that LM is of CF. Again, because LM is the same multiple of Book V. CF, that MN is of FD ; therefore LM is the same multiple a of CF, that LN is of CD: but LM was shown to be the same multiple of CF, that GK is of AB; GK therefore is the same multiple of AB, that LN is of CD; that is, GK, LN are equimultiples of AB, CD. Next, because HK is the same multiple of EB, that MN is of FD; and that KX is х also the same multiple of EB, that NP is of FD; therefore HX is the same multiple of EB, that MP is of FD. And because P b 2. 5. AB is to BE, as CD is to DF, and that of AB and CD, GK and LN are equimultiples, and of EB and FD, HX and MP are k equimultiples; if GK be greater than HX, N then LN is greater than MP; and if equal, equal; and if less, less c : but if GH be н: B c5. def.5. greater than KX, by adding the common part HK to both, GK is greater than HX; wherefore also LN is greater than MP; E and by taking away MN from both, LM F is greater than NP: therefore, if GH be greater than KX, LM is greater than NP. In like manner it may be demonstrated, G Å Ć L that if GH be equal to KX, LM likewise is equal to NP; and if less, less : and GH, LM are any equimultiples whatever of AE, CF, and KX, NP are any whatever of EB, FD. Thereforec, as AE is to EB so is CF to FD. If, then, magnitudes, &c. Q. E. D. DM PROP. XVIII. THEOR. IF magnitudes, taken separately, be proportionals, See N. they shall also be proportionals when taken jointly, that is, if the first be to the second, as the third to the fourth, the first and second together shall be to the second, as the third and fourth together to the fourth. Let AE, EB, CF, FD be proportionals; that is, as AE to EB, so is CF to FD; they shall also be proportionals when taken jointly ; that is, as AB to BE, so CD to DF. Take of AB, BE, CD, DF any equimultiples whatever GH, HK, LM, MN; and again, of BE, DF take any whatever equi Book V. multiples KO, NP: and because KO, NP are equimultiples of BE, DF; and that KH, NM are equimultiples likewise of M therefore NP is not greater than NM: pl and because GH, HK are equimultiples of AB, BE, and that AB is greater than K N a 3. Ax.5. BE, therefore GH is greater a than HK; but KO is not greater than KH, where- B D G A L b 5.5. Next, Let KO be greater than KH: therefore, as has been shown, NP is greater than NM: and because the whole GH is the same multiple of the whole AB, that HK is of BE, the re. mainder GK is the same multiple of O the remainder AE that GH is of AB b: which is the same that LM is of CD. H M In like manner, because LM is the same multiple of CD, that MN is of pl DF, the remainder LN is the same multiple of the remainder CF, that K NL the whole LM is of the whole CD b: but it was shown that LM is the same in ultiple of CD, that GK is of AE; therefore GK is the same multiple of B 'AE, that LN is of CF; that is, GK, El D LN are equimultiples of AE, CF: and because ko, 'NP are equimultiples of BE, DF, if from KO, NP G А С L there be taken KH, NM, which are likewise equimultiples of BE, DF, the remainders HO, MP are either equal to BE, DF, or equimultiples of them c. First, Let HO, MP be equal to BE, DF; and because AE is 10 EB, as ÇF to FD, and F € 6.5. 143 that GK, LN are equimultiples of AE, CF; GK shall be to Book V. EB, as LN to FD d: but HO is equal to EB, and MP to FD; wherefore GK is to HO as LN to MP. If, therefore, GK be d Cor.4.5. greater than HO, LN is greater than MP; and if equal, equal; and if lesse, less. e A.5. 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 O is greater than MP; and if equal, equal ; and if less, less f; which was f 5. def.5. likewise shown in the preceding case. If, therefore, GH be greater H P than Ko, taking KH from both, GK is greater than HO; wherefore also M LN is greater than MP; and, consequently, adding NM to both, LM is K greater than NP: therefore, if GH N be greater than KO, LM is great B er than NP. In like manner it D may be shown, that if GH be equal E to KO, LM is equal to NP; and if less, less. And in the case in which G A с L 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 of AB, CD, and KO, NP are any whatever of BÉ, DF; therefore f, as AB is to BE so is CD to DF. If then magnitudes, &c. Q. E. D. PROP. XIX. THEOR. IF a whole magnitude be to a whole, as a magnitude See N. 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 reinainder EB shall 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, a 16. 5. Book V. BA is to AE as DC to CF: and because, if mag- А nitudes, taken jointly, be proportionals, they are 17.5. also proportionals b when taken separately; therefore, as BE is to DF so is EA to FC; and alter с E F Cor. If the whole be to the whole, as a mag- 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. А с Let AB be to BE as CD to DF; then BA is to AE as DC to CF. Because AB is to BE as CD to DF, by division a, AE is to EB as CF to FD; and by inver. sion b, BE is to EA as DF to FC. Wherefore, by composition c, BA is to AE, as DC is to CF. If, therefore, four, &c. Q. E. D. a 17.5. b B. 5. c 18. 5. E F B PROP. XX. THEOR. See N. 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 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 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, therefore A has to B a greater ratio a 8.5. than C has to B: but as D is to E, so is A to B; therefore b D has to E a greater ratio than C to b 13. 5. A 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 D E F to have to E a greater ratio than C to B; therefore D has to E a greater ratio than F to Ec: but c Cor. the magnitude which has a greater ratio than 13. 5. another to the same magnitude, is the greater of the twod: D is therefore greater than F. d 10. 5. 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 Be: e 7.5. but A is to B, as D to E; and C is to B, as F to E; wherefore D is to E, as F to Ef; and therefore D is f11.5. equal to Fr. A A B Next, Let A be less than C; D shall be less than F: for C is great D E F D E F er than A, and, as was shown 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. D. 8 9.5. PROP. XXI. THEOR. IF there be three magnitudes, and other three, See N. 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. T |