Introduction of Complex (Imaginary) Hitherto only real elements have been admitted. We now extend the generalized concept of measurement to complex elements, points or lines, for which the co-ordinates s and s' will be of the form z=et where a, a' ,13, ft' are real and i= V —1. Now iB = r e • and z' = r' e , whence — titt.
z'= (r r') a = Hence distance (angle) se=c log 2/.'=c log (pel i (1)::27.2n7i)) i sb) = c log (pe = c log (pie ± c log in2st e = c log where n is an integer. It so appears that in the generic concept of measurement in hand distance (angle) is not a one-valued function but is an infinitely many-valued function, viz., a periodic function of period 2ctri. The periodicity of the angle in the ordinary system of measurement is a special case of the preceding. Let # be the angle be tween the lines y =mx b and y = m's b' ; then tan # = (m—m') : (1 i-mm'). As tan 0 = tan (0 ±-ner), we have tan (0n r)=--(en—m') (1 -{-mm'), whence 0 ± nor = tan—' I (m—en') : (1 + mm') ; hence [ (m — m') : (1 + mm') ) ± nn.
Infinite Distances and Angles in the Gen eralized System of Measurement.— We will premise a brief account of the infinite elements m the ordinary system of measurement. For this system the scale-generating transformation for the range is s'==z-l-c=(s+c) : (0z-1- I) ; for the pencil it is s'= (s+tan y): (1 — at tan y). In the former case the fixed elements, given by Oz= Oz c = 0 are z= oo and z=--- co ; i.e., the range's infinite (Desarguesian) point count ed twice. (The symbol z is here taken to repre sent distance from a finite origin). In ordinary
distance measurement, the range, or right line, contains two points from which all other points are equally and infinitely (algebraically) distant —a fact of which the average carpenter is not aware. In this case the two points happen to coincide but what is here important to note is the fact that the point of coincidence is double. How stands the mat ter in case of ordinary angle measurement? The fixed elements (lines) are given by making z'=2, i.e., by the equation tan y za-1- tan y = 0, or — l, as tan y O. Hence the fixed (absolute) lines of the pencil make with the origin (initial or referee line) angles whose tangents are + i and — i. Such lines are of course imaginary. Are the angles co ? And does every other line (real) of the pencil make the same angle with these fixed lines? To answer, take any finite point for vertex of the pencil and choose this vertex for origin of Cartesian axes. Then the angle 0 of any real line y=mx of the pencil with the line y=ix is given by tan # =(m—i) : (1 + im)=1 : —i. Hence # is independent of m, and hence all real lines are equi-inclined to the line y=ix. The same is true for line y= —ix. Hence as y = ens rotates about the vertex of the pencil, it keeps a constant angle with each of two imaginary lines of the pencil. This fact sug gests that tan•'i and taw' (—i) are each in finite. That they are oo is readily established as follows: From Euler's equations e = cos s i sin z, cos z —i sin 2, we have tan z=—i —1) : (ezei + 1).
In the present case tan z i, whence z=00,