M Book V. of A, C; therefore MA, MC are equimultiples of B; Da; Mthat is, they contain B, D equally; as many DC a 3. 5. times, therefore, as any multiple of A con tains B, so many times does the same mul tiple of C contain D: Wherefore, as A is to b 5. Def. 5. B, so b is C to D. Next, Let A be the same part of B, that A B C D preceding cafe, B is to A, as D to C; and in- (A, B, C, D. | c B. 5. versely C, A is to B, as C is to D. Therefore, &c, Q. E. D. PROP. D. THEOR. IF F the first be to the second, as the third to the fourth ; and if the first be a multiple, or a part of the second ; the third is the same multiple, or the fame part of the fourth. Let A be to B, as C is to D; and let A be a part of B; C.is the same part of D. Take MA equal to B, and take MC the same M multiple of C, that MA or B is of A: and be M cause A is to B, as C to D, and MA, MC are equimultiples of A, C; therefore MA is to B, a Cor. 4. 5. as MC to D a : and the first MA is equal to the second B; therefore the third MC is equal to the b A. 5. fourth D b: and C is the same part of MC, that A B CD A is of B; therefore C is the same part of D, that A is of B. Next, Let A be to B, as C to D; and A a mul- (A, B, C, D. tiple of B; C is the same multiple of D. cB. 5. Because A is to B, as C to D; then, inversely , B is to A, as D to C: but B is a part of A, therefore, by the preceding case, D is the fame part of C; that is, C is the same multiple of D, that A is of B. Therefore, &c. Q. E. D. PROP. E. THEOR. IF proportionals by conversion; that is, the first is to its excefs above the fecond, as the third to its excess above the fourth. Let Let AB be to BE, as CD to DF; then, by conversion, BA is Book V, to AE, as DC to CF. Because AB is to BE, 'as CD to DF, by division, C T a 17.5 b B. 5. 9 C 18. 5. a 8. Se C D which, taken two and two, have the same ratio ; if the first be greater than the third, the fourth thall 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, fo 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 magnitude, A has to B a greater ratio than C has to B a : But as D is to E, fo is A to B ; therefore D has to E a A, B, C, greater ratio than C to Bb: and because B is to C, D, E, F. b 13. 5. 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 Ed: But the dCor.13.5. magnitude which has a greater ratio than another to the same magnitude, is the greater of the two * ; D is therefore greater e 10. 5. than F. Secondly, Let A be equal to C; D shall be equal to F: Because A is equal to C, A is to B, as C is to Bf: But A is to B, f 7. 5. as D to E ; and C is to B, as F to E; wherefore D is to E, as F to E 8; and therefore D is equal to Fh. gil. 5. Next, Let A be less than C; D shall be less than F: For, 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; and C is greater than A, therefore F is greater than D, by the first case ; that is, D is less than F. Therefore, &c. Q. E. D. c B. 5. PROP. XXI. THEOR. If there be three magnitudes, and other three, but in a cross order; if the first be greater than the third, the fourth shall be greater than the fixth ; and if equal, equal; and if less, less. R Let a 8. 5. Book V. 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 Because A is greater than C, A has to B a greater ratio than C has to B a: But E is to F, as A to B ; therefore E has to F b 13. 5. a greater ratio than C to B b: and because B is to c B. 5. C, as D to E, by inversion , C is to B, as E to D: A, B, c, and E was shown to have to F a greater ratio than D, E, F. C to B; therefore E has to F a greater ratio than dCor.13.5. E to D d: But the magnitude to which the same has a greater e 19. 5. ratio than has to another, is the lesser of the two o; F there. fore is less than D; that is, D is greater than F. Secondly, Let A be equal to C; D shall be equal to F. Bef 7. 5. cause A and C are equal, A is to B, as C is to B*: But A is to B, as E to F; and C to B, as E to D; wherefore E is to F, as & 11. 5. E to Ds; and therefore D is equal to Fh. Next, Let A be less than C; D shall be less than F: For, as was shown, C is to B, as E to D, and, in like manner, B is to A, as F to E: and C is greater than A ; therefore, by case firft, F is greater than D; that is, D is less than F: Therefore, &c. Q.E.D. h 9. 5. PROP. XXII. THEOR. Z... F there be any number of magnitudes, and as many others, which, taken two and two in order, N M A B c. HT and that MA, ME are equimultiples of A, a Cor. 4.5. E; therefore MA is to B, as ME to Fa: M and Et اح с and because B is to CD, as F to GH, and NC, NG are equi- Book V. multiples of CD, GH, B is to NC, as F to NG a: and because w MA is to B, as ME to F, and B to NC, as F to NG ; and a Cor.4. 5o that MA is less than NC; therefore ME is less than NG 6. b 20. 5. In like manner, because MA is not less than ND, it may be proved, that ME is not less than NH: Wherefore MA, ME contain CD, GH the fame number of times that ND, NH contain them, that is, equally: and MA, ME are any equimultiples of A, E; as many times, therefore, as any multiple of A contains CD, so many times does the same multiple of E contain GH: Therefore as A is to CD, so is E to GH. C 5.Def. 5. Next, Let there be four magnitudes A, B, C, D, and other four E, F, G, H, which two and two have A, B, C, D, the same ratio, viz. as A is to B, so is E to F; as B to C, so F to G; and as C to D, so G to E, F, G, H. 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; there. fore, by the first case, A is to C, as E to G: But C is to D, as G to H; therefore again, by the first case, A is to D, as E to H; and so on, whatever be the number of inaguitudes, Wherefore, &c. 0. E. D. PROP. XXIII. THEOR. F 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 First, Let there be three magnitudes A, B, CD, and other three E, F, GH, which, taken two and two in a cross order, have the fame ratio, that is, such that A is to B, as F to GH, and as B is to CD, fo is E to F: A shall be to CD, as E is to GH. А Take MA, MB, ME any equimultiples of A, B, E, so that MA be greater than CD; and take NC the least multiple of CD that is greater than MA, and NF, NG the fame multiples of F, GH: and because MA, MB are equimultiples of A, B; A is to B, as MA to Ñ MB a : and, for the same reason, F is to GH, as NF to NG: But as A is to B, so is F to R2 N * N. B. This is said to be by Perturbate Equality. Book V. to GH;" as therefore MA is to MB, so is NF to NG 6. And because B is to CD, as E to F, and that MB, ME are equi. b 11. 5. multiples of B, E, and NC, NF of CD, F; as MB is to NC, C4. 5. fo is ME to NFC: Therefore MA, MB, NC are three magnitudes, and ME, NF, NG other three, which have the faine ratio, taken two and two in a cross order : and MA is less than d 21. 5. NC; therefore ME is less than NG 4: In like manner, be caufe MA is not less than ND, it may be proved, that ME is not less than NH: and ND, NH contain CD, GH equally ; therefore MA, ME contain them equally; as many times, therefore, as any multiple of A contains CD, so many times does the same multiple of E contain GH; therefore, as A is to e s.Def, 5. CD, so is E to GH. 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, A, B, C, D, as G to H; B to C, as F to G; and C to D, as E, F, G, H. E to F: A is to D, as E to H. Because A, B, C are three magnitudes, and F, G, H other three, which, taken two and two in a cross order, have the same ratio ; by the first case, A is to C, as F to H: But C is to D, as E to F; wherefore again, by the first case, A is to D, as E to H: and so on, whatever be the number of magnitudes, Wherefore, &c. Q. E. D. PROP. XXIV. THEOR. IF F the first has the same ratio to the second, which the third has to the fourth; and the fifth to the second, the same ratio that the fixth has to the fourth; the first and fifth together shall have to the second, the same ratio which the third and fixth together have to the fourth. Let AB the first, have to C the second, the same ratio which H В Because BG is to C, as EH to F; by in A ( DF a B. s. version, ·, C is to BG, as F to EH : and because, |