Book V. that AB is of CD; therefore EB is the fame multiple of FD, that AB is of CD. Therefore, if any magnitude, &c. Q. E. D. See No a 1. Ax. S. 162.50 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 fame E, F; the remainders GB, HD are either equal to E, F, or equimultiples of them. АК C GH B DEF Ki A First, Let GB be equal to E; HD is e qual to F: Make CK equal to F; and becaufe AG is the fame multiple of E, that CH is of F, and that GB is equal to E, and CK to F; therefore AB is the fame multiple of E, that KH is of F. But AB, by the hypothefis, is the fame multiple of E that CD is of F; therefore KH is the fame multiple of F, that CD is of F; wherefore KH is equal to CD: Take away the common magnitude CH, then the remainder KC is equal to the remainder HD: But KC is equal to F; HD therefore is equal to F. But let GB be a multiple of E; then HD is the fame multiple of F: Make CK the fame multiple of F, that GB is of E: And because AG is the fame multiple of E, that CH is of F; and GB the fame multiple of E, that CK is of F; therefore AB is the fame multiple of E, that KH is of Fb: But AB is the fame multiple of E, that CD is of F; therefore KH is the fame multiple of F, that CD is of it j wherefore KH is equal to CD': Take away CH from both; therefore the remainder KC is equal to the remainder HD: And because GB is the fame multiple of E, that KC is of F, and that KC is equal to HD; therefore HD is the fame multiple of F, that GB is of E: If therefore two magnitudes, &c. Q. E. D. C H B DE F PROP. 1 PROP. A. THEOR. Book V. IF the first of four magnitudes has to the fecond, the see N. fame ratio which the third has to the fourth; then, if the first be greater than the fecond, the third is alfo greater than the fourth; and, if equal, equal; if lefs, lefs. Take any equimultiples of each of them, as the doubles of each; then, by def. 5th of this book, if the double of the first be greater than the double of the fecond, the double of the third is greater than the double of the fourth; but, if the first be greater than the fecond, the double of the first is greater than the double of the second; wherefore also the double of the third is greater than the double of the fourth; therefore the third is greater than the fourth: In like manner, if the first be equal to the second, or less than it, the third can be proved to be equal to the fourth, or less than it. Therefore, if the first, &c. Q. E. D. IF four magnitudes are proportionals, they are propor- see N. If the magnitude A be to B, as C is to D, then alfo inversely Take of B and D any equimultiples whatever E and F ; and of A and C any equimultiples whatever G and H. First, Let E be greater than G, then G is lefs than E; and, because A is to B, as C is to D, and of A and C, the firft and third, G and H are equimultiples; and of B and D, the fecond and fourth, E and F are equimultiples; and that G is lefs than E, H is alfo lefs than F; that is, F is greater than H; if therefore E be greater than G, F is greater than H: In like manner, if E be equal to G, F may he fhewn to be equal to H; and, if lefs, lefs; and E, F are any equimultiples whatever of B and D, and G, H any whatever of A and C; therefore as B G A BE a 5. def. 5 Book V. in is to A, fo is D to C. If then four magnitudes, &c. See N. 8 3.5. IF PROP. C. THEOR. the firft be the fame multiple of the fecond, or the fame part of it, that the third is of the fourth; the firft is to the fecond, as the third is to the fourth. Let the firft A be the fame multiple of B the fecond, that C the third is of the fourth D: A is to B as C is to D. Take of A and C any equimultiples whatever E and F; and of B and D any equimultiples whatever G and H: Then, because A is the fame multiple of B that C is of D; and that E is the fame multiple of A, that F is of C; E is the fame multiple of B, that F is of D; therefore E and F are the fame multiples of B and D: But G and H are equimultiples of B and D; therefore, if E be a greater multiple of B, than G is, F is a greater multiple of D, than H is of D; that is, if E be greater than G, F is greater than H: In like manner, if E be equal to G, or lefs; F is equal to H, or lefs than it. But E, F are equimultiples, any whatever, of A, C, and G, Hany equimultiples whatever of B, bs. def. 5. D. Therefore A is to B, as C is to Db. c B. & Next, Let the firft A be the fame part of the fecond B, that the third C is of the fourth D: A is to B, as C is to D: For B is the fame multiple of A, that D is of C; wherefore, by the preceeding cafe, B is to A, as D is to C; and inverfely A is to B, as C is to D. There c fore, if the first be the fame multiple, &c. A B C D Q. E. D. PROP. Book V. PROP. D. THEOR. IF the first be to the fecond 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 firft let A be a multiple of B; C is the fame multiple of D. Take E equal to A, and whatever multiple A or É 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 Fb: And F is the fame multiple of D, that A is of B. A B Wherefore C is the 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, 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 preceeding cafe, D is the fame E C D F 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. EQUAL 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 I 2 of Book V. of C any multiple whatever F: Then, because D is the fame multiple of A, that E is of B, and that A is a 1. Ax. 5. equal to B; D is equal to E: Therefore, if D be greater than F, E is greater than F; and if equal, equal; if lefs, lefs: And D, E are any equimultiples of A, B, and F is any mulbs.def. 5. tiple of C. Therefore bas 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 conftruction, D may in like manner be fhewn equal to E: Therefore, if F be greater than D, it is likewife greater than E; and if equal, equal; if lefs, lefs: And F is any multiple whatever of C, and D, E are any equimultiples whatever of A, B. Therefore C is to A, as C is to B. Therefore equal magnitudes, &c. Q. E. D. Ο DA E PROP. VIII. THEOR. B 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 lefs than it has to the greater. Fig. 1. Let AB, BC be unequal magnitudes, of which AB is the greater, and let D be any magnitude whatever: AB has a greater ratio to D than BC to D: And D has a greater ra- E tio to BC than unto AB. If the magnitude which is not the greater of the two AC, CB, be not lefs 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: Therefore EF and FG are each of them greater than |