manner, because LM is the same multiple of E P M CD, that MN is of DF, the remainder LN is the same multiple of the remainder CF, that the whole LM is of the whole CD: (5. 5.) But it was shown that LM is the same multiple of CD, that GK is of AE; therefore GK is the same multiple of AE, that LN is of CF; that is, GK, LN are equimultiples of AE, CF: And because KO, NP are equimultiples of BE, DF, if from KO, NP there be taken KH, NM, which are likewise equimultiples of BE, DF, the remain ders HO, MP are either equal to BE, DF, or equimultiples of them. (6. 5.) First, let HO, MP, be equal to BE, DF; and because AE is to EB, as CF to FD, and that GK, LN are equimultiples of AE, CF; GK shall be to EB, as LN to FD: (Cor. 4. 5.) But HO is equal to EB, and MP to FD; wherefore GK is to HO, as LN to MP. If therefore GK be greater than HO, LN is greater than MP; and if equal, equal; and if less, (A. 5.) less. Or But let HO, MP be equimultiples of EB, FD; and 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; if GK be greater than HO, LN is greater than MP; and if equal, equal; and if less, less; (5. Def. 5.) which was likewise shown in the preceding case. If therefore GH be greater than KO, taking KH from Both, GK is greater than HO; wherefore alse 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 be shewn, that if GH be equal to KO, LM is equal to NP; and it less, less. And in the case in which KO is not greater than KH, it has been showu that GH is always greater than KO, and likewise LM than NP: But GH, LM are any equimultiples of AB, CD, and KO, NP, are any whatever of BE, DF; therefore, (5. Def. 5.) as AB is to BE, so is CD to DF. If then, magnitudes, &c. Q. E. D. PROP. XIX. THEOR. H K B P M E Ꮮ GA 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, to CF, a magnitude taken 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 is to AE, as DC to CF: And because, if magnitudes, taken jointly, be proportionals, they are also proportionals (17. 5.) when ta CF; likewise, alternately, (16. 5.) ken separately; therefore, as BE is to DF, so is EA to FC; and alternately, as BE is to EA, so is DF to FC: But, as AE to CF, so by the hypothesis, is AB to CD; therefore also BE, the remainder, shall be to the remainder DF, as the whole AB to the whole CD: 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 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. 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 i fless, 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; (8. 5.) 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 (13. 5.) 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 therefore D is to E, as F to E; (11. 5.) and therefore D is equal to F. А В С (9. 5.) Next, let A be less than C; D shall be less 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, &c. Q. E. P. PROP. XXI. THEOR. If there be three magnituaes, 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, 80 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 Bis any other magnitude, A has to B a greater ratio (8. 5.) than C has to B: But as E to F, so is A to B : therefore (13. 5.) 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 E to D; (COR. 13. 5.) but the magnitude to which the same has a greater ratio than it has to another, is the lesser of the two: (10. 5.) F therefore A C D E F is less than D; that is, D is greater than F. to B: But A is to B, as E to F; and C is to B, as E to D; wherefore E is to Fas E to D; (11.5.) and 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 shewn, C is to B, as E to D, and in like manner B is to A, as F to E; therefore F is greater than D, by case first ; and therefore D is less than F. Therefore, if ther be three, &c. Q. E. D. 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 magnitudes the same ratio which the first of the others usually cited by the words " ex N. B. This is has to the last. 66 First, let there be three magnitudes A, B, C, and as many 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; 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 G, H are equimultiples of A, D, and K, L egimultiples of B, E; as G is to K, so is (4. 5.) H to L: For the same reason, K is to M, as L to N; and because there are A G KM HLN three magnitudes, G, K, M, and other three H, L, 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 if less, less; (20.5.) and G, H are any equimultiples whatever of A, D, and M, N are any equimultiples whatever of C, F: Therefore, (5. Def. 5.) as A is to C, so is D to F. A. B. C. D. Next, let there be four magnitudes, A, B, C, D, and other four E, F, G, II, 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: 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. PROP. XXIII. THEOR. 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 magni- E, F, which, taken two and two, in tudes, A, B, C, and other three D, a cross order, have the same ratio, that is, such that A is to B, as E to F, and as B is to C, so is D to E. A is to C, as D to F. Take of A, B, D any equimultiples whatever, G, H, K; and of C, E, F any equimultiples whatever L, M, N: And because G, H are equimultiples of A, B, and that magnitudes have DE F GHL KM N B, D, and L, M of C, E; as H is to L, so is (4. 5.) K to M: And it has been shown that G is to H, as M to N: Then, 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 N; and if equal, equal; and if less, less: (21. 5.) and G, K are any equimultiples whatever of A, D; and L, 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 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; A. B. C. D. B to C, as F to G; and C to D, as Because A, B, C, are three magnitudes, and F, G, H, other three, which, taken two and two in a cross he same ratio which their equimul- order, have the same ratio; by the tiples have; (15. 5.) as A is to B, so first case, A is to C, as F to H: But is G to H: And for the same reason, C is to D, as E to F; wherefore again, as E is to F, so is M to N: But as A by the first case, A is to D, as E to to B, so is E to F; as therefore GH: And so on whatever be the num to H, so is M to N. (11. 5.) And ber of magnitudes. Therefore, if there because as B is to C, so is D to E, be any number, &c. Q. E. D. and that H, K are equimultiples of PROP. XXIV. THEOR. 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. |