than F. Whence a B: A:: D: C. PROP. V. Gr C FID Same Multiple of another CD, as a part AE taken from the one, is of a part CF taken from the other, then fhall the remaining part EB be the fame Multiple of the remaining part FD, as the whole AB is of the whole CD. Take fome other Magnitude GA, such a Multiple of the remaining one FD, as the whole AB is of the whole CD, or the Part taken away AE of the part taken away CF. Then fhall the whole GA + AE be the fame Multiple of the whole CF + FD, as one Magnitude AE is of one (CF); that is, as AB is of CD. Therefore GE AB. And fo if the common Part AE be taken away, there fhall remain GA = EB; therefore, &c. с If two Magnitudes AB,CD, be Equimultiples of two Magnitudes E, F; and Bfome Magnitudes as AG, CH Equimultiples of E, F, be taken a way; then the refidues GB, HD are either equal to thofe Magnitudes E, F, or elfe Equimultiples of them. For fince the Number of Parts in AB equal to E is fuppofed equal to the Number of Parts in CD equal to F; aifo the number of Parts in AG equal to the Number of parts in CH: If from hence you take away AG, and from thence CH, there will remain a a number of Parts in a 3 ax. GB equal to the number of Parts in HD. Therefore if GB be E; then fhall HD De alfo F. And if GB be any Multiple of E; HD fhall also be the fame Multiple of F. QE. D. = the fame Mag nitude C; and the fame C to the equal ones A and B. Take D and F Equimultiples of the equal Magnitudes A and B, and let E be any Multi-. ple of C; then fhall DF. Whence if D be 6 ax. ==, or than E, then fhall F be =,=, с с or than E. Therefore A:C:: B: C. And 6 def. 5. inversely C: A::C: B. QE. D. d SCHOL. If you take two Equimultiples inftead of the Multiple F, it may be proved after the fame manner that equal Magnitudes have the fame Proportion to equal ones. PROP. તા cor. 4, 5. VIII. Magnitude D than a leffer AC, and that third Magnitude D, has a greater Ratio to AC the leffer of thofe Magnitudes than to AB the greater. Take EF, EG Equimultipics of AB, AC, fo that EH a Multiple of D be greater than EG, and less than EF, (which will eafiy fall out, if EG, GF be taken greater than D.) Then, by AB def. 8. 5. fhall ; and fo AB EG, but EH EF, def. 5. (as has been already faid) therefore: CAB. Q. E.D. с B PRO P. IX. Magnitudes that have the fame Proportion to one and the fame Magnitude, are equal to one another; and if a Magnitude has the Same proportion to other Magnitudes, thefe Magnitudes are equal to one another. 1. Hyp. Let A:C:: B: C; I fay A = B. For if A be, or than B, then shall be A B or than C Which is contrary to the Suppofition. 2. Hyp. Let C: B:: C: A; I fay A =B. For contrary to the Suppofition. Which is PROP. A C B PROP. X. Of Magnitudes having Proportion to the fame Magnitude, that which has the greater Proportion is the greater Magnitude, and that Magnitude to which the fame bears a greater Proportion, is the leffer Magnitude. A B ; I fay AB. For 1. Hyp. Let if you fay A=B; then fhall which is contrary to the Hyp. B then fhall b contra. Hyp. C C 2. Hyp. Let I fay BA. For if с you fay that A=B; then C:B::C:A. 7.5. which is contrary to the Hyp. Or if you say that BA. then fhall d སང་་-iamns----་---ས་ནས་ས་"ན HCDL GABK PROP. XI. contra. Hyp. Let A B :: E: F. alfo C: D:: E: F. I fay that' A: B::C: D. Take G, H, I, Equimultiples of A, C, E; and K, L, M of B, D, F. Then because A:BE: F. If G be ,, or than K, in like manner fhall f I be, 6 def. 5. or than M. So likeIEFM wife fince E: F::C: D; if I,, or than M, then fhall H be likewife fore if G,,-K, in like manner shall H a 6 def. 5. ——, or then L. Whence a A:B::C: a 1..5. D. Q. E.D. SCHOL. Proportions that are the fame to the fame. Proportions, are the fame to each other. HCDL PROP XII. If any Number of Magnitudes A, B ; C and D; E, F; be Proportional: as one of the Antecedents A is to one of the Confequents B; fo are all the Antecedents A, C, E to all. the Confequents B, D, F. Take G, H, I, Equimultiples of the Antecedents, and K, L, M, of the Confequents. Now because one of the Magnitudes as G is the fame Multiple of anoIEFM ther as A, as all of them, viz. G, H, I, is a of all A, C, E; and likewife.one of them as. K is the fame Multiple of another as B, as all of them K, L, M, are of all B, D, F. and moreover fince A: B:: C: D:: E: F. If G be,, or than K, in like manner fhall H be, or than L, and I, or than M, and therefore if G=,=,K, in like manner shall G+ H+I, be =,=, or than K+L+M. 6 def. 5. Whence A: B:: A+ C+E:B+D+ F. Q. E.D. byp GABK C CORO L. b Hence if like Proportionals be added to like ones, the Wholes fhall be Proportionals. PROP |