Hence Theor. 12 may be thus expressed:

If there are two sets of ratios equal to one another, each to each, the ratio compounded of the ratios of the first set is equal to that compounded of the ratios of the other set.

DEF. II. When two ratios are equal, the ratio compounded of them is called the duplicate ratio of either of the original ratios. DEF. 12. When three ratios are equal, the ratio compounded of them is called the triplicate ratio of any one of the original ratios.

THEOR. 14. If two ratios are equal, their duplicate ratios are equal; and conversely, if the duplicate ratios of two ratio are equal, the ratios themselves are equal.

then the dupăcate ratio of A: B is equal to that of P : Q.

Let C, R be the third proportionals to A, B and to P, Q respectively, so that

and

then because

therefore

whence, ex æquali,

A: BBC,

P:Q:Q:R;

A: B:: P:Q,

B:C::Q:R;

A: CPR,

IV. I.

that is, the duplicate ratio of A: B is equal to that of P: Q. Next let the duplicate ratio of A: B be equal to that of P : Q, that is, let

A: C:P : R.

or S is the mean proportional between P and R,

IV. 1.

IV. I.

Q.E.D.

Ex. 8. The duplicate ratio of m:n, m and n being two numbers, is m2: no.

Ex. 9. The ratio compounded of m:n and p: q, m, n, p, q, being numbers, is mp : nq.

SECTION II

FUNDAMENTAL GEOMETRICAL PROPOSITIONS.

THEOR. 15. If two straight lines are cut by a series of parallel straight lines, the intercepts on the one have to one another the same ratios as the corresponding intercepts on the other have.

Let a series of parallel lines cut one line in the points A, B, C, D . . . and another in the points A', B', C', D', etc., so that AB and A'B' are corresponding intercepts, as also CD, C'D':

then shall AB be to CD as A'B' to C'D'.

On the line ABC take AM=m.AB and AN=n.CD, M and N being on the same side of A,

and draw MM',NN' parallel to any line of the series, and therefore parallel to all and to one another.

· I. 24.

Then if AM were divided into m parts, each equal to AB, and lines were drawn through the points of division parallel to AA' or MM', the corresponding intercepts (m in number) would be each equal to A'B,' I. 34.

Then according as MM' lies on the same side of NN' as AA', coincides with NN', or lies on the other side of NN',

THEOR. 16. A given finite straight line can be divided internally into segments having any given ratio, and also externally into segments having any given ratio except the ratio of equality; and in each case there is only one such point of division.

For demonstration, See Part I., Theor. 10.

THEOR. 17. A straight line which divides the sides of a triangle proportionally is parallel to the base of the triangle.

For demonstration, See Part I., Theor. II.

ΙΟ

THEOR. 18. Rectangles of equal altitude are to one another in the same ratio as their bases.

Let AC, BD be two rectangles of equal altitude on the bases AO, BQ,

then shall the rectangle AC be to the rectangle BD as the base AO is to the base BQ.

Since the rectangles are of equal altitude, QD is equal to OC, and if DQ be placed on CO, D upon C, and therefore Q on O, and B on the same side of CO as A, Q B will fall along OA. Take OM m.OA and ON = n.OB,

and complete the rectangles MC, NC.

Then if OM were divided into m parts, each equal to OA, and rectangles of altitudes equal to OC described on each part, these rectangles would be all equal,

II. 1, Cor. 2.

and they would together make up the rectangle MC,
therefore the rectangle MC = m. the rectangle AC.
In like manner the rectangle NC = n. the rectangle BC.
Then according as the base OM > = or < ON,

so is the rectangle MC > = or < NC;

II. 1, Cor. 2.