10. If there be four proportionals, the difference of the two first, to the difference of the two last, will be as the first antecedent is to the second antecedent, or as the first consequent to the second consequent. For if A. B :: C. D be proportionals, then by division A - B. B :: C - D. D will be proportionals; and by permutation A - B. C - D :: B. D; that is, the differences are proportional to the consequents, and therefore they are so also to the antecedents. 11. Of four proportionals, if the first be greater than the second, the third also shall be greater than the fourth. For seeing the first is greater than the second, the proportion of the first to the second is the proportion of excess; but the proportion of the third to the fourth is the same with that of the first to the second; and therefore also the proportion of the third to the fourth is the proportion of excess; wherefore the third is greater than the fourth. In the same manner it may be proved, that whensoever the first is less than the second, the third also is less than the fourth; and when those are equal, that these also are equal. 12. If there be four proportionals whatsoever, A. B :: C.D, and the first and third be multiplied by any one number, as by 2; and again the second and fourth be multiplied by any one number, as by 3; and the product of the first 2 A, be greater than the product of the second 3 B; the product also of the third 2 C, will be greater than the product of the fourth 3 D. But if the product of the first be less than the product of the second, then the product of the third will be less than that of the fourth. And lastly, if the products of the first and second be equal, the products of the third and fourth shall also be equal. Now this theorem is all one with Euclid's definition of _the same proportion_; and it may be demonstrated thus. Seeing A. B :: C. D are proportionals, by permutation also (art. 6, coroll. I.) A. C :: B . D will be proportionals; wherefore (by coroll. IV. art. 6) 2 A. 2 C :: 3 B. 3 D will be proportionals; and again, by permutation, 2 A. 3 B :: 2 C. 3 D will be proportionals; and therefore, by the last article, if 2 A be greater than 3 B, then 2 C will be greater than 3 D; if less, less; and if equal, equal; which was to be demonstrated. [Sidenote: Composition of proportions.]