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 remainder as the whole to the whole. Let the whole AB be to the whole CD, as AE a magnitude taken from AB is to CF a magnitude taken from CD. Then the remainder EB shall be to the remainder FD, as the whole AB to the whole CD. Because AB is to CD, as AE to CF: therefore alternately, BA is to AE, as DC to CF: (v. 16.) and because if magnitudes taken jointly be proportionals, they are also proportionals, when taken separately; (v. 17.) therefore, as BE is to EA, so is DF to FC; and alternately, as BE is to DF, so is EA to FC: but, as AE to CF, so, by the hypothesis, is AB to CD; therefore also BE the remainder is to the remainder DF, as the whole AB to the whole CD. (v. 11.) Wherefore, if the whole, &c. Q.E.D. COR.-If the whole be to the whole, as a magnitude taken from the first is to a magnitude taken from the other; the remainder shall likewise be to the remainder, as the magnitude taken from the first to that taken from the other. The demonstration is contained in the preceding. If four magnitudes be proportionals, they are also proportionals by conversion; that is, the first is to its excess above the second, as the third to its excess above the fourth. Because AB is to BE, as CD to DF, therefore by division, AE is to EB, as CF to FD; (v. 17.) and by inversion; BE is to EA, as DF is to CF; (v. B.) wherefore, by composition, BA is to AE, as DC is to CF. (v. 18.) If therefore four, &c. Q.E.D. PROPOSITION XX. THEOREM. If there be three magnitudes, and other three, which, taken two and two, have the same ratio; then 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; Because A is greater than C, and B is any other magnitude, therefore A has to B a greater ratio than C has to B: therefore D has to E a greater ratio than C to B: (v. 13.) by inversion, C is to B, as Fis to E: (V. B.) and D was shewn to have to E a greater ratio than C to B: therefore D has to E a greater ratio than F' to 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.) therefore D is greater than F. Secondly, let A be equal to C. Then D shall be equal to F. Because A and C are equal to one another, wherefore D is to E, as F to E; (v. 11. and V. B.) Next, let A be less than C. Then D shall be less than F. and as was shewn in the first case, C is to B, as F to E, Therefore, if there be three, &c. Q.E.D. 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; then 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; Because A is greater than C, and B is any other magnitude, therefore E has to F a greater ratio than C to B: (v. 13.) by inversion, C is to B, as E to D: and E was shown to have to Fa greater ratio than C has to B; therefore E has to Fa greater ratio than E has to D: (v. 13. Cor.) but the magnitude to which the same has a greater ratio than it has to another, is the less of the two: (v. 10.) therefore Fis less than D; that is, D is greater than F. Secondly, let. A be equal to C; A D D shall be equal to F. PROPOSITION XXI. 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 has to the last of the others. 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, which taken two and two in order, 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. Then A shall be to C, as D to F. N Take of A and D any equimultiples whatever G and H; and because there are three magnitudes G, K, M, and other three and other four E, F, G, H, which two and two have the same ratio, viz. as A is to B, so is E to F; and as B to C, so Fto G; and as C to D, so G to H. Then A shall be to D, as E to H. A.B.C.D Because A, B, C are three magnitudes, and E, F, G other three, which taken two and two, have the same ratio; therefore 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: PROPOSITION XXIII. THEOREM. 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 has to the last of the others. N.B. This is usually cited by the words "ex æquali in proportione perturbatâ ;" 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 have the same ratio, that is, such that A is to B, as E to F; Take of A, B, D any equimultiples whatever G, H, K; and that magnitudes have the same ratio which their equimultiples have; (v. 15.) therefore 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: and it has been shewn that G is to H, as M to N: therefore, because there are three magnitudes G, H, L, and other three K, M, N, which have the same ratio taken two and two in a cross order; if G be greater than L, K is greater than No: but &, K are any equimultiples whatever of A, D; (constr.) therefore as A is to C, so is D to F. (v. def. 5.) Next, let there be four magnitudes A, B, C, D, and other four E, F, G, H, which taken two and two in a cross order have the same ratio, viz. A to B, as & to Ħ; B to C, as Fto G; and C to D, as E to F. Then A shall be to D, as E to H. A.B.C.D E.F.G.H Because A, B, C are three magnitudes, and F, G, H other three, which taken two and two in a cross order, have the same ratio ; by the first case, A is to C, as F to H; but Cis to D, as E is to F; wherefore again, by the first case, A is to D, as E to H; Q. E.D. 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 sixth together have to the fourth. Let AB 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 Ċ the second the same ratio which EH the sixth has to F the fourth. Then AG, the first and fifth together, shall have to C the second, the same ratio which DH, the third and sixth together, has to F the fourth. |