A tient â is greater than, equal to, or less than unity, D the quotient must also be greater than, equal to, F or less than unity, and therefore if A be greater than C, D will be greater than F; if equal, equal; and if less, less. PROP. XXII. Theor. 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 rank of 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. DEMONSTRATION. Let the first rank of magnitudes be A, B, C, D, and the second rank be E, F, G, H, so that by hypothesis A is to B as E to F, B to Cas F to G, and C to D as G to H; we are to show that A:D::E: H. Since A: B:: E: F, therefore we have in like manner we have A E A E which by reduction becomes D-H' and therefore A:D:: E:H. In like manner the truth of the proposition may be shown, whatever be the number of magnitudes. PROP. XXIII. theor. Q.E.D. 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; that is, by equality in perturbate proportion. DEMONSTRATION. Let the first rank of magnitudes be A, B, C, D, and the other rank E, F, G, H, so that, by hypothesis, A is to B as G to H; B to Cas F to G, and C to D as E to F; we are to prove, that A: D::E:H. Since A: B::,G: H, therefore and because B:C::F:G, therefore and because C: D:: E: F, therefore A G B H' B F C G D-F and therefore A: D:: E: H. In like manner we may proceed for any number of `magnitudes. Q.E.D. PROP. XXIV. 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 sixth together have to the fourth. DEMONSTRATION. By hypothesis we bave rA: ArB: B, and r'A: A:: r'B: B, in which rA is the first, A the second, rB the third, B the fourth, r'A the fifth, and r'B the sixth; r' denoting each of the two equal ratios when the fifth is divided by the second, and the sixth by the fourth; and we have to show, that rA+r'A: ArB+r'B: B. therefore, rA+r'A: A:: rB+r' B: B. 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. COR. 2. The prop. 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 which the corresponding one of the second rank has to a fourth magnitude. PROP. XXV. THEOR. If four magnitudes of the same kind be proportionals, the greatest and least of them together are greater than the other two together. DEMONSTRATION. Let the proportionals be rA, A, rB, B; and let the first rA be the greatest: then since by hypothesis rA is the greatest, rAA, therefore r>1. Again, since by hypothesis rA is the greatest, therefore rArB, and consequently A>B; since then r is greater than unity, and A is greater than B, it is manifest that B is the least; and we are to show that rA+BrB+A. Now because therefore by multiplication and A-BA-B, rA-BA-B; to each side of this equation add and we shall have rB+B, A similar mode of demonstration may be adopted whichever of the four proportionals be the greatest. Q.E.D. PROP. XXVI. THEOR. If there be any number of magnitudes of the same kind, the ratio compounded of the ratios of the first to the second, of the second to the third, and so on to the last, is equal to the ratio of the first to the last. DEMONSTRATION. Let the magnitudes of the same kind be A, B, C, D; we are to prove that the ratio compounded of the ratios of A to B, of B to C, and of C to D, according to the definition of compound ratio, is equal to the ratio of A to D. A, B, C, D, Take any magnitude P, and let A be to B as P to Q, and B to C as Q to R, and C to D as R to S; then by the definition of compound ratio, the ratio of P to S is the ratio compounded of the ratios of A to B, B to C, and of C to D; and it is to be proved that the ratio of A to D is the same with P to S. Now because A, B, C, D, are several magnitudes, and P, Q, R, S, as many others, which taken two and two in order, have the same ratio; that is, A is to B as P to Q; E to C as Q to R, and C to D as R to S; therefore ex æquali, prop. XXII,. A: D:: P: S. In like manner the proposition is proved for any number of magnitudes. PROP. XXVIII. THEOR. Q.E.D. If four magnitudes be proportionals according to the common algebraic definition, they will also be proportionals according to Euclid's definition. Let the four DEMONSTRATION. rA, A, rB, B, be the proportionals according to our 5th definition; that is, according to the common algebraic definition; it is to be proved that the same four rA, A, rB, B, are proportionals by Euclid's fifth def. of the fifth book. Let m and n be any two integers, each greater than unity, so that mrA, mrB, are any equimultiples whatever of the first and third; and nA, nB are any whatever of the second and fourth; and the four multiples are therefore mrA, nA, mrB, nB; Now the thing to be proved is, that according as the multiple mrA is greater than, equal to, or less than nA; the multiple mrB will also be greater than, equal to, or less than nB. then by division First let mrA>nA, and by multiplication Secondly, if then mrn, mrBnB. mrA=nA, mở ng If four magnitudes be proportionals by Euclid's fth definition, they will also be proportionals by the common algebraic definition. DEMONSTRATION. Let A', A, B', B, be any four magnitudes, such that m, n, being any integers greater than unity, and the equimultiples, mA'; mB', being taken, and likewise the equimultiples nA, nB; making the four multiples mA', nA, mB', nB ; |