or l(x − a) +1(x −b) + l( x + a + b)−l(x + a) − 1 ( x + b) — + 3 (x2 + qx) 3 x3+gx From which equation (x + a + b) can be computed, if we know the logarithms of the five preceding numbers, a b, x Thus, if we assume b, x + b, x + a. 1 + x = (x − 1)(x − 1)(x+2)=x3- 3x + 2, 1 z= = 203 3x - 2, +3 + &c. In which equation if x = 5, 17 can be found in terms of 76, 74, and 73; i. e. in terms of 12 and 13. Or if the logarithms of 2 and 3 have not been previously computed, substitute for a the numbers 5, 6, 7, 8, successively, and there will result four equations containing the unknown quantities 12, 13, 15, and 17, which, therefore, may be determined by elinination. 31. Haros' method. This consists in substituting biquadraticks for 1 + z and 1- 2. Assume1+=(x+3)(x − 3)(x+4)(x — 4) = x1 — 25x2 +144 x2. (x+5)(x-5)=x+-25x greater rapidity, see La Croix's Introduction Calcul. Diff. p. 50. The reader may also consult his 3d volume for the method of computing logarithms by interpolation. 32. It may be here observed, that logarithms enable us to generalize the demonstration of the binomial theorem. At least, they enable us to show, that if (1 + x)" is developed in a series of the form 1 + A + B + &c. by any analy tical process whatever, A shall =n, B=n. a n-1 &c. &c. even = in the cases where n is either a fraction or an imaginary quantity. For since (1+x)2=1+ Ax+ Bx2 + cx3 + &c., therefore n.l(1+x)=l{1 + x(A+BX+ &c.)}, and by developement, When n is a positive integer, the binomial theorem is strictly demonstrable either by the method of induction, or by the method of combinations given in the Algebra (Art. 232), the latter of which is strictly analytical, because it does not assume that the law of the series is known. When n is not a positive integer, the proof given in this article seems quite unexceptionable, since, as La Croix observes, in the developement of 7(1 + x) we do not assume the expansion of the binomial when n is fractional. 33. Imaginary functions. Imaginary functions are the links which connect circular functions with the exponential and the logarithmick, and as they are of great use in analytical science, we shall consider them in this place. It appears by induction that every algebraick function of a quantity of the form a ± b√ −1 is of the same form. It will be shown that all transcendentals may likewise be expressed in terms of✓ 1, and that they possess the same property as the algebraick functions. They are not to be regarded as representing the numerical value of quantity, but merely as analytical symbols, which show that by an extension of the common rules of algebra certain quantities may be made to assume certain analytical forms. If we are sufficiently cautious not to extend these rules beyond what the circumstances of the case will allow, our results will be as accurate and as much to be depended upon as those which are obtained by any other analytical process. We have already seen one instance (Art. 16.) where it was necessary to observe this caution. John Bernoulli contended that because (-a) = a2, therefore L.(-a) = L.(a)2 and 2L(-a)=2(a) and L(-a) = L(+ a); in which he errs at the very first step from not sufficiently attending to the nature of logarithms. In Art. 35. the developement of sin.r and cos. into series in terms of x will be there assumed; for these series are most conveniently demonstrated by the theorems of the fourth chapter. In the management of irrational and imaginary functions, it will be generally found convenient to reduce the former to the form of the latter, and we shall then only have to consider the rules for the involution and the evolution of ✓ — 1. 34. Bezout's demonstration of the proposition that a = a. a = √a, and generally va is ±a; but the a2 in this case has arisen from multiplying -a and not +a into itself, so that +a is excluded by the nature of the case. Otherwise. (c—b)a = (c.—b)a, therefore vc - UN a = √(c — b)a, which are numerically equal so long as c is greater than b; but suppose b=c+1, then we have a, and therefore✔ For See also Alg. (242.). b=- Vab. b= a=a.1, and b√√ −1, therefore -a.√b√a 6-11-√ab. Cor. 2. vax √ − b = √ab. √ —]. But if any integral number be divided by 4, the remainder is either 0, 1, 2, or 3; the four quantities 4n, 4n+1, 4n+2, 4n+ 3, by a proper assumption of n may be made to represent the integral numbers. The following four forms then include all the possible With respect to the evolution of 1; by substituting x for 1 it will appear that its square root = √1; 1, one of which is possible and = -1; its biquadratick roots are all impossible, &c. &c. Ex. 2.—a—b=2/ab/−12/=1=2/ab( √−1 √ = 1}} = Jab × (−1)3, which has three values, one of which is possible and ab. = +&c., for a substitute x-1, therefore we have 1.2.3.4.5 |