SECTION VI. Containing the Propositions to which no reference is made in Book VI. PROPOSITION XXIII. (Eucl. v. 5.) 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 must be the same multiple of the remainder, that the whole is of the whole. Let B and D be the magnitudes which are taken away, and A and C the magnitudes which remain, then A, B together, and C, D together will be the wholes. And let A, B together be the same multiple of C, D together, that B is of D. Then must A be the same multiple of C that A, B together is of C, D together. Take E the same multiple of C that B is of D, Then E, B together is the same multiple of C, D together that B is of D. V. 1. But A, B together is the same multiple of C, D together that B is of D. .. A is the same multiple of C that B is of D. V. Ax. 1. I. Ax. 3. Q. E. D. PROPOSITION XXIV. (Eucl. v. 6.) If 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 B and D be the magnitudes which are taken away, and A and C the magnitudes which remain ; then A, B together and C, D together will be the wholes. Let A, B together be the same multiple of P, that C, D together is of Q, and let B be the same multiple of P, that D is of Q. Then must A and C be equal respectively to P and Q, or A and C be equimultiples of P and Q. For let A, B together then C, D together = = P, P.........repeated m+n times, ..repeated m+n times. Also, let BP, P.........repeated n times, Q, Q. and if A be a multiple of P, C is the same multiple of Q. ..repeated m times, Q. E. D. PROPOSITION XXV. (Eucl. v. 17.) If magnitudes, taken jointly, be proportionals, they shall also be proportionals when taken separately; that is, if two magnitudes together have to one of them the same ratio which two others have to one of these, the remaining one of the first two must have to the other the same ratio which the remaining one of the last two has to the other of these. Let A, B together have the same ratio to B that C, D together have to D. Then must A be to B as C to D. Take of A, B, C, D any equimultiples mA, mB, mC, mD, and again of B, D take any equimultiples nB, nD. Then . mA is the same multiple of A that mB is of B, .. mA, mB together is the same multiple of A, B together that mA is of A. V. 1. And.mC is the same multiple of C that mD is of D, .. mC, mD together is the same multiple of C, D together that mC is of C. But mA is the same multiple of A that mC is of C. V. 1. .. mA, mB together is the same multiple of A, B together that mC, mD together is of C, D together. Again, mB, nB together is the same multiple of B that mD, nD together is of D. Now, since A, B together is to B as C, D together is to D, .. if mA, mB together be greater than mB, nB together, mC, mD together is greater than mD, nD together; and if equal, equal; if less, less. V. 4. That is, if mA be greater than nB, mC is greater than nD; and if equal, equal; if less, less. .. A is to B as C is to D. I. Ax. 3, 5. V. Def. 5. Q. E. D. PROPOSITION XXVI. (Eucl. v. 19.) If a whole magnitude be to a whole as a magnitude taken from the first is to a magnitude taken from the other, the remainder must be to the remainder as the whole is to the whole. Let A, B together have the same ratio to C, D together that B has to D. Then must A be to C as A, B together is to C, D together. For A, B together is to C, D together as B is to D, .. A, B together is to B as C, D together is to D, and.. A is to B as C is to D, Hence A is to C as B is to D. But A, B together is to C, D together as B is to D. .. A is to Cas A, B together is to C, D together. PROPOSITION XXVII. (Eucl. v. 21.) V. 15. V. 25. V. 15. Hyp. V. 5. Q. E. D. If there be three magnitudes, and other three, which have the same ratio, taken two and two, but in a cross order, then if the first be greater than the third, the fourth must be greater than the sixth; and if equal, equal; and if less, less. Let A, B, C be three magnitudes, and D, E, F other three, and let A be to B as E is to F, and B be to C as D is to E. Then if A be greater than C, D must be greater than F; and if equal, equal; and if less, less. First, if A be greater than C, A has to B a greater ratio than C has to B, and .. E has to F a greater ratio than C has to B. Now B is to C as D is to E, .. C is to B as E is to D. Hence E has to F a greater ratio than E has to D. .D is greater than F. Similarly the other cases may be proved. V. 7. V. 13. Hyp. V. 12. V. 9. Q. E. D. PROPOSITION XXVIII. (Eucl. v. 23.) If there be any number of magnitudes, and as many others, which have the same ratio, taken two and two in a cross order, the first must have to the last of the first magnitudes the same ratio which the first of the others has to the last of these. Let A, B, C be three magnitudes, and D, E, F other three, and let A be to B as E is to F, and B be to C as D is to E. Then must A be to C as D is to F. Of A, B, D take any equimultiples mA, mB, mD, and of C, E, F take any equimultiples nC, nE, nF. Hence, if m▲ be greater than nC, mD is greater than nF; and if equal, equal; and if less, less. .. A is to C as D is to F. V. 27. The proposition may be easily extended to any number of magnitudes. Q. E. D. |