The number of Parts in AB equal to Cis suppos'd equal to the number of Parts in DE equal to F. Also the number of Parts in BG is supposed equal to the Number of Parts in EH. Therefore the number of Parts in AB + BG is * equal to the number of Parts in DE + EH : That is, the whole AG is the fame Multiple of C, as the whole GH is of F. Q.E.D. If the first A be the fame Multiple of the fecond Bas the third C is of the fourth D, and there be taken Equimultiples EI, Hyp FM of the first and third; then will each of the Magnitudes taken be Equimultiples of the fecond B and fourth D. Let EG, GH, HI be parts of the Multiple El equal to A; and FK, KL, LM parts of the Multiple FM equal to C. The Number of these is equal to the Number of them. Moreover A, that is, EG or GH, or GI, is supposed the same Multiple of Bas C or FK, &c. is of D; therefore EG+GH is the same Multiple of the second B, as FK + KL is of the fourth D. By the same way of reasoning, EI (EH + AI) is the fame Multiple of B, as FM (FL + LN) is of D. Q. E.D. €2.5. 2.5. PROP. has the Same Proportion to the Second B as the third C to the fourth D; then alfo fhall E, F the Equimultiples of the first A and third C, have the Same Proportion to G, H, the Equimultiples of the second B and fourth D, according to any MultiplicaGL tion whatsoever, if they be HM so taken as to answer each other, (viz. as E: G :: F: H.) Take I and K; and L, M, Equimultiples of E,F; and of G, H. Then shall * I be the same Multiple * 3. 36 of A as K is of C; so likewife is L the fame Multiple of B, as M is of D. nce A: B:: CD; according to Hyp De, = orL; confequently in shall K =, =,, than M. nce I, K are taken Equimultiples of nd L, M, of G and H; therefore Def. 7. shall E:G::F: H. COROL. e may Inverse Ratio be demonftrated. afe A: B:: C: D, if E be, =, G, in like manner shall F be, a 6 def. 3 an H. Therefore it is manifest that * 6 def. 5. = or QE. D. a than F. Whence B:A::D:C PROP. V. 1. 5. 6 ax. 3 ax. C F E B If one Magnitude AB be the Same Multiple of ID another CD, as a part AE taken from the one, is of a part CF taken from the other, then shall the remaining part EB be the Same Multiple of the remaining part FD, as the whole AB is of the whole CD. Take some 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 shall the whole GA + AE be the same 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 shall remain GA = EB; therefore, &c. F, be taken a way; then the refidues GB, HD are either equal to those Magnitudes E, F, or else Equimultiples of them. For fince the Number of Parts in AB equal to E is supposed equal to the Number of Parts in a CD equal to F; also 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 number of Parts in a 3 ax. GB equal to the number of Parts in HD. Therefore if GB be = E; then shall HD De alfo = F. And if GB be any Multiple of E; HD shall also be the same Multiple of F. Q.E. D. have the Same Proportion to nitude C; and the Same C to the equal ones A and B. Take Dand F Equimultiples of the equal Magnitudes A and B, and let E be any Multiple of C; then shall D = F. Whence if D be b 6 ax. ,, or than E, then shall F be, =, or than E. Therefore A:C:: B: C. And 6 def. 5. inversely C: A::C: B. Q.E. D. SCHOL. If you take two Equimultiples instead of the Multiple F, it may be proved after the fame manner that equal Magnitudes have the fame Proportion to equal ones. cor. 4, 5. ) Magnitude D than a leffer AC, and that third Magnitude D, has a greater Ratio to AC the leffer of those Magnitudes than to AB the greater. Take EF, EG Equimultiples of AB, AC, fo that EH a Multiple of D be greater than EG, and less than EF, (which will easiy fall out, if EG, GF be taken greater than D.) Then, by AB def. 8. 5. shall D Again, because EH AC ; and so D コ EG, but EH EF, *def.. 5. (as has been already said) therefore Q. E. D. DD Same proportion to other Magni tudes, these Magnitudes are equal to one another. 1. Hyp. Let A:C::B:C; I say A = B. A For if A be, or than B, then shall b be C B C Supposition. or than Which is contrary to the 2. Hyp. Let C: B:: C: A; I say A = B. For |