and if the firft Magnitude A be greater than the third C; then fhall the fourth D be greater than the fixth F. But if the A firft belequal to the third C; then the fourth will be equal to the fixth F; and if the first be less than the third, the fourth will be less than the fixth. 1. Hyp. If A be C. because E:F:: B: C. C B ope 8. 5. 2. Hyp. By the like Argument, if A be C, we prove that DF QE. D. 3, Hyp. If A Af: B:: D: E. QE. D. C. because F: E::C: B:: DF. A B C D E F PROP. XXI. If there be three Magnitudes A, B, C, and others D, E, F, equal to them in Number, which taken two and tu, are in the fame Proportion, (A: B:: E: F. and B:C:: D: E). and the Pro portion be perturbate, if the firft Magnitude A be greater than the third C, then will the fourth D be greater than the fixth F. But if the firft A be equal to the third B, then the fourth D fhall be equal to the fixth F; but if less, less. 1 Hyp. Let A be C. becaufe " D: E:: B: C, by Inverfion it fhall be as E: D:: C: B. But E 5° 10.5. F C Whence DF. Q. E. D. 2 Hyp. In like manner, if AC, fhall D be F. Q. E. D. e 3 Hyp. If AC. becaufe Dd: E: D:: C: B::AB:: E: F. Therefore is DF. Q.E.D. PROP. GIL HKM XXII. d с If there be any Number of Magnitudes A, B, C, and others D, E, F, equal to them in Number, which taken two and two, are in the fame Ratio, (viz. A: B:: D: E and B:C:: E:F) they shall be in the fame Proportion by equa lity, viz. A: C:: D: F. Take G, H; I, K ; L, M ; refpectivelyEquimultiples of A,D; B, E; C, F. h 7. 5. byp Ђура 9. 5. пЂуро i Because A: B::D: E. "hyp. 1 k k 20. 5. ABCNDEFO , or L, H fhall be likewife,, or M. Therefore A: C:: 16. def.5. D: F. in like manner. If moreover C:N::F: O, by equality fhall A: N:: D:O. Q. E. D. PROP. a 15. 5. b byp. 4. 5. D PROP. XXIII. E F If there be three Magnitudes A, B, C, and others D, E, F, equal to them in Number, which taken two and two are in the fame Proportion, and if their Proportion be perturbate, viz. A B :: E: F. and B: C:: D: E, then GHIKLM fhall they also be in the fame Proportion by equality A: C :: D: F. Take G, H, I; as alfo K, L, M, Equimultiples of A, B, D, and C, E, F. then shall G:H:: A: B:: EF:: a L: M. Again, becaufe B: C:: D: E. therefore H:I::K: L. Whence G, H, K; and I, L, M, have the Conditions according to Prop. 21. of this Book. Therefore if G be,=, or K, C * 6 def. S. in like manner will I —, or, M. therefore", A:C:: D:F. Q. E. D. After the fame manner, if there were more than three Magnitudes, &c. CORO L. From Prop. 22, 23, as alfo 5th, & 20th, of this Book, it follows that Ratio's compounded of the fame Ratio's, are the fame among themfelves. Alfo the fame parts of those Ratio's are the fame among themselves. PROP. If the first Mag- fourth F; and if the Second C, as the fixth EH to the fourth F; then shall the first compounded with the fifth, viz. AG, be to the fecond C, as the third compounded with the fixth viz. DH, is to the fourth F. b 22. 5. For becaufe AB: C:: DE: F; but from the a byp Hyp. and Inverfely C: BG:: F: EH ; therefore by equality AB: BG:: DE: EH. and fo by compounding AG: BG:: DH: EH. alfo BG: hyp C:: EH: F. therefore again by equality C::DH: F. Q.E. D. PROP. XXV. с AG: If four Magnitudes be Proportionals (AB: CD:: E: F) the greatest AB and the leaft F, ball be greater than CD and E, the other 免費 1 two. e с с Make AGE, and CH = F. • Because hyp. AB: CD:: E:F:: AG: CH. therefore is 7.5. AB: CD:: GB: HD. but ABCD. whence 19. $. f byp. 8 GB HD. but AG+FE+CH. there- schol. 14 fore AGFGBE+CH + HD, that s is, AB+FE+CD. End of the Fifth Book. K EU EUCLID's ELEMENTS. I. BOOK VI. DEFINITIONS. Imilar right-lined Figures, as ABC, about the equal Angles proportional. A B The Ang. B=DCE; and AB: BC :: DC: CE. alfo the Ang. AD; and BA: AC:: CD: DE. and finally the Ang. BC: CA :: CE: ED. ACB≈ E. and II |