IF PROP. D. THEOR. Book V. F the first be to the second as the third to the fourth, See N. and if the first be a multiple, or part of the second; 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 same 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 Fa. but A is equal to E, therefore C is equal to Fb. and F is the fame multiple of D, that A is of B. Wherefore C is the same 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, inversely 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, Dis the fame ABCD F multiple of C; that is, C is the fame part of D, that A is of B. Therefore if the first, &c. Q. E. D. EQUAL magnitudes have the fame ratio to the : same 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 same ratio to each of the magnitudes A and B. Take of A and B any equimultiples whatever D and E, and of a. Cor. 4. 5. b. A. 5. See the Fi gure at the foot of the preceding page. с. В. 5. Book V. Cany multiple whatever F. then because D is the fame multiple of A, that E is of B, and that A is equal to a. 1. A 5. B; Disa 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, so is B to C. See N. PROP. VIII. THEOR. CF F unequal magnitudes the greater has a greater ratio to the same than the less has. and the same 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 If the magnitude which is not the great- D, whether it be AC or CB. Let it be F A C GB LKHD. 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 less 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 less 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 AB2; wherefore EG and FG are equi- a. 1. 5. the same construction, it may be shewn, 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. a. 5. Def. 5. M PROP. IX. THEOR. 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 same 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 thewn in the preceding Propofition, there are some equimultiples of A and B, and fome multiple of C such, that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than that of C. Let such multiples be taken, and let D, E, be the equimultiples of A, B, and F the multiple of C so 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 Next, Let C have the fame ratio to A D F E each of the magnitudes A and B; A is B PROP. X. THEOR. Book V. T CHAT 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 lesser of the two. and D is greater 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 some equimultiples of A and B, and some 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 A A D C F b. 4. Ax. 5. than D. E therefore is less than D; and because E and D are equimultiples of B and A, therefore B is bless than A. That magnitude therefore, &c. Q. E. D. R PROP. XI. THEOR. 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, so 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 equimultiples whatever L, M, N. Therefore fince A is to B, as C to D, and of A, C are taken equimultiples G, |