as actual, for m, then, (since 4, is clearly positive) we obtain the correct equation, provided x is actual. If we now once for all denote the positive number 4, by e, this equation may be written thus provided that x is actual, so that e" has already received a meaning in the elements (sect. 28). And in this equation we have 1 2! 3! 4! This equation (No. 6) gives us now an opportunity of extending the idea of the power e" for imaginary values of x, by understanding the symbol e to denote from henceforth the infinite x x2 x3 + in which x is a mere supporter of the series 1 + + + ... symbols of operation, and may therefore just as well be actual as imaginary. For et now no longer differs from 4, and the equations (Nos. 2, 3, and 4) may now be written thus where x and z may be just as well actual as imaginary, while the last equation holds for any positive or negative whole m, and general x, or for any actual m but positive e. This power e is called the natural power, and it may be easily proved that it is always convergent for every actual and every imaginary value of x of the form p+q.i, and that it has therefore always an actual or imaginary value of the form p+q.i, and that it is at the same time single-meaning. Moreover it also follows that since by (No. III.) (IV.) '/(e2)='/1. e*: v (ex: V) V = e(x: V), V = e2, provided that be supposed positive whole, and /1 represents any of its different values, while x is conceived perfectly general. The formula (Nos. I.-IV.) are therefore those which can and may be applied for NATURAL POWERS, {under the restrictions assigned for (Nos. III. and IV.)}. SECTION 50. The natural logarithm follows instantly upon the natural power, if we mean by it the symbol log a which denotes any expression x that makes e = a, that is, 1+ + + +...=a. 1 2! 3! This equation from which x has to be found has the form of the higher algebraical equations, but of an infinite degree, and this circumstance leads us to suppose that log a (that is x) will have an infinite number of values, which however must all be of the form p+q.i. If we propose to ourselves the problem: to find the values of log a upon the supposition that a is actual or imaginary but of the form p+q.i, that is, if we desire to find all the values of log (p+q.i), we may represent them by a + B. i, where a and are conceived as actual and are most probably infinitely multiple-meaning, but are in any case yet to be found. We then have the equation ea +ß. i. = p + q i, But because ei is found from e* by putting i for x, it follows that if we set apart all the terms with even powers of ß, and represent the series (1) + 2! 4! 6! and also collect all the terms with uneven powers of ẞ and represent this second series, The above equation then separates into (A) ea. Kg = p, and (B) ea. Sg = q ; and it now only remains to find the unknown values of a and ß, which are conceived as actual, from these two equations. For this purpose a more intimate acquaintance with the two series represented by Kg and Se is required, and the result of this more intimate acquaintance is called analytical trigonometry. This more intimate acquaintance must be obtained *For those readers for whom these pages were put together the remark is needless, that Kg and Se are the general cos ẞ and sin ß, which here appear, as of themselves, on occasion of the natural power, and independently of geometry. Now we must never assume geometry in analysis, since mathematical analysis must according to these views precede any doctrine of magnitudes. before we can think of continuing the solution of the proposed problem. SECTION 51. First of all we easily prove that the series K and Sg are convergent for all actual and imaginary values of B (of the form p+q.), and have therefore always a value, and that they are also only single-meaning. By solving the equations (sect. 50, Nos. 3 and 4) algebraically with respect to Ks and Sp we obtain so that these series are exhibited as expressions in powers (commonly called exponential expressions) with which we can more easily calculate." Now if we take any equation between powers, as e. g. e(x±z)i = exi. e±zi, and substitute for the powers the expressions compounded of K and S (by sect. 50, Nos. 3 and 4) we immediately obtain equations between these series K and S, namely the equations, S2+ S. K. + K. S.. (I.) % K+K. K.-S. S. (III.) S.- S. K - K ̧ . S. (IV.) = K2-= K ̧ . K ̧ + S ̧ . S ̧. And if we multiply the equations (sect. 50, Nos. 3 and 4) together, we find (V.) 1 = (Kg)2 + (SB). Then it also follows from (Nos. I. and II. for z= x) (VI.) S2x=2Sr. Kr. K2x = (K ̧)2 – (S„)2 = 1 − 2 (S.)2 = 2 (K ̧)3 — 1, 2x and from (No. VII. for x = ક) (VIII.) S12 = √√√1-K,. (IX.) 2 By means of the formulæ (Nos. I.-IV.) the three products S. K„, K. K, and S. S, may be transformed into sums and differences, and also conversely the four sums or differences SaS and K, Kg may be again transformed into products. α SECTION 52. Upon now entering upon the values in cyphers of these general series denoted by K and S, we must first of all remark, that for actual values of x the values of K, and S, will be always actual, and (on account of No. V.) must always lie between +1 and -1. Moreover, if we write h for z and substitute for S and K2 the infinite series which are represented by these symbols, the formulæ (Nos. I. and II.) become (a) The actual values of the series S. and K, alter continuously with the actual values of x. - (b) The actual values of the series S, increase together with those of a, as long as K, is positive; but decrease continuously, while x is conceived as increasing continuously from ∞ through 0 to +∞, as soon and as long as K, is negative; they pass moreover from increasing to decreasing (that is, have a maximum) at the moment that K, becomes = 0, and S, is positive; and finally pass from decreasing to increasing (i. e. have a minimum) at the moment that K becomes = 0, and S, is negative. = (c) The actual values of the series K, on the other hand decrease, as those of x increase, as long as S, is positive; increase with the values of x as soon and as long as S, is negative; pass from in the moment increasing diminishing to diminishing that S ̧ becomes = 0 and K ̧ is at the same time{positive}. SECTION 53. If we suppose the value of S, (or of K) given, and the value of x required, the equation for determining a has always the form of an higher algebraical equation, but of an infinite degree, and this leads us to conjecture that there will be an infinite number of values of x of the form p + qi for which S, (or K ̧) will have one and the same value. All this points at a periodical recurrence in the values of K, and S. But we are confirmed in this view by a consideration of the equations (sect. 51, Nos. I. and II.) For they shew us that the values of S+ and S,, as also of K+, and K, are equal to one another whenever the difference x between the arguments (arcs) xz and z is such that K = 1 and S = 0. The whole investigation now turns upon the discovery of these values. = SECTION 54. = Now if we put 60 in (sect. 50, Nos. 1 and 2) we obtain, (1) K1 = 1, and (2) S 0. Hence, while x increases continuously from 0, S, will also increase continuously with it from 0 (by sect. 52, b), and K, simultaneously decrease continuously from 1 (by sect. 52, c). If we now denote the smallest positive value of x, for which K, has diminished down to 0, (and S, has therefore simultaneously increased up to 1,) and whose existence may be easily proved*, by 7, that is, the double of this positive number by π, then we have, (3) K=0 and (4) S = 1. And we now deduce by means of the equations (sect. 52, Nos. VI. VII.) for x ==π or x==π, and (sect. 52, Nos. I. and II.) for x=π, =π, the following results (15) K-=+ K ̧ If we now separate all positive (whole or broken, rational or irrational) contiguous numbers into equal sections (from 0 to, from 1 to π, from π to π, from 3 to 2π, from 2π to §π and so on), the limits of any one of which always exceed those of the previous one by, and if we call the collection of all positive numbers in any such section a quadrant, we shall see clearly from the formula (Nos. 1-16.) (by supposing all numbers in the first quadrant, i. e. from 0 to to be substituted for z), that: Within the four first quadrants, while the actual values of x are supposed to increase continuously from 0 to 2π, the values of the series K, and S, are as follows: And then we easily find from (sect. 52, Nos. I. and II.) (for X= 2π, 4π, 6π, ... and z=2′′), that when n is a positive whole number, (17) Kon=1 and (18) S2 = 0, We shew that for x=1, K, is still positive, but that for x=2, K, has become negative, consequently there is one value of a between 1 and 2, for which K,=0. Hence not only does this value exist, but it may even be found by the Newtonian method of approximation. |