IF PROP. D. THEOR. Book V. F the firft be to the second as the third to the fourth, See N. and if the first be a multiple, or part of the fecond; the third is the fame multiple, or the fame part of the fourth. Let A be to B, as C is to D; and first let A be a multiple of B; C is the fame multiple of D. Take E equal to A, and whatever multiple A or E is of B, make F the fame multiple of D. then because A is to B, as C is to D; and of B the second and D the fourth equimultiples have been taken E and F; A is to E, as C to F. but A is equal to E, therefore C is equal to F. and F is the fame multiple of D, that A is of B. Wherefore C is the A B C D fame multiple of D, that A is of B. Next, Let the first A be a part of the fecond B; C the third is the fame part of the fourth D. Because A is to B, as C is to D; then, inverfely B is to A, as D to C. but A is a part of B, therefore B is a multiple of A, and, by the preceding cafe, D is the fame E a. Cor. 4. 5. b. A. 5. F See the Fi multiple of C; that is, C is the fame part of D, that A is of B. Therefore if the firft, &c. Q. E. D. PROP. VII. THEOR, EQUAL m QUAL magnitudes have the fame ratio to the fame magnitude; and the fame has the fame ratio to equal magnitudes. Let A and B be equal magnitudes, and C any other. A and B have each of them the fame ratio to C. and C has the fame ratio to each of the magnitudes A and B. Take of A and B any equimultiples whatever D and E, and of gure at the foot of the preceding page. c. B. 5. Book V. C any multiple whatever F. then because D is the fame multiple of A, that E is of B, and that A is equal to a. I. Am 5. B; D is equal to E. therefore if D be greater than F, E is greater than F; and if equal, equal; if less, less. and D, E are any equimultiples of A, B, and F is any multiple of C. b. 5. Def. 5. Therefore b as A is to C, fo is B to C. See N. Likewife C has the fame ratio to A that it has to B. for, having made the fame con- D A CF F unequal magnitudes the greater has a greater ratio to the fame than the lefs has. and the fame magnitude has a greater ratio to the less than it has to the greater. Let AB, BC be unequal magnitudes of which AB is the greater, and let D be any magnitude whatever. AB E has a greater ratio to D than BC to D. and D has a greater ratio to BC than unto AB. If the magnitude which is not the greater of the two AC, CB, be not less than D, take EF, FG the doubles of AC, CB, as in Fig. 1. but if that which is not the greater of the two AC, CB be less than D (as in Fig. 2. and 3.) this magnitude can be multiplied fo as to become greater than D, whether it be AC or CB. Let it be multiplied until it become greater than D, and let the other be multiplied as often; and let EF be the multiple thus taken of AC, and FG the fame multiple of CB. F G L A B KHD. therefore EF and FG are each of them greater than D. and in Book V. every one of the cafes take H the double of D, K its triple, and so on, till the multiple of D be that which first becomes greater than FG. let L be that multiple of D which is first greater than FG, and K the multiple of D which is next lefs than L. Then because L is the multiple of D which is the first that becomes greater than FG, the next preceding multiple K is not greater than FG; that is, FG is not lefs than K. and fince EF is the fame multiple of AC, that FG is of CB; FG is the fame multiple of CB, that EG is of ABa; wherefore EG and FG are equi- a. x. 5. multiples of AB and CB. and it was fhewn that FG E was not less than K, and, by the construction, EF is greater than D; therefore the whole EG is greater than K and D together. but K together with D is equal to L; therefore EG is greater than L; but FG is not greater than L; and EG, FG are equimultiples of AB, BC, and L is a multiple of D; therefore b AB has to D a greater ratio than BC has to D. Alfo D has to BC a greater ratio than it has to AB, for, having made E F A A F G B L the fame construction, it may be fhewn, in like manner, that L is greater than FG, but that it is not greater than EG. and L is a multiple of D; and FG, EG are equimultiples of CB, AB. Therefore D has to CB a greater ratio than it has to AB. Wherefore of unequal magnitudes, &c. Q. E. D. Book V. See N. MAG AGNITUDES which have the fame ratio to the fame magnitude are equal to one another; and those to which the fame magnitude has the fame ratio are equal to one another. Let A, B have each of them the fame ratio to C; A is equal to B. for if they are not equal, one of them is greater than the other; let A be the greater; then, by what was fhewn in the preceding Propofition, there are fome equimultiples of A and B, and fome multiple of C fuch, that the multiple of A is greater than the mul tiple of C, but the multiple of B is not greater than that of C. Let fuch multiples be taken, and let D, E, be the equimultiples of A, B, and F the multiple of C fo that D may be greater than F, and E not greater than F. but because A is to C, as B is to C, and of A, B are taken equimultiples D, E, and of C is taken a multiple F; and that D is greater than F; a. 5. Def. 5. E fhall also be greater than Fa; but E is not greater than F, which is impoffible. A therefore and B are not unequal; that is, they are equal. Next, Let C have the fame ratio to A D F E each of the magnitudes A and B; A is B the greater, therefore, as was fhewn in Prop. 8th, there is fome multiple F of C, and fome equimultiples E and D of B and A such, that F is greater than E, and not greater than D. but because C is to B, as C is to A, and that F the multiple of the first is greater than E the multiple of the second; F the multiple of the third is greater than D the multiple of the fourth a. but F is not greater than D, which is impoffible. Therefore A is equal to B. Wherefore magnitudes which, &c. Q. E. D. THA PROP. X. THEOR. Book V. HAT magnitude which has a greater ratio than See N. another has unto the fame magnitude is the greater of the two. and that magnitude to which the fame has a greater ratio than it has unto another magnitude is the leffer of the two. Let A have to C a greater ratio than B has to C; A is greater than B. for because A has a greater ratio to C, than B has to C, there are a fome equimultiples of A and B, and fome multiple of a. 7.Def. 5. C fuch, that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than it. let them be taken, and let D, E be equimultiples of A, B, and F a multiple of Cfuch, that D is greater than F, but E is not greater than F. therefore D is greater than E. and because D and E are equimultiples of A and B, and D is greater than E; therefore A is greater than B. Next, Let C have a greater ratio to B than it has to A; B is lefs than A. for a there is fome multiple F of C, and fome equimultiples E and D of B and A fuch, that Fis greater than E, but is not greater B 이 F b. 4. Ax. 5. E than D. E therefore is lefs than D; and because E and D are equimultiples of B and A, therefore B is b lefs than A. nitude therefore, &c. Q. E. D. That mag RAT ATIOS that are the fame to the fame ratio, are Let A be to B, as C is to D; and as C to D, fo let E be to F; A is to B, as E to F. Take of A, C, E any equimultiples whatever G, H, K; and of B, D, F any equimultipies whatever L, M, N. Therefore fince A is to B, as C to D, and of A, C are taken equimultiples G, |