PROPOSITION XVI. (Eucl. v. 18.) If magnitudes taken separately be proportionals, they must be proportionals also when taken jointly. Let A have the same ratio to B that C has to D. Then must A, B together have the same ratio to B, that C, D together has to D. First, when all the magnitudes are of the same kind, A is to B as C is to D, .. A is to C as B is to D. .. A, B together is to C, D together as B is to D, and.. A, B together is to B as C, D together is to D. V. 15. V. 10. V. 15. Next, when all the magnitudes are not of the same kind, we may employ a method of proof which includes the former case: thus Take of A, B, C, D any equimultiples mA, mB, mC, mD, and of B and D take any equimultiples nB, nD. Then. A is to B as C is to D, ..if mA be greater than nB, mC is greater than nD; and if equal, equal; if less, less. V. 4. If then mA, mB together be greater than mB, nB together, mC, mD together is greater than mC, nD together; and if equal, equal; if less, less. I. Ax. 2, 4. Now mA, mB together is the same multiple of A, B together that mC, mD together is of C, D together; V. 1. and mB, nB together is the same multiple of B that mD, nD together is of D. V. 2. .. A, B together is to B as C, D together is to D. V. Def. 5. Q. E. D. SECTION V. Containing the Propositions occasionally referred to in Book VI. PROPOSITION XVII. (Eucl. v. 4.) If the first of four magnitudes has to the second the same ratio which the third has to the fourth, and any equimultiples of the first and third be taken, and also any equimultiples of the second and fourth, then must the multiple of the first have the same ratio to the multiple of the second which the multiple of the third has to that of the fourth. If A be to B as C is to D, and mA, mC be taken equimultiples of A and C, and nB, nD.............................. then must mA be to nВ as mC is to nD. of B and D, Take of mA, mC any equimultiples pmA, pmC, and of nB, nD...... .gnB, qnD. Then pmA, pmC are equimultiples of A and C, and qnB, qnD.... V. 3. of B and D. V. 3. Then and qnB, qnD.... pmA, pmO are equimultiples of mA, mC, .. mA is to nB as mC is to nD. .of nB, nD, V. Def. 5. Q. E. D. If the first of four magnitudes have the same ratio to the second that the third has to the fourth, then, if the first be greater than the second, the third must be greater than the fourth; and if equal, equal; and if less, less. Let A be to B as C is to D. Then if A be greater than B, C must be greater than D; and if equal, equal; and if less, less. Take any equimultiples of each, mA, mB, mC, mD. .. if mA be greater than mB, mC is greater than mD; and if equal, equal; and if less, less. First, suppose A greater than B, then mA is greater than mB, and.. mC is greater than mD, and .. C is greater than D. Similarly the other cases may be proved. V. 4 V. Ax. 3. V. Ax. 4. Q. E. D. PROPOSITION XIX. (Eucl. v. D.) If the first be to the second as the third is to the fourth, and if the first be a multiple, or a submultiple, of the second, the third must be the same multiple, or the same submultiple, of the fourth. Let A be to B as C is to D, and, first, let A be a multiple of B. Then must C be the same multiple of D. Let A=mB, and take mƊ the same multiple of D that A is of B. Next, let A be a submultiple of B. For. A is to B as C is to D, .. B is to A as D is to C, V. 12. Now B is a multiple of A, and.. D is the same multiple of C, by the first case. Q. E. D. PROPOSITION XX. (Eucl. v. 20.) If there be three magnitudes, and other three, which have the same ratio, taken two and two, then, if the first be greater than the third, the fourth must be greater than the sixth; and if equal, equal; if less, less. Let A, B, C be three magnitudes, and D, E, F other three, and let A be to B as D is to E, and B be to C as E is to F. Then if A be greater than C, D must be greater than F; and if equal, equal; if less, less. First, if A be greater than C, A has to B a greater ratio than C has to B. V. 7. But C has to B the same ratio that F has to E, Hyp. & V. 12. ...A has to B a greater ratio than F has to E. D has to E a greater ratio than F has to E. Similarly the other cases may be proved. V. 13. V. 9. Q. ED. PROPOSITION XXI. (Eucl. v. 22.) If there be any number of magnitudes, and as many others, which have the same ratio taken two and two in 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. First, let there be three magnitudes A, B, C, and other three D, E, F. And let A be to B as D is to E, and B be to Cas E is to F. Then must A be to C as D is to F. Take of A and D any equimultiples mA, MD, of B and E... of C and F... ...nB, nE, ..pC, pF. Then A is to B as D is to E, V. 17. .. mA is to nB as mD is to nE. So also, nB is to pC as nE is to pF. .. if mA be greater than pC, mD is greater than pF, and if equal, equal; if less, less. .. A is to C as D is to F. V. 20. V. Def. 5. The proposition may be easily extended to any number of magnitudes. PROPOSITION XXII. (Eucl. v. 24.) Q. E. D. If the first have to the second the same ratio which the third has to the fourth, and the fifth have to the second the same ratio which the sixth has to the fourth, then the first and fifth together must have to the second the same ratio which the third and sixth together have to the fourth. Let A be to B as C is to D, and E be to B as F is to D. Then must A, E together be to B as C, F together is to D. For. E is to B as F is to D, .. B is to E as D is to F. V. 12. And A is to B as C is to D, .. A is to E as C is to F. V. 21. .. A, E together is to E as C, F together is to F, V. 16. and E is to B as F is to D; ... A, E together is to B as C, F together is to D. V. 21. Q. E. D. |