In the arithmetical series, a, a + r, a + 2 r, a + 3 r, &c., r is sometimes called the common arithmetical ratio; and, in the geometrical series, a, ar, ar2, ars, &c., r is called the common ratio; in the first of these series, r has no right whatever to the term ratio; and, in the second, the ratio of the first term to the second, the second to the third, the third to the fourth, is expressed by and not by r.

1

IV.

Magnitudes are said to have a ratio to one another, when they are of the same kind; and the one which is not the greater can be multiplied so as to exceed the other

[The other definitions will be given throughout the book, where their aid is first required.]

AXIOM S.

I.

EQUIMULTIPLES or equisubmultiples of the same, or of equal magnitudes, are equal.

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

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

Let 2 A 2 B, then

AC B;

or, let 3 A3 B, then

AC B;

or, let m Am B, then

A B

PROP. I. THEO.

If any number of magnitudes be equimultiples of as many others, each of each: what multiple soever any one of the first is of its part, the same multiple shall of the first magnitudes taken together be of all the others taken together.

as there are in

The same demonstration holds in any number of magnitudes, which has here been applied to three.

.. If any number of magnitudes, &c.

Then the sum of 7, 42, 35, and 14 or 98, is the same multiple of 1, 6, 7, and 2 together, that 7 is of 1, that 42 is of 6, &c., i. e. 98 is seven times 14.

:.ma+mb + mc + md+,&c., or m (a+b+c+d+,&c.,) is the same multiple of a+b+c+d+, &c., that ma is of a, m b is of b, &c.

PROP. II. THEO.

If the first magnitude 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, then shall the first, together with the fifth, be the same multiple of the second that the third, together with the sixth, is of the fourth.

Let, the first, be the same multiple of, the second, that ▲▲▲, the third, is of A, the fourth; and let BOO, the fifth, be the same multiple of●, the second, that ▲▲▲▲, the sixth, is of A, the fourth.

ther, is the same multiple of, the second, that

the third and sixth together, is of the same multiple of ▲, the

fourth; because there are as many magnitudes in

{

=

Then it is evident, that 20 and 18 together is the same multiple of 2, that 350 and 315 together is of 35.

Algebraical Exposition.

m a the first is the same multiple of a the second that mb the third is of b the fourth, and n a the fifth is the same multiple of a the second, that n b the sixth is of b the fourth; then it is evident that m a + n a, or (m + n) a the first and fifth together, is the same multiple of a the second that (m b + n b), or (mn) b the third and sixth together, is of b the fourth.

PROP. III. THEO.

If the first of four magnitudes be the same multiple of the second that the third is of the fourth, and if any equimultiples whatever of the first and third be taken, those shall be equimultiples; one of the second, and the other of the fourth.

the same reasoning is applicable in all cases.

.. If the first of four, &c.