2. When two proportions have the same antecedents, the con. sequents may be put into a proportion; for, if we have

A: B:: C: D,

A: E:: C: F,

by changing the place of the means, these proportions will be

IV. Other changes, besides the transposition of terms, may be made among proportionals without destroying the equality of the product of the extremes to that of the means.

1. If to the consequent of a ratio we add the antecedent, and compare this sum with the antecedent, this last will be contained once more than it was in the first consequent; the new ratio then will be equal to the primitive ratio increased by unity. If the same operation be performed upon the two ratios of a proportion, there will evidently result from it two new ratios equal to each other, and consequently a new proportion.

Let there be, for example, the proportion

2. If from the consequent of a ratio we subtract the antecedent, and compare the difference with the antecedent, this last will be contained once less than it was in the first consequent ; the new ratio will be equal to the primitive ratio diminished by unity. If the same operation be performed upon the two ratios of a proportion, there will result from it two new ratios equal to each other, and consequently a new proportion.

There being a proportion among any magnitudes whatever designated by the letters

If we change the place of the means in these results, they will become

and, since the ratios A: C, B : D, are equal, we obtain

a result which may be thus enunciated.

In any proportion whatever, the sum of the two first terms is to the sum of the two last, and the difference of the two first terms is to the difference of the two last, as the first is to the third, or as the second is to the fourth.

Moreover the two ratios A: C, B: D, being common to the two proportions above obtained, it follows that the other ratios of the same proportions are equal, and that consequently

B+A:D+C:: B-A: D-C,

or, by changing the place of the means,

BABA::D+C: D-C;

that is, the sum of the two first terms of a proportion is to their difference, as the sum of the two last is to their difference.

For example,

A and B are the antecedents, C and D the consequents; and the proportions

B+A:D+ C:: A: Cor: : B : D,

B-A:D-C:: A: Cor: : B : D,

answer to the following enunciation;

The sum of the antecedents of a proportion is to the sum of the consequents, and the difference of the antecedents is to the difference of the consequents, as one antecedent is to its consequent ;

Whence it follows, that the sum of the antecedents is to their difference, as the sum of the consequents is to their difference. If we have a series of equal ratios

A:B::C:D :: E : F,

by considering only the two first, which form the proportion A: B: C: D,

we obtain by what precedes

A+ C: B+D::A: B;

and, since the third ratio E: F, is equal to the first A: B, we have

A+ C: B+ D : : E : F.

If we take the sum of the antecedents and that of the consequents in this last proportion, the result will be

A+ C+E: B+D+F::E: For:: A: B.

By proceeding in the same manner with any number of equal ratios, it will be seen, that the sum of any number whatever of antecedents is to the sum of their consequents, as one antecedent is to its consequent.

V. Let there be any two proportions

A: B:: C: D,

E: F:: G: H,

if we multiply them in order, that is, term by term, the products will form a proportion, thus

AXE:BXF:: CxG: D× H,

BXF DX H

This is evident, since the new ratios

AXE' CX G

are respec

whence it follows, that the squares of four proportional quantities form a new proportion.

that is, the cubes of four proportional quantities form a new pro portion.

VI. When a proportion is said to exist among certain magnitudes, these magnitudes are supposed to be represented, or to be capable of being represented by numbers; if, for example, in the proportion

A: B:: C: D,

A, B, C, D, denote certain lines, we can always suppose one of these lines, or a fifth, if we please, to answer as a common measure to the whole, and to be taken for unity; then A, B, C, D, will each represent a certain number of units, entire or fractional, commensurable or incommensurable, and the proportion among the lines A, B, C, D, becomes a proportion in numbers.

Hence the product of two lines A and D, which is called also their rectangle, is nothing else than the number of linear units contained in A multiplied by the number of linear units contained in B; and we can easily conceive this product to be equal to that which results from the multiplication of the lines B and C.

The magnitudes A and B in the proportion

A: B:: C: D,

may be of one kind, as lines, and the magnitudes C and D of

another kind, as surfaces; still these magnitudes are always to be regarded as numbers; A and B will be expressed in linear units, C and D in superficial units, and the product A× D will be a number, as also the product B× C.

Indeed, in all the operations, which are made upon proportional quantities, it is necessary to regard the terms of the proportion as so many numbers, each of its proper kind; then we shall have no difficulty in conceiving of these operations and of the consequences which result from them.