GK, LN are equimultiples of AE, CF: GK shall be to EB, Book V. as LN to FD; but HO is equal to EB, and MP to FD: wherefore, GK is to HO, as LN to MP. If therefore GK d Cor. 4. 5. be greater than HO, LN is greater than MP; and if equal, equal; and if less, less. Р But let HO, MP be equimultiples of EB, FD; and because AE is to EB, as CF to FD, and because of AE, CF are taken the equimultiples GK, LN; and of EB, FD, the equimultiples HO, MP; if GK be greater than HO, LN is greater than MP; and if equal, equal; and if less, less f; which was likewise shown in the preceding case. If therefore GH be H greater than KO, taking KH from both, GK is greater than HO; wherefore also, LN is greater than MP; and consequently, adding NM to both, LM is greater than NP: Therefore, if GH be greater than KO, LM is greater than NP. In like manner, it may he 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 likewise LM than NP: But GH, LM are any equimultiples whatever of AB, CD, and KO, NP are any equimultiples whatever of BE, DF; therefore f, as AB is to BE, so is CD to DF. 66 "If, then, magnitudes," &c. Q. E. D. M K B N D E F G A C L e A. 5. f 5. def. 5. PROP. XIX. THEOR. If a whole magnitude be to a whole, as a magni- See N. tude taken from the first, is to a magnitude taken from the other; the remainder will 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, to CF, a magnitude taken from CD; then will the remainder EB be to the remainder FD, as the whole AB to the whole CD. Because AB is to CD, as AE to CF; likewise alternately, a 16. 5. d 17. 5. d Book V. BA is to AE, as DC to CF: And because, if magnitudes, taken jointly, be proportionals, they are 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, as EA to FC, so, by the hypothesis, is AB to CD; therefore also BE, the remainder, will be to the remainder DF, as the whole AB to the whole CD: Wherefore, "if the whole," &c. Q. E. D. a 17. 5. b B. 5. B 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 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. PROP. E. THEOR. 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, by conversion, BA is AE, as DC to CF. Because AB is to BE, as CD to DF, by division 2, AE is to EB, as CF to FD; and by inversion, BE is to EA, as DF to FC. Wherec 18. 5. fore, by composition, BA is to AE, as DC is to CF: "If, therefore, four," &c. Q. E. D. A C E F B D See N. PROP. XX. THEOR. 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 will be greater than the sixth; if equal, equal; and if less, less. Let A, B, C be three magnitudes, and D, E, F other three, Book V. 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 will be greater than F; and if equal, equal; and if less, less. A B a 8. 5. Cb 13. 5. F 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; therefore A has to B a greater ratio than C has to B: But as D is to E, so is A to B; therefore 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 shown to have to E a greater ratio than C to B; therefore, D has to E a greater ratio than F to E But the magnitude which has a greater ratio than another to the same magnitude, is the greater of the two: D is therefore greater than F. Secondly, Let A be equal to C; then will D be equal to F: Because A and C are equal to and C is to B, as F to E; therefore DE c Cor. 13.5. d 10. 5. e 7. 5. E f 11. 5. g 9. 5. C A B C F DEF PROP. XXI. THEOR. If there be three magnitudes, and other three, which See N. have the same ratio taken two and two, but in a cross order; if the first magnitude be greater than the third, the fourth will be greater than the sixth; if equal, equal; and if less, less. 136 Book V. a 8. 5. b 13. 5. 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 will be greater than F; 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 a than C has to B: But E is to F, as A to B; therefore, 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 to D: And E was shown to have to F a greater ratio than C to B: therefore, E has to F a greater ratio than c Cor. 13.5. E to D: but the magnitude to which the same has a greater ratio than it has to another, is the less of the twod: F therefore is less than D; that is, D is greater than F. d 10. 5. e 7. 5. f 11. 5. g 9. 5, See N. A B C DE F Secondly, Let A be equal to C; then will D be equal to F. Because A and C are equal, A is 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 Df; and therefore D is equal to F8. Next, Let A be less than C; then will D be less than F: For, as was shown, C is to B as E to D; and, in like manner, B is to A, as F to E; and C is greater than A; therefore F is greater than D, by case first; consequently D is less than F. Therefore," if there be three," &c. Q. E. D. A B C A B C DE F Ꭰ E F 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 will have to the last of the first magnitudes the same ratio which the first of the others has to the last *. N. B.-This proposition is usually cited by the words, "ex æquali," or "ex æquo." First, Let there be three magnitudes, A, B, C, and as many Book V. others, D, E, F, 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; then will A be to C as D to F. A B C D E DE E For, of A and D take any equimultiples whatever G and H, and of B and E take any 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 G, H are equimultiples of A, D, and K, L equimultiples of B, E; as G is to K so is a H to L: For the same reason, K is to M, as L to N; and because there are three magnitudes, G, K, M, and other three, H, L, N, which, taken two and two, have the same ratio; if G be greater than M, H is greater than N; if equal, equal; GK M HL N a 4. 5. and if less, less; and G, H are any equimultiples whatever b 20. 5. of A, D, and M, N are any equimultiples whatever of C, F: Therefore, 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 E to F; and as B to C, so F to G; and as C to D, so G to H: then will A 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, 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. c 5. def. 5. |