FUNCTION. 
525 
78 7] 
Taking y as an angle, and defining as usual the sine and cosine as the ratios 
of the perpendicular and base respectively to the radius, the sine and cosine will be 
functions of y; and we obtain geometrically the foregoing fundamental equations for the 
sine and cosine; but in order to the truth of the foregoing equation exp iy = cos y + i sin y, 
it is further necessary that the angle should be measured in circular measure, that 
is, by the ratio of the arc to the radius; so that ir denoting as usual the number 
3T4159..., the measure of a right angle is =^7r. And this being so, the functions 
sine and cosine, obtained as above by consideration of the exponential function, have 
their ordinary geometrical significations. 
3. The foregoing investigation was given in detail in order to the completion of 
l 
the theory of the irrational function u m . We henceforth take the theory of the 
circular functions as known, and speak of tan#, &c., as the occasion may arise. 
We have 
x + iy = r (cos 6 + i sin 6), 
where, writing Vx- + y 2 to denote the positive value of the square root, we have 
tan 6 = y 
x ‘ 
Treating x, y as the rectangular coordinates of a point P, r is the distance (regarded 
as positive) of this point from the origin, and 6 is the inclination of r to the positive 
part of the axis of #; to fix the ideas 6 may be regarded as lying within the 
limits 0, 77-, or 0, — 7r, according as y is positive or negative ; 6 is thus completely 
determinate, except in the case, x negative, y = 0, for which 6 is =tt or —7r indifferently. 
And if u = x + iy, we hence have 
where r m is real and positive and s has any positive or negative integer value what 
ever: but we thus obtain for u m only the to values corresponding to the values 
0, 1, 2, ...,to — 1 of s. More generally we may, instead of the index — , take the index 
to be any rational fraction Supposing this to be in its least terms, and to to 
be positive, the number of distinct values is always = to. If instead of — we take 
TO 
the index to be the general real or complex quantity to, we have u m , no longer an 
algebraical function of u, and having in general an infinity of values.