PROPOSITION XIII.—THEOREM. If the first has to the second the same ratio which the third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the sixth, the first shall also have to the second a greater ratio than the fifth has to the sixth. LET a the first, have the same ratio to b the second, which c the third, has to d the fourth, but c the third, to d the fourth, a greater ratio than e the fifth, to f the sixth : also the first a shall have to the second b a greater ratio than the fifth e to the sixth f. Because c has a greater ratio to d, than e to f, there are some equimultiples of c and e, and some of d and f, such that the multiple of с is greater than the multiple of d, but the multiple of e is not greater than the multiple of f(v. def. 7): let such be taken, and of c, e, let g, h be & h k 1 equimultiples, and k, 1 equimultiples of d, f, so that go be greater than k, but h not greater than 1; and whatever multiple g is of c, take m the same multiple of a; and whatever multiple k is of d, take n the same multiple of b : then, because a is to b, as c to d, and of a and c, m and g are equimultiples : and of b and d n and k are equinsultiples ; if m be greater than n, g is greater than k; and if equal, equal ; and if less, less (v. def. 5); but g is greater than k; therefore m is greater than n : but h is not greater than 1; and m, h are equimultiples of a, e; and n, I equimultiples of bf: therefore a has a greater ratio to b than e has to f (v. def. 7). Wherefore, if the first, &c. Q. E. D. Cor. And if the first have a greater ratio to the second, than the third has to the fourth, but the third the same ratio to the fourth, which the fifth has to the sixth ; it may be demonstrated, in like manner, that the first has a greater ratio to the second, than the fifth has to the sixth. PROPOSITION XIV.—THEOREM. If the first has to the second the same ratio which the third has to the fourth ; then, if the first be greater than the third, the second shall be greater than the fourth ; and if equal, equal ; and if less, less. LET the first a have to the second b, the same ratio which the third C, has to the fourth d ; if a be greater than c, b is greater than d. Because a is greater than c, and b is any other magnitude, a has to b a greater ratio than c to b (v. 8): but, as a is to b so is c to d; therefore also c has to d a greater ratio than c has to b (v. 13). But of two magnitudes, that to which the same has the greater ratio is the lesser (v. 10). Wherefore d is less than b ; that is, b is greater than d. a b c d b c d fig. 3. Secondly, if a be equal to c, b is equal to d : for a is to b as c, that is a, is to d: b therefore is equal to d (v. 9). Thirdly, if a be less than c, b shall be less than d : for c is greater than a, and because c is to d as a is to b, d is greater than b, by the first case ; wherefore b is less than d. Therefore if the first, &c. Q. E. D. PROPOSITION XV.—THEOREM, Magnitudes have the same ratio to one another which their equimultiples have. LET a b be the same multiple of c that de is of f; c is to f as a b to de Because a b is the same multiple of c that de is of f; there are as many magnitudes in a b equal to C, as there are in de a equal to f: let a b be divided into magnitudes, each equal to c, viz. a g, gh, h b; and de into magnitudes, d each equal to f, viz. dk, kl, le : then the number of the first ag,gh, hb, shall be equal to the number of gt the last dk, kl, le: and because ag, gh, h b are all kH equal, and that dk, kl, le, are also equal to one another; is to dk as gh to kl, and as hb to le hh 1 (v. 7): and as one of the antecedents to its consequent, so are all the antecedents together to all the consequents together (v. 12); wherefore, as ag is to dk, so is a b to de: but ag is equal to c, and dk to f: therefore, ъ се f as c is to f so is a b to de. Therefore magnitudes, &c. Q. E. D. therefore ag PROPOSITION XVI.-THEOREM. If four magnitudes of the same kind be proportionals, they shall also be proportionals when taken alternately. Let the four magnitudes a, b, c, d be proportionals, viz. as a is to b, so is c to d: they shall also be proportionals when taken alternately; that is, a is to c, as b to d. Take of a and b any equimultiples whatever e and f; and of c and d take any equimultiples whatever and h: and because e is the same multiple of a that f is of b, and that magnitudes have the same ratio to one another which their equimultiples have (v. 15); therefore a is to b, as e is to f; but as a is to b, so is c to d; wherefore as c is to d, a so (v. 11) is e to f: again, be cause g, h are equimultiples of b d c, d, as c is to d so is to h f (v. 15): but as c is to do so is h e to f Wherefore, as e is to f so is g to h (v. 11). But when four magnitudes are proportionals, if the first be greater than the third, the second shall be greater than the fourth ; and if equal, equal ; if less, less (v. 14). Wherefore, if be greater than g, f likewise is greater than h ; and if equal, equal; if less, less; and e f are any equimultiples whatever of a, b; and g, h any whatever of c, d. Therefore a is to c as b to d (v. def. 5). If then four magnitudes, &c. Q. E. D. e PROPOSITION XVII.—THEOREM. If magnitudes, taken jointly, be proportionals, they shall also be proportionals when taken separately; that is, if two magnitudes together have to one of them the same ratio which two others have to one of these, the remaining one of the first two shall have to the other the same ratio which the remain ing one of the last two has to the other of these. LET a b, be, cd, df be the magnitudes taken jointly which are proportionals ; that is, as a b to be so is cd to df: they shall also be proportionals taken separately, viz. as a e to eb, so is cf to fd. Take of a e, eb, cf, fd any equimultiples whatever gh, hk, lm, mn: and again, of eb, fd take any equimultiples whatever kx, np: and because g h is the same multiple of a e that h k is of eb, wherefore gh is the same multiple (v. 1) of a e that gk is of a b: but g h is the same multiple of a e that lm is of cf; wherefore gk is the same multiple of a b that lm is of cf. Again, because lm is the same multiple of cf that mn is of fd; therefore im is the same multiple (v. 1) of cf that 1n is of cd: but lm was shewn to be the same multiple of cf that gk is of a b; & k therefore is the same multiple of a b that ln is of cd; that is, gk In are equimultiples of a b, cd. Next, because h k is the same multiple of e b that mn is of fd; and that kx is also the same multiple of e b that n p is of fd; therefore ha is the same multiple (v. 2) x P n of e b, that mp is of fd. And because a b is to be as cd is to df, and that of a b and (d, gk and in are equimultiples, and of eb and fd, hx and m p are equimultiples ; if g k be greater than hx, then I n is greater than mp; and if equal, kt equal ; and if less, less (v def. 5); but if g h be greater than kx, by adding the common part hk to both, gk is greater than hx; wherefore alson is greater than mp; and by taking away m n from h both, 1 m is greater than np: therefore, if g h be greater than k x, lm is greater than np. In like manner it may be demonstrated, that if g h be equal to kx, lm likewise is equal to n p; and if less, less : and gh, lm are any equimultiples whatever of a e, cf, and kx, n p are any whatever of e b, fd. Therefore (v. def. 5), as a e is to eb, so is cfto fd. 8 If then magnitudes, &c. Q. E. D. PROPOSITION XVIII.-THEOREM. If magnitudes, taken separately, be proportionals, they shall also be propor tionals when taken jointly, that is, if the first be to the second as the third to the fourth, the first and second together shall be to the second as the third and fourth together to the fourth. LET a e, e b, cf. fd be proportionals; that is, as a e to eb, so is cf to fd; they shall also be proportionals when taken jointly; that is, as a b to be, so cd to d f. Take of a b, be, cd, df any equimultiples whatever gh, hk, lm, mn: and again, of be, d f, take any equimultiples whatever ko, np: and because ko, np are equimultiples of be, df; and that k h, n m are equimultiples likewise of be, d f, if ko, the multiple of be, be greater than kh, which is a multiple of the same be, h np likewise the multiple of d f, shall be greater thân mn, the multiple of the same df; and if 0 ko be equal to k h, np shall be equal to nm; m and if less, less. First, let ko not be greater than kh, therefore np is not greater than nm: and because k gh, hk, are equimultiples of a b, be, and that åb is greater than be, therefore gh is greater (v. ax. 3) than hk; but ko is not greater than kh, wherefore gh is greater than ko. In like manner it be shewn, that lm is greater than b may np. Therefore, if ko be not greater than kh, d then gh, the multiple of a b, is always greater f than ko, the multiple of be; and likewise lm, the multiple of cd, greater than n p, the mul 1 tiple of df. g Next, let ko be greater than kh: therefore, as has been shewn, np 마 n is greater than nm: and because the whole gh is the same multiple of the whole a b, that h k is of be, the remainder gk is the same multiple of the remainder a e that gh is of a b (v. 5) : which is the same that lm is of cd. In like h manner, because Im is the same multiple of cd, P that m n is of d f, the remainder ln is the same multiple of the remainder cf, that the whole lm my is of the whole cd (v. 5): but it was shewn that lm is the same multiple of cd, that gk is of kr bi a e; therefore gk is the same multiple of a e, n d that ln is of cf; that is, gk, ln are equimul tiples of a e, cf: and because ko, np, are equif multiples of be, df, if from ko, np, there be taken k h, nm, which are likewise equimultiples of be, df, the remainders ho, mp are either 8 a 11 equal to be, d f, or equimultiples of them (v. 6). First, let ho, mp be equal to be, df; and because a e is to eb, as cf to fd, and that gk ln are equimultiples of a e, cf; gk shall be to eb, as ln to fd (v. 4 cor.) : but ho is equal to e b, and mp to fd; wherefore gk is to ho, as ln to mp. If therefore gk be greater than h o, 1 n is greater than mp; and if equal, equal ; and if less, less (5 ax.). But let ho, mp be equimultiples of eb, fd; and because a e is to eb, as cf to fd, and that of e, cf are taken equimultiples gk, ln; and of eb, fd, the equimultiples ho, mp; if g k be greater than ho ln is greater than mp; and if equal, equal ; and if less, less (v. def. 5) ; which was likewise shewn in the preceding case. If therefore gh be greater than ko, taking k h from both, gk is h p greater than ho; wherefore also l n is greater than mp; and consequently adding nm to both, im is greater than np: therefore, if g h be greater than ko, lm is greater than n p. k In like manner it may be shewn, that if g h be b equal to ko, lm is equal to np; and if less, less. And in the case in which ko is not d. greater than kh, it has been shewn that gh is eh always greater than ko, and likewise lm than f np: but gh, lm are any equimultiples of g al a b c d, and ko, n p are any whatever of be, C 1 df; therefore (v. def. 5), as a b is to be, so is cd to df. If then magnitudes, &c. Q. E. D. m n |