PROP. IV. THEOR. If 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 equimultiple of the • first shall have the same ratio to that of the second, which the • equimultiple 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 equiinultiples 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 same 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 is of C (3. 5.); for the same reason, M is the same multiple of B, that N is of D: and because, as A is to B, so is C to D (Hypoth.) § È Å Ġ À and of A and C have been taken certain equimultiples K, L; and of B L F C D H N and D have been taken certain equimultiples M, N; if therefore K be greater than M, L is greater than N; and if equal, equal; if less, less (5. def. 5.). And K, L are any equimultiples whatever of E, F; and M, N any whatever of G, H: as therefore E is to G, so is (5. def. 5.) F to H. Therefore, if the first, &c. Q. E. D. Cor. Likewise, if the first have the same ratio to the second, which the third has to the fourth, then also any equimultiples whatever of the first and third have the same ratio to the 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 same 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 be • See Note. fore, 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 (5. def. 5.): 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 is the other case demonstrated. PROP. V. THEOR. If one magnitude be the same multiple of another, 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 the whole AB is of the whole CD. Take AG the same multiple of FD, that AE is of CF: therefore AE is (1. 5.) 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 multiple of CD that AB is of CD; wherefore EG is equal to AB (1. Ax. 5.). Take from them the common magnitude AE; the remainder AG is equal to the remainder EB. Wherefore, since AE is the same multiple of CF, that AG is of FD, and that AG is equal to EB; therefore AE is the same B D multiple of CF, that EB is of FD: but AE is the same multiple of CF, that AB is of CD; therefore EB is the same multiple of FD, that AB is of CD. Therefore, if any magnitude, &c. Q. E. D. PROP. VI. THEOR. If two magnitudes be equimultiples of two others, and if equi. multiples 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. * See Note, First, let GB be equal to E, HD is equal A to F: make CK equal to F; and because AG is the same multiple of E, that CH is of F, and that GB is equal to E, and CK to F; therefore AB is the same multiple of E, that KH is of F. But AB, by the hypothesis, is the same multiple of E that CD is of F; therefore KH is the same multiple of F, that CD is of F; wherefore KH is equal to CD (1. Ax. 5.): 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 [ Ꭰ E F equal to F. But let GB be a multiple of E: then KI HD is the same multiple of F: make CK the same multiple of F, that GB is of E: and because AG is the same multiple of E, that GH is of F; and GB the same multiple of E that CK is of F; therefore AB is the same multiple of E, that KH is of F (2. 5.): but AB is the same multiple of E. That CD is of F, therefore KH is the H same multiple of F, that CD is of it: wherefore KH is equal to CD (1. Ax. 5.) : take away CH from both: therefore the remainder KC is equal to the remainder HD: and because GB is the same multiple of E, that KC is of F, and B D É É 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. PROP. A. THEOR. If the first of four magnitudes have to the second the same ratio which the third has to the fourth; then, if the first be greater than the second, the third is also greater than the fourth; and if equal, equal; if less, less.* 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 second, the double of the third is greater than the double of the fourth; but if the first be greater than the second, 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. * See Notes.. PROP. B. THEOR. If four magnitudes be proportionals they are poportionals also when taken inversely. * If the magnitude A be to B, as C is to D, then also inversely B is to A, as D, to C. 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 less than E; and because A is to B as C is to D, and of A and C, the first and third, G and H are equimultiples; and of B and D, the second and fourth, E and Fare equimultiples; and that G is less than E, H is also (5. def. 5.) less than F; that is, F is greater than Ġ À B É H; if therefore E be greater than G, F is H C D F greater than H; in like manner, if E be equal to G, F may be shown to be equal to H; and, if less, less; and E, F are any equimultiples whatever of B and D, and G, H any whatever of A and C; therefore as B is to A, so is D to C. If then, four magnitudes, &c. Q. E. D. PROP. C. THEOR. If the first be the same multiple of the second, or the same part of it, that the third is of the fourth; the first is to the second, as the third is to the fourth. * Let the first A be the same multiple of B the second 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 same multiple of B that C is of D; and that E is the same multiple of A that F is of C; E is the same multiple of B that F is of D (3. 5.); therefore E and F are the same 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 less; F is equal to H, or less than it. But E, F are any equimultiples whatever of A, C, and G, H any equimultiples whatever of B, D. Therefore A is to B, as C is to D (5. def.) * See Notes. Next, Let the first A be the same part of the second B, that the third C is of the fourth D: A is to B, as C is to D: for B is the same multiple of A, that D is of C: wherefore, by the preceding case, B is to A, as D is to C; and inversely (B. 5.) A is to B, as C is to D. Therefore, if the first be the same multiple, &c. Q. E. D. PROP. D. THEOR. If the first be to the second as the third to the fourth, and if the first be a multiple, or part of the second; the third is the same multiple, or the same 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 same 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 (Cor. 4. 5.): but A is equal to E, therefore C is equal to F (A. 5): and F is the same multiple of D, that A is of B. Wherefore C is the same À B C D multiple of D, that A is of B. Next, Let the first A be a part of the second E F Because A is to B, as C is to D; then inversely, B is (B. 5.) to A, as D to C: but A is a part of B, therefore P is a multiple of A; and, by the preceding case, D is the same multiple of C, that is, C is the same part of D, that A is of B; therefore, if the first, &c. Q. E. D. PROP. VII. THEOR. Equal magnitudes have the same ratio to the same magnilude; and the same has the same ratio to equal magnitudes. Let A and B be equal magnitudes, and C any other. A and B have each of them the same 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 C any multiple whatever F; then, because D is the same multi |