P. 38. 47. The Side bP
112. 03. and the Angle ZPb 45. To find Zb.
Therefore by the 8 Case of Oblique angled Spherical Triangles.
Whose Complement 6.55 is the ☉ Altitude required.
Having the Altitude of the Sun, his Distance from the Meridian, and Declination, to find his Azimuth.
The Azimuth of the Sun is an Arch of the
Horizon intercepted between the Meridian and
the Vertical Line passing by the Sun, being unHZC in Fig. 1. or Arch
HC. And in all the Varieties of the last Pro
Thus in the Triangle ZBP.
In the Triangle ZFP. I say.
s. ZF. s. ZPF
In the Triangle ZbP. I say.
Sine Zb. Sine ZPb :: Sine bP. Sine bZP.
The Poles Elevation, with the Sun's Altitude and Declination given, to find his Azimuth.
In the Oblique angled Spherical Triangle
FZP in Fig. 1. let there be given.
1. FP. 67. 97 the Complement of the ☉ De
2. ZP. 38. 47 the Complement of the Poles
Elevation.
3. FZ. 45. 46 the Complement of the ☉ Al
And let the Angle FZP the ☉ Azimuth be
required.
By the 11 Case of Oblique Angled Spherical Triangles.
As the Sine ZP x Sine FZ, Is to the Square of
Radius.
So is the Sine 1/2 Z of the Sides ZP x 1/2 Z cr—ZF.
To the Square of the Sine of half the Angle
FZP.
The Sum of the three Sides is 151.89
The half Sum is 75.945 from which
deduct PZ 38. 47. The difference is 37.475
And the Difference between 75.945 and FZ
is 30. 495.
The double whereof is 115.88 the ☉ Azimuth
from the North. And the Complement 64.12,
is the ☉ Azimuth from the South.
To find the Point of the Ecliptick Culminating, and its Altitude.
Before we can know what Sign and Degree of
the Ecliptick is in the Medium Coeli; we must
find the Right Ascension thereof, to do which,
we must add the Sun's Right Ascension to the
time afternoon, being reduced into Degrees and
Minutes of the AEquator, the Sum is the Right
Ascension of the Medium Coeli.
Example. Let the time given, be March the
20. 1674. at one of the Clock in the After
At which time the Sun's place is in ♈. 10 deg. 23 Centesms.
To find the Right Ascension thereof, in the
Right angled Spherical TADF in Fig.
1. we have given; The Angle of the Sun's greatDAF 23. 525 and the Sun's diAF 10. 23.
Therefore by the ninth Case of Right angled Spherical Triangles.
To which adding the Equinoctial Degrees anviz. 15. the Sum is 24.39
the Right Ascension of the Mid Heaven. Hence
to find the Point culminating; in the Right angled
Spherical Triangle ADF in Fig. 1. we have
given AD 24. 39 and DAF 23. 525 to find AF.
Therefore by the fifth Case of Right Angled Spherical Triangles.
Therefore the Point culminating is ♈ 26. 31.
To find the Altitude thereof above the HoriDAF
23. 525. and AF 26. 31. to find DF.
Therefore by the tenth Case of Right angled Spherical Triangles.
AEquator at London 38. 47 the
Sum is 48.66 the Altitude of the Mid Heaven as
was required.
Having the greatest obliquity of the Ecliptick together with the Distance of the Point given from the Equinoctial, to find the Meridian Angle, or Intersection of the Meridian with the Ecliptick.
Having drawn the Primitive Circle HZRN
in Fig. 5. representing the Meridian, and the two
Diameters HAR, and ZAN, set off the height
of the Pole from R to P. 51. 53, and from N to S,
and draw the Diameters PAS for the Axis of
the World, and AE AQ for the AEquator; this
done, the Right Ascension of the Mid Heaven
being given, as in the last Problem 24.39 with the
Point culminating. ♈. 26.31, and the Declination
thereof 10.19, if you set 10 deg. 19 Centesmes
from AE to F and e to X, you may draw the DiaFAX and cAd at Right Angles thereunImum Coeli is directly opposite
to the Point culminating, that is, in ♎ 26.31, if you
set 26.31 from X to b, a Ruler laid from c to b will
cut the Diameter FX in G, and then making
Xh Z Xb you have the three Points b G h, by
which to draw that Circle, which will cut the AEquator
AE AQ in ♎, and so you have the three
Points X ♎ F by which to describe the Arch of
the Ecliptick ♈ F ♎ X.
AEF we have given. The Angle AE ♈ F.
23.525 the Sun's greatest Declination, and ♈ F.
26. 31. the Point culminating, to find the Angle
♈ F AE.
Therefore by the seventh Case of Right ang
Which is the Angle of the Ecliptick with the Meridian.
To find the Angle Orient, or Altitude of the Nona
In Fig. 5. the Pole of the Ecliptick ♈ F ♎ X is
at m, and so you have the three Points Z m N
to draw the Vertical Circle Z k N cutting the
Ecliptick at Right Angles in the Point a: And
then in the Right angled Spherical Triangle
F a Z, we have given; FZ 41. 34 the CompleFH the Altitude of the Mid Heaven;
And the Angle a FZ 68. 68 the Angle of the
Ecliptick with the Meridian. To find Z a.
Therefore by the tenth Case of Right angled Spherical Triangles.
Radius.
Whose Complement is ak the Measure of the
Angle agk 52. 03 the Angle of the Ecliptick
with the Horizon, or Altitude of the Nonagesime
Degree.
To find the place of Nonagosime Degree of the Ecliptick.
In Fig. 5. F represents the Point of the EclipFZa, we have also given, FZ 41. 34
and the Angle ZFa. 68. 68. To find Fa.
Therefore by the ninth Case of Right angled Spherical Triangles.
Which being added to ♈ F 26. 31 the sum is
♈ a. 44. 04 the place of the Nonagesime Degree
of the Ecliptick at a.
The Mid Heaven being given, to find the Points of the Ecliptick Ascending and Descending.
Having found by the last Problem, the place
of Nonagesime Degree of the Ecliptick at a to
be in ♉. 14.04, if you add 90 Degrees or three
Signs thereto, the Ascendant at g will be in ♌ 14.
04, and the Point descending by adding of six
Signs will be in ♒ 14. 02. But these with the
Cusps of the other Houses of Heaven may be
otherwise found in this manner.
To the Right Ascension of the Medium Coeli
or the tenth House, add 30, it giveth the Ascen&c. According to which dire
Now because the Circles of
Position must according to these
Directions cut the AEquator at
30 and 30 Degrees above the
Horizon, if in Fig. 5. you set
30 Degrees from AE to n, and n
to r. A Ruler laid from P to n
and r, shall cut the AEquator at B and K, and then
you may describe the Circles of Position
HBR and HKR, make AT = AK and AV=
AB, and so you may describe the Circles HTR
and HVR, and where these Circles do cut the
F ♎ there are the Cusps
of the Coelestial Houses.
Thus a Ruler laid from m. the Pole of the Ect s. t. g.
will cut
the Primitive Circle in
and
being added to B will give you
the Cusps of the 11. 12. 1. 2 and 3 Houses, the o
Thus a Figure in Heaven may be erected by
Projection, the Arithmetical Computation now
followeth; In which the height of the Pole above
each Circle of Position is required, the which in
the Projection is easily found; as the Pole of the
Circle of Position HBR is at the Point D. and
so you have the three Points S, D, P, to describe
that Circle by, which will cut the Circle HBR
at Right Angles in the Point C. and the Arch PC
is the height of the Pole above that Circle of Po
In like manner the height of the Pole above
the Circle of Position HKR, will be the Arch
PE.
To compute the same Arithmetically in the
Right angled Spherical Triangle HAEB in Fig.
5. we have given AEH. 38. 47 the height of the
Equator. AEB 30. the difference of Ascension beHBAE the
Angle of that Equator with the Circle of Posi
Therefore by the first Case of Right angled Spherical Triangles.
Whose Measure in the Scheme is EC, and the
Complement thereof is CP. 32. 18374 the height
of the Pole required.
Therefore the height of the Pole above the
Circle of Position HKR. In the Triangle HAEK,
we have given, H AE as before, and AE K. 60 to
find HKAE. Therefore.
Whose Measure in the Scheme is GL, and the
Complement thereof is PL 47. 46692. the height
of the Pole required.
The height of the Pole above HDR is the
same with HBR, and the height of the Pole aHTR is the same with HKR.
Having found the Ascensions of the several
Houses together with the Elevation of the Pole
above their Circles of Position, in the Oblique
angled Spherical Triangle ♈ BS, we have given.
1. The Angle ♈ BS the Complement of
HBAE.
2. The Angle B ♈ S. 23. The Sun's greatest
Declination.
3. Their included Side ♈ B. 54. 39 the AsS the
But in my Trigonometria Britannica, Problem. 5.
for the resolving of Oblique angled Spherical
Triangles, I have shewed how this Case as to our
present purpose may be resolved, by these Pro
1. Arch. 22. 192. Their Sum is 70.803 the Point of the Ecliptick.
1. Arch—53. 296. Their Sum 121.693 is the Point of the Ecliptick for the Ascendant.
In the Oblique angled Spherical Triangle
♈ T
we have given,
♈
The An
1. Arch. 69. 306. Their Sum is 53.740 is the Point of the Ecliptick for the Second House.
In the Oblique angled Spherical Triangle
♈
1. ♈
For the Eleventh House.
In the Oblique angled Spherical Triangle
♈ KF, we have given.
♈ t.
1. Arch. 37. 625. Their Sum 113.6691 is the Point of the Ecliptick for the Twelfth House.
In the Oblique angled Spherical Triangle
♈ AG we have,
1. Arch. 86. 468. Their Sum 175.197 is the Point of the Ecliptick for the Third House.
And thus we have not only erected a Figure
for the Time given, but composed a Table for
the general erecting of a Figure in that Eatitude;
for by adding together the first and second NumAries, as in the preceding Ope
Only note, That if the Ascension of any House be more than a Semicircle, you must take the Tangent of half the Complement to a whole Circle. And to find the Cusp of the House, you must also take the Complement of the Sum of the Arches added together.
The Numbers according to the former OpeLondon. 51. 53 are as follow
The Six Oriental Houses, by the preceding Operations.
A Figure of the Twelve Coelestial Houses.
ASTRONOMY.
THE
Second Part:
OR,
AN ACCOUNT
OF THE
Civil Year,
With the Reason of the Difference
Between the
JULIAN & GREGORIAN
Calendars,
And the manner of CompuSVN
and MOON.
LONDON,
Printed for Thomas Passinger, at the Three Bibles
on London-Bridge. 1679.
Of the Year Civil and Astronomical.
HAving shewed the Motion of the Primum
Mobile, or Doctrine of the
Sphere, which I call the Absolute
Part of Astronomy; I come now un
2. And the Distinction of Time is to be con
4. The Sun doth pass through the Zodiack
in 365 Days, 5 Hours, and 49 Minutes. And the
Moon doth finish her course in the Zodiack, and
return into Conjunction with the Sun, in 29
Days, 12 hours, 44 Minutes, and 4 Seconds. And
from the Motion of these two Planets, the Civil
Year in every Nation doth receive its Institu
5. Twelve Moons or Moneths is the measure
of the Common Year, in Turkey in every Moneth
they have 29 or 30 Days, in the whole Year 354
Days, and in every third Year 355 Days.
6. The Persians and Egyptians do also account
12 Moneths to their Year; but their moneths
are proportioned to the Time of the Suns con
7. And the Iulian Year which is the Account
of all Christendom, doth differ from the other in
this; that by reason of the Sun's Excess in MotiRome to
make some further amendment of the Solar
8. This intercalation of one Day once in 4
Years, doth occasion the Sunday Letter still to al
9. This Intercalation of one day in four years,
doth occasion the Letter F to be twice repeated
in February, in which Moneth the day is added,
that is, the Letter F is set to the 24 and 25 days
of that Moneth, and in such a year S. Matthias day
is to be observed upon the 25 day, and the next
Sunday doth change or alter his Letter, from
which leaping or changing, such a year is called
Leap-year, aud the number of days in each
Moneth is well expressed by these old Verses.
Thirty days hath September, April, June and No
February hath 28 alone, All the rest have thirty and
one.
But when of Leap-year cometh the Time, Then days hath Februarytwenty and nine.
That this year is somewhat too long, is acIulius Caesar's time was upon the 24 of March,
is now in our time upon the 10 of March, 13
days backward, and somewhat more, and so if it
be let alone will go back to the first of March,
and first of February, and by degrees more and
more backward still.
10. To reform this difference, some of the
late Roman Bishops have earnestly endeavoured.
And the thing was brought to that perfection it
now standeth, by Gregory the Thirteenth, in the
year 1582. His Mathematicians, whereof Lilius
was the Chief, advised him thus: That conNice somewhat concerned in this
matter upon the motion of that Question, about
the Celebration of Easter. And that the FaMarch, and conMarch, had prevented so much,
as to begin in Gregorie's days at the 10 of the
same, they advised, that 10 days should be
cut off from the Calendar, which was done, and
the 10 days taken out of October in the year
1582. as being the moneth of that year in which
that Pope was born; so that when they came to
the fifth of the moneth they reckoned the 15,
and so the Equinox was come up to its place aMarch, as at
the Council of Nice.
But that Lilius should bring back the beginNicene
Council and no further, is to be marvelled at, he
should have brought it back to the Emperours
own time, where the mistake was first entered,
and instead of 10, cut off 13 days; however this
is the reason why these two Calendars differ the
space of 10 days from one another. And thus I
have given you an account of the year as it now
stands with us in England, and with the rest of the
Christian World in respect of the Sun, some o
11. Taking therefore the length of the year,
to be in several Nations as hath been before de
12. The most natural beginning of All AcRome, the Greeks from
their Olympicks, the Assyrians from Nabonassar,
and all Christians from the Birth of Christ: the
beginning of which and all other the most notaEpochaes, we have ascertained to their correScaliger
contrived by the continual Multiplication of
those Circles, all in former time of good use, and
two of them do yet remain; the Circles yet in
use are those of the Sun and Moon, the one, to
wit, the Sun, is a Circle of 28 years, and the Cir
I desire to know at what time in the Turkish
Account, the fifth of Iune in the year of our
Lord 1640. doth fall.
The Julian years complete are 1648, and are thus turned into days, by the Table of days in Julian years.
Now because the Turkish Account began Iuly
16. Anno Christi. 622. you must convert these
years into days also.
Therefore the fifth of Iune 1649. in our EnMahomet,
or the Turkish Hegira, the fourth day of
the moneth Giumadi. 11
I desire to know upon what day of our Julian
year the 17 day of the moneth in the 1069 year
complete of the Persian Account from Ieshagile
doth fall.
Therefore it falls out in the Julian year from
Christ 1700. the fifth day of Ianuary.
He that understands this may by the like meEpochas, into our
Julian years and the contrary.
The Anticipation of the Gregorian Calendar
is more easily obtained, for if you enter the Ta
Of the Cycle of the Moon, what it is, how
placed in the Calendar, and to what pur
THat the Civil Year in use with us and all
Christians, doth consist of 365 days, and
every fourth year of 366, hath been already shew
And by Observation of Meton an Athenian,
it was found out about 432 years before Christ,
that the Moon in nineteen years did return to be
in Conjunction with the Sun on the self same day,
and this Circle of nineteen years is called the Cy
2. And the reason why the Calendar begins
with the golden Number 3, not 1, is this. The
Christians in Alexandria had used this Circle of
the Moon two Years before the Nicene Council.
And in the first of these Years the new Moon next
to the Vernal Equinox was upon the 27th Day of
the Egyptian month Phamenoth answering to the
23d of our March, against that Day therefore
they placed the golden Number 1. And because
there are 29 Days and a half from one new
Moon to another, they made the distance bePhurmuthi
the Month following, and against the 26 Day of
the Month Pachon and so forward, and upon this
ground by the like progression was the golden
Number set in the Roman Calendar; and so the
golden Number 1 by their example was set aMarch 23. April 21. Iune 19. Iuly 19.
August 17. September 16. October 15. November
14. December. 13. But then because in the
following Year the golden Number was 2. reckonDecember, the
golden Number 2 was set to Innuary 12. February
10. March 12. April 10. May 10. Iune
8. Iuly 8. August 6. September 5. October 4.
November 3. December. 2. From whence reckonIanuary.
But that you may know how the golden Num
3. The Embolismical Years in the Cycle of the
Moon are properly these Seven. 3, 6, 9, 11,
14, 17, 19. because in the third Year 11 Days
being thrice reckoned do amount to 33 Days, that
is one Month of 30 Days and 3 Days over. A
4. But because there are 6939 Days and 18
Hours in 19 Solar Years, that is, 4 Days 18 Hours
more then in the common and Embolismical Lu
5. Now therefore in distributing the golden
Number throughout the Calendar. If the new
Moons should interchangeably consist of 30 and
29 days, and so but 228 Lunations in 19 Years;
we might proceed in the same order in which we
have begun, and by which as hath been shewed
the third Year of the Golden Number falls upon
the Calends of Ianuary. But for as much as
there are first six Lunations of 30 days apiece and
6. And because it was thought convenient, as
hath been shewed, to begin with the third Year of
the Cycle of the Moon, because the Golden NumIanuary, therefore
in this Cycle the Embolismical Years are, 2, 5,
8, 11, 13, 16, 19. But yet that it may ap
7. But as I said before, it was thought more
convenient to begin the account from the numIanuary, because by
so reckoning 30 and 29 days to each Lunation
interchangeably, the same Number 3 falls upon
Ianuary 31. March 1, and 31. April 29. May
29. Iune 27. Iuly 27. August 25. September 14.
October 23. November 22. December 21. As
if the Lunar years were compleated upon the 20
of December there remain just 11 Days, which the
Solar years doth exceed the Lunar.
8. And by ranking on and accounting 4 for
the Golden Number of the next year, you will
Ianuary 20, February 18, March
20, April 18, May 18, Iune 16, Iuly 16,
August 14, September
13, Octob. 12, Novemb.
11, Decemb. 10.
9. But in going on,
and taking 5 for the
Golden Number in
the third year, we
must remember that
that is an EmbolisIanuary 9, February
7, March 9, April
7, May 7, Iune 5,
Iuly 5, August 3, September
2, as also upOctober, and not upOctober, to make the
Lunation to consist
of 29 days, and to the thirtieth of November in
10. In like manner in the sixth Year, having
gone through the fourth and fifth as common
years, you may see the Golden Number 8 set to
the fifth of April, which should have been upon
the fourth, and in the ninth Year the Golden
Number 11 is set to the second of February which
should have been upon the first.
And there is a particular reason, for which
these numbers are otherwise placed from the
eighth of March to the fifth of April, namely, that
all the paschal Lunations may consist of 29 days:
For thus from the eighth of March to the sixth of
April, to both which days the Golden Number
is 16, there are but 29 days. And from the ninth
of March to the seventh of April, to both which
days the Golden Number is 5, there are also 29
days, and so of the rest till you come to the fifth
of April, which is the last Paschal Lunation, as
the eighth of March is the first, but at any other
time of the Year, the length of the Month in the
Embolismical Year, may be fixed as you please.
12. And in this manner in the 17 years, in
which the lunations of the whole Circle are fiIuly is taken at pleasure, to the thirAugust, that by the
two Lunations of 29 days together, it might be
understood, that the seventh Embolismical Month
consisting of 29 days is there inserted, instead
of a Month of 30 days. In which place the EmIanuary following. And for this reason the E
For Example, if you add 8 to the Golden
Number 3 set against the first of Ianuary, it ma
Of the Vse of the Golden Number in finding the Feast of Easter.
THe Cycle of the Moon or Golden Number is
a circle of 19 years, as hath been said alEaster
and other moveable Feasts depend. To this
purpose we must remember,
1. That the vernal Equinox is supposed to be
fixed to the twenty first day of March.
2. That the fourteenth day of the Moon on
which the Feast of Easter doth depend, can ne
3. That the Feast of Easter is never observed
upon the fourteenth day of the Moon, but upon
the Sunday following; so that if the fourteenth
day of the Moon be Sunday, the Sunday follow
4. That the Feast of Easter may fall upon the
fifteenth day of the Moon, or upon any other
day unto the twenty first, inclusively.
5. That the Paschal Sunday is discovered by
1. That the New Moon immediately preceEaster, cannot be before the
eighth day of March, for if you suppose it to be
upon March 6, the Moon will be 14 days old
March 19, which is before the Equinox, contraMarch there is no Golden Number
fixed; and therefore the Golden Number 16,
which standeth against March 8, is the first by
which the Paschal New Moon may be disco
2. It followeth hence, That the last Paschal
New Moon cannot happen beyond the fifth day of
April, because all the 19 Golden Numbers are
expressed from the eighth of March to that day.
And if a New Moon should happen upon the sixth
of April, there would be two Paschal New Moons
that year, one upon the eighth of March and aApril, the same Golden
Number 16 being proper to them both, but this
is absurd because Easter cannot be observed twice
in one year.
3. It followeth hence, That the Feast of Easter
can never happen before the twenty second day
of March, nor after the twenty fifth day of April:
For if the first New Moon be upon the eighth of
March, and that the Feast of Easter must be upon
the Sunday following the fourteenth day of the
Moon; it is plain that the fourteenth day of the
Moon must be March 21 at the soonest: So that
supposing the next day to be Sunday, Easter canMarch the twenty second. And
because the fourteenth day of the last Moon falApril, if that day
be Saturday, and the Dominical Letter D, Easter
shall be upon the nineteenth day, but if it be SunEaster cannot be till the twenty fifth.
4. It followeth hence, That although there
are but 19 days, on which the fourteenth day of
the Moon can happen, as there are but 19 Golden
Numbers, yet there are 35 days from the twenty
second of March to the twenty fifth of April, on
which the Feast of Easter may happen, because
there is no day within those Limits, but may be
the Sunday following the fourteenth day of the
Moon. And although the Feast of Easter can neMarch 22, but when the fourD, nor upon the twenty
fifth of April, but when the fourteenth day of the
Moon is upon April 18, and the Dominical Letter
C. Yet Easter may fall upon March 23, not only
when the fourteenth day of the Moon is upon the
twenty second day which is Saturday, but also if
it fall upon the twenty first which is Friday. In
like manner Easter may fall upon April 24, not
only when the fourteenth day of the Moon is upon
the eighteenth day which is Monday, but also if
it happen upon the seventeenth being Sunday.
And for the same reason it may fall oftner upon
other days that are further distant from the said
twenty second of March and twenty fifth of
April.
5. It followeth hence, That the Feast of Easter
may be easily found in any Year propounded: For
the Golden Number in any Year being given, if
April both inclusively, and reckon
14 days from that day, which answereth to the
Golden Number given, where your account doth
end is the fourteenth day of the Moon: Then
consider which is the Dominical Letter for that
Year, and that which followeth next after the fourD, which being sought in the
Calendar between the aforesaid limits, the fourApril the thirD next following is April 19.
And therefore Easter day that Year is April 19.
Otherwise thus.
In Marchafter the firstC,Look the Prime wherever it be, The third Sunday after Easterday shall be.And if the Prime on Sunday be, Reckon that for one of the Three.
6. Thus the Feast of Easter may be found in
the Calendar, and from thence a brief Table shewMarch
to the fifth of April, in the same order observing
the same distance. In the second Column set the
Dominical Letters in number 35 so disposed,
as that no Dominical Letter may stand against
the Golden Number 16, but setting the Letter D
against the Golden Number 5, write the rest in
E, F, G, A, B, &c. and when you
come to the Golden Number 8, set the Letter C,
and there continue the Letters till you come to C
again, because when the Golden Number is 16,
which in the Calendar is set to the eighth day of
March, is new Moon, and the fourteenth day of
that Moon doth fall upon the twenty first, to
which the Dominical Letter is C, upon which the
Feast of Easter cannot happen; and therefore in
the third Column containing the day in which the
Feast of Easter is to be observed, is also void. But
in the next place immediately following, to wit,
against the letter D is set March 22, because if
the fourteenth day of the Moon shall fall upon the
twenty first of March being Saturday, the next
day being Sunday, shall be the Feast of Easter.
To the Letters following, E, F, G, A, B, &c.
are set 23, 24, 25, and so orderly to the last of
March, and so forward till you come to the twenApril, by which Table thus made, the
Feast of Easter may be found until the Calendar
shall be reformed.
For having found the Golden Number in the
first Column, the Dominical Letter for the
Year next after it, doth shew the Feast of Easter,
as in the former Example, the Golden Number
is 3 and the Dominicall Letter D, therefore Easter
day is upon April 19. The other moveable Feasts
are thus found.
Advent Sunday is always the nearest Sunday to
St. Andrews, whether before or after.
Septuagesima Sunday is
Nine Weeks before
Easter.
Sexagesima Sunday is
Eight Weeks before
Easter.
Qainquagesima Sunday
is Seven Weeks beEaster.
Quadragesima Sunday is
Six Weeks before
Easter.
Rogation Sunday is
five Weeks after
Easter.
Ascension day is Forty
Days after Easter.
Whitsunday is Seven
Weeks after Easter.
Trinity Sunday is Eight
Weeks after Easter.
Of the Reformation of the Calendar by Pope Gregory the Thirteenth; and substituting a Cycle of Epacts in the room of the Golden Number.
HItherto we have spoken of the Calendar
which is in use with us, we will now shew you
for what reasons it is alter'd in the Church of Rome,
and how the Feast of Easter is by them observed.
The Year by the appointment of Iulius Caesar
consisting of 365 days 6 hours, whereas the Sun
doth finish his course in the Zodiack, in 365 days
5 hours 49 minutes or thereabouts, it cometh to
pass that in 134 Years or less, there is a whole
day in the Calendar more than there ought; in
268 years 2 days more; in 4002 years 3 days:
and so since Iulius Caesar's time the vernal Equinox
hath gone backward 13 or 14 days, namely from
the 24 of March to the tenth. Now because the EquiNicene Council upon the
twenty first of March, when the time for the obEaster was first universally established,
they thought it sufficient to bring the Equinox
back to that time, by cutting off 10 days in the
Calendar as hath been declared, and to prevent
any anticipation for the time to come, have apNicene Council,
which was March 21.
The Years then 1700, 1800, 1900, which
should have been Bissextile Years, are to he ac
2. Again, because it was supposed that the
Cycle of the Moon, or Golden Number was so
fixed, that the new and full Moons would in eveNicene Council,
the New and Full Moons do happen more than
4 days sooner than the Cycle of the Moon or GolEaster should be observed not
from the fifteenth day of the Moon to the twenty
first, but from the nineteenth to the twenty
fifth.
3. That the Moon therefore being once
brought into order, might not make any antici
4. For the better understanding whereof, to
the Calendar in use with us, we have annexed the
Gregorian Calendar also: In the first Column
whereof you have 30 numbers from 1 to 30,
save only that in the place of 30 you have this
Asterisk *, But they begin with the Calends
of Ianuary, and we continued and repeated af&c. and that for this cause especially,
that the number being given which sheweth the
New Moons in every Month for one Year, you
might by numbring 11 upwards exclusively find
the number which will shew the New Moons the
Year following, to wit, the Number which fal
5. And these Numbers are called Epacts, beDecember,
the same Numbers must be added to the
11 remaining days, which were added to the first
11 days in the Month of Ianuary.
6. And because 12 times 30 do make 360,
whereas from the first of Ianuary to the twentieth
of December inclusively, there are but 354 days,
you must know that to gain the other six days, the
numbers 25 and 24 are in every other Month both
placed against one day, namely, to February 5,
April 5, Iune 3, August 1, September 29, and
November 27. But why these two Numbers are
chosen rather then any other, and why in these 6
Months the number 25 is sometimes writ to XVI,
sometimes to XXV in a common character, and
why the number 19 is set to the last day of December
in a common Character, shall be declared
hereafter.
7. Here only note that this Asterisk * is set inDecember, it may someIanuary, because after the comDecember there
are 29 days, and for the like reason the Epact 28
is set against the third of Ianuary, because after
the compleat Lunation in the third of December
there are then 28 days over, and so the rest in
order till you come to the thirtieth of Ianuary,
where you find the Epact 1. because after the
compleat Lunation on the thirtieth day there is
only one day over.
8. And besides the shewing of the New Moons
in every Month, which is and may be done by the
Golden Number, the Epacts have this advanIanuary, should in 300 years be set
to the last of December, and so of the rest, but
the Epacts being once fixed shall not need any
such retraction or commutation. For as often as
the New Moons do change their day either by
Anticipation or by Suppression of the Bissextile
year, you shall not need to do any more than to
take another rank of 19 Epacts, insteed of those
which were before in use. For instance, the Epacts
which are and have been in use in the Church of
Rome since the year of reformation 1582, and
will continue till the year 1700, are these 10 fol
Of the Moons mean Motion, and how the Antici
THe Moon according to her middle motion
doth finish her course in the Zodiack in 29
days, 12 hours 44 minutes, three seconds or thereferé.
2. Hence it cometh to pass, that although the
New Moons do after 19 years return to the same
days; yet is there an Anticipation of 1 hour, 27
minutes, 33 seconds. And in twice 19 years, that
is, in 38 years, there is an Anticipation of 2 hours,
55 minutes, 6 seconds, and after 312 years and a
half, there is an Anticipation of one whole day
and some few Minutes. And therefore after 312
years no new Moon can happen upon the same
day it did 19 years before, but a day sooner.
Hence it comes to pass that in the Julian Calen
3. And hence it follows also, that if the Gol
4. And if this Anticipation would do the
whole work, nothing were more plain, then to
make that commutation of the 19 Epact once in
312 years: but because the detraction of the Bissex
4. First therefore there was made a Table calTabula Epactarum Expansa, in this manner.
First on the top were placed the 19 Golden
Numbers in order, beginning with the Number
3, which in the old Calendar is placed against
the Calends of Ianuary, and under every one of
these Golden Numbers there are placed 30 Epacts
all constituted from the lowest number in the first
rank in which the Epact is 1, and in that first
rank the Golden Number is 3, the rest from
5. And on the left hand of these Epacts before
those under the Golden Number 3. are set 30
Letters of the Alphabet, 19 in a small Character,
and 11 in a great, in which some are passed by, for
no other reason save only this, that their simili
6. Besides this Table there was another Table
made which is called Tabula AEquationis Epactarum,
in which there is a series of years, in which
the Moon, by reason of her mentioned anticipa
7. But it supposeth, that it was convenient to
suppress the Bissextiles once only in 100 years;
and the Moon to be aequated, or as far as concerns
her self, the rank of Epacts to be changed, once
only in 300 years, and the 12 years and a half
more, to be referred till after the years 2400,
they do amount unto 100 years, and then an ae
8. Again this Table supposeth, that seeing the
New Moon at the time of the Nicene Council
was upon the Calends of Ianuary, the golden
Number 3 being there placed, that it would
have been the same if the Epact * had been set to
the same Calends, that is if the Epacts had been
then in use. And therefore at that time the
highest or last rank of Epacts was to be used,
whose Index is P, and then after 300 years, the
lowest or first rank should succeed, whose Index
is a, (for the letters return in a Circle) and afIndex is b and so forward; but that it is conceiIanuary,
is more agreeable to the year of Christ
500, than the time of the Nicene Councel; and
therefore as if the rank of Epacts under the letl were sutable to the year 500, it seemed
a
in the year of Christ 800, and those under the letb, in the year 1100, and those under the lete in the year 1400.
9. Which being granted, because in the year
1582, ten days were cut off from the Calendar,
we must run backward, or in an inverted order
count 10 series, designed, suppose, by the letters
b. a. P. N. M. H. G. F. E. D. so that
from the year 1582 the series of Epacts whose
literal Index is D, is to be used, and this is that
rank of Epacts which is now used in the Church of
Rome.
10. And therefore as if this Table had its be&c.
And in the third Column every fourth hundred
year is marked for a Bissextile, that is, 1600.
2400. 2800, &c. and in the fourth Column to eveC, to
shew in what year the Moon by her AnticipatiCC, to sig
But in the first Column, or on the left hand of
these years are placed the Letters or Indices of
those ranks of Epacts in the former Table, which
are to be used in those years and when the LetD is continued, to shew that from that
C is set to the year 1700,
it sheweth that that rank of Epacts is then to
be used, which do belong thereto, and so of the
rest.
11. The reason why these Letters in the
first Column are sometimes changed in 100
years, sometimes in 200, sometimes not in
less then 300 Years, and that they are some
1. As often as the Bissextile is suppressed
without any aequation of the Moon, then the let
2. As often as the Moon needeth aequation,
without suppression of the Bissextile, then the
Letter which was in use to that time shall be chan
3. As often as there is a Suppression and an
aequation both, or when there is neither, the
Letter is not changed at all but that which ser
1. And this is enough to shew for what reaD used in the former Cen
In the year 1700, because there is a suppressiC descending.
In the Year 1800, because there is both a supC doth
still continue.
In the Year 2400, because there is an aequation
and no suppression, there is an ascension to the
Letter A.
And thus you see not only the construction of this Table, but how it may be continued to any other Year, as long as the World shall last.
series is proper whose Index is D.
Namely, 23, 4, 15, 26, &c. so in the two Ages
following, that is, from the Year 1700 to the Year
1900 exclusively, that series is proper whose Index
is C, namely these, 22, 3, 14, 25. and in the
three ages following thence, that is from the
Year 1900 to the Year 2100 exclusively, that
series is proper whose Index is B, namely these,
21, 2, 13, 24, &c. And so for any other.
Hence also it may be known, which of the 19
doth belong to any particular Year, for which
no more is necessary, than only to know the GolIndex
of that Age in the side, the common Angle, or
meeting of these two, will shew you the Epact
desired: As in the year 1674 the Golden Number
is 3 and the Index D; therefore in the common
Angle I find 23 for the Epact that year, and shew
And here it will not be unseasonable to give the
reason, for which the Epact 25 not XXV is writ
14. Hence it cometh to pass, 1. That in these
Years the Epacts 25 and XXIV do never meet on
the same day. 2. That there is no danger that
the Epacts 25 and XXVI should in these 6 Months
cause the same inconvenience, seeing that the EIndex is D. In which these two XXIV and
XXV are not both found. And in the two
following Ages, because the rank of Epacts in
use is that whose Index is C, in which there is
the Epact XXV, not XXIV, the New Moons
are shewed by the Epact XXV not by XXIV.
But because in three following Ages, the rank
of Epacts in use is that whose Index is B, in
which 25 and XXIV are both found, the New
15. And if it be asked why the Epact 19 in
the common Character is set with the Epact XX
against the last day of December; know that
for the reasons before declared, the last EmboDecember, shall end upon the 30 and not upIanuary, as if the Lunation of
30 days had been accomplished the Day be
How to find the Dominical Letter and Feast of Easter according to the Gregorian account.
HAving shewed for what reason, and in
what manner the Epacts are substituted
in the place of the golden Number, and how
the New Moons may be by them found in the
Calendar for ever; I shall now shew you how to
find the Feast of Easter and the other moveable
CB, and endeth in D. The second begins with
DC, and endeth in E. The third begins with
ED, and endeth in F &c. The first of these
Cycles began to be in use 1582, in which year
the dominical Letter according to the Julian acG, but upon the fifteenth day of
October, that Year was changed to C: for the
fifth of October being Friday and then called the
fifteenth, the Letter A became Friday, B SaturC Sunday, the remaining part of the
year, in which the Cycle of the Sun was 23, and
the second after the Bissextile or leap Year, and
so making C, which answereth to the fifteenth
year of that Circle, to be 23, the Circle will
end at D; and consequently CB, which in the
old account doth belong to the 21 year of the
Circle, hath ever since been called the first, and
so shall continue until the year 1700, in which the
Bissextile being suppressed, the next Cycle will
begin with DC as hath been said already. Under
the first rank or order of Dominical Letters are
written the years 1582 and 1600, under the se
1. And hence it appears, that the seven orG for the Sunday Letter of that year, in
like manner because the year 1750 is contained
under the Centenary 1700, the Cycle of the
Sun being 27, I find in the second rank the LetD answering to that Number, and that is
the Dominical Letter for that year, and so of the
rest.
3. Again for as much as the fifth Order is the same with that Table, which serves for the old account, therefore that order will serve the turn for ever where that Calendar is in use, and so this last will be of perpetual use to both the Calendars.
4. Now then to find the time in which the
Feast of Easter is to be observed, there is but little
to be added to that which hath been already said
concerning the Julian Calendar. For the Pas
5. For the terms of the Paschal New Moons
are always the eighth of March and the fifth of
April: but whereas there are 11 days within these
Limits to which no golden Number is affixed,
there is now one day to which an Epact is not
appointed, because there is no day within those
Limits, on which in process of time a New
Moon may not happen. And the reason for which
the two Epacts XXV and XXIV are both set to
the fifth of April, is first general, which was shew
6. The use of these Epacts in finding the Feast
of Easter, is the same with that which hath
been shewed concerning the golden Numbers.
For the Epact and the Sunday Letter for that
year propounded being given, the Feast of Easter
may be found in the Calendar after the same
G, and therefore reckMarch to
which the Epact is set, the Sunday following is
March 25, which is the day on which the Feast of
Easter is observed.
7. And hence as hath been shewed in the
third Chapter concerning the Julian Calendar,
a brief table may be made to shew the feast of
Easter and the other moveable Feasts for ever, in
which there is no other difference, save only that
the Epacts as they are in this new Calendar, are
to be used as the golden Numbers are, which
stand in the old Calendar. And a Table having
the golden Numbers of the old Calendar set in
one Column, and the Epacts as they are in the
new Calendar set in another, will indifferently
shew the movable Feasts in both accounts, as in
the Year 1674, the golden Number is 3 and
the Sunday Letter according to the Julian acD, according to the Gregorian G, and
the Epact 23, and therefore according to this
Table our Easter is April 19, and the other, to
wit, the Gregorian, is March 25. The like may
be done for any other year past or to come.
How to reduce Sexagenary numbers into Decimal, and the Contrary.
EVery Circle hath antiently, and is yet geH. Gellibrand, in which the Division of the
Circle into 360 degrees is retained, but every
degree is divided into 100 parts, is much bet
2. The parts of a Circle consisting of 360
degrees, may be reduced into the parts of a cir
As 360 is to 100, so is any other Number
of degrees, in the one, to the correspondent de
But if the sexagenary degrees have Minutes
and Seconds joyned with them, you must reduce
the whole Circle as well as the parts propoun
Example. Let it be required to convert 125
degrees of the Sexagenary Circle, into their
correspondent parts in the Decimal. I say, as
360 is to 100, so is 125 to 34, 722222, &c.
that is, 34 degrees and 722222 Parts.
2. Example. Let the Decimal of 238 de&c. In like manner if it were
required to convert the Hours and Minutes of
a Day into decimal Parts, say thus, if 24 Hours
give 100, what shall any other number of Hours
give. Thus if the Decimal of 18 hours were
required, the answer would be 75, and the De
But if it be required to convert the Decimal
Parts of a Circle into its correspondent Parts
in Sexagenary. The proportion is; as 100 is
Example. Let the Decimal given be 349
722222, if you multiply this Number given by
360, the Product will be 1249999992, that is
cutting off 7 Figures, 124 degrees and 9999992
parts of a degree. If Minutes be required, mul
But to avoid this trouble, I have here exhi
Let it be required to convert 258 degrees 34′.
47″, into the parts of a Circle decimally divi
The Table for this purpose doth consist of
two Leaves, the first Leaf is divided into 21
Columns, of which the 1. 3. 5. 7. 9. 11.
13. 15. 17. 19 doth contain the degrees in a
sexagenary Circle, the 2. 4. 6. 8. 10. 12. 14.
16. 18 and 20 doth contain the degrees of a
Circle Decimally divided, answering to the for
is the Decimal of 258 degrees, 34′. 47″ as was required.
In like mauner the Decimal of any Hours and Minutes may be found by the Table for that purpose.
Example. Let the Decimal of 7 Hours 28′ be required.
is the Decimal Sought.
To find the degrees and Minutes in a sexage
As if it were required to find the Degrees and
minutes answering to this decimal 71. 02776359,
the Degrees or Integers being sought in the
2. 4. 6 or 8 Columns &c. of the first Leaf of
that Table, right against 71. I find 256 and in
the last Column these parts 11111111, which
being less than the Decimal given, I proceed
Which is the Decimal of 47 seconds, and so
the degrees and Minutes answering to the De
Of the difference of Meridiens.
HAving in the first part shewed how the plaVraniburg to be in ♈. 3 d. 13′.
14″. March the fourteenth 1583 at what time
was the Sun in the same place at London? To reEngland from the City of London, and
of some places beyond the Seas also. The use
whereof is either to reduce the time given under
the Meridian of London to some other MeridiLondon.
1. If it be required to reduce the time given
under the Meridian of London to some other
Meridian, seek the place desired in the Catalogue,
and the difference of time there found, either
add to or subtract from the times given at London,
according as the Titles of Addition or SubLondon was in the
first Point of ♉, 6 Hours P. M. and it is required
to reduce the same to the Meridian of Vraniburg
I therefore seek in Vraniburg in the Catalogue of
places, against which I find 50′ with the Letter
A annexed, therefore I conclude, that the Sun
was that day at Vraniburg in the first point of ♉,
6 Hours 50′. P. M.
London, you must seek the place
given in the Catalogue, and the difference of
time there found, contrary to the Title is to be
added or subtracted from the time there gi
Example. Suppose the place of the Sun had
been at Vraniburg, at 6 Hours 50′. P. M. and
I would reduce the same to the Meridian of London;
against Vraniburg as before I find 50′ A.
therefore contrary to the Title I Subtract 50′
and the remainder 6 Hours is the time of the
Suns place in the Meridian of London.
Of the Theory of the Sun's or Earth's Motion.
IN the first part of this Treatise we have spok
1. And these Orbs are supposed to be EllipRepler, by the help of
Tycho's accurate observations, hath demonstraMars and Mercury, and
may therefore be conceived to be the Figure in
which the rest do move.
2. Here then we are to consider what an Ellipsis
is, how it may be drawn, and by what Me
3. What an Ellipsis is Apollonius Pergaeus in
Conicis, Claudius Mydorgius and others have well
defined and explained, but here I think it suffici
4. The usual and Mechanical way of drawing
this Ellipsis is thus; first draw a line to that length
which you would have the greatest Diameter to
be, as the Line AP in Figure 8, and from the
middle of this Line at X, set off with your comXM and XH.
5. Then take a piece of thred of the same
length with the Diameter AP and fasten one
end thereof in the point M and the other in
the point H, and with your Pen extend the
thred thus fastened to the point A, and from
thence towards P keeping the thread stiff upon
your Pen, draw a line from A by B to P, the
line so drawn shall be half an Ellipsis, and in like
manner you may draw the other half from P by
D to A. In which because the whole thred is
equal to the Diameter AP. therefore the two
Lines made by thred in drawing of the Ellipsis,
must in every point of the said Ellipsis be also eAP. They that deApollonius Pergaeus, Claudius Mydorgius
or others, in their treatises of Conical Sections,
this is sufficient for our present purpose, and
from the equality of these two Lines with the
Diameter, a brief Method of calculation of the
Ellipsis, is thus DemonstraWard now Bishop of Salisbury.
6. In this Ellipsis H denotes the place of the
Suns Center, to which the true motion of the
Planet is referred, M the other Focus whereunA
the Aphelion where the Planet is farthest distant
from the Sun and slowest in motion, P the Perihelion
where the Planet is nearest the Sun and
slowest in motion. In the points A and P the
Line of the mean and true motion do convene,
and therefore in either of these places the PlaP in aequality, but in all other points
the mean and true motion differ, and in D and
C is the greatest elliptick AEquation.
8. Now suppose the Planet in B, the line of
the middle motion according to this Figure is
MB, the line of the true motion HB. The
mean Anomaly AMB. The Eliptick aequatiProsthaphaeresis MBH, which in this ExamAMB, the remainer AHB
is the true Anomaly. And here note that in the
right lined Triangle MBH, the side MH is alFoci,
the other two sides MB and HB are together
equal to AP. Now then if you continue the side
MB till BE be equal to BH and draw the
line HE, in the right lined Triangle MEH,
we have given ME=AD and MH with the
Angle EMH, to find the Angles MEH and
MHE which in this case are equal, because
EB=BH by Contraction, and therefore the
double of BEH or BHE=MBH, which is
the Angle required.
And that which yet remaineth to be done, is
Aphelion, the true
Excentricity or distance of the umbilique points,
and the stating of the Planets middle motion.
Of the finding of the Suns Apogeon, quantity of Excentricity aend middle motion.
THe place of the Suns Apogaeon and quantity
of Excentricity may from the observatiEdward Wright
be obtained in this manner, in the years 1596,
and 1497, the Suns entrance into ♈ and ♎ and
into the midst of ♉. ♌. ♍. and ♒ were as in
the Table following expressed.
And hence the Suns continuance in the Nor
And the difference of the Suns continuance in
these Arks in the year 1596 and 1597 is 27′. and
therefore the mean time of his continuance in
those Arks is days 186. hours 18. minutes 51. se
And to find the middle motion answering to days 186. hours 18. Minutes 51. seconds 30 I say.
As 365 days, 6 hours, the length of the Julian, year is to 360, the degrees in a Circle.
So is 186 days, 18 hours, 51′. 30″ to 184 degrees. 03′. 56″.
In like manner the mean motion answering
h. 46′ is 183 degrees, 02′.09.
In 1596 from 15 ♒ to 15 ♌ there are days 185, hours 01, minutes 36. In 1597. days 135. hours 4. 02′.
And the mean motion answering thereunto is.
182d. 30′. 36″.
From 15 ♒ to 15 ♌ Days. 185. 04h. 02′
Now then in Fig. from PGC. 181. 32. 26
deduct NKD 180, the Remainer is DC+NP.
1. 32. 26. Therefore DC or NP. 46. 13, whose
Sine is HA.
And from XPG. 181. 15. 18 deduct TNK
180, the Remainer is KG+TX 1. 15. 18.
Therefore KG or TX 37. 39, whose Sine is
HR.
Now then to find the Apogaeon.
Hence to find the excentricity AR.
Or thus,
In the Triangle
we have given
and
Apogaeon 95 deg. 49′. 56″. as before.
Then for the Excentricity RA.
And this agreeth with the excentricity, used
Street in his Astron. Carolina, Pag. 23.
But Mr. Wing as well by observation in former
ages, as our own, in his Astron. Instaur. Pag.
39. doth find it to be 1788 or 1791. The
work by both observations as followeth.
2. And first in the time of Ptolemy, Anno
Christi 139 by comparing many observations to
Again from GEK 181. 56. 02. deduct the
Semicircle FED 180. the remainer is the summ
DK and FG. 1. 56. 2. and therefore DK=FG.
58′. 01″. whose sign is BC. 168755. L is the
place of the Aphelion, and AB the Excentricity.
Now then in the Triangle ABC. in the Fig.
6 we have given the two sides AC and BC. To
find the Angle BAC and the Hypotenuse AB.
Secondly for AB.
Therefore the Aphelion at that time was in II
10. 41. 10. And the excentricity. 1788.
3. Again Anno Christi 1652 the Suns place by
observation was found to be as followeth.
Hence it appeareth that the Sun is running through one Semicircle of the Ecliptick, that is from ♉ 15 to ♍ 15. 185 days 21 hours and 10′. And through the other Semicircle from ♒ 15 to ♌ 15, days 184. hours 5. therefore the Suns mean motion, according to the practice in the last example, from ♉ 15 to ♍ 15 is 181. 30. 26. and from ♒ 15 to ♌ 15. 181. 16. 30.
Now then in Fig. 7. if we subtract the semiKMH. 180. from WPV 181.
36. 26. the remainer is the sum of KW and HV
AC. 140252.
Again the mean motion of the Sun in his Orb
from ♒ 15 to ♉ 15 is the Arch SKP. 181. 16.
30. whose excess above the Semicircle being biCB. 111345. now
then in the Triangle ABC to find the Angle
BAC, the proportion is.
Which being deducted out of the Angle. 69
A ♌. 45 it leaveth the Angle 69 AL 6. 33. 39.
the place of the ☉ Aphelion sought, and this is
the quantity which we retain.
And for the excentricity BC.
And the difference of time is 1513 Julian years.
Hence to find the motion of the Aphelion for
2. years, say I, if 1513 years give 25. 52.29, what
shall one year give, and the answer is 00d. 01′
iv. 44v. that is in Decimal num
And the motion for. 1651 years. 7. 84298.
4208862, which being deducted from the place
of the Aphelion Anno Christi. 1652—26. 82245.
3703703. The remainer, viz. 18. 97946.
9494841 is the place thereof in the beginning
of the Christian AEra, which being reduced is,
68 deg. 19. min. 33. sec. 56. thirds.
4. The Earths middle motion, Aphelion and
Excentricity being thus found, we will now shew
how the same may be stated to any particular
time desired, and this must be done by help of
the Sun or Earths place taken by observation.
In the 178 year then from the death of Alexander,
Mechir the 27 at 11 hours P. M. Hipparcus
found in the Meridian of Alexand. that
the Sun entered ♈ 0. the which Vernal Equinox
happened in the Meridian of London according
to Mr. Wings computation at 9 hours 14′, and
the Suns Aphelion then may thus be found.
The motion of the Aphelion for one year, was
before found to be. 0. 00475. 04447. 0555. thereAEra began in the
4713 year compleat of the Julian Period, in
which there are days 1721423. The AEra Alexandri
began November the twelfth, in the
year 4390 of the Julian Period, in which there
are 1603397 days. And from the death of Alexander
to the 27 of Mechir 178, there are days
64781, therefore from the beginning of the
Julian Period, to the 178 year of the AEra Alexandri,
there are days 1668178 which being
deducted from the days in the Christian AEra,
Alexander, Mechir 27, and the beginning of
the Christian AEra.
Or thus. From the AEra Alexandri to the
AEra Christi there are 323 Julian years, and 51
days, that is 118026 days. And from the AEra
Alexandri to the time of the observation, there
are 64781 days, which being deducted from
the former, the remainer is 53245 as before.
Now then if you multiply the motion of the Aphelion
for one day, viz. 0. 00001. 3014917 by
53245, the product is 0. 69297. 9255665, which
being deducted from the place of the Aphelion
in the beginning of the Christian AEra, before
found. 18. 97946. 9494841. the remainer 18.
28649. 0239176 is the place of the Aphelion at
the time of the observation, that is in Sexage
5. The place of the Aphelion at the time of
the observation being thus found to be deg. 65.
49′. 53″. The Suns mean Longitude at that
time, may be thus computed.
In Fig. 8. In the Triangle EMH we have giME 200000, the side MH 3576, the
double excentricity before found, and the AnEMH 114. 10′. 07″. the complement of
the Aphelion to a Semicircle, to find the Angle
MEH, for which the proportion is,
As the Summ of the sides, is to the difference
of the sides, so is the Tangent of the half Summ
of the opposite Angles, to the Tangent of half
their difference.
The double whereof is the Angle MBH
1. 51. 10. which being Subtracted from 360 the reAphelion
before found, 65. 49. 53. the remain
Doubled 1. 54. 20, which added to the midHipparchus, to which if
you add the middle motion of the Sun for 53245
days, or for 323 AEgyptian years 131 days, 280.
46. 08′ the Summ, rejecting the whole Circles,
is 278. 51. 48 the Suns mean Longitude in the
beginning of the Christian AEra.
6. But one observation is not sufficient, whereEpocha, we will therefore examine the same by
another observation made by Albategnius at Aracta
in the year of Christ 882, March: 15.
hours 22. 21. but in the Meridian of London
at 18 hours. 58′.
The motion of the Aphelion for 881 years,
74 days is 3. 806068653737, which being adAEra, the place at the time of the obserd. 58′.
44″ is the estimate middle motion at that time,
from which deducting the Aphelion deg. 82. 01.
40. the remainer 275. 57. 04 is the mean anoviz. 80d. 06′. 10″. the reAEra.
Epocha, to more exactness, must take the pains
to compare the Collection thereof from sun
In the Ensuing Tables of the ☉ mean LonFlamsted, a little pains will fit the
Tables to any other measure.
Of the quantity of the Tropical and Sydereal Year.
THe year Natural or Tropical (so called
from the Greek word
2. That we may determine the true quanti
3. Albategnius, Anno Christi 882 observed
the ☉ entrance into the Autumnal AEquinox at
Aracta in Syria to be Sept. 19. 1 hour 15′ in the
Morning. But according to Mr. Wings corAstron. Instaur. Page 44, it was
at 1 hour 43′ in the Morning, and therefore acSept. 16. 12h.
14′. 25″. that is at London at 8h. 54′. 25″.
4. Again by sundry observations made in the
year 1650. the second from Bissextile as that
of Albategnius was, the true time of the ☉ inSept. 12. 14h. 40′.
and therefore his ingress according to his midSept. 10. 13h. 02.
5. Now the interval of these two observaEdward
Wright takes it to be 365d. 5 hours. 48′.
Mr. Iohn Flamsted, 5h. 29′. Mr. Tho. Street
5h. 49′. 01″. taking therefore the Tropical year
to consist of 365 days, 5 hours, 49 Minutes,
the Suns mean motion for one day is 0 deg. 59′.
8″. 19‴. 43iv. 47v. 21vi. 29vii. 23viii. or in de
6. The Sydereal or Starry year is found from the Solar by adding the Annual Motion of the eighth Orb or praecession of the AEquinoctial Points thereunto, that praecession being first converted into time.
7. Now the motion of the fixed Stars is found
to be about 50″. in a years time, as Mr. Wing
hath collected from the several observations of
Timocharis, Hipparchus, Tycho and others; and
to shew the manner of this Collection, I will
mention onely two, one in the time of Timocharis,
and another in the time of Tycho.
8. Timocharis then as Ptolemy hath it in his
Almagist, sets down the Virgins Spike more
northwardly than the AEquinoctial, 1 deg. 24′.
the time of this observation is supposed to be
about 291 years before Christ, the Latitude
1 deg. 59′ South, and therefore the place of
the Star was in ♍. 21d. 59′. And by the obTycho 1601 current, it was in ♎
iv. 48v. 47vi. 18vii. 30viii. 13ix. and
in decimal Numbers, the motion for a year is
00385. 80246. 91358. The motion for a day.
00001. 05626. 95938.
9. Now the time in which the Sun moveth
50″, is 20′. 17″. 28‴, therefore the length of
the sydereal year is 365 days, 6 hours, 9′. 17″.
28‴. And the Suns mean motion for a day 59′.
8″. 19‴. 43iv. 47v. 21vi. 29vii. 23viii. converted
into time is 00. 03′. 56″. 33‴. 18iv. 55v. 9vi.
23vii. 57viii. which being added to the AEquinoctiiv. 55. 9. 23. 57.
10. And the daily motion of the fixed Stars,
being converted into time is 32iv. 51v. 15vi.
9vii. 14viii. 24ix. and therefore the AEquinoctial
day being 24 hours, the sydereal day is 24 hours,
00′. 00″. 00‴. 32iv. 51. 15. 9. 14 24.
11. Hence to find the praecession of the AEqui
Example. The place of the first Star in ATycho in the year 1601 current,
was in ♈. 27d. 37′. 00. and I would know the
place thereof in the beginning of the Christian
AEra.
is the place thereof in the beginning of the
Christian d. 13′. 20″AEra.
12. Having thus found the ☉ middle motiAphelion and fixed Stars,
with their places, in the beginning of the ChriAEra; we will now set down the numAEra Christi. Mr. Wing
from the like observations, takes the ☉ motion
to be as followeth.
The which in decimal Numbers are
The mean motions for one year.
The ☉ mean motions for one day.
And according to these measures are the Ta
The Suns mean motions otherwise stated.
SOme there are in our present age, that will
not allow the Aphelion to have any motion,
or alteration, but what proceeds from the moAEra was found to be ♈.
5. 23d. 40.
Now then, if from the place of the Aphelion
Anno Christi. 1652 as was shewed in the tenth
Chapter, deg. 96. 33′. 39. we deduct the moAphelion; but Mr. Street in his Astronomia
Carolina Page 23, makes the constant place
of the Aphelion to be 68d. 20. 00, and the ☉ ex
And from the observation of Tycho 1590
March the eleventh. in the Meridian of Vraniburg,
but reduced to the Meridian of
London. March the tenth, hour 23. 2′. He
And so the mean Anomaly AEra Christi is 6.
23. 19. 56. But Mr. Flamsted according to
whose measure the ensuing Tables are compoAEra Christi. to be
6. 24. 07. 091. The place of the Aphelion
to be 8, 08. 23. 50. And so the Praecession of
the AEquinox and Aphelion in the beginning of
AEra. 8, 13. 47. 30. in decimal
Numbers.
How to Calculate the Suns true place by either of the Tables of middle motion.
VVRite out the Epocha next before the
given time, and severally under that
set the motions belonging to the years, months
and days compleat, to the hours, scruples, curFrebruary
the days must be increased by an unite)
then adding all together, the sum shall be the ☉
mean motion for the time given.
Example.
Let the given time be Anno Christi 1672. February
23. hours 11. 34′. 54″. by the Tables of
the ☉ mean Longitude and Anomaly, the num
By the Tables of the Suns mean Anomaly and praecession of the AEquinox, the numbers are these.
But because there is no Canon of Sines and Tan
Multiply the degrees and parts given by 36, the
Product, if you cut off one figure more towards
the right hand than there are parts in the num
Otherwise thus. Multiply the degrees and parts given by 6 continually, the second Product, if you cut off one figure more towards the right hand than are parts in the number given, shall be the degrees and parts of the common Circle. The third Product of the parts only shall give minutes, the fourth seconds, and so forward as far as you please. Example.
And thus the mean Anom. is deg. 246. 451742 or 27′. 06. The Prec. AEq. 97. 001964. or 00′. 07″.
Hence to find the Elliptical AEquation in degrees
and decimal parts: In Fig. 8. we have given in the
right lined plain Triangle EMH, the sides ME,
and MH, and the Angle EMH, 66. 451742. the
excess of the mean Anomaly above a Semicircle,
to find the Angle MEH.
MEH. 0. 917042 the double whereof is the
Angle MBH. 1. 834084 or Elliptick AEqua
But because the Elliptick AEquation thus found
doth not so exactly agree to observation as is deBullialdus in Chap. 3. of his Book entituled
Astronomiae Philolaicae fundamenta clarius explicata,
Printed at Paris, 1657. shews how to correct the
same by an Angle applied to the Focus of middle
motion, subtended by the part of the ordinate
line, intercepted between the Ellipsis and the CirStreet maketh
use of in his Astronomia Carolina, and this I thought
not amiss to add here.
In Fig. 9. let ABCPDF be supposed an Ellipsis,
and the Circle AGPK described upon the
extremes of the transverse Diameter, and the OrKN and OB extended to G and M in
Apollonius.
XN. GX∷OB tang. OEB. OM tang. OEM.
And the Angle OEM-OEB=BEM=ETY,
the variation to be deducted from the Elliptick
AEquation ETH, the Remainer is the absolute
AEquation YTS in the first Quadrant.
In the second and third Quadrants, the variati
For XN. XG. QV. tang. QEV. the comp. m. Anom.
QR. t. QER. and the Angle VER=ECO is the vaECO+ECH=OCH is the absolute
AEquation sought in the second Quadrant.
Again, XN. SG∷a D, tang. a ED. a b, tang. aEB.
And aEB—aED=DEf the variation=EFO and
EfO+EfH=OfH the absolute AEquation sought
in the third Quadrant.
Lastly, in the fourth Quadrant of mean Ano
XN. XG∷ch. tang. eEH. eg. tang. eFg. and hEg is the
variation: And EFH—
the absolute
AEquation sought in the fourth Quadrant.
And to find XN the conjugate Semi-diameter,
in the right angled Triangle ENX, we have giEN=AX and EX the semi-distance of the
umbilick points. And Mr. Briggs in Chap. 19. of
his Arithm. Logar. hath shewed, that the half Sum
of the Logarith. of the sum and difference of the
Hypotenuse, and the given leg. shall be the Loga
Now then in the former Example the mean
Anomaly is 246 deg. 451741. and the excess aaED. 66. 451742.
Therefore.
aEB—aED=DEf .003544 the variation,
which being added to the Elliptick AEquation beX. 15. 17. 28.
To find the place of the fixed Stars.
THe annual motion of the fixed Stars is, as
hath been shewed, 50 Seconds, hence to
find their places at any time assigned, we have
exhibited a Table of the Longitude and Latitude
of some of the most fixed Stars, from the CataTycho for the year of our Lord
1600 compleat. Now then the motion of the
fixed Stars according to our Tables being com
Example. Let the given time be 1500, the
difference of time is 100 years, and the moti
2. Example.
Of the Theory of the Moon, and the finding the place of her Apogaeon, quantity of excentricity and middle motion.
THe Moon is a secondary Planet, moving aHipparchus, there arises a twofold,
but according to Tycho a three-fold Inequality
in the Moons Motion. The first is Periodical
and is to be obtained after the same manner, as
was the excentrick AEquation of the Sun or
Earth: in order whereunto, we will first shew
how the place of her Apogaeon and excentricity
may be found.
At Bononia in Italy, whose Longitude is 13
degrees Eastward from the Meridian of London,
Ricciolus and others observed the apparent times
of the middle of three lunar Eclipses to be as fol
The equal times reduced to the Meridian
of London, with the places of the Sun in these
three observations, according to Mr. Street in
the 25 Page of his Astronomia Carolina, are thus.
Hence the place of the Moon in the first obserBHDGFE denote the
Moons AEquant T the Center of the Earth, the
Semidiameters TD, TE and TF the apparent
places of the Moon, in the first, second and third
observations, C the Center of the Excentrick,
CD, CE and CF the Lines of middle motion.
Suppose 10.00000000 the Logarithm of
DC, continue FC to H, and with the other
right Lines compleat the Diagram.
1. In the Triangle DCH we have given the
Angle DCH 86. 13. 15. the complement of
DCF 93. 46. 45 to a Semicircle. The Angle
DHC 50. 02. 57. The half of the Arch DF
and the side CD 1000000. To find CH.
2 In the Triangle HCE we have given CH
as before, the Angle CHE 25. 00. 07. The half
of the Arch FE, the Angle HCE 133. 04. 17
the complement of FCE, and by consequence
the Angle CEH 21. 55. 36 To find the Side
CE.
3. In the Triangle DCE, we have given DC.
CE and the Angle DCE 140. 42. 28. whose
complement 39. 17. 32 is the Summ of the AnCED and DE,
4. In the Isosceles Triangle DTE we have giDE, the angle DTE 150. 06. 08
whose complement 29. 53. 52 is the Summ of
the other two angles, the half whereof is the
angle TDE 14. 56. 56 which being subtracted
from the angle CDE. 19. 51. 21 the remainer
is the angle CDT. 4. 54. 25.
5. In the Triangle CDT we have given DC.
DT and the angle CDT, to find CTD and CT.
And for the excentricity in such parts, as the
Radius of the AEquant is 100000 the Proporti
And this is the Method for finding the place
of the Moons Apogaeon and excentricity. And
from these and many other Eclipses as well Solar
as Lunar, Mr. Street limits the place of the Apogaeon to be at the time of the first observation
21′. 04″ more, and the mean Anomaly 20. 41″
less, and the excentricity 8765 such parts as the
Radius of the AEquant is 100000.
And by comparing sundry observations both
antient and modern, he collects the middle moApogaeon, to be in
the space of four Julian years or 146 days, 53
revolutions, 0 Signes, 7 degrees, 56 minutes,
45 Seconds. And the Apogaeon from the AEquid. 03′. 53″. 57‴. 09iv. 58v. 46vi.
Of her Apogaeon 0. 06. 41. 04. 03. 25. 33.
And according to these Measures, if you deApril
Of the finding of the place and motion of the Moons Nodes.
ANno Christi 1652, March 28, hour. 22. 16′,
the Sun and Moon being in conjunction,
Mr. Street in Page 33, computes the London to be in ♈. 19. 14. 18
with latitude North 46′. 15″.
And Anno Christi 1654 August 1. hour. 21.
19′. 30″ was the middle of a Solar Eclipse at
London. at which time the Moons true place was
found to be in ♌ 18. 58′. 12″ with North La
From the first observation to the second there are 27 years, 4 months, 5 days, 23 hours 03′. 30″.
Therefore in the oblique angled Spherical
Triangle DPB we have given BP. 89. 13. 45
the complement of the Moons Latitude in the
first Observation 2. PD 89. 27. 50 the compleDPB 165. 03. 35, whose
complement to a Semicircle is DPF 14. 56. 25.
The angle PBD is required.
Their Z is FPB 178. 40. 27. whose complement
Is the Arch FG 1. 19. 33.
BA☊ right angled at A we have given AB 046′.
15″ the Latitude in the first observation, and the
Angle AB☊=PBD 85. 02. 56. to find A☊
the Longitude of the Moon from the ascending
Node.
The retrograde motion whereof in 4 Julian
years or 2461 days, is by other observations
found to be Sign 2. deg. 17. 22′. 06″. and thereiv. 35v.
And the motion thereof for 1651 years, March
28. h. 22. 16′, viz. Sign 8. deg. 18. 26′. 58″
being added to the place of the Node before
found Sig. 0. 10. 25: 01. Their Sum is the
place thereof in the beginning of the Christian
AEra Sign 8. deg. 28. 51′. 59″.
Rudolphin Tables as they are corrected
by Mr. Horron and reduced to the Meridian of
London, do differ a little from these measures, for
according to these Tables, the Moons mean mo
And according to these measures, the Moons mean motions in decimal Numbers are.
And according to these measures are the TaApogaeon, Anomaly, and Node retrograde for
Years, Months, Days and Hours.
And hence to compute the Moons true place in
her Orbit, I shall make use of the Method, which
Mr. Horron in his Posthumas works lastly pubFlamsted, in which from the Rudolphin
Tables he sets down these Dimensi
These things premised his directions for com
How to Calculate the Moons true place in her Orbit.
TO the given time find the true place of the Sun, or his Longitude from the Vernal AEquinox, as hath been already shewed.
2. From the Tables of the Moons mean motiEpocha next before the giFebruary,
the days must be increased by one Unite)
then adding them all together, the Summ shall be
Radix or first number, and the Summ of the
remainer being deducted from the Radix, shall
be the mean place of her Node required.
3. Deduct the Moons Apogaeon from the ☉ true
place, the rest is the annual Augment, the tanApogaeon AEquation to be added to the
mean Apogaeon, in the first and third quadrants
of the annual Augment, and Subtracted in the
second and fourth, their Summ or difference is
the true Apogaeon.
4. The true Apogaeon being Deducted from
the
5. Double the annual Augment, and to the
Cosine thereof add the Logarithm of 1161. 75.
the difference between the Moons mean and exviz. 3. 06511268, the
Summ shall be the Logarithm of a number which
being added to the mean Excentricity, if the
double annual Augment be in the first or fourth
quadrants; or Subtracted from it, if in the second
or third quadrants; the Summ or difference shall
be the Moons true Excentricity.
6. The Moons true Excentricity being taken
for a natural Sine, the Arch answering thereto
shall be the
7. To the half of the Moons greatest Physical
AEquation add 45 deg. the cotagent of the Summ
is the artificial Logarithm of the Excentrick.
8. To the Logarithm of the Excentrick, add
the tangent of half the Excentrick Anomaly,
the Summ shall be the tangent of an Arch, whose
double shall be the Coequated Anomaly, and the
difference between this and the mean Anomaly is
the terrestrial Equation, which being added to
the Moons mean Longitude, if the mean Anoma
9. From the place of the Moon first Equated,
Deduct the true place of the Sun, and double
the remainer, and to the Sine of the double add
the Sine of the greatest variation 0. 36. 27, viz.
8. 02541571, the Summ shall be the Sine of the
true variation, at that time, which being added
to the Moons place first Equated, when her single
distance from the Sun is in the first or third qua
Example.
Let the given time be Anno Christi 1672. Feb.
23. h. 11. 34′. 54″ at which time the Suns true
place is in ♓ 15. 29133 and the Moons middle
motions are as followeth.
Secondly.
Or thus.
And hence it appears that working by the
mean Anomaly instead of the mean Longitude,
the true Apogaeon Equation must be added to the
mean Anomaly, in the second and fourth Qua
Thirdly.
Fourthly.
Fifthly.
Sixthly,
To compute the true Latitude of the Moon, and to reduce her place, from her Orbit to the Ecliptick.
THe greatest Obliquity of the Moon's Orb
with the Ecliptick or Angle A ☊ B Fig. 11.
is by many Observations confirmed to be 5 DeHorrox refers us to pag.
87. in the Rudolphin Tables, to find from thence
the Equation of the Nodes, and Inclination limitis
menstrui, in this manner.
1. From the mean place of the Node, deduct
limitis menstrui, which being added to
or subtracted from the Nodes mean place acA ☊.
2. With the Augment of Latitude, enter the
Table of the Moon's Latitude, and take thence
her Simple and Latitude and Increase answering
to it. Then say, as the whole excess of Latitude
18′, or in Decimals 30. is to the Inclination of the
Monethly limit: So is the increase of Latitude to the
Part Proportional; which being added to the sim
3. With the same Augment of Latitude, enas 18′. 00″. or 0. 30. is to the Inclination
of the Monethly limit: So is the increase of
Reduction, to the Part Proportional; which being
added to the simple Reduction, shall give the
true, to be added to, or subtracted from the
place of the Moon in the Ecliptick.
Tabula
AEquationis Nodorum Lunae. I find the
Node to need no Equation, and the Inclinatilimitis menstrui to be deg. 00. 30.
To find the Mean Conjunction and Opposition of the Sun and Moon.
TO this purpose we have here exhibited a Ta
In like manner the Moons mean distance from the Sun in a year or a day is thus found.
And according to these measures are the Ta
To the given year and Month gather the mid
Example, Anno Christi 1676. I would know
the time of the mean Conjunction or New Moon
in October.
Therefore the mean Conjunction in October,
1676. was the 26 day, 10 min. 8 seconds after
8 at night.
And to find the mean opposition. To the com
Example, Anno Christi, 1676. I desire to know
the mean opposition in November.
November the 10th,
Hours 2, 32′ 09″. The like may be done for a
And here I should proceed to shew the manner
of finding the true Conjunction or Opposition of
the Sun and Moon, but there being no decimal
Canon yet extant, suitable to the Tables of midStreet's Astronomia Carolina,
for instructions in that particular, and what else
shall be found wanting in this Subject.
Of the Nature and Division of Geography.
GEOGRAPHY is a Science concern
3. The Earthy part of this Globe is common
4. A Continent is a great quantity of Land
not separated by any Sea from the rest of the
World, as the whole Continent of Europe, Asia,
and Africa, or the Continents of France, Spain,
and Germany.
5. An Island is a part of Earth environed round
about with some Sea or other; as the Isle of Britain
with the Ocean, the Isle of Sicily with the MeInsula, because it is scituate in Salo, in the
Sea.
6. Both these are subdivided into Peninsula,
Isthmus, Promontorium.
7. Peninsula, quasi pene insula, is a tract of land
which being almost encompassed round by water,
is joyned to the main land by some little part of
Earth.
8. Isthmus is that narrow neck of Land which
joyneth the Peninsula to the Continent.
9. Promontorium is a high mountain which
shooteth it self into the Sea, the outmost end
whereof is called a Cape or Foreland, as the Cape
of Good Hope in Africk.
10. The Watry part of this Globe may be also distinguished by diverse Names, as Seas, Rivers, Ponds, Lakes, and such like.
11. And this Terrestrial Globe may be measu
12. The measure of this Earthly Globe in
whole, is either in respect of its circumference,
England,
and hence the circumference of the Earth is found,
by multiplying 360 by 15, to be 5400 German
miles, or multiplying 360 by 60, the circumfe
14. The circumference of the Earth being thus
obtained, the Diameter may be found by the
common proportion between the Circumference
and the Diameter of a Circle, the which accordArchimedes is as 22 to 7, or according to
Van Culen as 1 to 3. 14159. and to bring an U
As the circumference 3. 14159. is to 1 the Di
15. The measure of the Earth being thus found
in respect of its whole circumference and Diame
16. And this is either Primary or Secon
17. The Primary distinction of the Earthly
Globe into convenient spaces, is by Circles con
Teneriff, which is believed to be the highIohnson in his greatest Globe of the year 1616,
hath drawn the great Meridian in that place, and
it were to be wished, that this might be made the
common and unchangeable practice.
25. The Horizon is a great Circle, designing so great a Part of the Earth, as a quick sight can discern in an open field; it is twofold Rational and Sensible.
26. The Rational Horizon is that which is
supposed to pass through the Center of the
Earth, and is represented by the wooden Circle
in the Frame, as well of the Celestial, as the Ter
27. The Sensible Horizon is that before de
28. The great but less principal Circle upon
the Terrestrial Globe is the Zodiack, in which
the Sun doth always move. This Circle is de
29. The lesser Circles are those which do not
divide the Terrestial Globe into two equal, but
into two unequal Parts, and these by a general
name are called Parallels, or Circles aequidistant
AEquator to the Poles.
30. These Parallels are not of the same Magni
31. Again, a Parallel is either named or un
32. An unnamed Parallel is that which is drawn with small black Circular Lines.
33. A named Parallel is that which is drawn
upon the Globe with a more full ruddy and circuCancer and
Capricorn, with the Arctick and Antarctick Cir
Of the Distinction or Dimension of the Earthly Globe by Zones and Climates.
HAving shewed the primary distinction of
the Globe into convenient spaces by Cir
2. This secundary Dimension or Distinction of the terrestial Globe into Parts, is either a Zone or a Clime.
3. A Zone is a space of the Terrestial Globe
included either between two of the lesser nomi
4. The over hot or Torrid Zone, is between
the two Tropicks, continually scorched with
the presence of the Sun.
5. The two over cold or Frigid Zones, are
scituated between the two polar Circles and
the very Poles, continually wanting the neighbour
hood of the Sun.
6. The two temperate Zones, are one of
them between the Tropick of Cancer and the ArCapricorn and the Antarctick Circle, enjoyZone are the
hotter, and the parts next the Frigid Zone are
the Colder.
7. The Inhabitants of these Zones, in respect
of the diversity of their noon Shadows are diviAmphiscii, Heteroscii and
Periscii. Those that inhabit between the two
Tropicks are called Amphiscii, because that
their noon Shadows are diversly cast, sometimes
towards the South as when the Sun is more North
Cancer
and the Arctick Circle or between the TroCapricorn and the Antarctick Circle are
Heteroscii, because the Shadows at noon are
cast one only way, and that either North or
South. They that inhabit Northward of the
Tropick of Cancer have their Shadows always
towards the North, and they that inhabit SouthCapricorn, have their
noon Shadows always towards the South.
Those that inhabit between the Poles and the
Arctick or Antarctick Circles are called Periscii,
because that their Gnomons do cast their ShaHorizon
in his whole diurnal revolution.
8. The next secundary Dimension or distincti
9. And a Clime or Climate is a space of Earth
conteined between three Paralells, the middle
10. These Climates and the Parallels by
which they are conteined are none of them of
equal quantity, for the first Clime as also the Pa
11. The Antients reckoned but seven Cli
Ptolemy accounted the Paralells 38 each
way from the Equator, that is 38 towards the
North, and as many towards the South, 24 of
which he reckoned by the difference of one quar
13. The distances of all both Climates and Parallels, together with their Latitudes from the AEquator, and difference of the quantity of the longest days, are here fully exprest in the Table following.
Perioeci, Antoec
and Antipodes.
15. The Perioeci are such as dwell in the same
Parallel on the same side of the AEquator, how far
distant soever they be East and West, the sea
16. The Antoeci are such as dwell under the
same Meridian and in the same Latitude, or Pa
17. The Antipodes are such as dwell Feet to
Feet, so as a right Line drawn from the one
unto the other, passeth from North to South
through the Center of the World. These are
distant 180 degrees or half the compass of the
Earth, they differ in all things, as Seasons of the
year, length of days, rising and setting of the
Sun and such like. A matter reckoned so ridiBoniface
Arch-Bishop of Mentz seeing a Treatise
concerning these Antipodes written by Virgilius
Bishop of Salisburg, and not knowing what damBohemia, and after to Pope Zachary Anno 745
by whom the poor Bishop (unfortunate only in
being learned in such a time of Ignorance) was
condemned of Heresie, but God hath blest this
latter age of the World with more understand
18. The second part of the Terrestial Globe
is the Water which is commonly divided into
these parts, or distinguished by these Names,
Oceanus, Mare, Fretum, Sinus, Lacus and Flumen.
19. And first Oceanus or the Ocean is that
general Collection of all Waters, which encom
20. Mare the Sea, is a part of the main Ocean,
to which we cannot come but through some Fretum
or Strait, as Mare Mediterraneum. And
it takes it name first either from the adjacent
Shore, as Mare Adriaticum, from the City of
Adria; or secondly from the first discoverer, as
Mare Magellanicum, from Magellanus who first
found it, or thirdly from some remarkable acciMare Icarium from the drowning of
Icarus the Son of Daedalus.
21. Fretum, a Strait is a part of the Ocean
penned within some narrow Bounds, and opeHellespont, Gibralter,
&c.
22. Lacus, a Lake is a great body or collectiThrasymene in Italy, and Lacus Asphaltites, or
the dead Sea in the Land of C
23. Flumen or Fluvius is a water-course conStagnum
a standing Pool) issuing from some Spring
or Lake, and emptying it self into some part
of the Sea, or some other great River, the
mouth or outlet of which is called Ostium.
And thus we have gon over those particulars
both of Earth and Water, which appertain to
this Science of Geography in the general; We
will now proceed to a more particular Conside
Of Europe.
THe Terrestial Globe is divided into two parts, known or unknown.
2. The unknown or the parts of the World
not fully discovered, are distinguished into
North and South, the unknown parts of the
World towards the North, are those which
lie between the North part of Europe or America
and the North Pole; and the unknown parts of
the World toward the South, are those which
ly between the South part of America and the
South Pole.
3. The known parts of the World were
antiently these three, Europe, Asia and Africk,
to which in latter ages a fourth hath been adAmerica.
Europe is bounded on the North with the
Northern Ocean, and on the South with the MeTanais,
and on the West with the Western Ocean,
and is contained between the Tropick of Cancer,
and the Pole Arctick, or 44 degrees as most do
say, taking its beginning Southward from Sicily
where the Pole is elevated 36 degrees, and is
thence continued to 80 degrees of North LatiEurope is in
English miles 2640, but some allow to Europe 45
degrees of Latitude, that is in English miles
2700.
5. The Longitude of Europe is reckoned from
the furthest part of Spain and the Atlantick OceTanais, which some reckon to
be 60 Degrees, to one of which Degrees passing
through the middle of Europe, they allow fifteen
German miles almost, or sixty English, and so the
Longitude in German miles is 900, in English
3600.
6. Europe though the least of all the four
Quarters of the World, is yet of most renown
amongst us: First, because of the temperature of
the Air, and fertility of the Soil: Secondly,
from the study of Arts, both ingenuous and me
7. Europe may be considered as it stands diSpain. 2. France. 3. Germany.
4. Italy, and the Alpes. 5. Belgium. 6. Denmark▪
Swethland. 8. Russia. 9. Poland.
10. Hungary. 11. Sclavonia, 12. Dacia, and
13. Greece. Of each of which I will give some
short account; as also of the chief Islands as
they are dispersed, in the Greek, AEgaean, Cretan
and Io
8. Amongst these I give Spain the first place, as
being the most Western Part of all the ContiEurope environed on all sides with the
Sea, except towards France; from which it is
separated by the Pyrenaean Hills: but more partiGibralter,
on the East with the Mediterranean, and on the
North East with the said PyrenStrabo to an Oxes
hide spread upon the Ground; the Neck whereof
being that Isthmus which unites it to France.
9. The greatest length hereof, it reckoned at
800 miles, the breadth where it is broadest at
500, the whole Circumforence 2480 Italian
miles: but Mariana measuring the compass of it
by the bendings of the Pyrenaean Hills, and the
creeks and windings of the Sea, makes the full
circuit of it to be 2816 miles of Italian mea
10. It is situate in the more Southerly Part of
the Northern temperate Zone, and almost in the
midst of the fourth and sixth Climates; the
longest day being 15 hours and a quarter in
length in the most Northern Parts hereof: but
in the extream South near to Gibralter not above
11. This Continent is subdivided into the
Kingdoms of Navarr. 2. Biscay. 3. Guipuscoand Oviodo. 5. Gallicia. 6. Corduba.
7. Granada. 8. Murcia. 9. Toledo. 10. Castile.
11. Portugal. 12. Valentia. 13. Catalonia. 14.
Majorca. And 15. Aragon; but all of them are
now united in the Monarchy of Spain.
12. France according to the present dimensiAlpes which divide Dauphine and Piemont,
as also with the Countries of Savoy, Switzerland,
and some Parts of Germany and the Netherlands.
On the West with the Aquitanick Ocean, and a
Branch of the Pyrenaean Mountains which divide
it from Spain. On the North with the English
Ocean, and some Parts of Belgium, and on the
South with the rest of the Pyrenaean Mountains,
and the Mideterranean.
13. The Figure of it is almost square, each
side of the Quadrature being reckoned 600 miles
in length, but they that go more exactly to work
upon it, make the length thereof to be 660 Ita
14. The Principal Provinces in this flourishing
Country, are. 1. France specially so called.
2. Champagne. 3. Picardy. 4. Normandy. 5.
Bretagne. 6. The Estates of Angiou. 7. La BeThe Dukedom of BourThe Dutchy. 22.
The County of Burgundy. 23. The Islands in
the Aquitanick and Gallick Ocean: Those of most
note are these six. 1. Oleron. 2. Reon the shores
of Normandy, of which the four last are under the
Kings of England.
15. Italy once the Empress of the greatest
part of the then known World, is compassed
with the Adriatick, Ionian and Tyrrhenian Seas,
except it be towards France and Germany, from
which it is parted by the Alpes; so that it is in a
manner, a Peninsula, or a Demi-Island. But
more particularly it hath on the East the lower
part of the Adriatick, and the Ionian Sea, by
which it is divided from Greece; on the West
the River Varus, and some part of the Alpes,
by which it is parted from France, on the
North in some part the Alpes which divide it from
Germany; and on the other, part of the AdriaDalmaria; and on
the South the Tyrrhenian and Tuscan Seas, by
which it is separated from the main Land of
Africa.
16. It containeth in length from Augusta Praetoria,
now called Aost, at the foot of the Alpes,
unto Otranto in the most Eastern Point of the
Kingdom of Naples 1020 miles; in breadth from
Varo, which parts it from Provence, to
the mouth of the River Arsia in Friuli, where it
is broadest, 410 miles; about Otranto, where it
is narrowest not above 25 miles; and in the
middle parts from the mouth of Peseara in the
Adriatick or upper Sea to the mouth of Tiber in
the Tuscan or lower Sea, 126 miles. The whole
compass by Sea reckoning in the windings and
turnings of the shore, comes to 3038 miles;
which added to the 410 which it hath by Land,
make up in all 3448 miles: but if the Coasts on
each side be reckoned by a straight Line, then as
Castaldo computes it, it comes to no more than
2550 miles.
17. The whole Country lieth under the first
and sixth Climates of the Northern temperate
Zone, which it wholly taketh up: so that the
longest day in the most Northern Parts is 15
hours and three first parts of an hour; the long
18. Italy as it stands now is divided into the
Kingdoms of Naples, Sicily and Sardinia. 2. The
Land or Patrimony of the Church. 3. The
great Dukedom of Tuscany. 4. The CommonVenice, Genoa and Luca. 5. The ELombardy, that is the Dukedoms of
1. M
and the Principality of Piedmont.
19. To the Peninsula of Italy belong the Alpes,
aridge of Hills, wherewith as with a strong and
defensible Rampart Italy is assured against France
and Germany. They are said to be five days
Journey high, covered continually with Snow,
Savoy; the
Seigniory of Geneva; the Country of Wallisland,
Switzerland and the Grisons.
20. Belgium, or the Netherlands, is bounded
on the East with Westphalin, Gulick, Cleve, and the
Land of Triers, Provinces of the higher Germany;
on the West with the main Ocean, which
divides it from Britain; on the North with the
River Ems, which parts it from East-Friezeland;
on the South with Picardie and Campagne two
French Provinces; upon the South-East with the
Dukedom of Lorrain.
21. It is in compass 1000 Italian or 280 Ger
22. It containeth those Provinces which in
these later Ages were possessed by the House of
Burgundy, that is the Lordship of West-Friezeland,
given to the Earls of Holland by Charles the Bald;
the Earldom of Zutphen united unto that of Gelder
by Earl Otho of Nassau, and finally the Estate
of Groening, Over-Yssel, and some part of Vtrecht,
by Charles the Fifth. As it stands now divided
between the Spaniards and the States it containeth
the Provinces of 1. Flanders. 2. Artois. 3. Hainault.
4. The Bishoprick of Cambray. 5. Namur.
6. Luxemburg. 7. Limbourg. 8. Luyckland,
or the Bishoprick of Leige. 9. Brabant. 10.
Marquisate. 11. Meohlin. The rest of the Netherlands
Spain,
are 1. Holland. 2. Zeland. 3. West-Friezeland.
4. Vtrecht. 5. Over-Yssel. 6. Gelderland. 7.
Zutphen. 8. Groening.
23. Germany is bounded on the East with Prussia,
Poland, and Hungary; on the West with
France, Switzerland and Belgium; on the North
with the Baltick Seas, the Ocean, and some part
of Denmark; on the South with the Alps which
part it from Italy.
24. The length from East to West, that is
from the Vistula or Weissel to the Rhine, is estimaBrixen in Tyrol, 740 of the same miles. So that
the Figure of it being near a Square, it may take
up 3160 miles in compass, or thereabouts. Si
25. The Principal Parts of this great Continent,
are 1. Cleveland. 2. The Estates of the
three spiritual Electors, Colen, and Triers.
3. The Palatinate of the Rhine. 4. Alsatia. 5.
Lorrain. 6. Suevia or Schwaben. 7. Bavaria. 8.
Austria and its Appendices. 9. The Confederation
of Waderaw. 10. Franconia. 11. WirtenThe Palatinate of Northor the Upper Palatinate. 14. Bohemia and
the Incorporate Provinces. 15. Pomerania. 16.
Mecklenburg. 17. The Marquisate of Brandenand the Members of it. 19.
The Dukedom of Brunswick and Lunenburg. 20.
The Lantgravedom of Hassia. 21. Westphalen. 22.
East-Friezeland.
26. Denmark or Danemark, reckoning in the
Additions of the Dukedom of Holstein, and the
great Continent of Norway, with the Isles thereSweden; on the West with the
main Western Ocean; on the North-East with
a part of Sweden; full North with the main froGermany, from
which it is divided on the South-West by the
River Albis, and on the South-East by the Trave;
a little Isthmus or neck of Land uniting it to the
Continent.
27. It lieth partly in the Northern temperate
Zone, and partly within the Arctick Circle; exGermany, as far as the 71 degree where
it hath no other bound but the frozen O
28. The whole Body of the Estate consisteth
chiefly of three Members. viz. 1. The DukeHolstein; containing Waggerland, Dilmarsh,
Starmaria, and Holstein, especially so callDenmark; compreIuitlands, part of Scandia, and the
Hemodes, or Baltick Islands. 3. The Kingdom
of Norway consisting of Norway it self, and the
Islands of the Northern Ocean.
29. Swethland is bounded on the East with
Muscovy, on the West with the Doferine Hills,
which divide it from Norway; on the North
with the great frozen Ocean spoken of before;
on the South with Denmark, Liefland, and the
Baltick Sea.
30. It is situate under the same Parallels and
Degrees with Norway, that is, from the first
Parallel of the 12 Clime, where the Pole is ele
31. The whole Kingdom is divided into two
Parts, the one lying on the East, the other on the
West of the Bay or Gulf of Bodener, being a large
and spacious branch of the Baltick Sea, extending
from the most Southerly Point of Gothland, as
far as to Lapland on the North. According to
which Division we have the Provinces of 1. Gothland.
2. Sweden lying on the West side of the
Gulph. 3. Lapland shutting it up upon the
North. 4. Bodia or Bodden. 5. Finland on the
East side thereof. 6. The Sweedish Islands, where
it mingleth with the rest of the Baltick Seas.
32. Russia is bounded on the East by Tartary,
Livonia and Finland, from which
it is divided by great mountains and the River
Poln, on the North by the frozen Ocean, and
some part of Lapland, and on the South by Lituania
a Province of the Kingdom of Poland, and
the Crim Tartar inhabiting on the Banks of Palus
Maeotis, and the Euxine Sea. It standeth
partly in Europe and partly in Asia, the River
Tanais or Don running through it, the common
boundary of those great and noted parts of the
world.
33. It is scituate North within the Artick Cir
34. It is divided into the Provinces of 1. Mosspecially so called. 2. Snolthe great, 6. Corelia,
7. Blarmia, 8. Petzora, 9. Condora, 10 Obdora,
11. Iugria, 12. Severia, 13. Permia, 14. Rozan,
15. Wiathka, 16. Casau, 17. Astracan, 18. Novoinferiour, 10. The Morduits or Mordua,
20. Worotime, 21. Tuba, 22. Wolodomir, 23. Duina,
24. the Russian Islands.
35. Poland is bounded on the East with Russia,
and the Crim-Tartar, from whom it is parted by
the River Borysthenes; on the West with Germany,
on the North with the Baltick Sea and some
part of Russia, on the South with the Carpathian
Mountains, which divide it from Hungary, Transilvania,
and Moldavia. It is of figure round in
compass 2600 miles, scituate under the 8 and
12 Climates, so that the longest day in the
southern parts is but 16 hours, and about 18
36. The several Provinces of which this Kingdom
doth consist, are 1. Livonia, 2. Samogitia,
3. Lituania, 4. Volkinia, 5. Podolia, 6. Russia nigra,
7. Massovia, 8. Podlassia, 9. Prussia, 10. Pomerelspecially so called.
37. Hungary is bounded on the East with Transilvania
and Walachia, on the West with Sterria,
Austria and Moravia, on the North with the Carpathian
mountains which divide it from Poland, and
on the South with Sclavonia, and some part of
Dacia: it extendeth in length from Presburg aDanow to the borders of Transilvania,
for the space of 300 English miles, and 190 of
the same miles in breadth.
38. It lieth in the Northern temperate Zone,
betwixt the middle parallels of the 7 and 9 Cli
40. This Country is commonly divided into
the upper Hungary and the lower, the upper lyDanow, the lower
lying on the South of that River, comprehendPannonia inferior and part of Superior,
and is now possessed by the King of Hungary and
the Great Turk, who is Lord of the most part
by Arms and Conquest.
04. Sclavonia is bounded on the East with
Servia, Macedonia and Epirus, from which it is
parted by the River Drinus, and a line drawn
from thence unto the Adriatick, on the West
with Carniola in Germany, and Istria in the SeigVenice, from which last it is divided by
the River Arsia; on the North with Hungary, on
the South with the Adriatick Sea.
41. It is scituate in the Northern temperate Zone, between the middle Parallels of the sixth and seventh Climates, so that the longest day in Summer is about 15 hours and an half.
42. This Country as it came at last to be diHungary and the
State of Venice; is distinguished into 1. Windischland,
2. Croatia, 3. Bosnia, 4. Dalmatia, 5. Liburnia
or Cantado di Zara, and 6. The Sclavonian I
43. Dacia is bounded on the East with the
Euxine Sea and some part of Thrace; on the
West with Hungary and Sclavonia; on the North
with Podolia, and some other members of the
Realm of Poland, on the South with the rest of
Thrace and Macedonia.
44. It lieth on both sides of the Danow frontHungary,
and some part of Sclavonia; extended from the
7 Climate to the 10; so that the longest Summers
day in the most northern parts thereof, is near
17 hours, and in the most southern 15 hours 3
quarters.
45. The several Provinces comprehended unDacia, are 1. Transilvania, 2. Moldavia,
3. Walachia, 7. Rascia, 5. Servia, 6. Bulgaria,
the first four in old Dacia, on the North
side of the Danow; the two last in new Dacia,
on the South thereof.
45. Greece in the present Latitude and exHaemus which parteth it from Bulgaria,
Servia and some part of Illyricum; and on the
South with the Sea-Ionian; so that it is in a manPeninsula or Demi-Island, environed on three
sides by the Sea, on the fourth only united to the
rest of Europe.
46. It is scituate in the northern temperate Zone, under the fifth and sixth Climates, the longest day being 15 hours.
47. In this Country formerly so famous for
learning and government, the several ProvinPeloponnesus, 2. Achaia, 3. Epirus, 4. Albania,
5. Macedon, 6. Thrace, 7. The Islands of
the Propontick; 8. AEgean, and 9. The Ionian
Seas, and 10. finally the Isle of Crete.
And thus I have given you a brief description
of those Countries which are comprehended in
the Continent of Europe; the Islands in this part
of the world are many; I will mention only some
few. These two in the British and Northern OGreat Britain and
Ireland are the most famous, to which may be adGreenland. In the Mediterranaen Sea you
have the Islands of Sicilia, Sardinia, Corsica and
Crete, which is now called Candia the greater
and the less: As for the other Islands belonging
to this part of the world, the Reader may expect
a more particular description from them who
have or shall write more largely of this subject:
This we deem sufficient for our present purpose.
Let this then suffice for the description of the
first part of the World called Europe.
Of Asia.
ASia is bound on the West with the MediteHellespont,
Propontis, Thracian Bosphorus and the Euxine
Sea, the Palus Maeotis, the Rivers Tanais and Duina,
a Line being drawn from the first of the
two said Rivers unto the other, by all which it is
parted from Europe; on the North it hath the
main Scythick Ocean; but on the East the IndiMare del Eur by which it is sepaAmerica; on the South the MediterAnatolia,
and the main Southern Ocean, passing along the
Indian, Persian and Arabian Coasts: and finally
on the south-west, the red Sea or Bay of Arabia,
by which it is parted from Affrick. EnviIsthmus in the south-west,
which joyns it to Africk, and the space of
ground (whatsoever it be) between Duina and
Tanais, on the North-west which unites it to
Europe.
2. It is situated East and West, from the 52
to the 169 degree of Longitude; and North
and South from the 82 degree of Latitude to
the very AEquator; some of the Islands only ly
Eden, which he had for the first
place of his habitation. 3. Because here flouNoah after their vain
attempt at Babel.
4. This part of the World for the better unAnatolia or Asia
minor. 2. Cyprus. 3. Syria. 4. Arabia. 5. Chaldea.
6. Assyria. 7. Mesopotamia. 8. Turcomania.
9. Media. 10. Persia. 11. Tartaria. 12.
China. 13. India. and 14. the Oriental Islands.
Anatolia or Asia minor, is bounded on the East
with the River Euphrates, by which it is parAsia; on the West with
the Thracian Bosphorus, Propontis, Hellespont, and
Europe;
on the North with Pontus Euxinus, called
also the black Sea, and Mare Maggiore, and on
the South by the Rhodian, Lydian and PamPeninsula environed
on all sides with water, excepting a small Isthmus
or Neck of Land extending from the head of
Euphrates to the Euxine Sea, by which it is joyAsia.
It reacheth from the 51 to the 72 degree of
Longitude, and from the 36 to the 45 degree
of Latitude, and lyeth almost in the same posiItaly, extending from the middle Pa
The Provinces into which it was divided before
the Roman Conquest were 1. Bithynia. 2.
Pontus. 3. Paphlagonia. 4. Galatia. 5. Cappadocia.
6. Armenia Major & Minor. 7. Phrygia minor.
8. Phrygia major. 9. Mysia the greater and the
less. 10. Asia specially so called, comprehending
AEolis and Ionia. 11. Lydia. 12. Caria. 13. LyThe Province of
the Asian Isles, whereof the most principal are
1. Tenedos. 2. Chios. 3. Samos. 4. Choos. 5. Ica
Cyprus is situated in the Syrian and Cilician
Seas, extended in length from East to West 200
miles, in breadth 60 the whole compass reckCilicia in Asia minor, and about one
hundred from the main Land of Syria.
It is situated under the fourth Climate, so that the longest day in Summer is no more than 14 hours and a half.
Divided by Ptolemy into the 4 provinces of
1. Paphia. 2. Amathasia. 3. Lepathia. 4. Sala
Syria is bounded on the East with the River
Euphrates by which it is parted from Mesopotamia;
on the West with the Mediterranean Sea;
on the North with Cilicia and Armenia minor,
parted from the last by mount Taurus; and on
the South with Palestine, and some parts of Arabia.
The length hereof from Mount Taurus
to the Edge of Arabia, is said to be 525 Miles;
the breadth from the Mediterranean to the RiEuphrates 470 Miles, drawing somewhat
near unto a Square.
The whole Country was antiently divided inPhoenicia. 2. Palestine. 3. Syria
specially so called. 4. Comagena. 5. Palmyrene.
and Caelosyria, or Syria Cava.
Arabia hath on the East Chaldaea and the Bay
or Gulf of Persia; on the West Palestine, some
part of Egypt, and the whole course of the
red Sea, on the North the River Euphrates with
some parts of Syria and Palestine, and on the
South the main southern Ocean. It is in circuit
about 4000 Miles, but of so unequal and heteArabia Deserta,
2. Arabia Petraea. 3. Arabia Felix and
4. The Arabick Islands.
Chaldea is bounded on the East with Susiana
a Province of Persia; on the West with Arabia
deserta; on the North with Mesopotamia; and on
the South with the Persian Bay and the rest of
Deserta.
Assyria is bounded on the East with Media,
from which it is parted by the Mountain called
Coathras; on the West with Mesopotamia, from
which it is divided by the River Tygris; on the
South with Susiana; and on the North with some
part of Turcomania; it was antiently divided
into six parts. 1. Arraphachitis. 2. Adiabene.
3. Calacine. 4. Aobelites. 5. Apolloniates.
Mesopotamia is bounded on the East with the
River Tygris by which it is parted from Assyria;
on the West with Euphrates which divides it
from Comagena a Province of Syria; on the North
with Mount Taurus; by which it is separated
from Armenia major; and on the South with Chaldea
and Arabia deserta from which last it is parEuphrates also. It was
antiently divided into, 1. Anthemasia. 2. Chalcitis.
3. Caulanitis. 4. Acchabene. 5. Ancorabitis
and 6. Ingine.
Turcomania is bounded on the East with Media
and the Caspian Sea; on the West with the Euxine
Sea, Cappadocia and Armenia minor; on the
North with Tartary, and on the South with Mesopotamia
and Assyria. A Countrey which conArmenia major or
Turcomania properly and specially so called. 2.
Colchis. 3. Iberia. 4. Albania.
Media is bounded on the East with Parthia,
and some part of Otyrcania, Provinces of the
Persian Empire; on the West with Armenia major,
and some part of Assyria; on the North
with the Caspian Sea and those parts of Armenia
major, which now pass in the account of Iberia,
Georgia; and on the South with Persia. It is now
Atropatia. 2. Media
major.
Persia is bounded on the East with India; on
the West with Media, Assyria, and Chaldea; on
the North with Tartary, on the South with the
main Ocean.
It is divided into the particular Provinces of
1. Susiana. 2. Persis. 3. Ormur. 4. Carmaand 13. Bactria.
Tartaria is bounded on the East with China,
the Oriental Ocean, and the Straits of Anian,
by which it is parted from America, on the
West with Russia and Podolia, a Province of the
Realm of Poland; on the North with the main
Scythick or frozen Ocean; and on the South
with part of China, from which it is separated
by a mighty Wall, some part of India, the RiOxus parting it from Bactria and Margiana,
two Persian Provinces; the Caspian Sea which
separates it from Media and Hyrcania; the Caucasian
Mountains interposing between it and
Turcomania; and the Euxine Sea which divideth
it from Anatolia and Thrace.
It reacheth from the 50 degree of Longitude
to the 195 which is 145 degrees from West to
East; and from the 40 degree of Northern La
It is now divided into these five parts. 1. Tar
China is bounded on the North with Altay and
the Eastern Tartars, from which it is separated
by a continued Chain of Hills, part of those of
Ararat, and where that chain is broken off or
interrupted, with a great wall extended 400
Leagues in length; on the South partly with
Cauchin China a Province of India, partly with
the Ocean; on the East with the oriental Ocean,
and on the West with part of India and Cathay.
It reacheth from the 130 to the 160 degree
of Longitude, and from the Tropick of Cancer
to the 53 degree of Latitude; so that it lieth
under all the Climes from the third to the ninth
inclusively. The longest summers day in the
southern parts being 13 hours and 40 Minutes
increased in the most northern parts to 16 hours
and 3 quarters.
It containeth no fewer than 15 Provinces. 1.
Canton. 2. Foquien. 3. Olam. 4. Sisnam 5.
Tolenchia. 6. Causay. 7. Minchian. 8. OchiBesides the provinces of Suehuen, the
Island of Chorea and the Island of Cheaxan.
India is bounded on the East with the OrienChina; on the West
with the Persian Empire; on the North with
some Branches of Mount Taurus, which divide
it from Tartary; on the South with the In
Extended from 106 to 159 degrees of LonAEquator to the 44th de
The whole Country is divided into two main
parts, India intra Gangem, and India extra Gangem.
The Oriental Islands are 1. Iapan. 2. The
Philippine and Isles adjoyning. 3. The Islands
of Bantam. 4. The Moluccoes. 5. Those calSinda or the Celebes. 6. Iava. 7. Borneo.
8. Sumatra. 9. Ceilan. and 10. others of less
note.
Of Africk.
AFrick is bounded on the East by the Red
Sea, and Bay of Arabia, by which it is
parted from Asia; on the West by the main AtAmerica;
on the North by the Mediterranean Sea,
which divides it from Europe and Anatolia; and
on the South with the AEthiopick Ocean, seTerra Australis incognita or the
southern continent, parted from all the rest of
the World except Asia only, to which it is joyIsthmus not above 60 miles in
length.
It is situate for the most part under the TorAEquator crossing it almost in the
midst. It is now commonly divided into these
seven parts. 1. AEgypt. 2. Barbary or the
Roman Africk. 3. Numidia. 4. Lybia. 5. Terra
Nigritarum. 6. AEthiopia superior. and 7. AEthiopia
rinferior.
AEgypt is bounded on the East with Idumea,
and the Bay of Arabia; on the West with Barbary,
Numidia, and part of Lybia; on the North
with the Mediterranean Sea; on the South with
AEthiopia superior, or the Abyssyn Emperor; it is
situate under the second and fifth Climates, so
that the longest day in Summer is but thirteen
hours and a half.
Barbary is bounded on the East with Cyrenaica;
on the West with the Atlantick Ocean; on
the North with the Mediterranean Sea, the
Straits of Gibralter and some part of the AtlanAtlas, by
which it is separated from Lybia inferior or the
Desarts of Lybia.
It is situated under the third and fourth CliTunis. 2. Tremesch
or Algiers. 3. Fesse and 4. Morocco.
Numidia is bounded on the East with Egypt,
on the West with the Atlantick Ocean; on the
North with Mount Atlas, which parteth it from
Barbary and Cyrene; on the South with Lybia
Deserta.
Lybia is either Interior or Deserta, Libia interior
is bounded on the North with Mount Atlas
by which it is parted from Barbary and Cyrenaica;
on the East with Lybia Marmarica interpoEgypt, and part of AEthiopia
superior, or the Habassine Empire; on the South
with AEthiopia inferior, and the Land of the Ne
Lybia deserta is bounded on the North with
Numidia or Biledulgerid; on the South with the
Land of the Negroes; and on the West with Gulata
another Province of the Negroes interposed
between it and the Atlantick.
Terra Nigritarum or the Land of the Negroes
is bounded on the East with AEthiopia Superior;
on the West with the Atlantick Ocean; on the
North with Lybia deserta and on the South with
the Ethiopick Ocean, and part of AEthiopia
Inferior.
AEthiopia Superior is bounded on the East with
the Red Sea and the Sinus Barbaricus; on the
West with Lybia Interior, the Realm of Nubia
in the Land of the Negroes and part of the
Kingdoms of Congo in the other AEthiopia; on
the North with Egypt and Lybia Marmarica,
and on the South with the Mountains of the
Moon, by which it is parted from the main Body
of AEthiopia Inferior.
It is situate on both sides of the AEquinoctial,
extending from the South Parallel of seven deAEthiopia to the Northern end of the Isle of
Meroz situated under the fifth Parallel on the
North of that Circle.
AEthiopia inferior is bounded on the East with
the Red Sea; on the West with the Ethiopick
Ocean; on the North with Terra Nigritarum,
and the higher AEthiopia; and on the South
where it endeth, is a point of a Conus, with the
main Ocean parting it from the Southern undisPtolemyes time went
under the name of Terra incognita.
Of America.
AMerica the fourth and last part of the World
is bounded on the East with the Atlantick
Ocean and the Vergivian Seas, by which it is parEurope and Africa; on the West with
the Pacifick Ocean, which divides it from Asia;
on the South with some part of Terra Australis
incognita, from which it is separated by a long,
but narrow Strait, called the Straits of Magellan;
the North bounds of it hither to not
so well discovered, as that we can certainly af
It is called by some and that most aptly, The
new World; New for the late discovery, and
World for the vast greatness of it. The whole
is naturally divided into two great Peninsules,
whereof that towards the North is called Mexicana.
That towards the South hath the name
of Peruana: the Isthmus which joyneth these two
The Northern Peninsula called Mexicana, may
be most properly divided into the Continent and
Islands: The Continent again into the several
Provinces of 1. Estotiland, 2. Nova Francia,
3. Virginia, 4. Florida, 5. California, 6. Nova
Gallicia, 7. Nova Hispania, 8. Guntimala. The
Southern Peninsula called Peruana, taking in some
part of the Isthmus, hath on the Continent the
Provinces of 1. Castella Aurea, 2. Nova Granada,
3. Peru, 4. Chile, 5. Paraguay, 6. Brasil,
7. Guiana, and 8. Paria. The Islands which
belong to both, are dispersed either in the SouthMare del Zur, where there is
not any one of Note but those called Los Ladrones
and the Islands of Solomon. Or in the NorthMare del Noords, reduced unto
the Caribes, Porto-Rico, Hispaniola, Cuba and
Iamaica. And thus much concerning the real
and known parts of the Terrestrial Globe.
Of the Description of the Terrestrial Globe by Maps Vniversal and Particular.
HItherto we have spoken of the true and real
Terrestrial Globe, and of the measure
thereof by Circles, Zones, and Climates, as it is
usually represented by a Sphere or Globe; which
must be confessed to be the nearest and the most
2. The description thereof by Parallelogram
is thus, the Parallelogram is divided in the midst
by a line drawn from North to South, passing by
the Azores or Canaries for the great Meridian.
Cross to this and at eight Angles, another line
is drawn from East to West for the AEquator;
then two parallels to each to comprehend the
figure, in the squares whereof there are set down
four parts of the world rather than the whole:
And this way of description though not exact or
near to the natural, hath yet been followed by
such as ought still to be accounted excellent, and
is the form of our plain Charts, and in places
near the AEquinoctial may be used without com
3. Hence it followeth, that if the picture of
the earth be drawn upon a Parallelogram, so that
the Meridians be equally distant throughout, and
the Parallels equally extended, the Parellel of 60
degrees shall be as great as the line of the AEquator
it self is, and he that coasteth about the world
AEquator,
though the way be but half as long. For
the longitude of the Earth in the AEquator it self,
is 21600; but in the Parallel of 60 but 10800
miles. So two Cities under the same parallel of
60, shall be of equal Longitude to other two unAEquator at 60
miles distance, and coming up to the Parallel of
60, shall be thirty miles nearer, and yet each of
them keep the same Meridians and sail by this
Card upon the very points of the Compass at
which they set forth. This was complained of
by Martin Cortez and others, and the learned
Mercator considering well of it, caused the deWright in the second Chapter
of his Correction of Errors in Navigation, hath
sought by the inscription of a Planisphere into a
Concave Cylinder. And this description of
the Earth upon a Parallelogram, may indeed be
so ordered by Art, as to give a true account of
the scituation and distance of the parts, but can
4. The description therefore of the whole by Planisphere is much better, because it represents the face of the Earth upon a plain, in its own proper Spherical Figure as upon the Globe it self. This description cannot well be contrived upon so few as one Circle or more than two.
Suppose then the Globe to be divided into
AEquator or Meridian (for
the Colure is all one with the Meridian) the Ho
5. Suppose then the Globe to be flatted upon
the plain of the AEquator, and you have the
first way of projection dividing the Globe into
the North and South Hemispheres.
In this projection the Pole is the Centre, the
AEquator is the Circumference divided into 360
degrees of Longitude, the Paralles are whole
Circles, the Meridians are streight lines, the
Parallels are Parallels indeed, and the Meridians
equi-distantly concur, and therefore all the dePtolemy describes that part of the habitable world
which was discovered to his time.
6. Suppose the Globe to be flatted upon the
plain of the Meridian, and you have the other
way of projection; the AEquator here is a streight
line, the great Meridian a whole Circle, in this
Section the Meridians do not equi-distantly con
However, this latter way is that which is now most and indeed altogether in use.
7. Particular Maps are but limbs of the Globe, and therefore though they are drawn asunder, yet are they still to be done with that proportion, as a remembring eye may suddenly acknowledge, and joyn them to the whole Body.
The difference of Miles in several Countries is great, but it will be enough to know that the Italian and English, are reckoned for all one, and four of these do make a German Mile; two a French League. The Swedish or Danish Mile consisteth of 5 Miles English and somewhat more. Sixty common English and Italian Miles answer to a degree of a great Circle.
Now as the Miles of several Countries do veMagna, Mediocria and
Parva, to shew the difference. In some Maps
you shall find the Miles thus hiddenly set down,
and the meaning is, that you should measure the
Milliaria magna upon the lowermost Line, the
Parva upon the uppermost, and the Mediocria
upon the middlemost.
The use of the Scale is for the measuring the
distances of places in the Map, by setting one
foot of your Compasses in the little circle repre
In Anno
THe Elements of the Mathematical Art,
commonly called Algebra, expounded in
four Books by Iohn Kersey, in two Vol. fol.
A mirror or Looking-glass for Saints and SinSam. Clark, in
two Vol. fol.
The Mariners Magazine by Capt. Sam. Sturmy,
fol.
Military and Maritime Discipline in three
Books, by Capt. Tho. Kent, fol.
Dr. Cudworth's universal Systeme.
The Triumphs of Gods Revenge against the
Crying and Execrable sin of wilful and premediIohn Reynolds, fol.
Royal and Practical Chymistry by Oswaldus
Crollius and Iohn Hartman, faithfully rendred into
English, fol.
Practical Navigation by Iohn Seller. Quarto.
The History of the Church of Great Britain
from the Birth of our Saviour until the Year of
our Lord 1667. quarto.
The Ecclesiastical History of France from the
first plantation of Christianity there unto this
time, quarto.
The book of Architecture by Andrea Palladio,
quarto.
Vincent Scammozi
quarto.
Trigonometry, on the Doctrine of Triangles,
by Rich. Norwood, quarto.
Markham's Master-piece Revived, containing
all knowledge belonging to the Smith, Farrier,
or Horse-Leach, touching the curing of all Diseaquarto.
Collins Sector on a Quadrant, quarto.
The famous History of the destruction of Troy,
in three books, quarto.
Safeguard of Sailers, quarto.
Norwood's Seamans Companion, quarto.
Geometrical Seaman, quarto.
A plain and familiar Exposition of the Ten
Commandments, by Iohn Dod, quarto.
The Mariners new Calendar, quarto.
The Seamans Calendar, quarto.
The Seamans Practice, quarto.
The honour of Chivalry do the famous and deDon Belianus of Greece, quarto.
The History of Amadis de Gaul, the fifth part,
quarto.
The Seamans Dictionary, quarto.
The complete Canonier, quarto.
Seamans Glass, quarto.
Complete Shipwright, quarto.
The History of Valentine and Orson, quarto.
The Complete Modellist, quarto.
The Boat-swains Art, quarto.
Pilots Sea-mirror, quarto.
The famous History of Montelion Knight of the
Oracle, quarto.
The History of Palladine of England, quarto.
Cleocretron and Clori
The Arralgnment of lower, idle, froward and
unconstant Women, quarto.
The pleasant History of Iack of Newb
Philips Mathematical Manual, Octavo.
A prospect of Heaven, or a Treatise of the hapoct.
Etymologicunt parvum, oct.
Thesaurus Astrologiae, or an Astrological Treasury
by Iohn Gadbury, oct.
Gellibrand's Epitome, oct.
The English Academy or a brief Introduction
to the seven Liberal Arts, by Iohn Newton, D. D. oct.
The best exercise for Christians in the worst
times, by I. H. oct.
A seasonable discourse of the right use and aoct.
The Mariners Compass rectified, oct.
Norwood's Epitome, oct.
Chymical Essays by Iohn Beguinus, oct.
A spiritual Antidote against sinful Contagions,
by Tho. Doolittle, oct.
Monastieon Fevershamiense; or a description of
the Abby of Feversham, oct.
Scarborough's Spaw, oct.
French Schoolmaster, oct.
The Poems of Ben. Iohnson, junior, oct.
A book of Knowledge in three parts, oct.
The Book of Palmestry, oct.
Farnaby's Epigramms, oct.
The Huswifes Companion, and the Husbandoct.
Jovial Garland, oct.
Cocker's Arithmetick, twelves.
The Path Way to Health, twelves.
Hall's Soliloquies, twelves.
twelves.
Newton's Introduction to the Art of Logick, twelves.
Newton's Introduction to the Art of Rhetorick, twelves.
The Anatomy of Popery, or a Catalogue of Popish ertwelves.
The famous History of the five wise Philosophers, conIehosophat the Hermit. twelves,
The exact Constable with his Original and Power in all
cases belonging to his Office, twelves.
The Complete Academy or a Nursery of Complements,
twelves.
Heart salve for a wounded Soul, and Eye salve for a blind
World, by Tho. Calvert. twelves.
Pilgrims Port, or the weary mans rest in the Grave, twelves.
Christian Devotion or a manual of Prayers, twelves.
The Mariners divine Mate, twelves.
At Cherry Garden Stairs on Rotherhith Wall, are taught
these Mathematical Sciences, viz. Arithmetick, Algebra,
Geometry, Trigonometry, Surveying, Navigation, Dyalling,
Astronomy, Gauging, Gunnery and Fortification: The use
of the Globes, and other Mathematical Instruments, the
projection of the Sphere on any circle, &c. He maketh and
selleth all sorts of Mathematical Instruments in Wood and
Brass, for Sea and Land, with Books to shew the use of them:
Where you may have all sorts of Maps, Plats, Sea-Charts, in
Plain and Mercator, on reasonable Terms.