PROPOSITION II. THEOREM. If the first be the fame Multiple of the second, as the third is of the fourth ; and if the fifth be the same Multiple of the second, as the fixth is of the fourth; then all the first, added to the fifth, be the same Multiple of the second, as the third, added to the fixth, is of tbe fourtb. E T the firft A B be the fame Multiple of the se cond C, as the third D E is of the fourth F; and let the fifth BG be the same Multiple of the second C, as the sixth Α. D B + E second C, as the third added to + the fixth, viz. DH, is of the fourth G C С НЕ For, because A B is the same Multipile of C, as D E is of F; there are as many Magnitudes equal to C in A B, as there are Magnitudes equal to F in D E. And, for the same Realon, there are as many Magnitudes equal to C in B G, as there are Magnitudes equal to Fin E H. Therefore there are as many Magnitudes equal to C, in the whole A G, as there are Magnitudes equal to F in , DH. Wherefore A G is the same Multiple of C, as DH is of F. And so the first, added to the fifth, A G, is the same Multiple of the second C, as the third, added to the sixth, DH, is of the fourth F. Therefore, if the first be the same Multiple of the second, as the third is of the fourth ; and if the fifth be the same Multiple of the second, as the fixth is of the fourth; then Mall the first, added to the fifih, be the same Multiple of the second, as tbe third, added to the sixth, is of the fourth ; which was to be demonstrated. PRO PROPOSITION III. . THEOREM. the third is of ibe fourth, and there be taken L ET the first A be the same Multiple of the second B, as the third C is of the fourth D; and let E F, GH, be Equimultiples of A F and C. I say, EF, is the same H Multiple of B as C H is of D. For, because E F is the same Multiple of A, as G H is of C, there are as many Magnitudes K+ L + equal to A in EF, as there are Magnitudes equal to Cin GH. Now divide E F into the Magnitudes E K, K F, each equal to A, and G H into the Mag EAB G CD nitudes GL, LH, each equal to C. Then the Number of the Magnitudes E K, K F, will be equal to the Number of the Magnitudes GL, L H. And because A is the same Multiple of B, as C is of D, and E K is equal to A, and G L to C; E K will be the same Muliiple of B, as GL is of D. For the same Reason K F shall be the same Multiple of B, as L H is of D. Therefore because the first E K is the same Multiple of the second B, as the third GL is of the fourth D, and K F the fifth is the same Multiple of B, the second, that LH, the fixth, is of D the fourth : Therefore the first added to the fifth, EF, shall be * the same Multiple of the second B, as * 2 of ebis. the third added to the sixth, G H, is of the fourth D. If, therefore, the first be the same Multiple of the second, as the third is of the fourth, and there be taken Equimultiples of the first and third i then will the Magnitudes to taken, be Equimultiples of the second and fourth ; which was to be demonstrated, PRO to H. PROPOSITION IV. THEOREM. as the third to the fourtb; then also fall ibe ever, if they be to taken as to answer each other. LET the firft A have the same Proportion to the second B, as the third Chath to the fourth D; and let E and F, the Equimultiples of A and C, be any how taken; as also G and H, the Equimultiples of B and D. I fay, E is to G as F is For take K and L, any Equimul- Then, because E is the same and L are taken Equimultiples of E * 3 of tbis. and F; therefore K will be * the same Multiple of A, as L is of C. multiples of B and D; if K exceeds Defo 5 of M, then + L will exceed N; if K be equal to M, L will be equal to ples of G and H. Therefore, as I Dif.5. E is to G, lo shall I F be to H. multiples this. multiples of the forft and third have the same Proportion to the Equimultiples of the second and fourth, according to any Multiplication whatsoever, if they be so taken as to answer each other ; which was to be demonstrated. Because it is demonstrated, if K exceeds M, then L will exceed N; and if K be equal to M, L will be equal to N ; and if K be less than M, L will be less than N: It is manifeft, likewise, if M exceeds K, that N fall exceed L; if equal, equal ; but if less, less. And therefore, as G is to E, so is * H to F. Dof. 5. Coroll. From hence it is manifeft, if four Magnitudes be proportional, that they will be also inverfely proportional. P R O POSITION V. THEOREM. Magnitude, as a Part taken from the one is of E+1 LET the Magnitude A B be the same Multiple of the Magnitude CD, as the Part taken away A E, is of the Part taken away CF. I say, that the Residue E B is the same Mul. B tiple of the Refidue FD, as the whole A B is of the whole CD. 1 For, let E B be such a Multiple of С + F * 3 of this. multiples of C F and CD: Therefore A B is the fame Multiple of GF, as of CD; and so GF is + equal to CD. Now let CF, which is + Ax. 2.cf common, be taken away, then the Residue G C is ibis. equal equal to the Refidue DF. And then, because A E is PROPOSITION VI. THEORE M. nitudes, and some Magnitudes Equimultiples of Equimultiples of them. of two Magnitudes E, F; and let the Magni- For, first, Let G B be equal to E. I say, HD is EF away * 1 of tbis. B) |