1 = I + 7/ and hence x ?-1.
Now it has been demonstrated in the introduction to the article FLuxions, § 12, that n being supposed infinitely great, n (v 1)= Nap. log. v, therefore Nap. log. v = x and hence c denoting the basis of the system, (ALGEBRA, X?-1, § 535 and § 356) we have v = e that is, Cos. x + sin. x 41-1 = e This is another imaginary formula of great value, because it exhibits under a finite form a relation between an arc or angle and its co-sine, sine, Sec. It was first observed by Euler, and is justly regarded as one of the most important analytic inventions of the last century. Other investiga tions of this formula have been given in ARITHMETIC or SINES, § 29, and FLuxioss, § 124. Observing, as before, that the square root of a quantity may be considered as negative as well as positive, we have from the formula, x V-1 ; Cos. x sin. x V-1 = e and from the two expressions, by addition and subtrac tion, 41-1} Cos. = + e 1 { 1 -- 1,/-1 Sin 2V-1 e e These formula, in their present state, are illusive ; fur the arc and sine, or co-sine, cannot, by means of them, be found, the one from the other. However, by expanding the exponentials into series, we have, (ALGEBRA,§ 357.) x ?-1 =1 r -f-)&c 1.2 1.2..1 ?-1 e +, Etc.
1.2.3 These being substituted in the formula, we get Cos. x = 1 -- ' 1.2.3.4 " 3Sin. x = x x xs Szc 1.2.3 " expressions which are altogether free from the imaginary symbol.
An example of the application of the same formula to the determination of the arc in terms of the tangent, is given in the ARITHMETIC or SINES. § 30. They also ad mit of various applications to the investigation of rules in plain and spherical trigonometry. (See Legcnclre, Ele mens de Geometric, in App. to 'firm.) Because Cos. x + Sin. x .V-1 = 1, therefore, Log. (Cos. x + Sin. x St) = x ?-1. Let -denote halt' the circumference oi a circle, of which the radius and suppose x = 1r, then Cos. x = 0, and Sin. x = 1; hence we get Log. = 1 7: v-1, and 4 log. v.-1 2 = This remarkable expression for the circumference of a cir cle was first found by John Bernoulli. It is of no use as a
rectification of the circle ; but it shews, that if the expres sion should occur in any investigation, we may substitute, instead of it, the real quantity 2 a.
The formula, Cos. x + Sin. x 1 = v-1, be ? comes, in the case of x= 11r, v-1 = e` Now, let both sides of this equation be raised to a power expressed by VI, then, observing that 1r ?-1 X 17, wc have, = e = 0.207872.
This very singular imaginary expression was, we believe, first noticed by Euler.
D'Alembert first demonstrated a remarkable property of imaginary expressions, namely, that however complicated they may be, they may always be reduced to the form A + B ; so that every function of an impossible quantity a + b has the form A + B 4/-1, where, however, B may equal 0.
Let r = V + and suppose s to be such an arc, that Cos. x , then Sin. x + , where r, Cos. x, and Sin. x will be all real quantities ; by substitution, the ima ginary formula a + b I will be transformed into r t Cos. a+Sin. a v--,1) If, now, a' 4- 6' ?-1 be another imaginary expression, by a like substitution it may be transformed into r' (Cos. x' + Sin. x' 4/I), where, as be fore. Cos. x' and Sin. x' are all real. Suppose, now, an imaginary expression to be the product of the formula a + b v 1, and a' + b' V I, this, by substitution, will be rr' (Cos. x + Sin. x V-1) (Cos. x' + Sin. x' 4-1).
By actual multiplication and substitution of Cos. (x + x') for Cos. x Cos. x' Sin, x Sin. x' ; also Sin. (x + x');for Sin x Cos. if + Cot. x Sin. if (ARITHMETIC OF SINES,) the product is equivalent to, rr' Cos."(x x') + Sin. (x + x') v-1 which has manifestly the form A + B v-1. From this it is evident, that the continual mi.clucC of any number of imaginary quantities, a + 6 V-1, a' + 6' v-1, &cc. will be an imaginary quantity of the same form.
a .
Consider now the fraction + b v-1 This, by sub "' i stituting the trigonometrical formula, will be changed to r (Cos. x + Sin. x ,/ I) (cos. + sal. if ,vt) .Multiply the numerator and denominator by r' (Cos. x' Sin. x' v-1), and substitute Cos. (x s') for Cos. x Cos.