not proportional, one of the antecedents, as A, must be contained in some multiple Q of its consequent B, oftener than C is contained in S, a like multiple of D (Def. 7.).

A B C D

P Q R S

Let P be the greatest multiple of A which does not exceed Q, and let R be a like multiple of C.

Then R must be greater than S, for R contains C as often as Q contains A, which, by hypothesis, is oftener than S contains C; so that equimultiples P, R of the antecedents, and equimultiples Q, S of the consequents may be found, such that R shall be greater than S, while P is not greater than Q, which contradicts the hypothesis.

PROPOSITION V. THEOREM.

If any number of homogeneous magnitudes be proportional, as one antecedent is to its consequent, so is the sum of the antecedents to the sum of the consequents. First, let there be four magnitudes, or the proportion A: B:: C: D, then also A: B:: A+C: B+D.

For let P, R be equimultiples of A, C, and Q, S equimultiples of B, D.

A B C D

P Q R S

Then (Prop. III.), if P>Q, R>S, or if R>S, P>Q; therefore if PQ, (P+R)>(Q+S), and if (P+R)>(Q+S), P>Q; for if, in this last case, P were not greater than Q, R could not be greater than S; and, therefore, P+R could not be greater than Q+S. Now, P and (P+R) are any equimultiples of A and (A+C) (Prop. I.), in like manner Q and (Q+S) are any equimultiples of B and (B+D); therefore (Prop. IV.)

A: B: A+C: B+D.

Let there be six magnitudes, A: B:: C: D:: E: F; then, with respect to the first four, there will be the proportion A: B: A+C: B+D, while the last four furnish the proportion C: D:: C+E: D+F, but A: B:: C: D; therefore (Prop. II.) A: B::C+E:D+F; hence, from what has been already demonstrated,

AB A+C+E: B+D+F, and so on for any number of proportionals.

Cor. 1. Since A: B:: A: B :: A: B, &c., it follows that A: B :: A+ A+ A+ &c. : B+ B+ B+ &c., that is, two magnitudes and their like multiples are proportional.

Cor. 2. Hence also two magnitudes and their like submultiples are proportional.

Cor. 3. Wherefore, in any proportion, one antecedent is to its consequent as any multiple or submultiple of the other antecedent is to a like multiple or submultiple of its consequent (Prop. II.).

Cor. 4. And moreover, if in any proportion like multiples or like submultiples of either the two first, or the two last terms be taken, and like multiples or submultiples of the others, the results will be proportional.

If in any proportion like multiples of the antecedents and like multiples of the consequents be taken, the results will be proportional.

In the proportion A: B:: C: D, let P, R be equimultiples of A, C, and Q, S equimultiples of B, D; then P: Q : : R : S. For let P', R' be any equimultiples of P, R; and Q', S' any equimultiples of Q, S;

P Q R S

P'Q' R' S'

Then it is obvious that P', R' must be equimultiples of A, C, and Q'S' equimultiples of B, D; therefore (Prop. III.) if P'>Q', then R'>S', and if R'> S', then P'>Q', and P', R' are any equimultiples of P, R, while Q', S are any equimultiples of Q, S; consequently (Prop. IV.),

P : Q :: R : S.

Cor. 1. In any proportion the first term is to any multiple of the second as the third is to a like multiple of the fourth.

For, as above, let P, R be any equimultiples of A, C; and Q, S any equimultiples of B, D; while Q', S are any equimultiples of Q, S; these last will obviously be equimultiples of B, D, and consequently (Prop. III.) if P>Q then R>S, and if RS then P>Q, and P, R are any equimultiples of A, C, while Q', S' are any equimultiples of Q, S; therefore (Prop. IV.)

AQ C: S.

:

Cor. 2. It follows moreover that any submultiple of the first term is to the second as a like submultiple of the third is to the

fourth, for, in the last proportion, Q, S are any equimultiples of B, D; and if P, R be the same submultiples of A, C, we have, by Cor. 4. Prop. V.

PB:: R: D.

PROPOSITION VII. THEOREM.

If in a proportion, an antecedent be either greater or less than its consequent, the other antecedent will, in like manner, be either greater or less than its consequent.

Let the proportion be A: B:: C: D; and suppose first that A>B; then also C> D.

For let Q, S be any equimultiples of B, D.

Then because A > B, Q contains B oftener than it contains A; and, since S contains D as often as Q contains B, it follows that S contains D oftener than Q contains A; but (Def. 7.) Q contains A as often as S contains C, therefore S contains Ď oftener than it contains C, and consequently C>D.

Next let A <B, then also C<D.

For, whatever be the difference between A and B, a multiple thereof may be taken, so great as to exceed A; and if the same multiple of B be taken, it must evidently contain A oftener than it contains B. Let Q be this multiple of B, and let S be an equimultiple of D. Then since S contains C as often as Q contains A, S must contain C oftener than Q contains B; but Q contains B as often as S contains D, consequently S contains C oftener than it contains D; hence C< D. Cor. Therefore if one antecedent be equal to its consequent, the other antecedent will be equal to its consequent.

PROPOSITION VIII. THEOREM.

The terms of a proportion are proportional when taken inversely, that is, as the second is to the first, so is the fourth to the third.

Let the proportion be A: B:: C: D, then also B: A::

D: C.

For let P, R be any equimultiples of A, C, and Q, S any equimultiples of B, D.

BA D C

Q P PS Ꭱ

:

Then (Prop. VI.) P Q R S; therefore (Prop. VII.), if Q>P, then S> R, and if S>R, then Q>P, consequently, (Prop. IV.)

BA: D: C.

Cor. 1. In any proportion, a multiple of the first term is to the second, as a like multiple of the third term is to the fourth (Cor. 1. Prop. VI.); also the first term is to a submultiple of the second as the third is to a like submultiple of the fourth.

Cor. 2. It follows from this and Cor. 2. Prop. VI. that if in a proportion like submultiples of the antecedents and like submultiples of the consequents be taken, the results will be proportional.

PROPOSITION IX. THEOREM.

In a proportion consisting of homogeneous magnitudes, if one antecedent be greater than the other, the consequent of the former will be greater than the consequent of the latter.

In the proportion A: B:: C: D, let A > C, then also B>D. By inversion (Prop. VIII.) B: A :: D: C, and whatever be the difference between A and C, it is possible for a multiple thereof to exceed D, and consequently such a multiple of A must contain D oftener than an equimultiple of C. Let P, R be these equimultiples of A, C.

Then (Def. 7.) P does not contain B oftener than R. contains D, but P does contain D oftener than R contains it; consequently P contains D oftener than it contains B; therefore B>D.

Cor. 1. It follows by inversion, that if one consequent be greater than the other, the antecedent of the former will be greater than that of the latter.

Cor. 2. Consequently if the antecedents be equal, the consequents will be equal, and if the consequents be equal the antecedents will be equal.

Cor. 3. Hence, and from Prop. II. if in two proportions there be three corresponding terms in each respectively equal, the fourth terms will be equal.

PROPOSITION X. THEOREM.

If the terms of a proportion are all of the same kind, they are proportional when taken alternately, that is, as the first is to the third, so is the second to the fourth.

Let the proportion be A: B :: C: D, then also A : C :: B: D.

For let P, Q be any equimultiples of A, B, and R, S any equimultiples of C, D.

D

A CB
P R Q S

:

Then (Prop. V. Cor. 4.) P Q R S; therefore, (Prop. IX.) if P >R, then Q>S, and if Q>S, then P>R; consequently (Prop. IV.)

A: C: B : D.

Cor. 1. Hence, and from Cor. 3. to Proposition V. it follows that, in such a proportion, the first term is to the third as any multiple or submultiple of the second to a like multiple or submultiple of the fourth.

Cor. 2. Likewise (Cor. 4. Prop. V.) like multiples or like submultiples of the first and third terms are to each other as like multiples or submultiples of the second and fourth terms.

PROPOSITION XI. THEOREM.

If, in a proportion, an antecedent be a multiple or submultiple of its consequent, the other antecedent will be a like multiple or submultiple of its consequent. In the proportion A: B:: C: D, let A be a multiple of B, then C will be the same multiple of D.

Take Q equal to A, and let S be the same multiple of D that Q or A is of B.

Then (Prop. VI. Cor.) A: Q :: C: S, but A is equal to Q ; therefore (Prop. VII. Cor.) C is equal to S, but S is the same multiple of D that A is fof B; therefore C is the same multiple of D that A is of B.

Again, let A be a submultiple of B, then will C be a like submultiple of D; for, by inversion (Prop. VIII.), B: A:: D: C; therefore, as just shown, D is the same multiple of C that B is of A: in other words, C is the same submultiple of D that A is of B.