equals, we obtain A + C+E = r(B+D+F), and dividing is equal to either of the three fractions, and that A+ C + E is the sum of all the numerators, and B + D + F is the sum of all the denominators; if any number of fractions be equal, the sum of the numerators divided by the sum of the denominators is a fraction equal in value to either of the given fractions. PROPOSITION XIII. THEOREM. (EUCLID XIII.) If the first has to the second the same ratio which the third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the sixth, the first shall have to the second a greater ratio than the fifth has to the sixth. Given that A: B:: C: D, but that C: DE: F; to prove that A:BE:F. PROPOSITION XIV. THEOREM. (EUCLID XIV.) If the first term of an analogy be greater than the third, the second shall be greater than the fourth; and if equal, equal; and if less, less. Given A: B:: C: D; to prove that $ if AC, BD, if A = C, B = D, and if AC, B ≤ D. PROPOSITION XV. THEOREM. (EUCLID XV.) Magnitudes have the same ratio to one another which their equimultiples have. Given A and B two magnitudes, and m any number; to prove that A:B:: mA: mB. (Dem.) By the Arithmetical Principles (5) at the beginning of and therefore A: B:: mA: mB. this Book, A B = mA mB PROPOSITION XVI. THEOREM. (EUCLID XVI.) If four magnitudes of the same kind be proportionals, they shall also be proportionals when taken alternately. Given four magnitudes, A, B, C, and D, of the same kind, such that A: B:: C: D; to prove that (Dem.) Since A:B:: C:D, A:C::B:D. PROPOSITION XVII. THEOREM. (EUCLID XVII.) If magnitudes, taken jointly, be proportionals, they shall also be proportionals when taken separately; that is, if the first, together with the second, have to the second the same ratio which the third, together with the fourth, has to the fourth, the first shall have to the second the same ratio which the third has to the fourth. Given that A+B:B::C+D:D; to prove that A:B::C:D. B Ꭰ but = are each equal to and one, and taking away these equals, there remains therefore A: B:: C: D. PROPOSITION XVIII. THEOREM. (EUCLID XVIII.) The terms of an analogy are proportional by composition. Given that A: B:: C: D; to prove that, A+B:B:: C+D: D. by composition, If four magnitudes of the same kind be proportionals, the difference of the antecedents is to the difference of the consequents as either antecedent to its consequent. Given four magnitudes of the same kind, such that A: B:: C: D; to prove that if A be greater than C, AC: B A: B, or as C: D. D:: (Dem.) Since the four magnitudes are of the same kind (V. 16), A-C:C::B-D:D, and alternately (V. 16), A-C:B-D::C: D. But A:B:: C: D, therefore (V. 11) A — C: B-D :: A: B. - In the same manner it may be demonstrated that if C is greater than A, CA:D—B::A: B, or as C: D. PROPOSITION XX. THEOREM. (EUCLID D.) The terms of an analogy are proportional by conversion. Given that A: B:: C: D; to prove that, A:AB:: C: C-D. by conversion, ABA::C-D: C, and again, by inversion (Prop. 4), A: A B::C:C- D. PROPOSITION XXI. THEOREM. (EUCLID F.) The terms of an analogy are proportional by addition. Given that A: B::C:D; to prove that (Dem.) Since A: B:: CD, A:A+B::C:C+D. by inversion (Prop. 4), PROPOSITION XXII. THEOREM. (EUCLID XXII.) If there be any number of magnitudes, and as many others, which, taken two and two in order, have the same ratio; the first shall have to the last of the first magnitudes the same ratio, which the first of the others has to the last. First, let there be given three magnitudes, A, B, and C, and other three, D, E, and F, which, taken two and two in order, have the same ratio; namely, A: B:: D: E, and B:C::E:F; to prove that A:C:: D:F, Again, let there be given four magnitudes, A, B, C, and D, and other four, E, F, G, and H; which, taken two and two in order, have the same ratio; namely, A: B:: E: F, B: C :: F: G, and CD::G: H; to prove that A:D:: E: H. and of the and therefore H' tor and denominator of the first side by B XC, A E second by F× G (Arith. Prin. 5), A:DE H. In the same manner the demonstration may be extended to any number of such magnitudes. PROPOSITION XXIII. THEOREM. (EUCLID XXIII.) If there be any number of magnitudes, and as many others, which, taken two and two in a cross order, have the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. First, let there be given three magnitudes, A, B, and C, and other three, D, E, and F, which, taken two and two in a cross order, have the same ratio; namely, A: B :: E: F, and B:C:: D: E; to prove that A:C:: D:F. |