PRO P. VII. THE O R. Book V. QUAL magnitudes have the same proportion to the same one and the same magnitude has the same proportion to equal magnitudes. E Let A, B, be two equal magnitudes, and C any third magnitude, then A, B, will have the same proportion to C. For, let D, E, be any equimultiples of A, B, and F any equimultiple of C; then, if D is equal to F, E is likewise equal to F; if greater, greater; and, if less, less; therefore A is to C as B is to C. Again, C is to A as C is to B; for, the same construction re. a def. S. maining, if F is equal to D, it is likewise equal to E ; wherefore C is to B as C is to A. Wherefore, &c. PRO P. VIII. THE O R. THE greater of two unequal magnitudes has a greater propor, tion to some third magnitude than the less has, and that third magnitude has a greater proportion to the leser magnitude than it has to the greater. Let AB, C, be two unequal magnitudes, of which AB is the greater; and let D be any third magnitude ; then AB will have a greater proportion to D than C has to D; but D has a greater proportion to C than it has to AB. For, because AB is greater than C, make BE equal to C, then AB will exceed C by AE; if AE is not greater than D, let. it be multiplied till it exceed D; and let this multiple be FG; let GH be the same multiple of EB that FG is of AE ; then FH will be the same multiple of AB that GH is of EB?; and make a r. K the same multiple of C that GH is of EB; but EB is equal to C; therefore K is equal to GHb; wherefore FH is the same b Ax re multiple of AB that K is of C. Now, of D let L be taken a double, M triple, and so on, till a multiple of D is found next greater than K, which ler be N; and let M be the next multiple of D, less than N; then M will not be greater than K, that is, K will not be less than M; but M and D are equal to N; and because FG 'exceeds D, and GH is equal to K, FH will be greater than N; that is, FH, the multiple of AB, exceeds N, the multiple of D; but K, the multiple of C, does not exceed N, the multiple of D; therefore AB has to D a greater proportion Def. 7. than C has to DC; but likewise D has to Ca greater proportion than it has to AB; for, the same construction remaining, N, the multipie of D, exceeds K, the multiple of C; but does not exceed FH, the multiple of AB. Wherefore, &c. PROP. Book. V. PRO P. IX. THE OR. AGNITUDES which have the same proportion to one and the same magnitude are equal to one another; and if a magnitude has the same proportion to other magnitudes, these magnitudes are equal to one another. a 8.) Let the magnitudes A, B, have the same proportion, to C, then A is equal to B. If not, let A be greater or less than B; if greater, then A has a greater proportion to C than B has to Ca; but it has not ; therefore A is not greater than B; if less, then B has a greater proportion to C than it has to C; but it has not“; therefore A is not less than B ; and, since neither greater nor less, it must be equal. Again, if C have the same proportion to A that it has to B, A is equal to B; if not, C will have a greater proportion to the lefser magnitude, than it has to the greater“; but it has not ; therefore A is not greater than B, or B is not greater than A; therefore A is equal to B. Wherefore, &c. PRO P. X. THE O R. F magnitudes having proportion to the same magnitude, that which has the greater proportion to some third is the greater magnitude ; and that magnitude to which the same has a greater proportion is the lefser magnitude. Of the magnitudes A, B, if A have a greater proportion to a third magnitude C, than B has to C, A is greater than B; for, if not, it will be either equal or less. If A is equal to B, then they would have the same proportion to Ca; but they have not; therefore A is not equal to B; neither is it,less; for then B would have a greater proportion to C than it has to Cb; but it has not: Therefore, since A is neither equal nor less than B, it must be greater Again, if C have a greater proportion to B than it has to A, then B is less than A; if not, let it be equal or greater ; if equal, then C has the fame proportion to B that it has to AC but it has not; therefore B is not equal to A. If greater, then C will have a greater proportion to A than it has to B; but it has not; therefore, since B is not equal or greater than A, it must be less. Wherefore, &c. go Book V. PRO P. XI. THEOR. ROPORTIONS that are the same to any third, are the same Let A be to B, as C is to D, and C to Das E to F; then A will be to B as E to F. For, let G, H, K, be any equimultiples of A, C, E; and L, M, N, any other equimultiples of B, D, F; now, because A is to B as C is to D; and G, H are equimultiples of A, C; and L, M any other equimultiples of B, D); if G is equal to L, H will be equal to Mo; if greater, greater; and, if less, less : Like- a Def. 5. wise, because C is to D as E is to F; if H be equal to M, K will be equal to N°; if greater, greater; and, if less, less. Wherefore, if G be equal to L, K will be equal to N; for they are e. qual to H, Mó; wherefore A is to B as E to F? Where. b Ax. 1. 1. fore, &c. PRO P. XII. T H E O R. F any number of magnitudes be proportional, as one of the an tecedents is to one of the consequents, so are all the antecedents to all the consequents. Let the magnitudes be A, B, C, D, E, F; and, as A is to B, fo is C to D, and E to F; then as A is to B, so are A, C, E, all the antecedents, to B, D, F, all the consequents. For, let let G, H, K be equimultiples of A, C, E ; and L, M, N be any other equimultiples of B, D, F; then, because A is to B as C is to D, and C to D as E to F, and G, H, K equimultiples of A, C, E, and L, M, N any other equimultiples of B, D, F; if G be equal to L, H will be equal to M, and K to N°; if a Def, so greater, greater ; and, if less, less ; wherefore, if G be equal to L; G, H, K, will be equal to L, M, N; if greater, greater ; and, if less, less b; but G; G, H, K, are equimultiples of A; A, b . C, E, together; and L; L, M, N, equimultiples of B; B, D, F, together ; such, that, if G be equal to L; G, H, K will be equal to L, M, N, together ; wherefore A, C, E, together, are 10 B, D, F, together, as A is to Bo Wherefore, &c. K PROP, Book V. PRO P. XIII. THEO R. TF the first is in the same proportion to the second, as the third to the fourth; and if the third has a greater proportion to the fourth, than the fifth to the sixth ; then shall also the first have a greater proportion to the second, than the fifth to the sixth. 7. Let the first A have the same proportion to the fecond B, that the third C has to the fourth D; but let the third C have a greater proportion to the fourth D, than the fifth E to the sixth F; then the first A will have a greater proportion to the second B, than the fifth E has to the fixth F. For, because C has a greater proportion to D, then E to F; let G, H be equimultiples of C, E; and K, L equimultiples of D, F; such, that, if Ġ exceeds K, but H does not exceed L"; and let M be the same multiple of A that G is of C; and N the fame multiple of B that K is of D; then, because G exceeds K, M will exceed Nb; but G exceeds K, and H does not exceed L; and M exceeds N, and H does not exceed L; and M, H are equimultiples of A, E; and N, L, of B, F; wherefore A has a greater proportion to B, than E has to F*. Wherefore, &c. PRO P. XIV. THEO R. IF the first has the same proportion to the second, that the third has to the fourth; if the first be greater than the third, the second will be greater than the fourth ; but, if the first be equal to the third, the second will be equal to the fourth; if the first is less than the third, the second will be less than the fourth. a 8. Let the first A have the same proportion to the second B, that the third C has to the fourth D; if A is greater than C, then B is greater than D. For, if A is greater than C, and B any third magnitude, A has a greater proportion to B than C has to B* ; but A is to B as C is to D; therefore C has to D a greater proportion than C has to Bd; therefore D is less than Bo; that is, B is greater than D. In the same manner it is proved, that, if A is equal to b 13 € IO. Book V. PRO P. XV. THEOR. ARTS have the same proportion as their like multiples, if taken correspondently. Let AB be the same multiple of C that DE is of F; then C will be to F as AB is to DE. For, let AG, GH, HB be each equal to C; and DK, KL, LE, each equal to F; then AG, GH, HB are equal to one a. nother ; and likewise DK, KL, LE equal to one another ; a Ax. 1, 5. therefore AG is to DK as GH is to KL, and as HB is to LEb; b 11. therefore AB is to DE as AG is to DK °; that is, as C to c 12. F. Wherefore, &c. PRO P. XVI. THE O R. F four magnitudes of the same kind are proportional, theyßball they also be alternately proportional. Let the four magnitudes A, B, C, D, be proportional, viz. as A is to B, so is C to D; they will likewise be proportional when taken alternately; that is, as A is to C, fo is B to D; for, take E, F equimultiples 6f A, B; and G, H any equimultiples of C and D; then, because E is the same multiple of A that F is of B, and G the same multiple of C that H is of D, A is to B as E is to Fa; but A is to B, as C is to D; therefore C is to a 15. Das E is to Fb; and as C is to D, fo is G to Hà; therefore b 11. E is to F as G is to Hb; therefore, since E, F, G, H are four magnitudes proportional, equimultiples of other four, A, B, C, D; therefore, if E is equal to G, F is equal to H; if greater, greater; and, if less, less; wherefore A is to C as B is to D d; c 14. Wherefore, &c. d Def. s. PROP. XVII. THEO R. If magnitudes compounded are proportional, they shall also be proportional when divided.. |