Book I.

XV.
Rectilineal figures are those which are contained by straight

lines.

XVI.
Trilateral figures, or triangles, by three straight lines.

XVII.
Quadrilateral, by four straight lines.

XVIII.
Multilateral figures, or polygons, by more than four straight

lines.

XIX.
Of three sided figures, an equilateral triangle is that which

has three equal sides.

XX.
An isosceles triangle is that which has only two sides equal.

AAN

XXI.
A scalenę triangle is that which has three unequal sides.

XXII.
A right angled triangle is that which has a right angle.

XXIII.
An obtuse angled triangle is that which has an obtuse angle.

Book I.

XXIV.
An acute angled triangle is that which has three acute angles.

Xxv.
Of four sided figures, a square is that which has all its sides

equal, and all its angles right angles.

XXVI.
An oblong is that which has all its angles right angles, but

has not all its sides equal.

XXVII.
A rhombus is that which has all its sides equal, but its angles

are not right angles.

XXVIII.
A rhomboid is that which has its opposite sides equal to one

another, but all its sides are not equal, nor its angles right
angles.

XXIX.
All other four sided figures besides these are called trapez

ziums,

1

Book I.

XXX.,
Parallel straight lines are such as are in the same plane, and

which, being produced ever so far both ways, do not meet.

POSTULATES.

I.
LET it be granted that a straight 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 straight line.

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

distance from that centre.

AXIOMS.

I.
N. THINGS which are equal to the same thing are equal to one

another.

II.
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.

Book I.

VI.
Things which are doubles of the same thing are equal to one

another.

VII.
Things which are halves of the same thing 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. The whole is greater than its part.

X. All right angles are equal to one another. “ Two straight lines, which intersect one another, cannot be

“ both parallel to the same straight line.”

equal to AB; but things which

Book I.

PROPOSITION I. PROBLEM.
TO describe an equilateral triangle upon a given
finite straight line.

Let AB be the given straight line; it is required to des-
cribe an equilateral triangle upon it.

From the centre A, at

the distance AB, desa 3. Postu- scribea the circle BCD, late.

and from the centre B, at
the distance BA, describe

D A

В. E
the circle ACE; and
from the point c, in
which the circles cut one

another, draw the straight b 1. Post. linesb CA, CB, to the

points A, B: ABC shall
be an equilateral triangle.

Because the point A is the centre of the circle BCD, AC c 11. Defi. is equal to AB; and because the point B is the centre of the nition.

circle ACE, BC is equal to BA: but it has been proved that

CA is equal to AB; therefore CA, CB are each of them d 1. Axi- equal to one anotherd; therefore CA is equal to CB; where

fore CA, AB, BC are equal to one another; and the triangle
ABC is therefore equilateral, and it is described upon the
given straight line AB. Which was required to be done.

om.

PROP. II. PROB.

FROM a given point to draw a straight line equal to a given straight 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,