PROPOSITION II.—THEOREM. If the first magnitude be the same multiple of the second that the third is of the fourth, and the fifth the same multiple of the second that the sixth is of the fourth ; then shall the first together with the fifth be the same multiple of the second, that the third together with the sixth is of the fourth. LET a b the first, be the same multiple of c the second, that d e the third is of f the fourth ; and bg the fifth, the same multiple of c the second, that e h the sixth is off the fourth : then is ag, the first, together with the fifth, the same multiple of c the second, that d h, the third, together with the sixth, is of f the fourth. di a h 1 f 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 equal to f. In like manner, as many as there are in bg equal to c, so many are there in e h equal to f: as many, then, as are in the whole a g. equal to c, so many are there in the whole d h equal to f; therefore a g is the same multiple of c, that dh is of f; that is, ag the first and fifth together, is the same multiple of the second c, that dh the third and sixth together is of the fourth f. If, therefore, the first be the same multiple, &c. Q. E. D. COR. From this it is plain, that if any number of magnitudes a b, bg; gh, be multiples of another c; and as many de ek, kl be the same multiples of f, each of each ; the whole of the first, viz. a h, is the same multiple of c, that the whole of the last, viz. dl, is of f. PROPOSITION III.--THEOREM. If the first be the same multiple of the second, which the third is of the fourth ; and if of the first and third there be taken equimultiples, these shall be equimultiples, the one of the second, and the other of the fourth. LET a the first, be the same multiple of b the second, that c the third is of d the fourth ; and of a, c, let the equimultiples e f g h be taken : then ef is the same multiple of b, that g h is of d. Because ef is the same multiple of a, that gh is of c, there are as many magnitudes in ef equal to a, as are in gh equal to c. Let ef be di- fi vided into the magnitudes e k, kf, each equal to a, and gh into g. 1, 1 h, each equal to c: the number therefore of the magnitudes ek, kf, shall be equal hi to the number of the others g 1, 1h: and because a is the same multiple of k b, that c is of d, and that ek is equal to a, and gl to c; therefore e k is the 14 same multiple of b, that gl is of d. For the same reason, kf is the same multiple of b, that lh is of d ; and so, if there be more parts in ef, gh, equal to a, c: because, therefore, the first ek is the same multiple of the second e a to & C d b, which the third g1 is of the fourth d, and that the fifth kf is the same multiple of the second b, which the sixth lh is of the fourth d; e f the first together with the fifth, is the same multiple (ii. 5) of the second b, which gh the third together with the sixth, is of the fourth d. If, therefore, the first, &c. Q. E. D. PROPOSITION IV.—THEOREM. If the first of four magnitudes has the same ratio to the second which the third has to the fourth ; then any equimultiples whatever of the first and third shall have the same ratio to any equimultiples of the second and fourth, viz. the equimultiple of the first shall have the same ratio to that of the second, which the equimultiple of the third has to that of the fourth. LET a the first have to b the second, the same ratio which the third chas to the fourth d ; and of a and c let there be taken any equimultiples whatever e, f; and of b and d any equimultiples whatever h: then e has the same ratio to g, which f has to h. Take of e and f any equimultiples whatever k, l, and of g, h, any equimultiples whatever m, n : then, because e is the same multiple of a that f is of c: and of e and f have been taken equimultiples k, 1 ; therefore k is the same multiple of a, that 1 is of c (v. 3). For the same reason, m is the same multiple of b, that n is of d : and because, as a is to b, so is c to d (hypoth.), and of a and chave been taken certain equimultiples k, 1: and of b and d have been taken certain equimultiples m, n; if therefore k be greater than m, 1 is greater than n: and if equal, equal ; if less, less (v. def. 5). And k, 1 are any equimultiples whatever of e, f; and m, n any whatever of g, h: as therefore e is to g, so is (v. def. 5) f to h. Therefore, if the first, &c. Q. E. D. COR. Likewise, if the first has the same ratio to the second, which the third has to the fourth, then also any equimultiples whatever of the first and third have the same ratio to the second and fourth. And in like manner, the first and the third have the same ratio to any equimultiples whatever of the second and fourth. Let a the first have to b the second, the same ration which the third c has to the fourth d, and of a and c let e and f be any equimultiples whatever; then e is to b, as f to d. Take of e, f any equimultiples whatever k, l, and of b, d any equimultiples whatever g h; then it may be demonstrated, as before, that k is the same multiple of a, that 1 is of C. And because a is to b, as c is to d, and of 'a and c certain equimultiples have been taken, viz. k and l; and of b and d certain equimultiples g, h; therefore, if k be greater than g, 1 is greater than h; and if equal , equal ; if less, less (v. def . 5). And k. 1 are any equimultiples of e, f, and g h any whatever of b, d; as therefore e is to b, so is f to d. And in the same way the other case is demonstrated. PROPOSITION V.-THEOREM. If one magnitude be the same multiple of another which a magnitude taken from the first is of a magnitude taken from the other, the remainder shall be the same multiple of the remainder that the whole is of the whole. g LET the magnitude ab be the same multiple of cd, that a e taken from the first, is of cf taken from the other ; the remainder eb shall be the same multiple of the remainder fd, that the whole a b is of the whole cd. a. Take the same multiple of fd that a e is of cf: therefore ae is (v. 1) the same multiple of cf, that is of cd. But a e, by the hypothesis, is the same multiple of cf, that a b is of cd: therefore eg is the same multiple of cd that a b is of cd; wherefore eg is equal to a b (v. ax. 1). Take from them the common magnitude a e; the remainder e ag is equal to the remainder eb. Wherefore, since a e is f the same multiple of cf, that ag is of fd, and that ag is equal to eb; therefore a e is the same multiple of cf, that eb is of fd. But a e is the same multiple of cf that a b ъ is of cd; therefore eb is the same multiple of fd, that a b is of cd. Therefore, if one magnitude, &c. Q. E. D. eg PROPOSITION VI.-THEOREM. a If two magnitudes be equimultiples of two others, and if equimultiples of these be taken from the first two, the remainders are either equal to these others, or equimultiples of them. LET the two magnitudes a b, cd be equimultiples of the two e, f, and ag, ch taken from the first two be equimultiples of k the same e, f; the remainders gb, hd are either equal to e, f, or equimultiples of them. First, let gb be equal to e; hd is equal to f. C Make ck equal to f; and because ag is the same multiple of e, that ch is of f, and that gb is equal to e, and ck to f; therefore a b is the same multiple of e that kh is of f. But a b, by the hypothesis, is the same multiple of e that cd is of f; therefore k h is the same multiple of f that cd is b of f; wherefore kh is equal to cd ((v. ax. 1). Take d e f away the common magnitude ch, then the remainder kc is equal to the remainder hd. But kc is equal to f; hd therefore is equal to f. But let gb be a multiple of e; then hd is the same multiple of f. Make ck the same multiple of f, k that gb is of e. And because a g is the same multiple of e that ch is of f; and gb the same multiple of e that ck is of f; therefore a b is the same multiple of e that k h is of f (v. 2). But a b is the same multiple of e that cd is of f; therefore k h is the same multiple of f that cd is of it: wherefore 8 h kh is equal to cd (v. ax. 1). Take away ch from both; therefore the remainder kc is equal to the remainder hd. And because gb is the same multiple of e that k c is of f, and that kc is equal to hd ; ъ а f therefore hd is the same multiple of f that gb is of e. If, therefore, two magnitudes, &c. Q. E. D. PROPOSITION A.-THEOREM. If the first of four magnitudes has to the second the same ratio which the third has to the fourth; then, if the first be greater than the second, the third is also greater than the fourth; and if equal, equal; if less, less. TAKE any equimultiples of each of them, as the doubles of each ; then, by def. 5th of this book, if the double of the first be greater than the double of the second, the double of the third is greater than the double of the fourth : but if the first be greater than the second, the double of the first is greater than the double of the second ; wherefore also the double of the third is greater than the double of the fourth ; therefore the third is greater than the fourth. In like manner, if the first be equal to the second, or less than it, the third can be proved to be equal to the fourth, or less than it. Therefore, if the first, &c. Q. E. D. PROPOSITION B.—THEOREM. If four magnitudes are proportionals, they are proportionals also when taken inversely. IF the magnitude a be to b as c is to d, then also inversely b is to a as d to c. Take of b and d any equimultiples whatever e and f; and of a and c any equimultiples whatever g and h. First, let e be greater than g, then g is less than e; and because a is to b as c is to d, and of a and c, the first and third, g. and h are equimultiples; and of b and d, the second and fourth, e and f are equimultiples ; a be and that g is less than e, h is also (v. def. 5) less than h с d ff; that is, f is greater than h; if therefore e be greater than f is greater than h. In like manner, if é be equal to f may be shewn to be equal to h; and if less, less; and e, f are any equimultiples whatever of b and d, and g whatever of a and c: therefore, as b is to a, so is d to c. If then four magnitudes, &c. Q. E. D. h any PROPOSITION C.-THEOREM. If the first be the same multiple of the second, or the same part of it, that the third is of the fourth, the first is to the second as the third is to the fourth. LET the first a be the same multiple of b the second, that c the third is of the fourth d : a is to b as c is to d. Take of a and c any equimultiples whatever e and f; and of b and d any equimultiples whatever g and h. Then, because a is the same multiple of b that c is of d ; and that e is the same multiple of a that f is of e g fh а ъ C d a b c d c: e is the same multiple of b that f is of d (v. 3); therefore e and f are the same multiples of b and d. But g and h are equimultiples of b and d: therefore, if e be a greater multiple of b than g is, f is a greater multiple of d than h is of d ; that is, if e be greater than g, f is greater |