PROPOSITION XIX.—THEOREM. If a whole magnitude be to a whole as a magnitude taken from the first is to a magnitude taken from the other, the remainder shall be to the re mainder as the whole to the whole. LET the whole a b be to the whole cd, as a e, a magnitude taken from ab, to cf, a magnitude taken from cd; the remainder eb 2 shall be to the remainder fd, as the whole a b to the whole cd. Because a b is to cd, as a e to cf; likewise, alternately (v. 16), b a is to a e, as d c to cf; and because if magnitudes, taken jointly, be proportionals, they are also proportionals e(v. 17) when taken separately; therefore, as be is to ea, so is f df to fc, and alternately, as be is to d f, so is ea to fc; but, as a e to cf, so, by the hypothesis, is a b to cd.: therefore also be, the remainder, shall be to the remainder df as the whole a b to the whole cd. Wherefore, if the whole, &c. Q. E. D. Cor. If the whole be to the whole, as a magnitude taken b a from the first is to a magnitude taken from the other, the remainder likewise is to the remainder, as the magnitude taken from the first to that taken from the other. The demonstration is contained in the preceding PROPOSITION E.-THEOREM. If four magnitudes be proportionals, they are also proportionals by conver sion : that is, the first is to its excess above the second, as the third to its excess above the fourth. a LET a b be to be, as cd to df; then ba is to a e, as do to cf. f Because a b is to be, as cd to df, by division (v. 17), a e is to eb, as cfto fd; and by inversion (v. B.), be is to ea as df to fc. Wherefore, by composition (v. 18), ba is to a e as dc is to cf. If, therefore, four magnitudes, &c. Q. E. D. PROPOSITION XX.-THEOREM. e If there be three magnitudes, and other three, which, taken two and two, have the same ratio ; if the first be greater than the third, the fourth shall be greater than the sixth, and if equal, equal ; and if less, less. LET a, b, c be three magnitudes, and d, e, f other three, which, taken two and two, have the same ratio, viz. as a is to b, so is d to e; and as b to c, so is e to f. If a be greater than c, d shall be greater than f; and if equal, equal ; and if less, less. Because a is greater than c, and b is any other magnitude, and that the greater has to the same magnitude a greater ratio than the less has to it (v. 8); therefore a has a b C to b a greater ratio than c has to b; but as d is to e, so is d f to b; therefore (v. 13) d has to e a greater ratio than c to b; and because b is to c, as e to f, by inversion, c is to b, as f is to e; and d was shewn to have to e a greater ratio than c to b; therefore d has to e a greater ratio than fto e (v. 13. cor.). But the magnitude which has a greater ratio than another to the same magnitude, is the greater of the two (v. 10); d is therefore greater than f. Secondly, let a be equal to c; d shall be equal to f. Because a and c are equal to one another, a is to b as c is to b (v.7): but a is to b as d to e; and c is to b as f to e; wherefore d is to e as f to e (v. 11); and a b therefore d is equal to f (v. 9). a b c d f Next, let a be less than c; d shall be less d f than f; for c is greater than a, and, as was shewn in the first case, c is to b, as f to e, and in like manner, b is to a, as e to d; therefore f is greater than d, by the first case ; and therefore d is less than f. Therefore, if there be three magnitudes, &c. Q. E. D. e PROPOSITION XXI.—THEOREM. If there be three magnitudes, and other three, which have the same ratio taken two and two, but in a cross order; if the first magnitude be greater than the third, the fourth shall be greater than the sixth ; and if equal, equal ; and if less, less. LET a, b, c be three magnitudes, and d, e, f other three, which have the same ratio, taken two and two, but in a cross order, viz. as a is to b, so is e to f, and as b is to c, so is d to e. If a be greater than c, d shall be greater than f; and if equal, equal ; and if less, less. Because a is greater than c, and b is any other magnitude, a has to b a greater ratio (v. 8) than c has to b: but as e is to f, so is a to b: therefore (v. 13) e has to f a greater ratio than c to b. And because b is to c as d to e, by inversion, c is to b as e tod: and e was shewn to have to f a greater ratio than c to b; therefore e has to f a greater ratio than e to d (v. 13. cor.) ; but the magnitude to which the same has a greater ratio than it has to another, is the lesser of the two (v. 10): f therefore is less than d ; that is, d is greater than f. Secondly, let a be equal to c; d shall be equal to f. Because a and c are equal, a is (v. 7) to b, as c is to b: but a is to b, as e to f; and c is to b, as e to d; wherefore e is to f, as e to d (v. 11); and therefore d is equal to f (v. 9). Next, let a be less than c:d shall be less than f: for c is greater than a, and, as was shewn, c is to b, as e to d, and in like manner b is to an as f to e; therefore f is greater than d, by case first ; and therefore, d is less than f. Therefore, if there be three magnitudes, &c. Q. E. D. PROPOSITION XXII.-THEOREM. If there be any number of magnitudes, and as many others, which, taken two and two in order, have the same ratio , the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. N.B.—This is usually cited by the words "ex æquali,” or “ex æquo.” FIRST, let there be three magnitudes a, b, c, and as many others d, e, f, 8 km h 1 !!!.. which, taken two and two, have the same ratio : that is, such that a is to b as d to e; and as b is to c, so is e to f; a shall be to c, as d to f. Take of a and d any equimultiples whatever g and h; and of b and e any equimultiples whatever k and l; and of c and f any whatever m and n: then, because a is to b as d to e, and that & h are equimultiples of a, d, and k, 1 equimultiples of b, e; as g is to k, so is (v. 4) h to 1. For the same reason, k is to m, as I is to n; and because there are three magnitudes g, k, m, and other three h 1, n, which, two and two, have the same ratio ; if g be greater than m, h is greater than n; and if equal, equal; and it less, less (v. 20); and g, h are any equimultiples whatever of a, d, and m, n are any equimultiples whatever of cf: therefore (v. def. 5), as a is to c, so is d to f. Next, let there be four magnitudes a, b, c, d, and other a, b, c, d, four e, f, g, h, which, two and two, have the same ratio, e, f, g, h, viz. as a is to b, so is e tof; and as b to c, so f to g; and as c to d, so g to h: a shall be to d, as e to h. Because a, b, c, are three magnitudes, and e, f, g, other three, which, taken two and two, have the same ratio ; by the foregoing case, a is to c, as e to g: but c is to d, as g is to h; wherefore again, by the first case, a is to d, as e to h; and so on, whatever be the number of magnitudes. Therefore, if there be any number, &c. Q. E. D. PROPOSITION XXIII.-THEOREM. If there be any number of magnitudes, and as many others, which, taken two and two, in a cross order, have the same ratio ; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. N.B. This is usually cited by the words “ ex æquali in proportione perturbata," or "ex æquo perturbato.” FIRST, let there be three magnitudes a, b, c, and other three, d, e, f, which, taken two and two, in a cross order, and of c, e, f any equimultiples a b с d e f whatever 1, m, n: and because g, hare 8 h 1 n equimultiples of a b, and that magnitudes have the same ratio which their equimultiples have (v. 15): as a is to b, so is g to h : and for the same reason, as e is to f, so is m to n: but as a is to b, so is e to f; therefore 8 is to h, as m is to n (v. 11). And because as b is to c, so is d to e, and that h, k are equimultiples of b, d, and l, m of c, e; as h is to l, so is (v. 4) k to m: and it has been shewn that g is to h, as m to n: then, because there are three magnitudes h, 1, and other three k, m, n, which ever h, k; k m have the same ratio taken two and two in a cross order ; if g be greater than 1, k is greater than n: and if equal, equal ; and if less, less (v. 21); and k are any equimultiples whatever of a, d ; and 1, n any whatever of c, f; as therefore a is to c, so is d to f. Next, let there be four magnitudes a, b, c, d, and other a, b, c, d, four e, f, g, h, which, taken two and two in a cross order, have the same ratio, viz. a to b as g to h; b to c as f to e, f, g, h, and c to d as e to f. a is to d as e to h. Because a, b, c are three magnitudes, and f, h other three, which, taken two and two in a cross order, have the same ratio ; by the first case, à is to c, as f to h; but c is to d, as e is to f: wherefore again, by the first case, a is to d, as e to h: and so on, whatever be the number of magnitudes. Therefore, if there be any number, &c. Q. E. D. g; PROPOSITION XXIV.-THEOREM. If the first has to the second the same ratio which the third has to the fourth, and the fifth to the second the same ratio which the sixth has to the fourth ; the first and fifth together shall have to the second the same ratio which the third and sicth together have to the fourth. LET a b the first, have to c the second, the same ratio which de the third has to f the fourth ; and let bg the fifth have to c the second, the same ratio which e h the sixth has 8 to f the fourth ; ag, the first and fifth together, shall h have to c the second, the same ratio which d h, the third and sixth together, has to f the fourth. Because bg is to c, as eh to f; by inversion, c b ъ is to bg, as f to eh : and because, as a b is to c, so e ag g b, so is dh to he: but as gb to c, so is he to f. Therefore ex æquali (v. 22), as c d f ag is to C, so is dh to f. Wherefore, if the first, &c. Q. E. D. CoR. 1. If the same hypothesis be made as in the proposition, the excess of the first and fifth shall be to the second as the excess of the third and sixth to the fourth. The demonstration of this is the same with that of the proposition, if division be used instead of composition. Cor. 2. The proposition holds true of two ranks of magnitudes, whatever be their number, of which each of the first rank has to the second magnitude the same ratio that the corresponding one of the second rank has to a fourth magnitude ; as is manifest. |