BOOK V. PROP. VII. THEOR. QUAL magnitudes have the same ratio to the same magnitude; and the same has the same ratio to equal magnitudes. Let A and B be equal magnitudes, and C any other; A has the same ratio to C that B has to C; and C has the same ratio to A that it has to B. Take MA, MB any equimultiples of A, B: and because A is equal to B, and MA, MB are equimultiples of them, MA is 2 a 4. Ax. 5. equal to MB: and equal magnitudes contain V N € 5.Def. 5. tain C. Therefore, as A is to C, so is B to C. AC B Likewise, C has the fame ratio to A that it has to B. Take NC any multiple of C greater than A: and because A is equal to B, NC contains each of them the fame number of times b; that is, any multiple of C contains A the fame number of times that it contains B; therefore C is to A, as Cisto B. Wherefore, &c. Q. E. D. PROP. VIII, THEOR. Funequal magnitudes, the greater has a greater ratio to the same than the less has; and the fame 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 has a greater ratio to D than BC to D: and Dhas a greater ratio to BC than to AB. Take EF, FG such equimultiples of AC, CB, that each of them may be greater than D: and because EF is E the same multiple of AC that FG is of CB; 21.5. the whole EG is the fame multiple a of the F whole AB that FG is of CB: but, because EF is greater than D, EG contains D a greater number of times than FG contains it. Wherefore a multiple of AB is found, viz. EG, which contains Da greater number of times than FG, the C A M GB D the same multiple of BC contains D; therefore AB has to D BOOK V. a greater ratio than BC has to D. Also D has to BC a greater ratio than it has to AB. For, b7. def. 5. having made the same construction, take also MD the least multiple of D, that is not less than FG: therefore MD is either equal to FG, or exceeds it by a magnitude less than D: but EF is greater than D; therefore MD is less than EG: but EG, FG contain AB, BC the same number of times exactly; therefore MD does not contain AB so many times as it contains BC; that is, a multiple of D is taken, which contains BC a greater number of times than it contains BA: therefore D has to BC a greater ratio than it has to AB. Wherefore, &c. Q. E. D. M PROP. IX. THEOR. AGNITUDES which have the fame ratio to the same magnitude, are equal to one another; and those to which the same magnitude has the fame ratio, are equal to one another. Let A have the same ratio to C that B has to C; A is equal to B: For if not, let A be the greater; then, by what was shown in the preceding proposition, there is some multiple of A which contains C a greater number of times than the sante multiple of B does. Let MA be that multiple of A which contains C a greater number of times than MB, the same multiple of B contains C. But, because A is to Cas B is to C, and MA, MB are equimultiples of A, B; MA, MB contain C the same number of times a but MA contains C a greater number of times than MB does; which is impossible; A therefore and B are not unequal; that is, they are equal. N M M 25. Def. 5. ; A C B Next, let C have the fame ratio to A that it has to B; A is equal to B. For, if not, let A be the greater; therefore, as was shewn in Prop. VIII. there is some multiple NC of C, which contains B a greater number of times than it contains A: but because C is to B, as C is to A, and that NC is a multiple of C; NC contains B and A the fame number of times: but it contains B oftener; which is impoffible. Therefore A is equal to B. Wherefore, &c. Q. E. D. PROP. BOOK V. T PROP. X. THEOR. HAT magnitude which has a greater ratio than another has to the fame magnitude, is the greater of the two: and that magnitude to which the fame has a greater ratio than it has to another, is the leffer of the two. Let A have to C a greater ratio than Bhas to C; A is greater a 7. Def. 5. multiple of Ba: Let MA be that multiple of A c 5. Ax. 5. greater than B.. C M N AC M B Next, let C have a greater ratio to B than it has to A; Bis less than A: For, there is some multiple NC of C, which contains B a greater number of times than it contains A; therefore d 3. Ax. 5. B is less d than A. Wherefore, &c. Q. E. D. R PROP. XI. THEOR. ATIOS that are equal to the same ratio, are equal to one another. 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. M M T M Take MA, MC, ME any equimultiples of A, C, E: and because A is to Bas C to D, and MA, MC are equimultiples of A, C; MA a 5. Def. 5. contains B the fame number of times a that MC contains D: and because C is to Das E is to F, and MC, ΜΕ are equimultiples of C, E; MC contains D the fame number of times, a that ME contains F: But MC con- A B CDEF tains D the same number of times that MA contains B; therefore MA contains B the same number of times that ME contains F: and MA, ME are any equimultiples of A. E; as many times, therefore, as any multiple of A contains B, fo many times does the fame multiple of E contain F: therefore, as A is to B, so is E to F. Wherefore, &c. Q. E. D. PROP. PROP. XII. THEOR. F any number of magnitudes be proportionals, as one of the antecedents is to its consequent, fo shall all the antecedents taken together be to all the confequents. Let any number of magnitudes A, B, C, D, E, F, be proportionals; that is, as A is to B, so C to D, and E to F; as A is to B, so shall A, C, E together be to B, D, F together. Take MA, MC, ME any equimultiples of A, C, E: therefore MA is the same multiple of A, that MA, MC, ME together is of A, C, E together 2: And be- M M BOOK V. a 1. 5. cause A is to B, as C is to D, and as E M to F; and that MA, MC, ME are equimultiples of A, C, E; MA contains B as many times b as MC contains D, and ME contains F: Wherefore, as b 5. Def. 5. many times as MÁ contains B, so many times shall MA, MC, ME together A B CDEF F together: therefore, as A is to B, fo bare A, C, E together to B, D, F together. Wherefore, &c. Q. E. D. PROP. XIII. THEOR. F the first has to the second the fame ratio that the third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the fixth; the first shall also have to the fecond a greater ratio than the fifth has to the fixth. Let A the first have the fame ratio to B the second, which C the third has to D the fourth, but C the third, to D the fourth, a greater ratio than E the fifth, to F the fixth: Alío the first A thall have to the fecond B a greater ratio than the fifth E to the fixth F. Because C has a greater ratio to D, than E to F, there is some multiple of C which contains D, a greater number of times a than the fame multiple of E contains F: Let MC be that multiple of C, which contains D a greater number of times than ME, a 7. Def. 5. : M M BOOK V. ME, the fame multiple of E, contains F; and whatever multiple therefore some multiple of A which contains B a greater number of times than the same multiple of E contains F: therea 7. Def. 5. fore A has a greater ratio to Ba, than E has to F. Wherefore, &c. Q. E. D. Cor. And if the first has a greater ratio to the second, than the third has to the fourth, but the third the same ratio to the fourth, which the fifth has to the fixth; it may be demonstrated in like manner, that the first has a greater ratio to the fecond, than the fifth has to the fixth. I F the first has to the fecond the fame ratio which the third has to the fourth; then, if the first be greater than the third, the second shall be greater than the fourth; and if equal, equal; and if less, lefs. Let the first A have to the second B, the fame ratio which the third C has to the fourth D; if A be greater than C, B is greater than D. Because A is greater than C, and B is any other magnitude, 28. 5. A has to B a greater ratio than C to Ba: But, as A is to B, so is C to D; therefore alfo C has to D a greater ratio than C has to b 13.5. Bb: But of two magnitudes, that to which C 10.5. the fame has the greater ratio is the leffer : Wherefore D is less than B; that is, B is greater than D. AB CD Secondly, If A be equal to C, B is equal to D: For A is to d 9. 5. B, as C, that is A, is to D; therefore B is equal to D d. Thirdly, If A be less than C, B shall be less than D: For C is greater than A; and because C is to D, as A to B, and C greater than A, D is greater than B, by the first cafe; therefore B is less than D. Wherefore, &c. Q. E. D. PROP. |