Book. V. II. Those magnitudes of which the same, or equal magnitudes, are equimultiples, are equal to one another. III. IV. multiple of another, is greater than that other magnitude. IF any number of magnitudes be equimultiples of as many, each of each ; what multiple soever any one of them is of its part, the same multiple fhall all the first magnitudes be of all the other. Let any number of magnitudes AB, CD be equimultiples of as many others E, F, each of each ; whatsoever multiple AB is of E, the fame multiple shall AB and CD together be of E and F together. Becaute AB is the same multiple of E that CD is of F, as many magnitudes as are in AB equal to E, so many are there in CD equal to F. Divide AB into magnitudes equal to E, viz. AG, GB; and CD in A. to CH, HD equal each of them to F: The number therefore of the magnitudes CH, HD shall E be equal to the number of the others AG, G GB : And because AG is equal to E, and CH to F, therefore AG and CH together are B a Ax. 2c 1. equal to E and F together : For the same reafon, because GB is equal to E, and HD to F; GB and HD together are equal to E and F C F Therefore, if any magnitudes, how many soever, be equimultiples of as many, cach of each, whatsoever multiple any one of them is of its part, the same multiple shall all the firit magnitudes be of all the other: For the same demonstration holds Book V. holds in any number of magnitudes, which was here applied mito two.' 0. E. D. PRO P. II. THEOR. cond that the third is of the fourth, and the fifth the same multiple of the second that the sixth is of the fourth; then shall the first together with the fifth be the fame multiple of the second, that the third together with the fixth is of the fourth, Let AB the first be the same multiple of C the second, that DE the third is of F the fourth ; and BG the fifth the same multiple of the second, that EH D B D of the fourth F. If, therefore, the first be Α. E B K C, that the whole of the last, viz. DL, HC LF PROP. G Book V. PROP. III. THE O R. the first be the same multiple of the second, which the third is of the fourth ; and if of the first and third there be taken equimultiples, these shall be equimultiples the one of the second, and the other of the fourth. Let A the first be the same multiple of B the second, that C the third is of D the fourth; and of A, C let the equimultiples EF, GH be taken : Then EF is the fame multiple of B, that GH is of D. Because EF is the same multiple of A, that GH is of C, there are as many magnitudes in EF equal to A, as are in GH equal to C: Let EF be di vided into the magnitudes F H EK, KF, each equal to A, and GH into GL, LH, cach equal to C:The number therefore of the magnirudes EK, KF, shall be e K+ qual to the number of the L+ others GL, LH: And becaufe A is the same multiple of B, that C is of D, and that EK is equal to A, and GL to C; therefore EK is the fame multiple of E A B G C D B, that GL is of D: For the same reason, KF is the same multiple of B, that LH is of D; and so, if there be more parts in FF, GH equal to A, C: Because, therefore, the first EK is the same multiple of the fecond B, which the third GL is of the fourth D, and that the fifth KF is the same multiple of the second B, which the fixth LH is of the fourth D; EF the first together with the fifth is the fame multiple of the second B, which GH the third together a 2. s. with the fixth is of the fourth D. If, therefore, the first, &c. Q. E. D. PROP. Book V. PRO P. IV. THE O R. Sec N. TF the first of four magnitudes has the same ratio to the second which the third has to the fourth ; then any equimultiples whatever of the first and third fhall have the fame ratio to any equimultiples of the second and fourth, viz. ' the equimultiple of the first shall have • the same ratio to that of the second, which the equi. multiple of the third has to that of the fourth.' Let A the first have to B the second, the same ratio which the third C has to the fourth D; and of A and C let there be taken any equimultiples whatever E, F; and of B and D any equimultiples whatever G, H: Then E has the same ratio to G, which F has to H. Take of E and F any equimultiples whatever K, L, and of G, H, any equimultiples whatever M, N: Then, because E is the fame multiple of A, that F is of C; and of E and F have been taken equimultiples K, L; therefore K is the same multiple of A, that L a 3.5. is of C: For the same reafon, M K E A B Ġ M is the same multiple of B, that N L F C DH N is of D: And because as A is to b Hypoth. B, fo is C to Db, and of A and Chave been taken certain equi. N; and if equal, equal; if less, multiples whatever of E, É ; and Cor. Likewise, if the first has the same ratio to the second, which the third has to the fourth, then also any equimulti p.es ples whatever of the first and third have the same ratio to the Book V. second and fourth: And in like manner, the first and the third have the same ratio to any equimultiples whatever of the second and fourth. Let A the first have to B the second, the fame ratio which the third C has to the fourth D, and of A and C let E and F be any equimultiples whatever; then E is to B, as F to D. Take of E, F any equimultiples whatever K, L, and of B, D any equimultiples whatever G, H; then it may be demonstrated, as before, that K is the same multiple of A, that L is of C: And because A is to B, as C is to D, and of A and C certain equimultiples have been taken, viz. K and L; and of B and D certain equimultiples G, H; therefore, if K be greater than G, L is greater than H ; and if equal, equal; if less, less : cs. def. so And K, L are any equimultiples of E, F, and G, H any whatever of B, D, as therefore E is to B, so is F to D: And in the same way the other case is demonstrated. F one magnitude be the fame multiple of another, Sec N. which a magnitude taken from the first is of a magnitude taken from the other; the remainder shall be the same multiple of the remainder, that the whole is of the whole. Let the magnitude AB be the same multiple of CD, that AE taken from the first, is of CF taken from the other ; the remainder EB shall be the same multiple of the remainder FD, that she whole AB is of the whole CD. AL Take AG the same multiple of FD, that AE is of CF: Therefore AE is the same multiple of CF, that EG is of CD: But AE, by the hypothesis, is the same multiple of CF, that AB is of CD: Therefore EG is the same muls E tiple of CD that AB is of CD; wherefore EG F is equal to AB6: Take from them the common bs. AX. 56 magnitude AE ; the remainder AG is equal to the remainder EB. Wherefore, fince AE is B D the same multiple of CF, that AG is of FD, and that AG is equal to EB; therefore AE is the same multiple of CF, that EB is of FD : But AL is the same multiple of CF, that |