Book v. PROP. XV. THEOR. MAGNITUDES have the same ratio to one another wbich their equimultiples have. Let AB be the same multiple of C, that DE is of F; O is to F, as AB to DE. Because AB is the same multiple of C, that DE is of F; there are as many magnitudes in AB equal to C, as there are in DE equal to F: Let AB be divided into magnitudes, each equal to C, viz. AG, GH, HB; and DE into magnitudes, each equal to F, viz. DK, KL, LE: Then the number of the first AG, GH, HB, shall be equal to the al number of the last DK, KL, LE: And because AG, GH, HB are all equal, and that DK, KL, LE, are also equal to one I another; therefore AG is to DK as GH to B C E F KL, and as HB to LEa: And as one of the antecedents to a 7. 5. its consequent, so are all the antecedents together to all the consequents together b; wherefore, as AG is to DK, so is 512 5. AB to DE: But AG is equal to C, and DK to F: Therefore, as C is to F, so is AB to DE. Therefore magnitudes, &c. Q. E. D. PROP. XVI. THEOR. If four magnitudes of the same kind be proportionals, they shall also be proportionals when taken alternately, Let the four magnitudes A, B, C, D, be proportionals, viz, as A to B, so C to D: They shall also be proportionals when taken alternately; that is, A is to C, as B to D. Take of A and B any equimultiples whatever E and F; and of C and D take any equimultiples whatever G and H: Book V. and because E is the same multiple of A, that F is of B, hour and that magnitudes have the same ratio to one another " " 15. 5. which their equimultiples have a; therefore A is to B, as E is to F; But as A D; Wherefore as 111. 5. C is to D, sob is E to F: Again, B- greater than the third, the second shall be greater than the *14. 5. fourth ; and if equal, equal; if less, less c. Wherefore, if E be greater than G, F likewise is greater than H: and if equal, equal; if less, less: And E, F are any equimultiples whatever of A, B; and G, H any whatever of C, D. There-' 4 5 Def. 5. fore A is to C as B to Dd If then four magnitudes, &c. Q.E.D. PROP. XVII. THEOR. See n. If magnitudes, taken jointly, be proportionals, they shall also be proportionals when taken separately; that is, if two magnitudes together have to one of them the same ratio which two others have to one of these, the remaining one of the first two shall have to the other the same ratio which the remaining one of the last two has to the other of these. Let AB, BE, CD, DF be the magnitudes taken jointly which are proportionals; that is, as AB to BE, so is CD to DF: they shall also be proportionals taken separately, viz, as AE to EB, so CF to FD. Take of AE, EB, CF, FD any equimultiples whatever GH, AK, LM, MN: and again, of EB, FD take any equimultiples whatever KX, NP: And because GH is the same multiple of AE, 'that HK is of EB, wherefore GH is the * 3. 5. same multiplea of AE, that GK is of AB: But GH is the same multiple of AE, that LM is of CF; wherefore GK is the same multiple of AB, that LM is of CF. Again, be- Beox l. . cause LM is the same multiple of CF, that MN is of war FD; therefore LM is the same multiplea of CF, that']. 5. 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; aud that KX is also tlie same multiple of EB, that NP is of TD; therefore HX is the same multiple of b . 3. EB, that MP is of FD. And because 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 equimultiples; if GK be greater than HX, then LN is greater than MP; and if equal, equal; and if less, less"; , lej Def... But if GH be greater than KX, by adding the common part HK to both, GK is greater than HX; wherefore also LN is greater than MP; and by taking away MN from both, LM is greater than NP: Therefore, if GH be greater than KX, LM is greater than G A C L NP. In like manner it may be demonstrated, 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, só is CF to FD. If then magni. tudes, &c. Q. E. D. PROP. XVIII. THEOR. If magnitudes, taken separately, be proportionals, Sec 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 equimultiples KO, NP: And because KO, NP CF, FD be bey shall albe, so CD BL D, Book V. are equimultiples of BE, DF; and that KH, NM are equi multiples likewise of BE, DF, if KO, the multiple of BE, First, let KO not be greater than BE, and that AB is greater than * 3 Ax. 5. BE, therefore GH is greater a than HK; but KO is not greater E F 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 5. 5. remainder AE that GH is of ABb: which is the same that LM is of op the remainders HO, MP are either equal to BE, DF, or 6. 5. equimultiples of them. First, let HO, MP be equal to BE, DF; and because AE is to EB, as CF to FD, and BUD Ed Cor. 4.5. that GK, LN are equimultiples of AE, CF; GK shall be Book V. to EB, as LN to FDd: But HO is equal to EB, and MPS 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 lesse, less. 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 is greater than MP; and if equal, oi equal; and if less, lessf; which was f5 Def. 5. likewise shown in the preceding case. If therefore GH be greater than KO, taking KH from both, H GK is greater than HO; wherefore also LN is greater than MP; and consequently adding NM to both, LM is greater than NP: KTherefore, if GH be greater than KO, LM is greater than NP. In like manner it may be shown, that ifGH be equal to KO, LMis equal to NP; and if less, less. And in the case in which KO is not great- o er 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; thereforef, 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 magni- See N. tude 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 ED, as the whole AB to the whole CD. Because AB is to CD, as AE to CF; likewise, alternate |