3. A multiple of a greater magnitude is greater than the same multiple of a less.

4. That magnitude, of which a multiple is greater than the same multiple of another, is greater than that other magnitude.

NOTE 1. If A and B be two commensurable magnitudes, it is easy to show that there is some multiple of A, which is equal to some multiple of B.

For let M be a common measure of A and B; then the scale of multiples of M is

M, 2M, 3M,.

V. Ax. 1.

Now one of the multiples in this scale, suppose pM, is equal to A, and one ...........suppose qM,...........................................B. Hence the multiple qpM is equal to qA, and the same multiple is equal to pB; and therefore q▲ = pB.

PROPOSITION I. (Eucl. v. 1.)

I. Ax. 1.

If any number of magnitudes be equimultiples of as many, each of each; whatever multiple any one of them is of its submultiple, the same multiple must all the first magnitudes, taken together, be of all the other, taken together.

Let A be the same multiple of C that B is of D.

Then must A, B together be the same multiple of C, D together that A is of C.

Let A
Then B

=

C, C, C............repeated m times.
D,D,D.. ....repeated m times.

.. A, B together = C, D ; C, D; C, D;......repeated m times. .. A, B together is the same multiple of C, D together that A is of C.

E. D.

PROPOSITION II. (Eucl. v. 2.)

If the first be the same multiple of the second that the third is of the fourth, and the fifth the same multiple of the second that the sixth is of the fourth; the first together with the fifth must be the same multiple of the second, that the third together with the sixth is of the fourth.

Let A, B, C, D, E, F be six magnitudes, such that

A is the same multiple of B, that C is of D, and E is the same multiple of B, that F is of D. Then must A, E together be the same multiple of B, that C, F together is of D.

then CD, D, D,............repeated m times.

Also, let E = B, B, B,..

then FD,D,D,.

repeated m times;

repeated n times;

..repeated n times.

.repeated m+n times,

and C, F together = D, D, D,................................repeated m+n times. A, E together is the same multiple of B,

that C, F together is of D.

PROPOSITION III. (Eucl. v. 3.)

Q. E. D.

If the first be the same multiple of the second that the third is of the fourth; and if of the first and third there be taken equimultiples, these must be equimultiples, the one of the second, and the other of the fourth.

Let A be the same multiple of B that C is of D ; and let E and F be taken equimultiples of A and C. Then must E and F be equimultiples of B and D.

.. E is the same multiple of B that F is of D.

Q. E. D.

SECTION II.

On Ratio and Proportion.

DEF. III. If A and B be magnitudes of the same kind, the relative greatness of A with respect to B is called the RATIO of A to B.

NOTE 2. When A and B are commensurable, we can estimate their relative greatness by considering what multiples they are of some common standard. But as this method is not applicable when A and B are incommensurable, we have to adopt a more general method, applicable both to commensurable and incommensurable magnitudes.

If A and B be magnitudes of the same kind, commensurable or incommensurable, the scale of multiples of A is

A, 2A...mA, (m+1)A.....2mA, (2m+1)A...3mA...nmA...

and the Ratio of B to A is estimated by considering the position which B, or some multiple of B, occupies among the multiples of A.

If A and B be commensurable, a multiple of B can be found, such that it would occupy the same place among the multiples of A, which is occupied by some one of the multiples of A; that is, this particular multiple of B represents the same magnitude as that, which is represented by some one of the multiples of A. See Note 1, p. 213.

If, for example, the 7th multiple in the scale of B represents the same magnitude as that which is represented by the 5th multiple in the scale of A, or in other words, if 7B = 5A, we are enabled to form an exact notion of the greatness of B relatively to A.

When A and B are incommensurable, the relation mA=nB can have no existence; that is, no pair of multiples, one in each of the scales of multiples of A and B, represent the same magnitude. But we can always determine whether a particular multiple of B be greater or less than some one of the multiples of A; that is, we can always find between what two successive multiples of A any given multiple of B lies.

Hence, whether A and B be commensurable or incommensurable, we can always form a third scale, in which the multiples of B are distributed among the multiples of A. Suppose, for example, we discover the following relations between particular multiples of A and B :

B greater than A and less than 24, 2B greater than 3A and less than 44, 3B greater than 6A and less than 74,

and so on; the third scale will commence thus

A, B, 2A, 3A, 2B, 4A, 5A, 6A, 3B, 7A,

and so on; the scale not being formed by any law, but constructed by special calculations for each term.

Such a scale we call the SCALE OF RELATION of A and B, and we give the following DEFINITION :—

The Scale of Relation of two magnitudes of the same kind is a list of the multiples of both ad infinitum, all arranged in order of magnitude, so that any multiple of either magnitude being assigned, the scale of relation points out between which multiples of the other it lies.

NOTE 3. It may here be remarked that, if A and B be two finite magnitudes of the same kind, however small B may be, we may, by continuing the scale of multiples of B sufficiently far, at length obtain a multiple of B greater than A.

Also, if B be less than A, one multiple at least of the scale of B will lie between each two consecutive multiples of the scale of A. From these considerations we shall be justified in assuming

(1.) That we can always take mB greater than A or than pA.

(2.) That we can always take nB such that it is greater than pA but not greater than q4, provided that B is less than A, and p than q.

We can now make an important addition to Definition III., so that it will run thus :

If A and B be magnitudes of the same kind, the relative greatness of A with respect to B is called the Ratio of A to B, and this Ratio is determined by, that is, depends solely upon, the order in which the multiples of A and B occur in the Scale of Relation of A and B.

DEF. IV. Magnitudes are said to have a Ratio to each other, which can, being multiplied, exceed each the other.

This definition is inserted to point out that a ratio cannot exist between two magnitudes unless two conditions be fulfilled :- first, the magnitudes must be of the same kind; secondly, neither of them may be infinitely large or infinitely small. See Note 3.

DEF. V. When there are four magnitudes, and when any equimultiples of the first and third being taken, and any equimultiples of the second and fourth, if, when the multiple of the first is greater than that of the second, the multiple of the third is greater than that of the fourth, and when the multiple of the first is equal to that of the second, the multiple of the third is equal to that of the fourth, and when the multiple of the first is less than that of the second, the multiple of the third is less than that of the fourth, then the first of the original four magnitudes is said to have to the second the same ratio which the third has to the fourth.