For, if the first have not the same ratio to the se cond that the third has to the fourth, it will have to the second the same ratio that the third has to a mag nitude, either greater or less, than the fourth. But if the first have the same ratio to the second that the third has to a magnitude greater than the fourth, then, by article 10, certain equimultiples can be taken of the first and third, and certain equimultiples of the second and fourth, such, that the multiple of the first shall be less than the multiple of the second, but the multiple of the third not less than the multiple of the fourth ; and this would be contrary to the first of the above, suppositions.
Again, if the first has the same ratio to the second, that the third has to a magnitude less than the fourth, then, by article 9, certain equimultiplies can be taken of the first and third, and certain equimultiples of the second and fourth ; such, that the multiple of the first shall he greater than the multiple of the second, but the multiple of the third not greater than the multiple of the fourth ; and this would be contrary to the last of the three suppositions.
Lastly, if the multiple of the first be equal to the multiple of the second, and the multiple of the third to the multiple of the fourth, then the multiple of the first will have the same ratio to that of the second, that the multiple of the third has to that of the fourth ; and, con sequently. by article 7, the first will have the same ratio to the second that the third has to the fourth.
REMARK.—The fifth definition of the 5th book of Euclid, having been considered as a proposition, and established as such by demonstration, the doctrine of ratio. and proportion may be extended as in that book. The same extension, however, may be effected by, means, of the first seven of the preceding articles, as a. :oundation.Fonnpcted with this evident truth, that two magnitudes of the same must have the same ratio to one another, as the numbers which measure them, or express their relative values. Whatever is proved
as to the proportionality of the numbers, must be ap plicable to the magnitudes to which they arc strictly analogous.
XII. If four numbers be proportionals, the product of the first and fourth is equal to the product of the se cond and third. Thus if N, P, M, Q, he four num bers, and if it be N : : M : Q, then N x Q = P x M. For dividing the first and third of the proportion als by N, and the second and fourth by P, we have, ac Q cording to article 7, I : I : : 7 and, therefore, by article 5,— — Q , and N x x M.
XIII. If there be four numbers, such that the proXiii. If there be four numbers, such that the pro- duct of the first and fourth is equal to the product of the second and third, the first has the same ratio to the second, that the third has to the fourth. Thus, if N, P, M, Q, be four numbers, and if N x Q = P x M, then. N : P : : M : Q. For, let R be a fourth propor tional number to N, P, M ; and then, by the last arti cle, N x R=P x M. But, by hypothesis, P x M=N X Q; and, therefore, N x Q=N x R. Consequently, N: P : : M : Q.
In the following articles, let the small letters, a, b, c. &c. denote the numbers which express the relative va lues of the magnitudes A. B, C, &c., and then the subsequent explanation applies to them all. The large letters are used in the data and assertions, the demon strations are effected by the small letters, and the large are put instead of the small in the conclusion, thereby intimating that the assertion, has been proved.
XIV. If four magnitudes of the same kind be pro portionals, they will also be proportionals when taken alternately. Thus, if it be A : B : : C: D, then A : C : : B: D.
For it being a : b : :c:d, by article 12, a X d= b X c; and, therefore, by article 13, a :c b : d, that is