Arithmetical Exposition.

Suppose the first of four numbers to be three times the second, and the third three times the fourth, as 33, 11, 24, and 8.

Then, if we take four times the first (132) and four times the third (96), it is evident that 132 is the same multiple of 11 the second, that 96 is of 8 the fourth, for 132 is 12 times 11, and 96 is 12 times 8.

Algebraical Exposition.

Let the four magnitudes be m a, a, m b, and b, take equimultiples of the first and third, as, n times the first, and n times the third; then it is evident that nm a, or n m times a, is the same multiple of a, that n m b, or n m times b, is of b.

Four magnitudes,

DEFINITION V.

▲, and ▲, are said to be propor

tionals when every equimultiple of the first and third be taken, and every equimultiple of the second and fourth, as,

Then taking every pair of equimultiples of the first and third, and every pair of equimultiples of the second and fourth,

That is, if twice the first be greater, equal, or less than twice the second, twice the third will be greater, equal, or less than twice the fourth; or, if twice the first be greater, equal, or less than three times the second, twice the third will be greater, equal, or less than three times the fourth, and so on, as above expressed.

In other terms, if three times the first be greater, equal, or less than twice the second, three times the third will be greater, equal, or less than twice the fourth; or, if three times the first be greater, equal, or less than three times the second, then will three times the third be greater, equal, or less than three times the fourth; or if three times the first be greater, equal, or less than four times the second, then will three times the third be greater, equal, or less than four times the fourth, and so on. Again,

And so on, with any other equimultiples of the four magnitudes, taken in the same manner.

Euclid expresses this definition as follows:

The first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth; if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth; or, if the multiple of the first be equal to that of the second, the multiple of the third is also equal to that of the fourth; or, if the multiple of the first be greater than that of the second, the multiple of the third is also greater than that of the fourth.

In future we shall express this definition generally, thus :If M

orm, when MA, or ms.

Then we infer, that the first, has the same ratio to, the second, which▲, the third, has to ▲ the fourth: expressed in the succeeding demonstrations, thus:

And if

infer if M●—‚— or—mo, That is, if the first be to the

▲▲ we shall then will M▲, or m s. second, as the third is to the fourth; then, if M times the first be greater than, equal to, or less than m times the second, then shall M times the third be greater than, equal to, or less than m times the fourth, in which M and m are not to be considered particular multiples, but every pair of multiples whatever; nor are such marks as A,, &c., to be considered any more than representatives of geometrical magnitudes.

The student should thoroughly understand this definition before proceeding further.

c 2

PROP. IV. THEO.

If the first of four magnitudes have the same ratio to the second, which the third has to the fourth, then any equimultiples whatever of the first and third shall have the same ratio to any equimultiples of the second and fourth; viz., the equimultiple of the first shall have the same ratio to that of the second, which the equimultiple of the third has to that of the fourth.

Let : 0 multiple of 3

2 A., every equiand ▲, and every

▲ ▲, then 3 : 2:3▲ and 3 ▲ are equimultiples of equimultiple of 2□ and 2 ▲, are equimultiples of □ and ▲ : (3. B. V).

That is, M times 3 and M times 3 ▲ are equimultiples of and ▲, and m 2 and m 2 ▲ are equimultiples of 2 and 2▲; but : 0 :: ▲ ▲ (hyp.); .. if M 3 —, —, or — m2, then M 3 ▲, 2▲ (by the fifth definition), and, therefore 3:2:3▲: 2 A (by the fifth definition).

The same reasoning holds good if any other equimultiple of the first and third be taken, and any other equimultiples of the second and fourth.

.. If the first of four magnitudes, &c.

Arithmetical Illustration.

Let 3:59 15 be four numbers that are proportionals; and let us multiply the first and third of these numbers by any other, say 5, and the second and fourth by any number, say 2, then we have 15 10 :: 45: 30, for it is easily observed that 13 = 43, or 15 contains 10 as often as 45 contains 30.

Cor. Likewise, if the first have the same ratio to the second, which the third has to the fourth, then also any equimultiples of the first and third have the same ratio to the second and fourth; and, in like manner, the first and the third have the same ratio to any equimultiples whatever of the second and fourth.

That is, if ab :: cd, then ma: b:: mc : d, and a ; n b :: c: n d.