Book V. PROP. II. THEOR. second that the third is of the fourth, and the fifth the fame multiple of the second that the fixth is of the fourth; then shall the first together with the.fifth be the same 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 C the second, that EH the fixth is of F the fourth. Then is AG the first together with the fifth the D E B G HIF D А E GH H ČLF d Book v. PRO P. III. THEO R. F 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 fame multiple of B the second, that the third is of D the fourth ; and of A, C let the equimultiples EF, GH be taken. then EF is the same 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 divided into the F magnitudes EK, KF, each e H qual to A, and GH into GL, LH, each equal to C. the number therefore of the magnitudes EK, KF, shall be e K qual to the number of the L others GL, LH. and because 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 E A B G G C D multiple of 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 EF, GH equal to A, C, because therefore the firft EK is the same multiple of the second 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 same mul. tiple • of the second B, which GH the third together with the 7, 2052 Sixth is of the fourth D. If therefore the first, &c. Q. E. D. IF Book V. PROP. IV. THEOR. SE N. F 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 shall have the same ratio to any equimultiples of the second and fourth, viz. ' the equiinultiple of the first shall - have the fame ratio to that of the second, which the equimultiple of the third has to that of the fourthi’ Let A the first have to B the second, the same ratio which the Take of E and F any equimul. that N is of D. and because as A KE A B GM Þ. Hypoth. is to B, fo is C to D5, and of ALF CD HN and C have been taken certain equi- er than M, L is greater than N; c. $.Def s. and if equal, equal; if lefs, less . And K, L are any equimultiples Cor. Likewise if the first has the same ratio to the second, which the first and third have the same ratio to the second and fourth. Book v. Let A the first have to B the second, the fame ratio which the Take of E, F any equimultiples whatever K, L, and of B, D any equimultiples whatever G, H; then it may be demonstrated, as be-. fore, that Ķ 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". 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 demonstrated, C. IF PROP. V. THE OR which a magnitude taken from the first is of a mag- Let the magnitude AB be the same multiple of CD, that AE G A 2. Bol E F is of CD; wherefore EG is equal to AB6. take b. 1. AX. from them the common magnitude AE; the remainder AG is equal to the remainder EB, Wherefore since AEis the same multiple of CF, B D that AG is of FD, and that AG is equal to EB; therefore AE is the fame multiple of CF, that EB is of FD. but AE is the fame HA IF Book V. multiple of CF, that AB is of CD; therefore EB is the same mul tiple of FD, that AB is of CD. Therefore if one magnitude, &c. Q. E. D. PROP. VI. THEOR. See N. F two magnitudes be equimultiples of two others, and if equimultiples of these be taken from the first two, the remainders are either equal to these others, or equimultiples of them. Let the two magnitudes AB, CD be equimultiples of the two E, F, and AG, CH taken from the first two be equimultiples of the same E, F; the remainders GB, HD are either equal to E, F, or equimultiples of them. First, Let GB be equal to E; HD is equal to F. make CK equal CH multiple of F, that CD is of F; wherefore Gt at 2. 1. Ax. 5. KH is equal to CD'. take away the common magnitude CH, then the remainder KC is B DEF to F, HD therefore is equal to F. But let GB be a multiple E; then HD is the same multiple of F. Make CK the fame multiple of F, that GB is of E. and because AG is the fame mul K A fore AB is the same multiple of E, that KH E, that CD is of F; therefore KH is the G B DEF and that KC is equal to HD; therefore HD is the same multiple of F, that GB is of E. If therefore two magnitudes, &c. Q. E. D. |