straight lines drawn joining the two pairs of opposite angular points, is called a diagonal of the parallel

ogram.

OBS. In the figure the parallelogram is denoted either by ABCD, or by AC, or by BD; AC, which joins the opposite angular points A, C, is one diagonal; and BD, which joins the opposite angular points B, D, is the other diagonal.

This def1., and Bk. ii. Def. 1, have superseded Defs. 31, 32, 33.

POSTULATES.

I.

Let it be granted that a straight line may be drawn from any one given point to any other given point.

II.

Let it be granted that a given terminated straight line may be produced to any length required either way in a straight line.

III.

Let it be granted that a circle may be described from any given point as centre, at any given distance from that centre; or, what is the same thing, with any given point as centre, and with any given finite straight line drawn from that point as radius.

AXIOMS.

I.

Things that are equal to the same thing are equal to one another.

II.

If equals, or the same thing, be added to equals, the sums are equal.

III.

If equals, or the same thing, be taken from equals, and if equals be taken from the same thing, the remainders are equal.

IV.

If equals, or the same thing, be added to unequals, the sums are unequal in the same kind of inequality.

V.

If equals, or the same thing, be taken from unequals, the remainders are unequal in the same kind of inequality.

VI.

Things that are double of the same thing, or of equals, are equal to one another.

VII.

Things that are halves of the same thing, or of equals, are equal to one another.

VIII.

Magnitudes that coincide with one another, i.e. exactly fill up the same space, are equal to one another.

IX.

The whole is greater than its part.

X.

Two straight lines cannot enclose a space.

XI.

All right angles are equal to one another.

XII.

If a straight line cut two straight lines, so as to make the two interior angles on the same side of it, when taken together, less than two right angles, then those two straight lines, being continually produced, shall at length meet on that side of the cutting line on which are the angles that are together less than two right angles.

OBS. 1. This axiom will be made clearer by illustration.

(1) Let the straight line EFGH (Fig. 1) cut the two straight lines AB, CD in the points F, G, so as to make the two interior angles BFG, FGD on the same side of EH (viz. that side towards B, D), when taken together, less than two right angles; then the 12th Axiom asserts that the two straight lines AB, CD, being continually produced (as represented by the dotted lines) shall at length meet in some point K upon the side of EH towards B, D, that being the side on which the angles BFG, FGD are, which are together less than two right angles.

(2) Let the straight line EFGH (Fig. 2) cut the two straight lines AB, CD in the points F, G, so as to make the two interior angles AFG, FGC on the same side of EH (viz. that side towards A, C), when taken together, less than two right angles; then the 12th

Axiom asserts that the two straight lines BA, DC, being continually produced (as represented by the dotted lines) shall at length meet in some point L on the side of EH towards A, C, that being the side on which the angles AFG, FGC are that are together less than two right angles.

OBS. 2. When, as in the figures in Obs. 1, a straight line EFGH cuts two other straight lines AB, CD in F, G, it makes with them eight angles, four on one side of EF, viz. AFE, AFG, FGC, CGH, and four on the other side of EF, viz. EFB, BFG, FGD, DGH. Of these eight angles:

:

(1) the angles AFE, EFB, CGH, HGD, are called exterior angles;

(2) the angles AFG, FGC, DGF, GFB, are called interior angles;

(3) any angle at F is said to be opposite to any angle at G; (4) the interior angles which are opposite to one another, and on different sides of the cutting line, are called alternate interior, or alternate angles: thus AFG, FGD are alternate angles; BFG, FGC are alternate angles.

These remarks will render clearer the enunciations of Props. 27, 28, 29.

PROPOSITIONS.

PROP. I. PROBLEM.

To describe an equilateral triangle on a given finite straight line.

Let AB be the given finite straight line. It is required to describe an equilateral triangle on AB.

D

G

E

B

With one of the extremities A of AB as centre, and with AB as radius, describe (Post. 3) the circle BCD; with the other extremity B of AB as centre, and BA as radius, describe the circle AEF; and from the point & where these circles cut one another, draw (Post. 1) the straight lines GA, GB to the points A, B. Then ABG shall be the equilateral triangle required.

C

F

Because A is the centre of the circle BCD, AB is equal to AG by def" (Def. 15); and because B is the centre of the circle AEF, BG is equal to BA for the same reason. Hence AG, BG are each of them equal to AB; and things that are equal to the same thing are equal to one another (Ax. 1) therefore AG is equal to BG, and the three straight lines AB, BG, GA, are all equal. Hence the triangle GAB is equilateral (Def. 24), and it has been described on the given straight line AB. Which was to be done.

:

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 ▲ a straight line equal

to BC.