Foucson HCDL GABK PROP. XIII. If the firft A has the fame Proportion to the Second B, as the third C to the fourth D; and if the third C has a greater Proportion to the fourth D, than the fifth E to the fixth F then fball the firft A have a greater Proportion to the Second B, than the fifth E has to the fixth F. Take G, H, I, Equimultiples of A, C, E, and K, L, M of B, D, F. Because IEFM A:B:: C: D; if a H, a 6 def. 5. fible that H may be than L, and I not greater than M. Therefore it is poffible that G may be greater than K, and I not greater than M. A E Q. E. D. A C D B PROP. XIV. If the first Magnitude A has the fame Proportion to the Second B, as the third C to the fourth D; and if the first A be greater than the third C; then fball the fecond B be greater than the fourth D. But if the firft A be equal to the third C, then shall the Second B be equal to the fourth D; and if the first be lefs than the third, the fecond will be lefs than the fourth. € 119.5° A C Ab B C C D B U Whence BdD; d C, then fhall BD. In like manner if A be e C: B::A:B:: D. QE. D. SCHOL, By a more powerful Reason if C and B AC. fhall B be D. Alfo if AB; C fhall be D. And if A be, or fhall be likewife or D. Let AG, GK, KB, be parts of the Multiple AB equal to C; and DH, HL, LE parts of the Multiple hyp. Multiple DE equal to F. Now the Number of thofe is equal to the Number of these. There a fore fince AG: DH:: C: F:: GK: HL.: 7.5. KB: LE; then shall AG+GK+KB: (AB): 12.5. DH+ HL + LE (DE) :: C: F. Q. E.D. с PROP. XVI. If four Magnitudes A, B, C, D are proportional, they shall be alternately proportiona?, (viz. A: C:: B: D.) Take E and F Equimultiples of A and B; and G, H Equimultiples of C and D. Now E: b e byp F:: A: B::C: De :: Gf: H. Whence if Ed 15. 5. be,,G, then fhall likewife F =,=, , H. Whence & A: C:: B: D.. Q.E. D. SCHOL. Here you must observe that alternate Ratio only takes place, where the four Proportionals are all of the fame kind; for if they are not, a Comparison cannot be admitted. f. 5. Ego 14. 5. 86 def. 5. PROP. : M E Κ PRO P. XVII. 3 If Magnitudes compounded are Proportional (AB: CB:: DE: FE) they fhall also be Proportional when divided, (AC: CB:: DF: FE). Take GH, HL, IK, KM refpectively Equimultiples of AC, CB, DF, FE. Alfo LN, MO K Equimultiples of CB, FE. Then the whole GL is the fame Multiple of the whole AB, as one of the Magnitudes GH is of AC another of them; that is as IK is of DF; that is, as the whole IM is of the whole DE: -alfo HN (HL+LN) is the fame Multiple of CB, as KO (KM +MO) is of FE. Therefore fince, by the hyp. AB : BC :: 6 def. 5. HN, in like f 5 ax. e с h GADI DE: EF. if GL be,, 1 manner fhall IM —, —, KO. Therefore taking from both, the common parts HL, KM. if the remainder GH be, than LN, in like manner fhall fIK be, than MO. Whence AC: CB :: DF: FE. Q. E. D. g PROP. XVIII. or or B C A D+ If Magnitudes divided be Proportional (AB: BC :: GEGF DE: EF) thefe fhall be Proportional when cam-i pounded, (viz., AC: CB :: DF:FE). h For if poffible, let AC: CB :: DF: FG k is a is abfurd. The like abfurdity will follow if it 29 ax. be faid that AC: CB:: DF: GFFE. b DE as a part c AC taken away, is to the part DF taken away; d e CORO L. then by bbyp. Therefore whence 16. 5. d 1. Hence if fimilar Proportionals be taken from fimilar Proportionals, the remainders fhall be Proportional. 2. Hence alfo is converfe Ratio demonftrated. : e g 17. 5. byp. & 11. 5. For let AB CB :: DE: FE. then fhall AB: AC:: DE: DF. for by Permutation AB: £ 16. 5. DE: CB FE. therefore AB : DE:: AC: & 19. 5. DF. and fo again by Permutation, AB : AC:: DE: DF. Q.E. D. A B C D E F : PROP. XX. If there be three Mag- |