[ i ]

The Definition of a POINT.

THE Point, is that which hath not any Part.

By this Definition, it is easie to conceive, that the Point hath neither length, nor breadth, nor depth, and that also it is not fenfible, but only Intellectual; seeing that there is nothing which falleth under fenfe, which hath not a quantity, and that there is no quantity without Parts, which would altogether contradict this Definition. Nevertheless, as none can make any Operation, but by the Interpolation of Corporal things, they represent therefore the Mathematical Point, by the Point Physical, which is the Object of the fight, the smallest and least fenfible, which hath no Geometrical greatnefs, divifible to our fenfe, and is made with the Point of a Needle, or with the end of the Point of the Compafs of a Pen, or Penfil, as the Point noted by A.

The Point Central, or Center, is a Point by which a Circle is drawn, or a Circumference, or rather it is the midft of a Figure, as C.

The Point Secant, is a Point where the Lines do interdivide themselves, and which is ordinarily called a Section, as C D.

See the Ace of Diamonds.

B

The

[ 2 ]

The Definition of a LINE.

The *Line*, is a Length without Breadth:
The *Line* is no other thing, then the paſ-
fing of the Point from one place to ano-
ther, and it would be unperceivable, if one
should not fet it forth by a Point Physical, the
which by its flowing, doth reprefent to us, as
AB, CD, EF.

There are many forts of *Lines*, as the Point,
which is the Original thereof, is able to receive
different Motions; nevertheless, they take into
consideration only two Single and Principal
ones; the Straight, and the Crooked; and a
third alfo, which they call the Mixt, because
that it is compoſed of the two firſt.

The right, or ſtraight *Line* is, that which is e-
qually comprized within its extremities.

Otherwife, it is that which floweth from one
Point to another, without any turnings afide,
as A B.

The Crooked *Line*, is that which turneth, or
wandreth from its Extremities, by one, or more
Turnings afide, as C D.

When as this *Line* is deſcribed with a Com-
pafs, they call it Circular, as E.

The Mixt *Line*, is that which is Straight and
Crooked, as the *Line* V.

See the Ace of Diamonds.

The

[ 3 ]

*The LINE is distinguished into Finite and In-finite, into Apparent and Occult, or Hidden.*

The Line Finite, is a Line Bounded, which containeth, or fuppofeth, a length ne-ceffary, as A.

The Infinite, is a Line undetermined, which hath no precife length, as B.

The Apparent, or Traced out, is that which is defcribed with Ink or Pencil, as A B.

The Occult, or White, which is drawn only with the Point of the Compaſs, or marked with Points, and therefore they call it, the Pointed, or Line with Points, as C.

*Here endeth the Ace of Diamonds.*

*The Line receiveth also divers Denominations, according to its divers Positions and Proprieties.*

A Perpendicular, is a right Line, which fal-leth, or lifteth it felf up upon another maketh the Angles of the one part, and, the other, equal between themſelves, as A B.

A Plumb Line, is that which goeth from high to low without bending, either to the right, or to the left, and which would paſs through the Center of the World, if it were prolonged in-finitely, as C.

*See the Duce of Diamonds.*

B 2

A

[ 4 ]

A Line Horizontal, is a Line in an equal poize,
which inclineth it self equally on the one part,
and the other, as D E.

Lines Parallels, are those which follow each
other by an equal distance, as H.

An Oblique Line, which is neither Horizon-
tal, nor a Plumb-line, but of a Bias, as F G.

The Basis is the Line, upon which the Figure
repoſeth it ſelf, as I L.

Sides, are those Lines which encompafs a Fi-
gure, as I N. N M.

The Diagonal, is a right Line that traver-
feth a Figure, and that which endeth at
two Angles oppofite, as A B. The Dia-
meter, is a right Line, which traverſeth, or paſ-
feth through a Circular Figure by its Center,
and which ends at the Circumference.

A Spiral Line, is a Crooked Line, which par-
teth from its Center, and groweth farther off in
proportion as it turneth about, as E E.

The Cord, or Subtendant is a right Line,
which is join'd to an Arch, or Bow by its ends,
as G H.

The Arch, or Bow, is a Part of a Circumfe-
rence, as G I H. A Line Tangent, is that which
toucheth any Figure, without dividing it, and
without being able to divide, or paſs through it,
although it were prolonged, as L M.

See the Duce of Diamonds.

A

[ 5 ]

A Line Secant, which Crofteth, which divideth,
or traverfeth, as L O. M O.

The Definition of the ANGLE.

AN Angle, is the indirect meeting of two
Lines, at one and the fame Point, or ra-
ther it is the lpace encompafled between
the indirect meeting, or concourse of two lines,
joining together in one Point, as A B C. When
as this Concourse is made of two right Lines, the
Angle is called Rectilinear, and when it is made
of two crooked Lines, it is called Curvi-linear ;
but when it is made of one right Line, and one
crooked Line, it is called Mixtilinear.

A. The Angle Rectilinear.

B. The Angle Curti-linear.

C. The Angle Mixt-linear, or Composite.

The Angle Rectilinear, according as it is more
or lefs open, it receiveth particular Denomina-
tions; as Right, Sharp, Obtufe, or Blunt : So
that theſe Terms of Rectilinear, Curvi-linear,
and Mixt, are in reſpect of the Quality of the
Lines, and theſe of Right, Sharp and Obtufe, are
in reſpect of the quantity of the ſpace incloſed
within the ſaid Lines.

It is a right Angle, when one of the Lines is
Perpendicular upon the other, as E D F.

See the Tray of Diamonds.

B 3

The

[ 6 ]

The Angle is sharp, when as it is less open than the right Angle, as E DG.

The Angle is Obtuse, when it is more open than the Right Angle, as FDG.

The Letter D in the midst sheweth the Angle.
Here endeth the Tray of Diamonds.

## The Definition of the Superficies.

The Superficies is that which hath length, and breadth without depth.

According to the Geometricians, the Superficies is a Production of the flowing forth of the Line, as the Line is a Production of the Point: And thus we must conceive, that the Line E F. flowing towards G H. doth make the Superficies E F G H. which is an extension bounded with Lines, which hath nothing but length and breadth without any depth or thickness, which is called the Surface, or Figure, if one consider it in respect of its extremities, which are the Lines that enclose it.

If the Superficies be on the upper part, it is called a Convex, if it be in the inner, or hollow part, it is called a Concave, if it be plane and united, it is called a Plane.

B. A Convex Superficies.

C. A Concave Superficies.

A. A Plane Superficies.

See the Four of Diamonds.

The

[ 7 ]

The first Part teacheth only the Construction,
a framing of plain Superficies.

A Term, or Bound, is the extremity of any
thing. The Point, is the Term, or Bound of the
Line. The Line, is the Term of the Superficies.
And the Superficies, is the Term, or Bound, of
a Body.
Here endeth the Four of Diamonds.

Of Superficies, or Figures Rectilinear.

The Superficies do take particular Names, ac-
cording to the Number of their Sides, as

A. A Trigone, or Triangle, a Figure of 3 sides.

B. A Tetragone, or Square, a Figure of 4 sides.

C. A Pentagone, or Figure of 5 sides.

D. An Exagone, or Figure of 6 sides.

E. Eptagone, a Figure of 7 sides.

F. Octogone, a Figure of 8 sides.

G. Enneagone, a Figure of 9 sides.

H. Decagone, a Figure of 10 sides.

I. Endecagone, a Figure of 11 sides.

L. Dodecagone, a figure of 12 sides.

All these Figures are called likewise by one general
Name Poligones.

Of TRIANGLES.

The Triangles are also distinguished by the quality of
their Angles, and by the disposition of their sides, as

M. A Triangle Rectangle, which hath a Right
Angle.

See the Five of Diamouds.

B 4

N. A.

[ 8 ]

N. A Triangle Ambligone, which hath an Obтуſe Angle.

O. A Triangle Oxigone, which hath Three Angles ſharp.

P. A Triangle Equilateral, which hath its Three ſides equal.

Q. A Triangle Iſo ſele, which hath two ſides equal only.

R. A Triangle Scalene, which hath his three ſides unequal.

Here endeth the Five of Diamonds.

Of Figures of Four Sides.

A. T he Square, is a Figure compoſed of four equal Sides, and four right Angles.

B. A Long Square, is a Superncies Rectangle, that is to ſay, which hath its Angles right, but hath not its Sides equal.

A B C. A Parallelo-gramme, is a Square-fide figure, whereof the oppoſite ſides are Parallels.

D. A Rhombus, or Lozange, is a ſquare fide figure, which hath the four ſides equal, but not the four Angles.

E. A Rhomboid, which hath the Angles and the fide oppoſite equal, without being equal-an-gled, or equal-fided.

F. A Trapeze, which hath only two ſides oppoſite Parallel, and the other two equal.

See the Six of Diamonds.

G. A

[ 9 ]

G. A Trapezoide, or Tablett, which hath its Sides and its Angles unequal.

H. A Gnomon, is the excefis of a Parallelo-gram
upon another Parallo-gram, framed upon the
fame Diagonal.

*All other Figures, of more than four Sides, are cal-
led by one general Name, Multi-lateres, or,
Many-Sizes.*

*Here endeth the Six of Diamonds.*

*Of Figures Crooked, or Curvi-linear.*

A. A Circle, is a Superficies, or Figure, perfect-
ly round, defcribed, or drawn from a
Center, from which the whole Circumference
is of equal diſtance.

A B C D. The Circumference is the Extremity,
or outmoſt part of the Circle, otherwife it is
the Circular-line that encloſeth it.

B. An Oval, is a crooked Figure drawn from many Centers, and which all the Diameters divide into two equally.

C. An Eclipſe, is alſo a crooked Figure drawn
from many Centers, but in ſhape of an Egg,
within the which there is but one only Dia-
meter, which divideth it into two equally.

D. A Volute, is a Figure, or Superficies, encom-
pafled by a Line Spiral.

*See the Seven of Diamonds.*

Of

[ 10 ]

Of FIGURES Composite.

A. A Demi-Circle, is a Figure contained in the Diameter, with the half of the Circumference.

B. A part of a Circle, is a Figure contained within a right Line, and a part of a Circle.

F. The great Portion of the Circle, is that which containeth more than the half of the Circle.

G. The small Portion of a Circle, is that which containeth less than the half of the Circle.

C. A Sector, is a Figure contained within two Semi-Diameters, with more, or less, than the half of the Circle.

There is likewise the great and the small Sector.

D. Figures Concentrical, are those which have one and the same Center.

E. Figures Excentrical, are those which are contained in others of divers Centers.

Here endeth the Seven of Diamonds.

Of Figures Regular and Irregular.

A. A Figure Regular, it that which hath its opposite parts like an Equal.

B. An Irregular Figure, is that which is composed of Angles and Sides unlike.

E E. Figures a-like, are those whereof all the Parts of one, are proportionable to all the See the Eight of Diamonds.

Parts

[ 15 ]

## PROPOSITION I.

To elevate a Perpendicular from a Point pro-
pounded within the midft of a straight
Line.

The Position.

Let C be the Point propounded within the
midft of the Line A B, from which a Perpen-
dicular muſt be elevated.

The Practice.

From the Point given C. draw at pleafure
the half-circle D E, from the Points D and E.
Make the Section I. from the Point C. Draw
the right Line demanded C O. by Section I.
This Line C O. ſhall be Perpendicular to the
Line given A B. and elevated from the Point
propounded C.

## PROPOSITION II.

To elevate a Perpendicular at the end of a right
Line propounded.

Let A be the end propounded of the Line
A B. upon the which a Perpendicular muſt be
elevated.

The Practice.

Set at pleafure the Point C. above the Line
A B. From this Point C. and the Interval C A.
draw the portion of the Circle E A D. Bring
the right Line D C E. by the Points D and C.
Draw the Line demanded A E. it ſhall be Per-
[ See the King of Diamonds.

pen-

[ 16 ]

pendicular to A B. and to the End propounded A:

*Otherwise.*

<table><tr><td>From the Point A, draw the Arch</td><td>G H M.</td></tr><tr><td>From the Point G, draw the Arch</td><td>A H.</td></tr><tr><td>From the Point H, draw the Arch</td><td>A M N.</td></tr><tr><td>From the Point M, draw the Arch</td><td>H N.</td></tr><tr><td>Draw the Line demanded</td><td>A N.</td></tr></table>

[ Here endeth the King of Diamonds:

## PROPOSITION III.

Upon an Angle given, to elevate a right Line, which inclineth neither to the right hand, nor to the left.

*The Position.*

Let B A C be the Angle, upon the which a right Line must be elevated, which inclineth neither to the right hand, nor left.

*The Practice.*

From the Angle given A. draw at pleafure the Arch B C. from the Points, or Ends B and C. Make the Section D. from the Point, or Angle given A. Draw the right Line required A D. by the Section D. This right Line A D. shall be elevated upon the Angle B A C. without inclining either to the right hand, or to the left.

[ See the Queen of Diamonds.

[ 17 ]

## PROPOSITION IV.

To depreſs, or bring down a Perpendicular-Line up-
on a right Line given, and from a Point without
the ſame.

The Position.

Let C. be the Point given, from which a Per-
pendicular Line is to be brought down upon
the Line A B.

The Practice.

From the Point given C. draw, at pleafure,
the Arch D E. cutting the Line A B. at the
Points D and E. From the Points D and E.
make the Section F. Draw the Line C F. The
Line C O. will be the Line Demanded.

## PROPOSITION V.

By a Point given to draw a Line Parallel to a right
Line given.

The Position.

Let A be the Point, by the which we muſt
draw a Line, which may be Parallel to the
Line B C.

The Practice.

Draw at pleafure the oblique Line A D. from
the Point A. Draw the Arch D E. from the
Point D. Draw the Arch A F. Make the Arch
D G. equal to the Arch A F. Bring the Line
demanded M N. by the Points A and G.

See the Queen of Diamonds.

C

Other-

[ 18 ]

Otherwise.

From the Point A. draw the Arch E F G,
touching the Line B C.

*Without changing the opening of the Compasses.*

From the Point H. draw the Arch L R I.

*The Point H. is placed at pleasure within the line BC.*

Draw the Line demanded O P. by the Point A.
and grating upon the Arch L R I.

*Here endeth the Queen of Diamonds.*

## PROPOSITION VI.

*To cut a right Line given, and bounded into two
equally.*

*The Position.*

Let A B. be the right Line propounded, to
be cut into two equally.

*The Practice.*

From the point, or end A. draw the Arch
C D.

From the point, or end B. draw the Arch
E F.

*These two Arches must divide each other.*

Draw the right Line G H. by the Sections
G and H. A B shall be divided into two equal-
ly, at the Point O.

*See the Knave of Diamonds.*

PROPO-

[ 19 ]

## PROPOSITION VII.

To cut an Angle Rectilinear given into two equally.

The Position.

Let B A C. be the Angle propounded, to be
cut into two equally.

The Practice.

From the Angle A. draw, at pleafure, the
Arch D E. From the Points D and E. make
the Section O. Draw the Line A O.

This Line A O. shall divide the Angle gi-
ven B A C. into two equally.

Here endeth the Knave of Diamonds.

## PROPOSITION VIII.

At the end of a right Line to make an Angle Re-
ctilinear, equal to an Angle Rectilinear proposed.

The Position.

Let A. be the end of the Line A B. to which
we must make an Angle, equal to the Angle
Rectilinear given C D G.

The Practice.

From the Angle D. draw, at pleafure, the
Arch C G.

From the Point, or End A. draw the Arch
H O. Make the Arch H E. equal to the Arch
C G. Draw the Line A E. The Angle B A E.

See the Ace of Harts.

C 2

shall

[ 20 ]

shall be equal to the Angle CD G, the which
was propounded to be done.

## PROPOSITION IX.

To divide a straight Line given, into as many equal Parts as one will.

The Position.

Let A B be the Line proposed to be divided
into Six equal Parts.

The Practice.

From the end A. draw, at discretion, the
Line A C. From the end B. draw the Line B
D. parallel to the Line A C. from the Points A
and B. and upon the Lines A C B D. Bring,
at discretion, fix equal Parts, viz. E F G H I L.
upon the Line A C. R Q P O N M upon the
Line B D. Draw the Lines E N. F O. G P.
H Q. L R. The Line A B. shall be divided in-
to fix equal Parts, by the Sections S. T. V. X. Y.

## PROPOSITION X.

From a Point given to draw a straight Line, which
toucheth a Circle propounded.

The Position.

Let A. be the Point, from which we must
draw a Line which toucheth the Circle D O P.

The Practice.

From the Center of the Circle B. draw the
Line Secant B A. Divide this Line B A. into
See the Ace of Harts.

two

[ 21 ]

two equally in C. from the point C. and the
Interval C A. Draw the Demicircle ADB.
cutting the Circle in D. from the point given
A. Draw the right Line AE by the point D.
This right Line A E. shall be the Line touching
the required.
[Here endeth the Ace of Harts,

## PROPOSITION XI.

To draw a right Line, which toucheth a Circle at a Point propounded.

The Pofition.

Let ABC. be the Circle given, within the
Circumference of which, is the Point A. propounded.

The Practice.

From the Point, or Center D. draw the
Line D F. by the Point propounded A. at the
Point propounded A. and upon the Line D F.
Draw the Perpendicular A H. prolonged towards I.
This Line Tangent H I. shall touch
the Circle at the Point propounded A, the
which is required by the Propofition.

## PROPOSITION XII.

A Circle being given, and a straight Line that toucheth it, to find the Point where it toucheth.

The Pofition.

Let ABC. be the Circle touched by the
Line G H. we must find the Point where it
toucheth,
See the Duce of Harts.

C 3

The

[ 22 ]

The Practice,

From the Center of the Circle F. let down
the Perpendicular F C. upon the Line touching
D E. The Section C, shall be the Point of
Touching demanded.

PROPOSITION XIII.

To describe a Line Spiral upon a straight Line given,

The Position.

Let I L. be the Line, upon which we would
describe a Line Spiral.

The Practice.

Page 18. Provide the half of the Line I L. into as many equal Parts, as you would describe the Revolution.

Example.

If you would divide it into Four.

Divide the half B I. into four equal parts
B C E G I. Divide also B C. into two
Page 12. equally at A. From the Point A.
draw the Demi-circles B C. D E.
F G. H I, From the point B, draw the Demi-
circles C D. E F. G H. I L, and you shall have
the Spiral Line required.

Here endeth the Duce of Harts,

PRO-

[ 23 ]

## PROPOSITION XIV.

Between the Points given, to find two other directly interpoſed.

### The Pofition.

Let A and B. be the Points given, between the which we muſt find two other Points di-rectly interpoſed, by the means whereof we may draw a ſtraight Line from the Point A. to the point B. with a ſhort Ruler,

### The Practice.

From the Points A and B. make the Sections C and D. From the Points C and D. make the Sections G and H. The Points G and H. ſhall be the demanded; by the means of which, one may draw three ways a right Line from the point A. to the point B. the which could not be done in one, with a Ruler which ſhould be ſhorter than the ſpace between A and A.

The Second Part of the Conſtruction of Plain Figures.

## PROPOSITION I.

To frame a Triangle Equilateral upon a right Line given, and bounded.

### The Pofition.

Let A B. be the Line given, upon the which we muſt frame a Triangle Equilateral.

### The Practice.

From the end A. and the Interval A B. deſcribe the Arch B D. from the end B. and the See the Tray of Harts.

C 4

Inter-

[ 24 ]

Interval B A. Describe the Arch A E. from the
Section C. Draw the Lines C A. C B.

A B C shall be the Triangle Equilateral de-
manded.

## PROPOSITION II.

To make a Triangle of three straight Lines, equal to
three straight Lines given.

The Position.

Let A B C, be the three Lines given, we
must make a Triangle of three right Lines equal
to them.

The Practice.

Draw the right Line D E. equal to the Line
A A. from the Point D. and from the Interval
B B. Describe the Arch G F. from the point E.
and the Interval C C. Describe the Arch H I.
from the Section O. Draw the Lines O E, O D.
The Triangle D E O. shall be comprifed of
three right Lines, equal to the three right
Lines given A A, B B, C C.

[ Here endeth the Tray of Harts,

## PROPOSITION III.

To frame a Square upon one right Line given, and
bounded.

The Position.

Let A B be the right Line given and boun-
ded, upon the which we must frame a Square.

See the Four of Harts.

The

[ 25 ]

The Practice.

Elevate the Perpendicular A C,
from the point A. Describe the Arch
B C. from the Points B and C. and
from the Interval A B. Make the Section D.
from the point D. Draw the Lines D C, D B.
A B C D shall be the Square demanded, framed
upon the right Line given A B.

PROPOSITION IV.

To frame a Pentagone Regular upon a right Line
given.

The Position.

Let A B be the Line given, upon the which
we must frame a Pentagone.

The Practice.

From the end A. and from the Interval A B.
Describe the Arch B D F. Elevate the Perpen-
dicular A C. Divide the Arch B C. into five e-
qual Parts I D L M. Draw the right line A D.
Cut the Bases A B. into two equally in O.
Elevate the Perpendicular O E. from the Secti-
on E. and from the Interval E A. Describe the
Circle A B F G H. Bring five times the Line
A B. within the circumference of the Circle,
and you shall have a Pentagone Regular, Equi-
angle, Equilateral A B F G H.

See the Four of Harts.

PRO-

[ 26 ]

## PROPOSITION V.

To frame an Exagone Regular upon a right Line given.

### The Position.

Let A B. be the right Line, upon the which we must frame an Exagone.

### The Practice.

From the end A and B. and from the Interval A B. Describe the Arches A C, B C. from the Section C. Describe the Circle A B E F G. Bring Six times the Line given A B, within the circumference, and you shall have an Exagone Regular A B E F G D. framed upon the Line given A B.

[Here endeth the Four of Harts.]

## PROPOSITION VI.

Upon a right Line given, to describe such a Poligone, as you would have from the Exagone unto the Dodecagone.

### The Position.

Let A B be the Line, upon the which that must frame an Exagone, or an Eptagone, or an Octogone, &c.

### The Practice.

Page 10. Cut the Line A B. into two equal parts. In O. Elevate the Perpendicular OI. From the Point B. describe the Arch A C. Dividing the Five of Harts.

vide

[ 27 ]

vide A C. into fix equal parts M N. P Q R.
this may make an Eptagone if you will. From
the Point C. and the Interval of one part C M.
describe the Arch M D. D shall be the Cen-
ter to describe a Circle, capable of containing
seven times the Line A B.

If you would make an Octogone.

From the point C. and the Interval of two
parts C N. describe the Arch N E. E. shall be
the Center, to describe a Circle capable of
containing eight times the Line A B,

If you would make an Enneagone.

You must take the three parts C P.

And so likewise of others, always augmenting it by
one part.

[ Here endeth the Five of Harts.

## PROPOSITION VII.

Upon a right Line given to frame such a Poli-
gone, as one would have from 12 to 24 sides,

The Position.

Let A B. be the Line, upon the which one
would frame any Poligone.

The Practice.

Divide the Arch A C. into twelve equal
parts, from the point C. Take as many parts
upon C A. as there is need above the Twelve,
to have as many parts as one requireth of the
Sides.

[See the Six of Harts.

Example

[ 28 ]

Example.

If you make a Figure of fifteen Sides.

From the point C. and from the Interval of
three parts C E. describe the Arch E O.

AC of 12, CO of 3, will make together 15.

From the point O, and the Interval O B.
describe the Arch B E. From the point F. and
the Interval F A. describe a Circumference, it
shall contain fifteen times the Line given A B.

And so of other Poligones.

[ Here endeth the Six of Harts.

PROPOSITION VIII.

Upon a right Line given, to describe a Portion of a
Circle, capable of an Angle equal to an Angle
given.

The Position.

Let A B. be a Line, bounded upon that
which one would make a Portion of a Circle,
capable of containing an Angle equal to the
Angle given C.

The Practice.

Page 16. Make the Angle B A D. equal to
Page 2. the Angle C. Elevate upon A D. the
Page 2, Perpendicular A E. Divide the Line
A B. into two equally in H. Elevate
the Perpendicular H E. from the Section F.
and from the Interval F A. Describe the Por-
tion of the Circle A E B. All the Angles that
See the Seven of Harts.

you

[ 29 ]

you shall make within the Portion of a Circle,
and upon the Line given A B, shall be all equal
to the Angle C.

## PROPOSITION IX.

*To find the Center of a Circle given.*

*The Position.*

Let A B C. be a Circle propounded, where-
of we must find the Center.

*The Practice.*

Draw, at discretion, the right Line A B.
bounding it self at the Circumference B C.
Cut this right Line A B. into two by the Line
D C, Cut also this right Line C D. into two
equally in F. The point F. shall be the Center
demanded of the Circle A B C.

[ Here endeth the Seven of Harts. ]

## PROPOSITION X.

*To finish a Circumference begun, whereof the Center is lost.*

*The Position.*

Let A B C. be the part of the Circumference
given, we must find the Center, that we may
finish it.

*The Practice.*

Place, at discretion, the three Points A B C.
within the circumference begun. From the
See the Eight of Harts.

Points

[ 3ο ]

Points A and B. make the Sections E and F.
Draw the right Line E F. from the Points B and
C. Make the Sections G and H. Draw the right
Line G H. From the Interfection and Center I.
and from the Interval I A. finish the Circum-
ference began.

## PROPOSITION XI.

To describe a Circumference by three Points given.

The Position.

Let ABC. be the three Points, by the which
one would pass a Circumference.

The Practice.

From the Points given A B C. describe the
three Circles D E H. D E F. F G L. of the
fame Interval, dividing each other in the points
D and E, F and G. Draw the right Lines D
E, F G. until they meet each other in I.
From the point I. and the Interval I A. de-
scribe the Circumference required.

This Practice is like the former.

[ Here endeth the Eight of Harts.

## PROPOSITION XII.

To describe an Oval upon a Length given.

The Position.

Let A B. be the Length, upon the which
we must frame an Oval.

See the Nine of Harts.

The

[ 31 ]

The Practice.

Divide the length given A B. into
Page 18. three parts equal A C D B. from the
points C and D. and from the Inter-
val C A. Describe the Circles A E F, B E F.
from the Sections E and F. and from the In-
terval of the Diameter E H. Describe the Ar-
ches I H, O P. A I H B P O. shall be the Oval
required.

PROPOSITION XIII.

*To describe an Oval upon two Diameters given.*

*The Position.*

A B, C D. are the Diameters, upon the which
we must frame an Oval.

The Practice.

Make the Rule M O. equal to the great half
Diameter A E. upon the which you shall make
the length M N. equal to the half Diameter
C E.

*This Rule being so ordered.*

Place it so upon the Demands A B. C D.
That the point N. sliding upon the line A B.
the end O. may never leave the line C D. run-
ning on so the said Rule M O. Describe the
Oval by the end M.

See the Nine of Harts,

PRO-

[ 32 ]

PROPOSITION XIV.

To find the Center, and the two Diameters of an
Oval.

The Position.

Let ABCD. be the Oval propounded,
whereof we must find the Centers, and the
Diameters.

The Practice.

Within the Oval propounded ABCD. draw
at discretion, the Lines parallels A N.
pgae 10. H I. Cut these Lines AN. H I. into
two equally in L and M. Draw the Line, PL
MO. Cut it into two equally in E. and the
point E. shall be already the Center from the
Point E. Describe, at discretion, the Circle F
HQ. cutting the Oval in F and G. from the
Sections F and G. Draw the right Line F G.
Cut it into two equally in R. Draw the great
Diameter B D. by the Points E R. from the
Center E. Draw the les Diameter
A E C. parallel to the Line F G.
This is it which was propounded.

Here endeth the Nine of Harts.

PROPO-

[ 33 ]

## PROPOSITION XV.

To frame a Figure Rectilinear upon a right line bounded, like unto a Figure Rectilinear Proportioned.

Let A B. be the line upon the which we must frame a Figure like to the Figure C D E F.

*The Practice.*

Draw the Diagonal C E. make the Angle B A G. equal to the Angle F C E. make the Angle A B G. equal to the Angle C F E. the triangle A B G. shall be like to the Triangle C F E.

*The same.*

Make the Triangle A G H. like to the Triangle C E D, the whole Figure A B G H. shall be like to the whole Figure C D E F.

*See the Ten of Hearts.*

## PROPOSITION XVI.

Upon a right line Propounded to frame two Rectangles according to one Reason given,

A B is the line upon which we must frame two Rectangles, which may be between them, according to the reason of C to D.

*The Practice.*

Cut the line A B. at the point E according to the reason or proportion of C to D. make the square A B, H F. draw the line E I. parallel to the line A F. BE IH, A E I F shall be the rectangle D

[ 34 ]

angle required; the Rectangle A I. is to the
Rectangle E. H. as the line D. is to the line C.
*See the Ten of Hearts.*

## PROPOSITION I.

*Within a Circle given to inscribe a Triangle Equi-lateral, an Exagone and a Dodecagone.*

Let A BG. be the Circle, within which we
must inscribe a Triangle Equilateral, &c.
*The Practice.*

*Of the Triangle Equilateral.*

From the point, as A. and from the interval
of the half Diameter A B. describe the Arch
CBD. draw the right line DC. bring this
Interval CD. from the point C. to the point
F. Draw the lines FC, FD. CDF shall be
the Triangle required.
*Of the Exagone.*

Bring fix times the half Diameter A B. with-
in the circumference given.
*Of the Dodecagone.*

Cut the Arch of the Exagone A C. into
two equally in O. A O shall be the side of the
*Dodecagone*.

*Here ended the Ten of Hearts.*

## PROPOSITION II.

*Within a Circle given to inscribe a Square, and an Octogone.*

Let ABCD be the Circle, within the
which, one would inscribe a Square and an
*Octogone*.

*The*

[ 35 ]

### The Practice. Of the Square.

Draw the two Diameters A B C D. dividing each other at Right Angles, that is to say, draw the right line C D. by the center of the Circle O. from the Points or Ends C and D. Make the Sections I and L. draw the right line I L. passing also by the Center O. the Lines or Diameters A B, C D. shall divide themselves at right Angles being the lines A C, A D, B C, B D. And A C, B D. shall be the Square required.

#### Of the Octogone.

Subdivide every fourth of the Circle, you shall make the Octogone.

See the King of Hearts.

## PROPOSITION III.

Within a Circle given to inscribe a Pentagone and a Decagone.

Let A B, C D be the Circle propounded.

### The Practice. Of the Pentagone.

Draw the two Diameters A B C D. dividing themselves at right Angles in E. Divide the half Diameters C E. into two equally in F. from the point F. and from the Interval F A. Describe the Arch A G. from the point A. and from the Interval A G. describe the Arch G H. The right line A H. shall divide the circle into five equal parts.

D 2

Of

[ 36 ]

Of the Decagone.

Subdivide every part of the circle into two
equally.

*Here endeth the King of Hearts.*

## PROPOSITION IV.

*Within a Circle given to inscribe an Eptagone.*

Let ABC be the circle propounded, with-
in the which we must make an Eptagone.

*The Practice.*

Draw the half Diameter I A. from the end
A. and from the Interval A I. describe the
arch CIC. draw the right line C C. bear the half
CO. seven times within the circumference of
the circle, you shall have the *Eptagone* required.

*See the Queen of Hearts.*

## PROPOSITION V.

*Within a Circle given to describe an Enneagone.*

Let BCD be a circle propounded, within
which one would inscribe an Enneagone.

*The Practice.*

Draw the half Diameter A B. from the end
B. and from the Interval B A. Describe the
Arch C A D. draw the right line C D. en-
larged towards F. make the line E F. equal to
the line A B. from the point E. Describe the
Arch F G. from the Point F. Describe the
Arch E G. draw the right line A G. D H,
shall be the Ninth part of the circumference.

*Here endeth the Queen of Hearts.*

PRO

[ 37 ]

## PROPOSITION VI.

*Within a Circle given to inscribe an Endecagone.*

Let AEF be the Circle given, within which we must inscribe an Endecagone.

*The Practice.*

Draw the half Diameter A B. divide the half Diameter A B. into two equally in C. from the points A and C. and from the Interval A C. Describe the Arches CDI, A D. from the point I. and from the Interval ID. Describe the Arch DO. the Interval CO. shall be the side of the *Endecagone*, required very punctually.

*See the Knave of Hearts.*

## PROPOSITION VII.

*Within the Circle given to inscribe such a Poligone as one would.*

*The Practice.*

Draw the Diameter A B. describe the circle A BF. capable to contain seven times A B. as if you would frame upon A B. a Poligone like to that which you should inscribe within the circle given A B C. Draw the Diameter D E. parallel to the Diameter A B. draw the right lines DAG, FBH. by the ends DA, EB. GH shall divide the circle given ABC. into seven equal parts: And so of all other Poligones.

*Here endeth the Knave of Hearts.*

D 3 PRO-

[ 38 ]

## PROPOSITION VIII.

From a Circle given, to take a Portion
Capable of an Angle, equal to an
Angle Reetilinear propounded.

Let A C E be the Circle given from which
we must take a Portion capable to contain an
Angle equal to the Angle D.

The Practice.

Draw the half Diameter A B. bring the line
touching A F. make the Angle F A C. equal
to the Angle given D. All the Angles which
shall be framed upon the line A C. and with-
in the portion AEC. shall be all equal to the
Angle given D. for the portion AEC. is the
required.

See the Ace of Spades.

## PROPOSITION IX.

Within a Circle to inscribe a Triangle of equal
Angles to a Triangle given.

Let ABC be the Circle within the which
we must inscribe a Triangle like to the Tri-
angle DEF.

The Practice.

Bring the line touching G H. from the point
of the touching A. make the Angle H A C.
equal to the Angle E. make also the Angle G.
A B. equal to the Angle D. Draw the line B C.
ABC is the Triangle required like to the
Triangle given DEF.

See the Ace of Spades.

PRO-

[ 39 ]

## PROPOSITION X.

*To inscribe a Circle within a Triangle given.*

Let ABC be the Triangle, within the
which we must inscribe a circle.

*The Practice.*

Divide the two Angles B and C. each into two equally, by the right lines B D, C D. From the Section D. bring down the perpendicular D F. From the Section or Center D. and from the Interval D F. describe the circle demanded E F G.

*Here endeth the Ace of Spades.*

## PROPOSITION XI.

*To inscribe a Square within a Triangle given.*

Let ABC be the Triangle, within the
which we must inscribe a Square.

*The Practice.*

Elevate the perpendicular A D. at the end
of the Basis A B. make this perpendicular A D. equal to the basis A B. From the Angle C. draw the line C E. parallel to the line A D. Bring the oblique line D E. from the section F. draw the line F G. parallel to the basis A B. draw the lines F H, G I. parallel to the line C E, F G, H I shall be the Square required.

*See the Deux of Spades.*

D 4

PRO-

[ 40 ]

## PROPOSITION XII.

*To inscribe a Pentagone Regular within a Triangle Equilateral.*

Let ABC be the Triangle, within the which
one would inscribe a Pentagone.

*The Practice.*

Bring down the Perpendicular A I. from
the center A. describe the Arch BIM. divide
into five equal parts the Arch B I. bring the
sixth I M. draw the line A M. divide A M.
into two equally in L. from the point A. de-
scribe the Arch L D. draw the right line L
D, unto H. Make the part A G. equal to the
part B H. draw the right line DG, MC.
from the center D. and from the Interval of
the section N. describe the Arch NO. from
the points N and O. describe the Arches DQ,
DP. draw the lines OP, PO, NO, DOP
QN shall be the Pentagone required.

*See the Deux of Spades.*

## PROPOSITION XIII.

*To inscribe a Triangle Equilateral within a square.*

Let ABCD be the square, within the
which we must make a Triangle Equilateral.

*The Practice.*

Draw the Diagonals AC, BD. from the
Center E. and from the Interval E A. describe
the circle A B C D. from the point C. and the
Interval C E. Describe the Arch GEF. draw

the

[ 41 ]

the right lines A F, A G. bring the right line
HI. A H L shall be the Triangle Equilateral
required.

Here endeth the Deux of Spades.

## PROPOSITION XIV.

To inscribe a Triangle Equilateral within a
Pentagone.

Let ABCDE be the Pentagone, within
the which we must inscribe a Triangle Equi-
lateral.

The Practice.

Circumscribe the circle A B C D E.
from the point A. and from the Inter-
val of the half Diameter A F. describe
the Arch F L. divide this Arch F L. into two
equally in N. draw the line A N I. from the
point A. and from the Interval A I. describe
the Arch I O H. draw the lines A H, H I. A
H I shall be the Triangle demanded.

See the Trois of Spades.

## PROPOSITION XV.

To inscribe a Square within a Pentagone.

Let ABCDE be the Pentagone, within
the which we must inscribe a square.

The Practice.

Draw the right line B E. let down the Per-
pendicular E T. at the end of B E. make this
Perpendicular E T. equal to the line B E. draw
the line E T. from the section O. bring the
line

[ 42 ]

line O P. parallel to the side C D. at the end
O P. Elevate the perpendiculars O M, P I. draw
the line N M. N M O P shall be the square
required.

Here endeth the Trois of Spades.

## PROPOSITION I.

*About a Triangle given to Circumscribe a Circle.*

Let ABC be the Triangle, about the which
one would circumscribe a Circle.

*The Practice.*

Describe the circumference A B C. by the
three Points A, B, C. and you shall have the
demanded.

See the Four of Spades.

## PROPOSITION II.

*About a Square to Circumscribe a Circle.*

Let ABCD be the Square about which we
must circumscribe a Circle.

*The Practice.*

Draw the two Diagonals A B C D. from
the intersection or Center G. and from the Interval
G A. describe the circle demanded A B C D.

See the Four of Spades.

## PROPOSITION III.

*About a Circle to circumscribe a Triangle of equal
Angles to a Triangle given.*

Let DEV be the Circle, about the which
we must make a Triangle which may be like
to the Triangle F G H.

*The*

[ 43 ]

The Practice.

Draw the Diameter A B. by the center C.
make the Angle A CE. equal to the Angle H.
make the Angle B C D. equal to the Angle G.
prolong the lines E C, D C. towards R and S.
draw the line Tangent N O. parallel to the
line D R. draw the line Tangent O I. parallel
to the line E S. draw also the line Touchant
N I. parallel to the Diameter A B. INO shall
be the Triangle demanded, like to the Tri-ngle F G H, circumscribed about the circle,
DEV.

Here endeth the Four of Spades.

## PROPOSITION IV.

About a Circle to circumscribe a Square.

Let ABCD be the circle, about the which
we must describe a Square.

The Practice.

Draw the Diameters A B, C D. dividing
themselves at right Angles in O. From the points
A, C, B, D, and from the Interval A O. de-
scribe the Demicircles H OG, H OE, EOF,
FOG, draw the right lines EF, FG, GH,
HE, by the Section E, F, G, H. E F G H
shall be the square demanded.

See the Five of Spades.

PRO-

[ 44 ]

## PROPOSITION V.

*About a Circle given to circumscribe a Pentagone.*
*The Practice.*

Let ABCDE be the Circle given, about the which one would describe a Pentagone.

Inscribe the Pentagone ABCDE. from the center F. and by the midline of each of the sides draw the lines FO, FP, FQ, FR, FS. bring the line FA. draw the line Tangent PQ. by the point A. from the center F. and from the Interval F. P. describe the circle OPQRS. draw the sides of the Pentagone, demanded by the Section OPQRS.

*See the Five of Spades.*

## PROPOSITION VI.

*About a Poligone Regular to circumscribe the same Poligone.*

Let BCDEFG be the Poligone given, about the which we must circumscribe another Poligone like.

*The Practice.*

Prolong two sides as BGEF. unto the point of the meeting H. draw the line AH. draw the line FI. dividing the Angle GFH. into two equally from the center A. and from the Interval AI. describe the circle IMO. draw the Rays AL, AM, AN, AO; by the midline of each side, draw the sides of the outward Poligone demanded by the Sections IL, MNOP. *See the Five of Spades.*

PRO-

[ 45 ]

## PROPOSITION VII.

About a Triangle Equilateral to Circumframe
a Square.

Let ABC be a Triangle Equilateral, about
the which we must circumscribe a Square.

The Practice.

Divide the Basis B C. into two equally in E.
prolong this Basis B C. the one part and the
other towards D and D. Make the Lines E
D, E D. equal to the line E A. From the point
E. and from the Interval E C. describe the De-
my-circle B F C. draw the line A E F. from
the point F. draw the lines F C G, F B G.
A G F G shall be the Square demanded.

Here endeth the Five of Spades.

## PROPOSITION VIII.

About a Triangle given Equilateral to circumscribe
a Pentagone.

Let ABC be the Triangle given, about the
which we must describe a pentagone.

The Practice.

From the Point or Angle A B C. and with
the same opening of the Compasses, describe
at discretion the Arch DE, L P. divide the
Arch D O. into five equal parts, 1 2 3 4 5.
From the center or section O. and from the
Interval of 4 parts O N. describe the Arch N
M E. draw the right line A E F. divide the
Arch M P. equal to the Arch E N. draw the
right line F P C G. equal to the line F A.

make

[ 46 ]

make the arch D H. equal to the Arch D E.
draw the fides A I, I R. equal to the fides A F,
F G. the fide G R shall finish the Pentagone
demanded.

See the Sixth of Spades.

## PROPOSITION IX.

*About a Square to circumscribe a Triangle of equal Angles to a Triangle given.*

Let DEFG be the Square, about the
which we must circumscribe a Triangle like
to the Triangle ABC.

*The Practice.*

Make the Angle EFM equal to the Angle
A. make the Angle MEF equal to the Angle
B. prolong the lines ME, MF, DG, towards
I and H. MIH shall be the Triangle required,
like to the Triangle ABC. and circumscribe
about the square given DEFG.

*Here endeth the Six of Spades.*

## PROPOSITION X.

*About a Square to circumscribe a Pentagone.*

Let ABCD be the Square, about the which
we must circumscribe a Pentagone.

Prolong the fide C B. towards N. divide
the fide A B. into two equally in R. Elevate the
Perpendicular R V. from the points B, D, C.
and from the fame Interval B R. divide the
Arches R N, S T, ST. divide the Arch R N.
into five equal parts R H, G F E N. make the
Angle R BV. from the opening of two parts
R G. make the Angles S C T, S D T. from the
opening

[ 47 ]

opening of one part R H. prolong the lines
V B, C T in O. make the line O Q. equal to
the line O V. draw the other fides on the fame
manner, and you shall have the demanded.

See the Seven of Spades.

## PROPOSITION I.

To find a line which may be a mean proportional
between two others.

Let A and B be the lines, between the which
we must find a third which may be proportional to them.

### The Practice.

Draw a line Undeterminate G H. make C
E. equal to the line A. make E D. equal to the
line B. divide C D. into two equally in I. from
the point I. and from the Interval I C. describe
the Demy-circle C F D. elevate the Perpendicular E F.
This line E F. shall be the mean proportional between A and B. according as it
is propounded.

Here endeth the Seven of Spades.

## PROPOSITION II.

There being given the sum of the Extreams, and
the mean proportional to discern the Ends and Extreams.

Let A B be the sum of the Ends, that is to
say, two Grandeurs or (Greatnesses the one
at the end of the other without distinction)
whereof the line C is the mean proportional,
and

[ 48 ]

and by the means of which we must find the
point where the Extreams or Ends do join
themselves.

The Practice.

Divide the sum or the line A B. into two equal parts in G. from the point G. and from the Interval G A. describe the Demy-circle AEB. Elevate the perpendicular B D. equal to the mean C. draw the line D E. parallel to the line A B. From the section E. draw the line E F. parallel to the line B D. F shall be the point where the Extreams join themselves, and so C or his equal E F. shall be the mean between the Extreams A F, and F B.

See the Eighth of Spades.

## PROPOSITION III.

There being given the mean of three Proportionals, and the difference of the Extreams (or Ends) to find the Extreams.

Let G H be the mean Proportional, and A B the difference of the Extreams, we must find the length of the Extreams.

The Practice.

Elevate the perpendicular B C. at the end of the difference A B. and equal to the mean G H. divide the difference A B. into two equal parts in D. prolong it towards E and F. from the point D. and from the interval D C. describe the Demy-circle E C F. B E, B F shall be the Extreams demanded.

Here endeth the Eight of Spades.

PRO-

[ 49 ]

## PROPOSITION IV.

From a right line given, to cut off a part, which
may be a mean proportional between the reft,
and another right line Propounded.

Let AA be the line, from the which we
must cut off a part, which may be the mean
proportional between the part that shall re-
main, and the line propofed B B.

### The Practice.

Draw the line Indeterminate C D. divide
the lines D E, E C. equal to the lines A A and
B B. describe the Demy-circle C F D. elevate
the perpendicular E F. divide the line C E.
into two equally in B. from the point B. and
from the interval B F. describe the Arch F G.
divide the part demanded A G. equal to the
part E G. A H shall be the mean proportional
between the reft H I. and the other line pro-
pounded B B.

See the Nine of Spades.

## PROPOSITION V.

There being given two right lines, to find a
Third Proportional.

A B, A C are the two right lines given, we
must find a third, which may be proportional
to them.

### The Practice.

Make at discretion the Angle D N E. di-
vide the part N H. equal to the line A B. di-
vide

E

vide

[ 50 ]

divide the part NO. equal to the line A C. divide also H D. equal to the line A C. bring the line H O. draw the line D E. parallel to the line H O. E O shall be third proportional demanded.

Here endeth the Nine of Spades.

## PROPOSITION VI.

To find a fourth Proportional.

A, B, C, are the three lines proposed, we must find a fourth which may be to the third, as the second to the first.

The Practice.

Make at discretion the Angle G D H. divide the part D E. equal to the line A. divide the part D F. equal to the line B. divide the part E G. equal to the line C. bring the line E F. draw the line G H. parallel to the line E F. F H shall be fourth proportional demanded.

See the Ten of Spades.

## PROPOSITION VII.

Between two right lines given to find two means proportional.

Let I and H be the lines propounded, between the which we must find two mean proportional.

The Practice.

Draw the line A B. equal to the line H. let down the Perpendiculars B C. equal to the line

[ 52 ]

## PROPOSITION IX.

There being given the Exceſs of the Diagonal of a Square, about the ſide to find the greatneſs of the ſaid ſide.

Let A B be the Exceſs of the Diagonal of a Square, above its ſide whereof we must find the greatneſs.

### The Practice.

Elevate the perpendicular B C. equal to the Exceſs B A. draw the line A C. prolonged towards D. from the point C. and from the interval C B. describe the Arch B D. A D shall be the ſide of the Square, whereof A B is the Exceſs of the Diagonal A E, above the ſaid ſide A D.

See the Queen of Spades.

## PROPOSITION X.

To divide a right line Terminated within the mean and extream reason.

Let A B be the line, which we must divide in such manner, as the Rectangle compoſed of the whole line, and of one of the two parts, may be equal to the Square framed upon the other part.

### The Practice.

Elevate the perpendicular A D. prolong it towards D. make A C. equal to the half of A B. from the point C. and from the Interval C B. describe the Arch B D. from the point A. and

[ 55 ]

height of a fourth part of a Foot, and so of
the reft, be there never so much applied to it;
and this Principle cannot be denied, if we con-
fider, that the Effect ought to be proportioned
to the Action that is necessary for the Produ-
ction of it: So that if it be necessary to em-
ploy an Action, by which we may raise a
Weight of 100l. to the height of two foot,
for to raise one such to the height of one
foot only; this fame ought to weigh 200l.
for it's the fame thing to raise 100l. to the
height of one foot, and again, yet another
100l. to the height of one foot, as to raise
one of 200l. to the height of one foot, and
the fame, also, as to raise 100l. to the height
of two feet.

Now the Engines which serve to make this
Application of a Force, which acteth at a
great space upon a Weight which it causeth to
be raised by a leffer, are the Pulley, the incli-
ned Plane, the Wedge, the Capthen, or
Wheel, the Screw, the Lever, and some o-
thers; for if we will not apply or compare
them one to another, we cannot well number
more, and if we will apply them, we need
not instance in so many.

The Pulley, Trochlea.

Let ABC be a Chord put about the Pul-
ley D, to which let the Weight E be fastned ;

E 4

and

[ 56 ]

and first, supposing that two Men suftain or pull up equally each of them one of the ends of the said Chord, it is manifest, that if the Weight weigheth 200 l. each of those Men shall employ but the half thereof, that is to say, the force that is requisite for suftaining or raising of 100 l. for each of them shall bear but the half of it.

Afterwards, let us suppose that A, one of the ends of this Chord, being made fast to some Nail, the other C be again suftained by a Man; and it is manifest, that this Man in C, needs not (no more than before) for the suftaining the Weight E, more force than is requisite for the suftaining of 100 l. because the Nail at A doth the same Office as the Man which we supposed there before; in fine, let us suppose that this Man in C do pull the Chord to make the Weight E to rise, and it is manifest, that if he there employeth the force which is requisite for the raising of 100 l. to the height of two foot, he shall raise this Weight E of 200 l. to the height of one foot; for the Chord ABC being doubled, as it is, it must be pulled two feet by the end C, to make the Weight E rise as much, as if two men did draw it, the one by the end A, and the other by the end C, each of them the length of one foot only.

There's always one thing that hinders the exactness of the Calculation, that is the pon- perity

[ 57 ]

perofity of the Chord or Pulley, and the dif-
ficulty that we meet with in making the Chord
to flip, and in bearing it: But this is very small
in comparifon of that which raifeth it, and
cannot be eftimated, fave within a small mat-
ter.

*See the Ace of Clubs.*

Moreover, its neceffary to obſerve, that its
nothing but the redoubling of the Chord, and
not the Pulley, that cauſeth this force; for if
we faſten yet another Pulley towards A, about
which we paſs the Chord ABCH, there will
be required no leſs force to draw H towards
K, and ſo to lift up the Weight E, than there
was before to draw C towards G. But if to
thefe two Pulleys we add yet another towards
D, to which we faſten the Weight, and in
which we make the Chord to run or ſlip, juſt
as we did in the firſt, then we ſhall need no
more force to lift up this Weight of 200l. than
to lift up 50l. without the Pulley; becauſe
that in drawing four foot of Chord we lift it
up but one foot, and ſo in multiplying of the
Pulleys, one may raſe the greateſt Weights
with the leaſt Forces. Its requisite alſo to ob-
ſerve, that a little more Force is always neceſ-
fary for the raſing of a Weight than for the
ſuſtaining of it, which is the reaſon why I
have ſpoken here diſtinetly of the one and the
other.

The

[ 58 ]

The inclined Plane.

If not having more Force than sufficeth to
raife 100 l. one would nevertheless raife this
body F, that weigheth 200 l. to the height of
the line BA, there needs no more but to
draw or rowl it along the inclined Plane C A,
which I fuppoſe to be twice as long as the
line AB; for by this means, for to make it
arrive at the point A, we must there employ
the Force that is neceſsary for the raiſing 100 l.
twice as high; and the more inclined this
Plane ſhall be made, ſo much the leſs Force
ſhall there need to raiſe the Weight F. But
yet there is to be rebated from this Calculati-
on, the difficulty that there is in moving the
body F, along the plane AC, if that plane
were laid down upon the line BC, all the
parts of which I ſuppoſe to be equidiftant
from the Center of the Earth.

It is true, that this impediment being so
much leſs as the Plane is more united, more
hard, more even, and more polite, it cannot
likewise be eſtimated but by gueſs, and it is
not very confiderable. Neither need we much
to regard that the line B C being a part of a
Circle that hath the fame Center with the
Earth, the plane A C ought to be ( though
but very little ) curved, and to have the Fi-
gure of part of a Spiral, defcribed between two

Circles,

[ 59 ]

Circles, which also have for their Center that of the Earth, for that it is not any way flexible.

### The Wedge, Cuneus.

The force of the Wedge ABCD is easily understood after that which hath been spoken above of the inclined Plane, for the force wherewith we strike downwards, acts as if it were to make it move according to the line BD; and the Wood, or other thing and body that it cleaveth, openeth not, or the Weight that it raiseth doth not rive, fave only according to the line AC; infomuch that the force, wherewith one driveth or striketh this Wedge, ought to have the fame proportion to the resistance of this Wood or Weight, that A Chath to A B. Or else again, to be exact, it would be convenient that BD were a part of a Circle, and AD and CD two portions of Spirals that had the fame Center with the Earth, and that the Wedge were of a matter so perfectly hard and polite, and of so small weight, as that any little force would suffice to move it.

### The Crane, or the Capstan, Axis in Peritrochio.

We see also very easily, that the force wherewith the Wheel A or Cogg B is turned, which make the Axis or cylinder C to move, about which a Chord is rowled, to which the weight

D

[ 60 ]

D, which we would raise is fastned, ought
to have the fame proportion to the faid Weight,
as the circumference of the Cylinder hath to
the circumference of a Circle which that force
describeth, or that the Diameter of the one
hath unto the Diameter of the other; for that
the circumferences have the fame proportion,
as the Diameters; in so much that the Cy-
linder C, having no more but one foot in Di-
ameter, if the Wheel A B be fix feet in its Di-
ameter, and the Weight D do weigh 600 l. it
shall suffice that the force in B shall be capable
to raise 100 l. and so of others. One may al-
so instead of the Chord that rolleth about the
Cylinder C, place there a small Wheel with
Teeth or Coggs, that may turn another greater,
and by that means multiply the power of the
force as much as one shall please, without ha-
ving any thing to deduct of the fame, save
only the difficulty of moving the Machine,
as in the others.

Here endeth the Ace of Clubs.

The Screw, Cochlea.

When once the force of the Capften and of the
inclined Plane is underftood, that of the Screw
is eaſe to be computed, for it is compoſed on-
ly of a Plane much inclined, which windeth
about a Cylinder; and if this Plane be in ſuch
manner inclined, as that the Cylinder ought
to make v. gr. ten turns to advance forwards
the

[ 61 ]

the length of a foot in the Screw, and that the bigness of the circumference of the Circle which the force that turneth it about doth describe, be of ten feet, for as much as ten times ten are an Hundred; one Man alone shall be able to prees as strongly with this Instrument, or Screw, as one Hundred without it, provided always, that we rebate the force that is required to the turning of it.

Now I speak here of preesing rather than of raising, or removing, in regard that it is about this most commonly that the Screw is employed; But when we would make use of it for the raising of Weights, instead of making it to advance into a Female Screw, we join or apply unto it a Wheel of many Coggs, so made, that if v. gr. this Wheel have thirty Coggs, whilst the Screw makes one entire turn, it shall not cause the Wheel to make more than the 30th. part of a turn, and if the Weight be fastened to a Chord, that rowling about the Axis of this Wheel shall raise it but one foot in the time that the Wheel makes one entire revolution, and that the greatness of the circumference of the Circle that is described by the force that turneth the Screw about, be also of ten foot, by reason that ten times 30 makes 300; one single Man shall be able to raise a Weight of that bigness with this Instrument, which is called the perpetual Screw, as would require 300 Men without it.

Pro-

[ 62 ]

Provided as before, that we thence deduct
the difficulty that we meet with in turning of
it, which is not properly caused by the Ponde-
roliity of the Weight, but by the force or
matter of the Instrument, which difficulty is
more sensible in it than in those aforegoing,
for as much as it hath greater force.

The Leaver, Vectis.

I have deferred to speak of the Leaver until
the last, in regard that it is of all Engines for
raising of Weights, the most difficult to be ex-
plained.

Let us suppose that C H is a Leaver, in
such manner supported at the point O, ( by
means of an Iron Pin that passes through it a-
cross, or otherwise) that it may turn about
on this point O; its part C describing the fer-
micircle ABCDE, and its part H the semi-
circle FGHIK, and that the Weight which we
would raise by help of it were in H, and the
force in C, the line CO being supposed tri-
ple of OH. Then let us consider that in the
time whilst the force that moveth this Leaver,
describeth the whole fermicircle ABCDE,
and acteth according to the line ABCDE,
although that the Weight describeth likewife
the fermicircle FGHIK, yet it is not raised
to the length of this curved line FGHIK,
but only that of the line FOK, insofar that
the

[ 63 ]

the proportion that the Force which moveth
this Weight ought to have to its Ponderosity,
ought not to be meaured by that which is be-
tween the two Diameters of these Circles, or be-
tween their two circumferences, as it hath been
said above of the Wheel, but rather by that which
is betwixt the circumference of the greater,
and the Diameter of the lefter. Furthermore
let us confider, that there is a necessity that
this Force needeth not to be so great, at such
time as it is near to A, or near to E, for the
turning of the Leaver, as then when it is near
to B or to D; nor so great when it is near
to B or to D, as then when it is near to C. of
which the reason is, that the Weights do there
mount lefs, as it is easie to understand; if ha-
ving supposed that the line COH is parallel
to the Horizon, and that AOF cutteth it at
right Angles, we take the point G equidi-
stant from the point F and H, and the point
B equidistant from A and C. and that having
drawn GS perpendicular to FO, we obferve
that the line FS (which fheweth how much
the Weight mounteth in the time that the force
operates along the line AB) is much lefter
than the line SO, which fheweth how much
it mounteth in the time that the force ope-
rates along the line BC.

And to measure exactly what his force ought
to be in each point of the curved line ABC
DE, it is requisite to know that it operates
there

[ 64 ]

there just in the manner as if it drew the Weight along a plane circularly inclined, and that the inclination of each of the points of this circular plane were to be measured by that of the right line, that toucheth the circle in this point. As for Example, when the force is at the point B, for to find the proportion that it ought to have with the Ponderosity of the Weight which is at that time at the point G, its necessary to draw the Tangent line G M, and to account that the ponderosity of the Weight is to the force which is required to draw it along this plane, and consequently to raise it, according to the circle FGH, as the line G M is to S M. Again, for as much as BO is triple of OG, the force in B needs to be to the Weight in G, but as the 1/3 of the line S M is to the whole line G M. In the self same manner, when the force is at the point D, to know how much the Weight weigheth at I, its necessary to draw the Contingent line betwixt I and P, and the right line I N perpendicular to the Horizon, and from the point P taken at discreffion in the line I P, provided that it be below the point I, you must draw P N parallel to the same Horizon, to the end you may have the proportion that is betwixt the line I P and the 1/3 of I N, for that which is betwixt the ponderosity of the Weight, and the force that ought to be at the point D for the moving of it, and so of others:

[~65]

thers: Where, nevertheless you must except
the point H, at which the contingent line be-
ing perpendicular to the Horizon, the weight
can be no other than triple the force which
ought to be in C for the moving of it; in the
points F and K, at which the contingent line
being parallel to the Horizon it self, the least
force that one can align is sufficient to move
the Weight. Moreover, that you may be per-
fectly exact, you must observe that the lines
SM and PN ought to be parts of a circle that
have for their center that of the Earth; and
GM and IP part of Spirals drawn between
two such Circles, and lastly, that the right
lines SM and IN both tending towards the
center of the Earth are not exactly parallels:
And furthermore, that the point H where I
fuppose the contingent line to be perpendicular
to the Horizon, ought to be some small mat-
ter nearer to the point F than to K, at the
which F and K the contingent lines are paral-
lels to the said Horizon.

This done, we may easily resolve all the dif-
ficulties of the Balance, and shew, that then
when its most exact, and for instance, fuppo-
fing its center at O by which it is sustained, to
be no more but an indivisible point, like as I
have supposed here for the Leaver, if the arms
be declined one way or the other, that which
shall be the lowermost ought evermore to be
adjudged the heavier; so that the center of
Gravity is not fixt and immovable in each fe-

F

veral

[ 66 ]

veral body, as the Ancients have fuppoſed,
which no Perſon that I know of hath hitherto
obſerved.

But theſe laſt confiderations are of no mo-
ment in Practice, and it would be good for
thoſe who ſet themſelves to invent new Ma-
chines, that they knew nothing more of this
buſineſs, than this little which I have now
writen thereof, for then they would not be in
danger of deceiving themſelves in their com-
putation, as they frequently do in fuppoſing
other Principles.

*See the Deux of Clubs.*

A

[ 68 ]

from it would be yet less true; I would not
only to day defer lending you a more particu-
lar Explication. Its requisite above all things
to consider, that I did speak of the Force that
ferveth to raise a Weight to some height, the
which Force has overmore two Dimensions,
and not of that which serveth in each Point
to suftain it, which hath never more than one
Dimension, insofar that these two Forces
differ as much the one from the other, as a
Superficies differs from a Line; for the same
Force which a Nail ought to have for the
sustaining of a Weight of 100 l. one moment
of time, doth also suffice for to sustain it the
space of a Year, provided that it do not di-
minish, but the same Quantity of this Force
which serveth to raise the Weight to the
height of one foot, sufficeth not (eadem nu-
mero) to raise it two feet, and it is not more
manifest that 2 and 2 makes 4, than its mani-
feet that we are to employ double as much
therein.

Now, for as much as that this is nothing
but the same thing that I have supposed for a
Principle, I cannot guess on what the Scrue-
ple should be grounded, that Men make of
receiving it; but I shall in this place speak of
all such as I suspect, which for the most part
arise only from this, that Men are beforehand
overknowing in the Mechanicks; that is to say,
that they are pre-occupied with Principles
that

[ 69 ]

that others prove touching these matters,
which not being absolutely true, they deceive
the more, the more true they seem to be.

The first thing wherewith a Man may be
pre-occupied in this business, is, That they
many times confound the consideration of
Spaces with that of Time, or of the Velo-
city; so that, for Example, in the Leaver, or
(which is the fame) the Ballance A B C D,
having supposed that the Arm A B is double
to B C, and the Weight in C double to the
Weight in A, and also that they are in an Equi-
librium, instead of saying, that that which cauſ-
eth this Equilibrium, is, that if the Weight C
did fall, or was raised up by the Weight
A, it did not fall more than half so much
space as it; they say, that it did move forward
by the half, which is a fault so much the more
prejudicial, in that it is very difficult to be
known; for it is not the difference of the
Velocity that is the cause why these Weights
are to be one double to the other, but the dif-
ference of the Space, as appears by this;
that to raise, for Example, the Weight F with
the hand to G, it is not necessary to
employ a Force that is precisely double to that
which one should have therein employ'd the
first bout, to raise it twice as quickly, but its
requisite to employ therein either more or
less than the double, according to the diffe-
rent.

[ 70 ]

rent proportion of this Velocity may have to
the causes that refift it.

See the Deux of Clubs.

Instead of requiring a Force juſt double for
the raiſing of it with the fame Velocity twice
as high unto H, I ſay, that it is juſt double in
counting ( as two and two make four ) that
one and one make two, for it is requisite to
employ a certain quantity of this Force to
raife the Weight from F to G, and again al-
ſo, as much more of the fame Force to raiſe
it from G to H.

For if I had had a mind to have joyned
the consideration of the Velocity with that of
the Space, it had been neceſsary to have af-
ſigned three dimensions to the Force, whereas
I have aſſigned it no more but two, on pur-
poſe to exclude it. And if I have teſtified that
there is ſo little of worth in any part of this
ſmall Tract of the Staticks, yet I deſire that
Men ſhould know, that there is more in this
alone, than in all the reſt: For its impoſsible
to ſay any thing that is good and ſolid touch-
ing Velocity, without having rightely explain-
ed what we are to understand by Gravity, as
alſo the whole System of the World. Now
because I would not undertake it, I have
thought good to emit this consideration, and
in this manner to ſingle out theſe others that I
could explain without it; for though there
be no motion but hath ſome Velocity, never-
theleſs

[ 71 ]

thelefs it is only the Augmentations and Di-
minutions of this Velocity that are considera-
ble. And now that speaking of the motion
of a Body, we suppoſe that it is made accor-
ding to the Velocity which is moſt natural to
it, which is the fame as if we did not confi-
der it at all.

The other reaſon that may have hindred
Men from rightly understanding my Principle,
is, that they have thought that they could
demonſtrate without it ſome of thofe things
which I demonſtrate nor without it; As for
Example, touching the Pulley ABC, they
have thought that it was enough to know
that the Nail in A did ſuſtain the half of the
Weight B; to conclude that the hand in C
had need but of half ſo much Force to ſuſtain
or raife the Weight, thus wound about the
Pulley, as it would need for to ſuſtain or raife
it without it. But howbeit that this ex-
plaineth very well, how the application of
the Force at C is made unto a Weight dou-
ble to that which it could raife without a
Pulley, and that I my ſelf did make uſe there-
of, yet I deny that this is ſimply, becauſe that
the Nail A ſuſtaineth one part of the Weight
B, that the Force in C, which ſuſtaineth it,
might be leſs than if it had been ſo ſuſtained.
For if that had been true, the Rope C E be-
ing wound about the Pulley D, the Force in
E might by the fame reaſon be leſs than the

F 4

Force

[ 73 ]

considered according to its two Dimensions,
ought to be less for the raising in like manner
the Weight E, than for the raising it without
any Pulley.

Here endeth the Deux of Clubs.

The third thing which may make men imagine some obscurity in my Principle is, that they, it may be, have not had regard to all the words by which I explain it; for I do not say simply that the Force that can raise a Weight of 500. to the height of 4 feet can raise one of 200. to the height of one foot; but I say, that it may do it, if so be that it be applied to it: Now its impossible to apply the same, but by the means of some Machine, or other invention that shall cause this Weight to ascend but one, in the time whilst the Force passes the whole length of 4 feet, and so that it do transform the Quadrangle, by which the Force is represented, that is required to raise this Weight of 400. to the height of one foot into another, that is equal and like to that which represents the Force that is required for to raise a Weight of 500. to the height of 4 feet.

In fine, it may be that Men may have thought the worse of my Principle, because they have imagined that I have alledged the Examples of the Pulley, of the inclined Plane, and of the Leaver, to the end that I might better perfwade the truth thereof, as if it had been

[ 74 ]

been dubious, or else that I had so ill dif-
courfed as to offer to affume from thence a
Principle, which ought of it self to be so clear,
as not to need any proof by things that are
so difficult to comprehend as that; it may be,
they have never been well demonstrated by
any Man, but neither have I made use of them,
save only with a design to show that this Prin-
ciple extends it self to all matters of which
one treateth in the Staticks; or rather, I have
made use of this occasion for to insert them
into my Treatife, for that I conceived that it
would have been too dry and barren if I had
therein spoken of nothing else but of this Que-
tion, that is of no use, as of that of the Geo-
staticks, which I purposed to Examine.

Now one may perceive by what hath al-
ready been said, how the Forces of the Lea-
ver and Pulley are demonstrated by my prin-
ciple so well, that there only remains the in-
clined plane, of which you shall clearly see
the demonstration by this Figure, in which
G F represents the first Dimension of the
Force that the Rectangle F H describes, whilst
it draweth the Weight D along the plane BA,
by the means of a Chord parallel to this plane,
and passing about the Pulley E, in such sort
that HG, that is the height of this Rectangle,
is equal to BA, along which the Weight D
is to move, whilst it mounteth to the height
of the line CA. And NO represents the first
Di-

Di-

[ 75 ]

Dimension of such another Force, that is described by the Rectangle N P, in the time that it is raising the Weight L to M.

And I suppose that L M=BA=2C A, and that NO-. FG::OP-GH. This done, I consider that at such time as the Weight D is moved from B towards A, one may imagine its motion to be composed of two others, of which the one carrieth it from B R towards C A, ( to which operation there's no Force required, as all those suppose who treat of the Mechanicks) and the other raileth it from B C towards R A, for which alone the Force is required, in so much that it needs neither more nor less Force to move it along the inclined plane B A, than along the perpendicular C A. For I suppose that the unevenness, &c. of the Plane do not at all hinder it, like as it is always supposed in treating of this matter:

So then the whole Force F H is employed only about the raising of D to the height of C A, and for as much as it is exactly equal to the Force N P, that is required for the raising of L to the height of L M, double to C A, I conclude by my principle that the Weight D is double to the Weight L. For in regard that it is necessary to employ as much Force for the one as for the other, there is as much to be raised in the one as in the other, and no more knowledge is required than to count unto

[ 78 ]

the first place, the Definitions and Principles
of which we make use.

We in general call that quality a Force or Pow-
er, by means of which any thing whatever doth
tend or aspire into another place than that in which
it is, be it downwards, upwards, or side-ways,
whether this Quality naturally belongeth to the Bo-
dy, or be communicated to it from without. From
which definition it followeth, that all Weights
are a Species of Force, in regard that it is a
Quality, by means whereof Bodies do tend
downwards. We often also assign the name
of Force to that very thing to which the
Force belongeth, as a ponderous Body is call-
ed Weight; but with this pre-caution, that
this is in reference to the true Force, the
which augmenting or diminishing shall be
called a greater or lesser Force, albeit that the
thing to which it belongeth do remain al-
ways the same. If a Force be suspended or
fastened to a flexible Line that is without Gra-
vity, and that is fastened at one end to some
Fulciment or Stay, in such fort as that it fu-
stain the Force, drawing without impediment
by this line, the Force and the line shall take
some certain position in which they shall rest,
and the line shall of necessity be straight, let
the line be termed the Pendant, or line of Di-
rection of the Force; and let the point by which
its fastened to the Fulciment be called the point
of Suspension; which may sometimes be the
Arm

[ 79 ]

Arm of a Leaver or Ballance, and then let the line drawn from the Center of the Fulciment of the Leaver or Ballance to the point of suspension be named the *Distance or the Arm of the Force*, which we suppose to be a line fixed, and considered without Gravity.

Moreover, let the Ang. comprehended be twixt the Arm of the Force and the line of Direction be termed, *the Ang. of the Direction of the Force*.

### Axiom I.

After these Definitions we lay down for a Principle, that in the Leaver, and in the Ballance, equal Forces drawing by Arms that are equal, and at equal Angles of Direction, to draw equally. And if in this Position they draw one against the other, they shall make an Equilibrium; but if they draw together, or towards the same part, the Effect shall be double. If the Forces being equal, and the Angles of Direction also equal, the Arms be unequal, the Force that shall be suspended at the greater Arm shall work the greater Effect.

As in this Figure, the Center of the Ballance or Leaver being A, if the Arms AB and AC are equal, as also the Angles ABD, and ACE, the equal Forces D and E shall draw equally, and make an Equilibrium. So likewise the Arm AF being equal to AB, the Ang. AFG

10

[ 80 ]

to the Ang. A B D, and the Force G to D, these
* two Forces * G and D shall draw
  In M. S. equally, and in regard that they
  Copyy its C draw both one way, the Effect shall
  and D. be double. In the same manner
the Forces G and F shall make an Equilibrium,
  as also I and L shall counterpoise, if
  ( being equal ) the Arms A K and A H, and
  the S AHT and AKL be equal.

The same shall befall in the Forces P and R,
if all things be disposed as before. And in
this case we make no other distinction be-
twixt Weights and other Forces save only
this, that Weights all tend towards the Center of grave Bodies, and Forces may be understood to tend all towards all parts of the Universe, with so much greater or lesser impetus
than Weights; so that Weights and their parts
do draw by Lines of Direction, which all con-
cur in one and the same point, and Forces
and their parts may be understood to draw
in such fort, that all the Lines of Direction
are parallel to each other.

Axiom 2.

In the second place, we suppose that a Force
and its Line of Direction abiding always in the
same position, as also the Center of the Balance or Leaver, be the Arm what it will, that
is drawn from the Center of the Balance to
the

[ 81 ]

the line of Direction, the Force drawing always in the same fashion, will always produce the same effect.

As in the second Figure, the center of the Ballance being A, the Force B, and the line of Direction B F prolonged, as occasion shall require, in which the Arms AG, AC and AF do determine; in this position, let the line B F be fastened to the Arm AF or AC, or to another Arm drawn from the center A to the line of Direction * BF; we suppose that this Force B shall always work the same Effect upon the Ballance. And if drawing by the Arm AC it make an Equilibrium with the Force D drawing by the Arm AE, when ever it shall draw by the Arms AF or AG, it shall likewise make an Equilibrium with the Force D, drawing by the Arm AE. This Principle altho' it be not expreſly found in Authors, yet it is tacitly ſuppoſed by all thofe that have writ on this Argument, and experience constantly confirmeth it.

### Axiom 3.

If the Arms of a Ballance or Leaver are directly placed the one to the other, and that being equal they sustain equal Forces, of which the Angles of Direction are right Angles, in those

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these Forces do always weigh equally upon the center of the Ballance, whether that they be near the fame center, or far distant, or both conjoined in the center it self; as in this Figure the Ballance being E D, the center A, the equal Arms A D and A E, let us suftair equal Forces H and I, of which the Angles of direction A D H and A E I are right Angles, we fuppoſe that these two Forces I and H weigh alike upon the center A as if they were nearer to the center, at the equal distances A B and A C, and we also ſuppoſe the fame if these very Forces were ſuspended both together in A, the Angles of Directions being ſtill right Angles.

Here endeth the Trois of Clubs.

## PROPOSITION I.

These Principles agreed upon, we will easily demonstrate in imitation of Archimedes, that upon a straight Ballance the Forces of which and of all their parts, the lines of direction are parallel to one another, and perpendicular to the Ballance, shall counterpoife and make an Equilibrium, when the said Forces shall be to one an other in reciprocal proportion of their Arms, which we think to be to manifeft to you, that we thence shall derive the Demonſtration of this Univerſal Proposition, to which we haſten.

PRO-

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qual to the Force D, and H to E, the one, to wit, G, drawing up the Arm A F, and the other, to wit, H, upon the Arm A C; now, by the Propoſition, C and H ſhall make an Equilibrium upon the Ballance C A F, but by the first principle, the force D upon the Arm A B worketh the fame effect as the force G on the Arm A F; Therefore the force D upon the Arm A B maketh an Equilibrium with the force H upon A C, and the force H drawing in the fame manner upon the Arm A G, and the force H the fame first Axiom, the force D upon the Arm A B ſhall make an Equilibrium with the force E upon the Arm A C.

See the Four of Clubs.

Now, in the following Figure, let there be a center of the Ballance be A, the arms A B and A C, the lines of Direction B D and C E, which are not perpendicular to the arms, and the Forces D and E drawing likewife by the lines of Direction, upon which perpendiculars are erected unto the center A, that is A P upon B D, and A G upon E C, and that as A F is to A G, so is the force E to the force D; which forces draw one against the other, I say, that they will make an Equilibrium upon the Ballance C A B; for let the lines A F and A G be understood to be the two arms of a Ballance G A F, upon which the forces D and E do draw by the lines of Direction.