5. o BE 5. * Cor. 4. to FD:* but HO is equal to EB, and MP to FD; therefore GK is to HO, as LN to MP: therefore if * A. 5. GK be greater than HO, LN is greater* than MP; and if equal, equal; and if less, less. But let HO, MP be equimultiples of EB, FD; then * Hyp. because * AE is to EB, as CF to FD, and that of AE, CF are taken equimultiples GK, LN; and of EB, FD, the equimultiples HO, MP; therefore if GK be greater than HO, LN is greater than MP; and if equal, equal; and if less, * 5 Def. less ; * which was likewise shown in the preceding case. But if GH be greater than KO, taking KH from • 5 Ax. both, then GK is greater* than HO; therefore LN is greater than MP; and consequently, adding NM to * 4 As. both, LM is greater* than NP: there fore, if GH be greater than KO, LM is greater than NP. In like manner it may be shown, that if GH be equal to KO, LM is equal to NP; and if less, less. And in the case in which KO is not greater than KH, it has been shown that GH is always greater than KO, and LM greater than NP: but GH, LM are any equi* Const. multiples* of AB, CD, and KO, NP are any whatever * 5 Def. of BE, DF; therefore,* as AB is to BE, so is CD to DF. If, then, magnitudes, &c. Q. E. D. 1. 1. 5. PROP. XIX. THEOR, 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 EF from CD; 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; and because * 16. 5. if magnitudes taken jointly be proportionals, they are * 17. 5. also proportionals* when taken separately ; therefore, as BE is to EA, so is DF to FC; and alternately, as BE is to DF, so is EA to FC: but, by hypothesis, as AE is to CF, so is AB to CD; therefore the remainder BE is to the remainder DF, as the whole AB to the whole CD.* Wherefore, if the whole, &c. Q. E. D. * 11. 5. 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 be to the remainder, as the magnitude taken from the first to that taken from the other. The demonstration is contained in the preceding PROP. E. THEOR. D A 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. Let AB be to BE, as CD to DF; then BA shall be to AE, as DC to CF. Because AB is to BE, as CD to DF, there * 17. 5. fore by division,* AE is to EB, as CF to FD; and by inversion,* BE is to EA, as DF to FC; therefore, by composition, * BA is to AE, as DC is to CF. If, therefore, four, &c. Q. E. D. * B. 5. * 18. 5. PROP. XX. THEOR. If there be three magnitudes, and other three, which, taken two and two, have the same ratio; then if the first D 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: then 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 * 8. 5. than the less has to it;* therefore A has to * Hyp. B a greater ratio than C has to B: but* as D * 13. 5. is to E, so is A to B; therefore * D has to E a greater ratio than C has to B; and because B is to C, as E to * B. 5. F, by inversion,* C is to B, as F is to E; and D was shown to have to E a greater ratio than C to B; there* Cor.13. fore D has to E a greater ratio than F to E:* but the magnitude which has a greater ratio than another * 10.5. to the same magnitude, is the greater* of the two: therefore D is 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 * 7. 5. B:* but A is to B, as D to E; and C is to B, as F to E; wherefore D A * 11.5. is to E, as F to E;* and therefore * 9. 5. D is equal to F.* Next, let A be less than C; D shall be less than F. For C is greater than A, and, as was shown in the first case, C is to B, as F is to E, and in like manner B is to A, as E is to D: therefore by the first case F is greater than D; and therefore D is less than F. Therefore, if there be three, &c. Q. E. D. 5. D 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: then 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 * than C has to B: but as E to F, so is A to B: * 8. 5. therefore* E has to F a greater ratio than C has to B: * 13. 5. and because* B is to C, as D to E, by inversion, C is to Hyp. B, as E to D: and it was shown that E has to F a greater ratio than C to B; therefore E has to F a greater ratio than E to D;* but the magnitude which * Cor. has a greater ratio than another has to the same magnitude, is the greater of the two: * therefore D is * 10. 5. greater than F. Secondly, let A be equal to C; D shall be equal to P. Because A and C are equal, A is * to B, as C is * 7. 5. to B: but* A is to B, as E to F; and C is to B, as E to D; therefore E is to F as E to D;* and * 11.5. therefore D is equal* to F. * 9. 5. Next, let A be less than C; D shall be less than F. For C is greater than A, and, as was shown, C is to B, as E is to D, and in like manner B is to A, as F is to E; 13. 5. • Нур. . E F P therefore by the first case F is greater than D, that is, D is less than F. Therefore, if there be three, &c. Q. E. D. PROP. XXII. THEOR. 66 If there be any number of magnitudes, and as many others, which, taken two and two in order, hare the same ratio; the first shall have to the last of the first magnitudes, the same ratio which the first has to th: 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. Of A and D take 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 G, H are equimultiples of A, D, and K, L equimul tiples of B, E; therefore as G is * 4. 5. to K, so is * H to L. For the same ratio; therefore if G be greater than M, H is * 20 5. greater than N; and if equal, equal; and if less, less; * Const. but G, H are any equimultiples* whatever of A, D, and M, N are any equimultiples whatever of C, F: there* 5 Def. fore,* as A is to C, so is D to F. Next, let there be four magnitudes, A, B, C, D, and other four, E, F, G, H, which two and two have the same ratio, viz. as A is to B, so is EIGE K M 5. ABCD |