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XV.

A circle is a plane figure contained by one line, which is called the circumference, and is fuch that all ftraight lines drawn from a certain point within the figure to the circumference, are equal to one another.

*Book I.

XVI.

And this point is called the centre of the circle

XVII.

A diameter of a circle is a straight line drawn through the cen tre, and terminated both ways by the circumference.

XVIII.

A femicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter.

XIX.

"A fegment of a circle is the figure contained by a straight line, and the circumference it cuts off."

XX.

Rectilineal figures are thofe which are contained by straight lines.

XXI.

Trilateral figures, or triangles, by three ftraight lines.

XXII,

XXIII.

Quadrilateral, by four ftraight lines.

Multilateral figures, or polygons, by more than four straight

lines.

XXIV.

Of three fided figures, an equilateral triangle is that which has three equal fides.

XXV.

An ifofceles triangle is that which has only two fides equal.

XXVI.

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A fcalene triangle, is that which has three unequal fides.
XXVII.

A right angled triangle, is that which has a right angle,

XXVIII.

An obtufe angled triangle, is that which has an obtufe angle,

XXIX.

An acute angled triangle, is that which has three acute angles.
XXX.

Of four fided figures, a fquare is that which has all its fides
equal, and all its angles right angles,

XXXI.

An oblong, is that which has all its angles right angles, but has not all its fides equal.

XXXII.

A rhombus, is that which has all its fides equal, but its angles are not right angles.

ᄆᄆ

XXXIII,

A rhomboid, is that which has its oppofite fides equal to one another, but all its fides are not equal, nor its angles right angles.

XXXIV.

XXXIV.

All other four fided figures befides thefe, are called Trapeziums.

XXXV.

Parallel ftraight lines, are fuch as are in the fame plane, and which, being produced ever so far both ways, do not meet.

POSTULATES.

I.

LET it be granted that a fraight line may be drawn from any one point to, any other point.

II.

That a terminated straight line may be produced to any length in a ftraight line.

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III.

And that a circle may be described from any centre, at any diftance from that centre.

AXIOM S.

I.

THINGS which are equal to the fame are equal to one an ·

other.

11.

If equals be added to equals, the wholes are equal.

III.

If equals be taken from equals, the remainders are equal.

IV.

If equals be added to unequals, the wholes are unequal.

V.

If equals be taken from unequals, the remainders are unequal,

VI.

Things which are double of the fame, are equal to one another.
VII.

Things which are halves of the fame, are equal to one another.
VIII.

Magnitudes which coincide with one another, that is, which
exactly fill the same space, are equal to one another.

IX.

Book I.

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"If a straight line meets two straight lines, fo as to make the "two interior angles on the fame fide of it taken together "less than two right angles, these straight lines being con"tinually produced, fhall at length meet upon that fide on which are the angles which are lefs than two right angles. See the notes on Prop. 29. of Book I."

PROPO

T

PROPOSITION I. PROBLEM.

O defcribe an equilateral triangle upon a given
finite ftraight line.

Let AB be the given straight linë; it is required to describe

an equilateral triangle upon it.

From the centre A, at the diftance AB, defcribe a the circle BCD, and from the centre, B, at the distance BA, defcribe the

circle ACE; and from the point D
C, in which the circles cut one
nother, draw the straight linesb
CA, CB to the points A, B; ABC
hall be an equilateral triangle.

BE

Because the point A is the centre of the circle BCD, AC is

Book I.,

a. 3. Poftulate.

b. I. Poft.

equal c to AB; and because the point B is the centre of the c. 15. Deficircle ACE, BC is equal to BA: But it has been proved that CA nition. is equal to AB; therefore CA, CB are each of them equal to AB; but things which are equal to the fame are equal to one another d; therefore CA is equal to CB; wherefore CĂ, AB, BC d. 1ft Axiare equal to one another; and the triangle ABC is therefore om. equilateral, and it is described upon the given straight line AB. Which was required to be done.

F

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ROM a given point to draw a straight line equal to a given ftraight line.

Let A be the given point, and BC the given straight line; it is required to draw from the point A a straight line equal to BC.

From the point A to B draw a the ftraight line AB; and upon it defcribe b the equilateral triangle DAB, and produce the ftraight lines DA, DB, to E and F; from the centre B, at the diftance BC, defcribe d the circle CGH, and from the centre D, at the diftance DG, defcribe the circle GKL. AL hall be equal to BC.

D

a. I. Poft.

b. I. I.

K

H

c. 2. Poft.

d. 3. Poft.

E

F

Because

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