If A varies as B, and B as C, then A varies as C. In this proposition and the following, a, b, c, &c., represent corresponding values of A, B, C, &c.; supposing A, B, C, &c., to vary. In the same manner it may be shown, if A x B, and B o then If A x C, and B x C, then A + B ∞ C; and √A B ∞ C. By the supposition A: a :: Cc:: B: b, .. Aa :: B: 6 (11 B. v.); PROP. III. THEO. B If A x B, then A x m B, and A ∞ ;m being any number whatever. Because A m a :: B: b, If A vary as B. A is always equal some constant multiple or submultiple of B. Because A a :: B : b, .. A a :: m B: m ; and .. Am B: a: mb. Now, if m be taken so that A = m B, then must a = m b, in all cases; and if any constant values of A and B be known, m may be found: If A & B, then A" α B"; n being any number-whole or fractional. For A a B : 6 (hyp.) .. A1 : a1 :: Ba : bn (L. B. v.) PROP. VI. THEO. If one quantity vary as another, and each of them be multiplied or divided by any number, the products or quotients will vary as each other. Let A vary as B, and let T be any number-constant or variable. Then, by the supposition, A a B b, Cor. If A x B, by dividing both by B, we have a 1, that is, varying as one, but one is constant, therefore B A is constant. B PROP. VII. THEO. If A x B x C, then B varies directly as A, and inversely as C; and C varies directly as A, and inversely as B. Because A x B x C, by dividing by C, we have a B, or B varying inversely as C, and directly as A. The other part of the proposition may be readily proved, by dividing by B. PROP. VIII. THEO. If the product of two quantities be invariable, these quantities vary inversely as each other. Let B x P be constant, or B x P x 1; If four quantities be always proportionals, and one of them be invariable, PROP. X. THEO. If A a B, and C a D, then A C∞ B D. For A a: B: b, and Cc: D: d; .. AC ac :: BD: bd, PROP. XI. THEO. When the increase or decrease of one quantity depends on the increase or decrease of two others, and it appears that either of these latter be invariable, the first varies as the other; when they both vary, the first varies as their product. Let S a V, when T is constant, and Sa T, V is constant. When neither T nor V is constant, S a TV. If S be changed to s, let T be changed to t, Again, lets be changed to s', and V to v, .. Ssss :: TV: tv; or Ss TV : tv; This will be readily seen, if S be considered as a variable space, passed over by a body moving with a variable uniform velocity in a given time, T; and again, if S be considered as a variable space, passed over by a body moving with a given uniform velocity, V, in a variable time, T; then it is clear, when the time and velocity both vary, the space varies as the time multiplied by the velocity. PROP. XII. THEO. If A x P, when Q, R, S, &c., are constant; and if A x Q, when P, R, S, &c., are constant; and if A ∞ R, when P, Q, R, S, &c., all become variable; A x P x Q x R x S, &c. The truth of this proposition may be readily shown by employing the process used in the demonstration of the last proposition. DRURY, Printer, 17, Bridgewater Square, Barbican, London. |