M PROP. XV. THEOR. AGNITUDES have the fame ratio to one another Let AB be the fame multiple of C, that DE is of F C is to A K BOOK V. Because AB is the fame 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, G viz. DK, KL, LE: then the number of the first AG, GH, HB, fhall be equal to the II number of the laft DK, KL, LE: and because AG, GH, HB are all equal, and that DK, KL, LE are alfo equal to one another; therefore AG is to DK, as GH to KL, and as HB to LE 2; and as one of the antecedents to its confequent, fo are all the antecedents together to all the confequents together; where- b 12. 5. fore, as AG is to DK, fo is AB to DE: But AG is equal to C, and DK to F; therefore, as C is to F, fo is AB to DE. Therefore, &c. Q. E. D. PROP. XVI. THEOR. BU F four magnitudes of the fame kind be propor tionals, they fhall alfo be proportionals when taken alternately. Let the four magnitudes A, B, CD, EF be proportionals, viz. as A to B, fo CD to EF: They shall also be proportionals when taken alternately; that is, A fhall be to CD, as B to EF. M N: a 7.5: N a 15.5. Take MA, MB any equimultiples of A, B, and take NC the leaft multiple of CD that is greater than MA, and NE the fame multiple of EF: and because MA, MB are equimultiples of A, B, and that magnitudes have the fame ratio to M one another, which their equimultiples have a; therefore A is to B, as MA to MB: But as A is to B, fo is CD to EF; wherefore CD is to EF, as b MA to MB. Again, because NC, NE are equimultiples of CD, EF, as CD to EF, fo is NC to NE : but as CD to EF, fo is MA to MB; therefore MA is to MB, as NC to NE: and the first MA is lefs than the third NC; therefore the fecond D b 11. 5. A B C E MB BOOK V. MB is lefs than the fourth NE. In like manner, because MA is not less than ND, it may be proved, that MB is not lefs than 14 5 NF: Therefore MA, MB contain CD, EF the fame number of dA.Def.5. times that ND, NF contain them : But ND, NF contain CD, EF equally, for they are either equal to CD, EF, or equimultiples of them; therefore MA, MB contain CD, EF equally : and MA, MB are any equimultiples of A, B: as many times, therefore, as any multiple of A contains CD, fo many times does the fame multiple of B contain EF: therefore A is to CD, e 5. Def. 5. as B to EF. Wherefore, &c. Q. E. D. PROP. XVII. THEOR. F magnitudes, taken jointly, be proportionals, they fall alfo be proportionals when taken feparately; or divifion. Let AB, BE, CD, DF be the magnitudes taken jointly, which are proportionals; that is, as AB to BE, fo is CD to DF; they fhall also be proportionals taken feparately, viz. as AE to EB, fo CF to FD. Take of AE, EB, CF, FD any equimultiples MA, MN, MC, MP, fo that MA be greater than BE: and becaufe MA is the fame multiple of AE, that MN is of EB, the whole NA a 1. 5. is the fame multiple of AB, that MA is of AE a : N but MA is the fame multiple of AE, that MC is of CF; wherefore NA is the fame multiple of AB, that MC is of CF. Again, because MC is the fame multiple of CF, that MP is of FD; therefore PC is the fame multiple of CD, that MC is of CF but NA, MC are equimultiples of AB, CF; therefore NA, PC are equimultiples of AB, CD and because AB is to BE, as CD to DF, and that NA, PC are equimultiples of AB, CD, b5. Def. 5. NA contains BE the fame number of times b that c 6.5. M M B D F АС PC contains DF: And from NA, PC are taken PROP, IF PROP. XVIII. THEOR. F magnitudes, taken feparately, be proportionals, they shall also be proportionals when taken jointly; that is, by compofition. Let AE be to EB, as CF to FD; they fhall alfo be proportionals when taken jointly; that is, as AB to BE, fo CD to DF. N M Take of AB, BE, CD, DF any equimultiples MA, MN, MC, MP: And because the whole MA is the fame multiple of the whole AB, that MN is of BE; the remainder NA is the fame multiple of the remainder AE, that the whole MA is of the whole AB a. For the fame reason, PC is the fame multiple of CF, that MC is of CD: But MA, MC are equi- M multiples of AB, CD; therefore NA, PC are equimultiples of AE, CF: and because AE is to EB, as CF to FD, and that NA, PC are equimultiples of AE, CF: therefore NA contains BE the fame number of times that PC contains DF: and MN, MP are equimultiples of the fame BE, DF; therefore the whole MA contains BE the same number of times B that the whole MC contains DF: and MA, MC E are any equimultiples of AB, CD; as many times, therefore, as any multiple of AB contains BE, fo many times does the fame multiple of CD contain DF: Wherefore, as AB is to BE, fo is CD to DF. Wherefore, &c. Q. E. D. PROP. XIX. THEOR. F a whole magnitude be to a whole, as a magnitude taken from the firft is to a magnitude taken from the other; the remainder fhall be to the remainder, as the whole to the whole. Let the whole AB, be to the whole CD, as AE taken from AB, to CF taken from CD; the remainder EB fhall be to the remainder FD, as the whole AB to the whole CD. Because AB is to CD, as AE to CF; likewise, alternately, BA is to AE, as DC to CF: and becaufe, if magnitudes, taken jointly, be proportio- E nals, they are also proportionals when taken feparately; therefore, as BE is to EA, fo is DF to FC; and alternately, as BE is to DF, fo is EA to FC: But, as EA to FC, fo, by the hypothefis, is AB to b CD; Book V. a 5. 5. b5. Def. 5. C 2.5. a 16. 5. C b 17. 5. F B D Book. V. CD; therefore alfo BE, the remainder, fhall be to the remainder DF, as the whole AB to the whole CD: Wherefore, &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 likewife is to the remainder, as the magnitude taken from the first to that taken from the other: The demonftration is contained in the preceding. IF PROP. A. THEOR. F the first of four magnitudes has to the fecond, the fame ratio which the third has to the fourth; then, if the first be greater than the fecond, the third is alfo greater than the fourth; and if equal, equal; and if lefs, less. Let A be to B, as C is to D; if A be greater than B, C is greater than D. M Because A is greater than B, take, as in the eighth propofition, MA a multiple of A, that fhall contain B a greater number of times than it contains A, and take MC the fame M multiple of C: and becaufe A is to B, as C is to D; MC contains D the fame number of a 5. Def. 5. times that MA contains B: and this number is greater than the number of times that MA contains A, or MC contains C; therefore D is contained a greater number of times than C is in the fame magnitude MC: wherefore D is b 3. Ax. 5. less than C ; that is, C is greater than D. A B C D Take MA, MC Next, If A be lefs than B, C is lefs than D: any equimultiples of A, C: and because A is lefs than B, MA contains A a greater number of times than it contains B; therefore, as before, MC contains C a greater number of times than it contains D; C is therefore lefs than D . Laftly, If A be equal to B, C is equal to D: For, because Cis to D, as A is to B, if C were greater than D, or less than it, A would be greater than B, or less than it, by the former cafes; but it is not: Therefore C is equal to D. Wherefore, &c. Q. E. D. Otherwife, Take E equal to D: and let A be greater than B; then, as in the eighth propofition, fome multiple of A can be found, which contains B a greater number A, B, C, D, of times than the fame multiple of E contains D; E. D; therefore A has to B a greater ratio than E has to D: Book V. but C is to D, as A to B; therefore C has to D a greater ratio than E has to D; wherefore C is greater than E or D. In like manner, it may be proved, that if A be equal to B, Cis equal to D; and if lefs, lefs. IF PROP. B. THEOR. F four magnitudes be proportionals, they are alfo proportionals when taken inverfely. Let AB be to C, as DE is to F; then alfo, inversely, C is to AB, as F to DE. c7. Def. 5. d 13-5 e 10. N M M E 24. S. A C DF b A. Sa Take MC, MF any equimultiples of C, F; and take NA the least multiple of AB that is greater than MC, and ND the fame multiple of DE: and because AB is to C, as DE to F, and NA, ND are equimultiples of AB, DE, and MC, MF equimultiples of C, F; therefore NA is to MC, as ND to MF 2: B and the first NA is greater than the fecond MC, therefore the third ND is greater than the fourth MF b; that is, MF is lefs than ND. In like manner, because MC is not lefs than NB, it may be proved, that MF is not lefs than NE: Wherefore MC, MF contain AB, ED the fame number of times that NB, NE contain them: But NB, NE contain AB, DE equally, for they are either equal to AB, DE, or equimultiples of them; therefore MC, MF contain AB, DE equally: and MC, MF are any equimultiples of C, F; as many times therefore, as any multiple of C contains AB, fo many times does the fame multiple of F contain DE; therefore, as C is to AB, fo is F to c 5. Def. şi DE. Wherefore, &c. Q. E. D. IF PROP. C. THEOR. F the firft be the fame multiple of the second, or the fame part of it, that the third is of the fourth; the firft is to the fecond, as the third is to the fourth. Let the first A be the fame multiple of B the fecond, that C the third is of the fourth D: A is to B, as C is to D. Take MA, MC any equimultiples of A, C: and because A, Care equimultiples of B, D: and MA, MC are equimultiples of |
