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.....2m A, (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 2A, 2B greater than 34 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 qA, provided that B is less than A, and Ρ than 4.

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.

NOTE 4. To make Def. v. clearer we give the following illustration. Suppose A, B, C, D to be four magnitudes; the scales of their multiples will then be-

where mA, mC stand for any equimultiples of A and C, and nB, nD stand for any equimultiples of B and D: then the Definition may be stated more briefly thus:

A is said to have the same ratio to B which C has to D, when m▲ is found in the same position among the multiples of B, in which mC is found among the multiples of D; or, which is the same thing, when the order of the multiples of A and B in the Scale of Relation of A and B, is precisely the same as the order of the multiples of C and D in the Scale of Relation of C and D; or, when every multiple of A is found in the same position among the multiples of B, in which the same multiple of C is found among the multiples of D.

NOTE 5. The use of Def. v. will be better understood by the following application of it.

To show that rectangles of equal altitude are to one another

Let AC, ac be two rectangles of equal altitude.

Let B, B' and R, R' stand for the bases and the areas of

these rectangles respectively.

Take AD, DE, EF,........
And ad, de, ef, fg, gh,....

.m in number, and all equal,

n in number, and all equal.

Complete the rectangles, as in the diagram.

Then base AF = mB,

base ahnB'.

rectangle AP m.R,

=

rectangle ap = nR',

Now we can prove, by superposition, that if AF be greater than ah, AP will be greater than ap, and if equal, equal; and if less, less.

That is, if mB be greater than nB', mR is greater than nR'; and if equal, equal; and if less, less.

Hence, by Def. v.,

B is to B' as R is to R'.

Hence we deduce two Corollaries, which are the foundation of the proofs in Book VI.

COR. I. Parallelograms of equal altitude are to one another as their bases.

For the parallelograms are equal to rectangles, on the same bases and between the same parallels.

COR. II. Triangles of equal altitude are to one another as their bases.

For the triangles are equal to the halves of the rectangles, on the same bases and between the same parallels.

N.B.-These Corollaries are proved as a direct Proposition in Eucl. vI. 1. Cor. II. could not, consistently with Euclid's method, be introduced in this place, for it assumes Proposition XI. of Book v.

DEF. VI. Magnitudes which have the same ratio are called Proportionals.

If A, B, C, D be proportionals, it is usually expressed by saying, A is to B as C is to D.

The magnitudes A and C are called the Antecedents of the ratios. B and D..................Consequents .

The antecedents are said to be homologous to one another, that is, occupying the same position in the ratios (óμóλoyoi), and the consequents are said to be homologous to one another.