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of, equal between themselves,

each of these equal Angles is a Right one ; and that Right Line, wbich stands upon the other, is called a Perpendicular to that whereon it stands. XI. An Obtuse Angle is that which is greater - than a Right one. XII. An Acute Angle is that which is less than a

Right one. XIII. A Term (or Bound) is that which is the

Extreme of any Thing. XIV. A Figure is tbai which is contained under

one or more Terms XV. A Circle is a plain Figure, contained under

one Line, called the Circumference ; to whicb all Right Lines, drawn from a certain Point

within the Figure, are equal. XVI. And that Point is called the Centre of the

Circle. XVII. A Diameter of a Circle is a Right Line

drawn ibrough the Centre, and terminated on both Sides by the Circumference, and divides the

Circle into two equal Paris. XVIII. A Semicircle is a Figure contained under : a Diameter, and that part of the Circumfe

rence of a Circle cut off by that Diameter. XIX. A Segment of a Circle is a Figure contain

ed under a Right Line, and Part of tbe Circumference of the Circle (which is cut off by ibat

Right Line.] XX. Right-lined Figures are such as are con

tained under Right Lines. XXI. Three-fided Figures are such as are con

tained under three Lines, XXII. Four sided Figures are such as are con

tained under four Lines. XXIII. Many fided Figures are those that are contained under more iban four Right Lines.

1

XXIV. Of three-sided Figures, that is an Equi

lateral Triangle, which bath three equal Sides. XXV. That an Isoceles, or Equicrural one,

which hath only two Sides equal. XXVI. And a Scalene one, is that which bath

ibree unequal Sides. XXVII. Also of three fided Figures, that is a

Right-angled Triangle, which hath a Right

Angle. XXVIII. That an Obtufe-angled one, which

bath an Obtufe Angle. XXIX. And that an Acute-angled one, which

baib tbree Acute Angles. XXX. Of four-sided Figures, that is a Square,

whose four Sides are equal, and its Angles all

Right ones. XXXI. Tbat an Oblong, or ReEtangle, which is

longer than broad; but its opposite fides are

equal, and all its Angles Right ones. XXXII. Thai a Rbombus, which haib four equal

Sides, but not Right Angles. XXXIII. That a Rhomboides, whose opposite

Sides and Angles only are equal. XXXIV. All Quadrilateral Figures, besides

these, are called Trapezia. XXXV. Parallels are such Right Lines, in the fame Plane, which, if infinitely produc'd both Ways, would never meet.

POSTULAT E S.

1.
1. GI

RANT that a Right Line may be drawn

from any one point to another. II. That a finite Right Line may be continued di

really forwards. III. And that a Circle may be described about any Centre with any Distance.

AXIOMS.

B 2

A X I 0 MS.

1. THING $ equal to one and the fame Thing,

are equal to one another. II. If to equal Things are added equal Things,

the Wholes will be equal. III. If from equal Things equal Things be taken

away, the Remainders will be equal. IV. If equal Things be added to unequal Things,

the Wholes will be unequal. V. If equal Things be taken from unequal Things,

the Remainders will be unequal.
VI. Things which are double to one and the same

Tbing, are equal between tbemselves.
VII. Things which are balf one and the same

Tbing, are equal between themselves.
VIII. Things which mutually agree together, are

equal lo one another.
IX. The Whole is greater than its Part.
X. Two Right Lines do not contain a Space.
XI. All Right Angles are equal between tbem-

Jelves.
Xi. If a Right Line, falling upon two other

Right Lines, makes the inward Angles on the
Jame Side thereof, both together, lejs than two
Right Angles, i boje two Right Lines, infinitely
produc’d, will meet each other on that Side,

where the Angles are less than Rigbt ones.
Nute, When there are several Angles at one point,

any one of them is express'd by three Letters, of
which that at the Verlex of the Angle is plac'd in
the Middle. For Example; in the Figure of Prop.
XIII. Lib. I. the Angle contained under the Right
Lines A B, BC, is called the Angle A BC; and
the Angle contained under the Right Lines A B,
BE, is called the Angle A BE.

PRO

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PROPOSITION I.

PROBLEM.

To describe an Equilateral Triangle upon a given

finite Right Line.

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ET A B be the given finite Right Line, upon
which it is required to describe an equilateral

Triangle. About the Centre A, with the Distance A B, defcribe the Circle BCD*; and about the Centre B, **Poß. 3. with the same Distance B A, describe the Circle A CE*; and from the Point C, where the iwo Circles cut each other, draw the Right Lines CA, CBt.

+ Poß. 1. Then because A is the Centre of the Circle DBC, A C shall be equal to A B I.

And because B is the t Def. 15. Centre of the Circle C A E, B C shall be equal to BA: But C A hath been proved to be equal to A B; therefore beth C A and C B are each equal to A B. But Things 'equal to one and the same Thing, are equal between themselves *, and consequently A C is * Ax. 1, equal to CB; therefore the three Sides C A, A B, B C, are equal between themselves.

And so, the Triangle B A C is an Equilateral one, and is described upon the given finite Right Line A B; which was to be done.

PROPOSITION II.

PROBLEM.

At a given Point to put a Right Line equal to a

Right Line given.
ET the Point given be A, and the given Right

Line BC; it is required to put a Right Line at
the Point A, equal to the given Right Line BC.
B 3

Draw

Poft. 3.

Poft. I. Draw the Right Line AC from the Point A to C *; + 1 of tbis. upon it describe the Equilateral Triangle DAC+; | Poft, 2. produce D A and DC directly forwards to E and GI;

about the Centre C, with the Distance B C, describe
the Circle B GH*; and about the Centre D, with
the Distance D G, describe the Circle KGL.

Now because the Point C is the Centre of the Circle + Def. 15. BGH, B C will be equal to CG t; and because D

is the Centre of the Circle KGL, the Whole DL
will be equal to the Whole D G, the Parts whereof
D A and D C are equal; therefore the Remainders
A L, CG, are also equal f. But it has been demon-
strated, that B C is equal to CG; wherefore both AL
and B C are each of them equal to CG. But Things

that are equal to one and the same Thing, are equal * Ax, 1.

to one another *; and therefore likewile A L is equal
to B C.

Whence the Right Line A L is put at the given
Point A, equal to the given Right Line B C, which
was to be done.

| Ax. 3.

PROPOSITION III.

PROBLEM

2 of tbis.

Two unequal Right Lines being given to cut off a

Part from ihe greater, equal to the leffer.
ET A B and C be the two equal Right Lines

given, the greater whereof is A B; it is required
to cut off a Line from the greater A B equal to the
leffer C.

Put * a Right Line A D at the Point A, equal to the Line C; and about the Centre A, with the Diftance AD, describe a Circle D E F +.

Then because A is the Centre of the Circle DEF, A E is equal to AD; and so both A E and Care each equal to AD; therefore A E is likewise equal to C I.

And so, there is cut off from A B the greater of two given Right Lines A B ana C, a Line A E equal to the leffer Line C; which was to be done.

PRO.

& Poft. 3.

Ax. 1.

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