multiples KO, NP: and because KO, NP are equimultiples of BE, DF; and that KH, NM are equimultiples likewise of BE, DF, if KO, the multiple of BE, be greater than KH, which is a multiple of the same BE, NP, likewise the multiple of DF, shall be greater than NM, the multiple of the same DF: and if KO be equal to H First, Let KO not be greater than KH, therefore NP is not greater than NM; and because GH, HK are equimultiples of AB, BE, and that AB is greater than BE, therefore GH is greater (3. Ax. 5.) than KH: but KO is not greater than KH, wherefore GH is greater than KO. In like manner it may be shown, that LM is greater than NP. Therefore if KO be not greater than KH, then GH, the multiple of AB, is always greater than KO, the multiple of BE; and likewise LM, the multiple of CD, greater than NP, the multiple of DF. ་ K B E D M P! ΝΙ G A C L P M 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 remainder GK is the same multiple of the remainder AE that GH is of AB (5. 5.): which is the same that LM is of CD. In like manner, because LM is the same multiple of 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 K N B D E F KH, NM, which are likewise equimultiples of BE, DF, the remainders HO, MP are either equal to BE, DF, or equi multiples 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 MH 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 (Ax. 5.), less. K P M N 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 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 be shown, that if GH be equal to KO, G 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 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. B D E A C L Ir 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, See Note.. A Because AB is to CD, as AE to CF, likewise, alternately (16. 5.) BA is to AE, as DC to CF; and because, if magnitudes, taken jointly, be proportionals, they are also proportionals (17. 5.) when taken 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. E C F B 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 BA is to A AE, as DC to CF. Because AB is to BE, as CD to DF, by division (17. 5.), AE is to EB, as CF to FD, and by inversion (B. 5.), BE is to EA, as DF to FC. Wherefore, by composition (18. 5.), BA is to AE, as DC is to CF. If, therefore, four, &c. Q. E. D. PROP. XX. THEOR. E C 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, which, taken two and two, have the same ratio, viz. as A is to B, • See Note. 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. A B E F D 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 Č, 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 (Cor. 13. 5.); but the magnitude which has a greater ratio than another to the same magnitude, is the greater of the two (10. 5.): D is therefore greater than F. Secondly, Let A be equal to C; D shall be equal to F: be cause A and C are equal to one another, A is to B as C is to B (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 E (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 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 great A D BE E F DEF er than D, by the first case; and therefore D is less than F. Therefore, if there be three, &c. Q. E. D. PROP. XXI. THEOR. If there be three magnitudes, 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, • See Note. 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; 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 (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 is less than D, that is, D is greater than F. A D B E F Be Secondly, Let A be equal to C, D shall be equal to F. cause A and C are equal, A is (7. 5.) 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 D (11. 5.); and therefore D is equal to F (9. 5.). A Next, Let A be less than C; D shall be less than F; for C is greater than A, and, as was shown, C D E 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 there 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 has to the last. 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 • See Note. |