1 Def. 5. shall † F be to H. Wherefore, if the first have the same Proportion to the second, as the third to the fourth; then also, Mall the Equimultiples of the firf and third have the fame Proportion to the Equimultiples of the second and fourth, according to any Multiplication whatsoever, if they be fo taken as te answer each; which was to be demonstrated. Because it is demonstrated, if K exceeds M, then L will exceed N; and if it be equal to it, it will be equal; and if less, lefser. It is manifest likewise, if M exceeds K, that N shall exceed L; if equal, equal; but if lefs, less. And therefore as G is to E, * Def: 5. fo is * H to F., Coroll. From hence it is manifest, if four Magnitudes be proportional, that they will be also inversely proportional. PROPOSITION V. THE O RE M. Magnitude, as a Part taken from the one is of a of the Magnitude CD, as the Part taken away G whole AB is of the whole CD. For let EB be such a Multiple of E F as AB is of GF. But AE and AB A D are put Equimultiples of CF and CD. Therefore AB is the fame Multiple of GF as of +2 exism CD; and fo G F is t equal to CD. Now let CF, which is common, bę takon away, and the Residue GC of Ibis. GC is equal to the Residue DF. And then becaufe AE is the same Multiple of CF, as EB is of CG, and CG is equal to DF; AE shall be the fame Multiple of CF, as EB is of FD. But AE is put the fame Multiple of CF as AB is of CD. Therefore EB is the fame Multiple of FD, as AB is of CD: and so the Residue EB is the same Multiple of the Residue. FD, as the whole AB is of the whole CD. Wherefore, if one Magnitude be the same Multiple of another Magnitude, as a Part taken from the one is of a Part taken from the other; then the Residue of the one shall be the same Multiple of the Residue of the other, as the whole is of the whole ; which was to be demonstrated. PROPOSITION VI. . THEORE M. nitudes, and some Magnitudes Equimultiples of ples of two Magnitudes E, F, and let the Magnitudes AG, CH, Equimultiples of the fame E, F, be taken from AB, CD. I say, the Residues GB, HD, are either equal to E, F, or are Equimultiples of them. For first, Let GB be equal to E. I say, HD is also equal to F. For let CK be equal to F. Then because AG A is the same Multiple of E, as CH K is of F; and GB is equal to E; and CK to F; AB will be * the с * 1 of this. fame Multiple of E, as KH is of F. But A B and CD are put G H Equimultiples of E and F. 1 And because KH and CD are Take away CH which is com- K A с G | H PROPOSITION VII. PROBLEM. Tame Magnitude ; and one and the same Magni- any other Magnitude. I say, A and B have the same Proportion to C; and likewise Chas the same Proportion to A as to B. For take D, E, Equimultiples D A E B C F А Déf. 5o A as to B. For the same Construction remaining, we prove, in like manner, that D is equal to E. Therefore, if F exceeds D, it will also exceed E; if it be equal to D, it will be equal to E; and if it be less than D, it will be less than E. But if F is Multiple of C; and D, E, any other Equimultiples of A, B. Therefore as C is to A, fo shall * C be to B: Where- * Def. go fore equal Magnitudes have the same Proportion to the same Magnitude, and the fame Magnitude to équal ones; which was to be demonstrated. PROPOSITION VIII. THEOREM. a greater Proportion to some third Magnitude, is let Because AB is greater than C, make BE equal to C, that is, let AB exceed C by AE; then AE multiplied some Number of Times, will be greater than D. Now let G AE be multiplied until it ex A ceeds D, and let that Multiple of AE, greater than D,be FG. E Now because N is the than thän K, M will not be greater than K, that is, ķ will not be less than M. And since FG is the same Multiple of AE; às ĠH is of EB; FG shall be * 1 of this. * the fame Multiple of AE; as FH is of AB, but FG is the same Multiple of A E, as K is of C; wherefore FH is the same Multiple of AB, as K is of C; that is, FH, K, are Equimultiples of AB and C. Again, because GH is the same Multiple of EB, as K is of C, and EB is equal to C; GH shall if Ax. I. be f equal to K. But K is not less than M. Therefore GH shall not be less than M; but FG is greater than D. Therefore the whole FH will be greater than M and D; but M and D together, are equal to N; because M is a Multiple of D, the nearest lesser than N: Wherefore FH is greater than N. And so finĉe FH exceeds N, and K does not, and FH and K are Equimultiples of A B and C, and N is another Def. 7. Multiple of D; therefore A B will have I a greater Ratio to D, than C has to D. I say, moreover, that D has a greater Ratio to C, than it has to AB; for the fame Construction remaining, we demonstrate, as béfore, that N exceeds K, but not FH. And N is a Multiple of D, and FH, K, are Equimultiples of AB and C. Therefore D has I a greater Proportion to C, than D hath to B. Wherefore the greater of any two unequal Magnitudes, has a greater Proportion to some third Magnitude, than the lefs has; and that third Magnitude hath a greater Proportion to the lefser of the tws Magnitudes, than it has to the greater. PROPOSITION IX. . THEOREM. and the fame Magnitude, are equal to one an- fame Proportion to Č. I say, A is equal to B. For |