Def. When any number of magnitudes A, B, C, D, E, &c., are so related that

A: BB: C = C : D = D: E = &c.,

then the magnitudes are said to be in continued proportion; B is said to be a mean proportional between A and C; B and C are said to be two mean proportionals between A and D; B, C, D are said to be three mean proportionals between A and E; and so on.

Also A is said to have to C the duplicate ratio of that which A has to B; A is said to have to D the triplicate ratio of that which A has to B; and so on.

THEOREM 9-If two ratios are equal, their duplicates are also equal; and conversely.

Let A, B, C, D, X, Y be magnitudes such that

..

ex æquali ▲ : X = C : Y.

i. e. the duplicate of A : B = the duplicate of C : D.

Note-From Theorem 8 and its Corollary, it follows that, of a set of ratios of magnitudes of the same kind, the ratio

first antecedent : last consequent,

is independent of the other antecedents and consequents, or their order; provided only that each magnitude which occurs anywhere as antecedent also occurs somewhere else as consequent.

Hence, under such conditions, it produces the same effect to alter a magnitude successively in any number of ratios, as to alter it at once in the ratio first antecedent to last consequent. Reference to this very important principle will be facilitated by the following definition.

Def. When there are any number of magnitudes of the same kind, the first is said to have to the last a ratio which is compounded of the ratio of the first to the second, of the ratio of the second to the third, and so on up to the ratio of the last but one to the last.

Thus if A, B, C, &c., X, Y, Z represent the magnitudes, then the ratio A: Z is said to be compounded of the ratios

A: B, B: C, &c. X: Y, Y: Z.

We shall denote this, for brevity, by the notation

A: Z = (A : B) (B : C) (C : D) &c. (X : Y) (Y: Z).

then, in accordance with the foregoing definition and notation, (A : B) (C : D) will be properly interpreted to mean the ratio which is compounded of ratios, of the form a: B and B : Y, that are the same with the ratios A: B and C : D. Similarly for the rest of the magnitudes.

This may be briefly expressed as follows

(A : B) (CD) (E : F) &c. (W: X) (Y : Z)

Note (1) By this definition of compound ratio, we see that Theorem 8 and its Cor. can be both included in this brief enunciation-Two ratios which are compounded of two sets of equal ratios are themselves equal.

Note (2) The ratio which is compounded of reciprocal ratios is a ratio of equality, that is unity.

For (AB) (B : A) = A : A.

Note (3)-Duplicate ratio is the ratio compounded of two equal ratios; triplicate ratio is the ratio compounded of three equal ratios; and so on.

ADDENDA TO BOOK v.

When four magnitudes are of the same kind, the principle alternando can be used; and, in that case, ex æquali and componendo can be much more easily proved than when the second pair of magnitudes are of a different kind from the first pair. X:Y,

=

Thus let A: B
and B: C Y: Z;

=

Again, let A, B, C, D be four magnitudes of the same kind, such that

It has been stated, in Book v, that pairs of magnitudes of the same kind exist, which are incapable of being measured by any (the same) unit: or, in other words, that there is no unit of measurement which is contained in each of such a pair an exact number of times.

We proceed to demonstrate this in two particular cases.

LEMMA-If a magnitude X measures each of two magnitudes A and B, then X also measures the difference of A and B.

THEOREM (1)—The segments of a line divided in medial section (ii. 11) are incommensurable.

It was shown Cor. to ii. 11 (p. 99) that if a line AB is divided in medial

A

that AY

=

BX, then AX is divided in medial section in Y, and also BY in X.

Simrly. if Z is taken in BX, so that BZ divided in medial section.

=

XY, then BX and YZ are each

Whence, 1o, BX does not measure AX;

and, 2o, if there is any finite line M which measures AX and BX, then (by the Lemma) M measures YX and ZX, the parts of a line divided in medial section, and which < 1⁄2 AB.

i. e. M measures ZX, which < AB.

And, after p repetitions of this process, we should get that M measures a line

which <

I

4P

AB; i. e. that a finite line measures a line which can be made

as small as we please. But this is absurd.

So that the ratio (greater segt. : lesser segt.) is more and more nearly approximated to by taking the ratio of each successive term of the following series to the one that precedes it—

I, I, 2, 3, 5, 8, 13, 21, 34, 55, 89, &c.,

where it will be found that each term is the sum of the two terms preceding. Thus, if 89 represents the whole line,

THEOREM (2)—A side and a diagonal of a square are incommensurable.

Then the following geometric facts are either obvious, or easily proved—

Whence, 1o, AB does not measure BD;

and, 2o, if there is any finite line M which measures AB and BD, then (by the Lemma) M measures BX and BY, the side and diag. of a sq. whose side < AB.

Simrly. M measures BZ, which < BX, i. e. which < & AB.

And, after p repetitions of the process, we should get that M measures a line

AB; i. e. that a finite line measures a line which can be made

The ratio (diagonal : side) is in fact what is denoted by the symbol √2.