Generation of Ordinary Scale by Linear It is important to note that the displacement property enables us to con struct the ordinary scale by means of a special linear transformation. Thus if x denote dis tance from a point assumed as origin in the range, then the transformation x'=x + c will serve to generate the distance-measuring scale in which the unit, or interval, is c. If x be the distance from the origin to the point 1 chosen for initial point of the scale, then x' will denote the second division point 2. A second applica tion of the transformation will convert point 1 into point 2 and point 2 into a third point 3 of the scale ; and so on. The transformation for generating the angle-measuring scale is x' (x + tan y) (1— x tan y), y being the unit of angle and x being tangent of the angle of any line of the pencil with a line assumed as Generalization by Means of Transforma tion s' (az b): (as+ d) —The ordinary scales being generable by special linear trans formations, the possibility. is suggested of gen erating by more general transformations more general scales that shall involve and disclose moregeneral concepts of distance and angle. We will first suppose the fixed elements of the transformation to be distinct (real or imagi nary). Taking these as elements of reference 0, and Os for homogeneous co-ordinates xs and xs in range or pencil, the transformation will assume the form r'= AZ, where z= xs, and where A is the characteristic constant of the transformation. The constant A will be subject to the condition that real elements (points or lines) z shall be converted into real elements a' and that, in case the reference ele ments 2= 0 and z= c0 are real, A shall be positive.
If now we apply the transformation suc cessively to an arbitrarily chosen element we shall obtain a series of elements A's,, . ... This series of elements con stitutes our scale. Obviously this scale is con verted into itself by the transformation z'=A2 by which it was generated just as the old scales are converted respectively into themselves by the transformations generating them. just as the interval between two successive divisions of the old scale is called unit of distance or angle because one application of the generating transformation carries us over that interval, so for the same reason we name unit of distance or angle the magnitude extending from any division to the next in the new scale. Accord ingly the distances or angles from the points or lines z,, Az,, RAzi, .. to point or line are 0, 1, 2, 3, . . . respectively. By means of the new scale we can measure distances and angles whose magnitudes reckoned from a are expressible as whole numbers. In order to render the scale available for measuring dis tances and angles whose magnitudes are any rational numbers, we have merely to subdivide the scale intervals already obtained into n equal parts. This is done by applying n-1 times to
the elements at the beginning of each interval, a transformation which repeated n times repro duces the transformation zi=?,x. The trans formation required is therefore z'=al'isis where that one of the n roots of A must be used that secures that the element X'Ins shall fall be tween s and Az. We can then measure all distances or angles from z, to z where s is of a the form a= A a, a and # being integers.
The distance or angle from point or line a +#/n a to a, is precisely a+#/n.
If now we suppose the subdivision extended indefinitely and if we admit the notion of irra tional number, then obviously we must regard the distance or angle from the point or line Si to the point or fine a s/as being expressed by a where a may be any positive real number whatever, whole or fractional, rational or ir rational.
From the last equation we have a -=logz/so log A. Hence: The distance or angle of a point or line s from the point or line z, is the loga rithm of the ratio z/z, divided by the constant log A.
The point or line z, was arbitrarily taken as initial element of our scale and our formula gives the distance or angle from a to any other point or line. The general linear transforma tion z' = (az b): (cz d) involves three parameters, two of which are determined by requiring the elements Oi and O. to remain fixed. By means of the third parameter, the element a, may be brought to any desired posi tion z' of range or pencil. Accordingly: The distance or angle from point or line a to any other point or line a' is log 2/2': log A, or c log ea', where c=1: log A.
It is readily seen that, as ought to be the case, the addition property of the old magnitude concept is preserved in the new concept just defined. That is to say, Distance (angle) szt distance (angle) Az' = Distance (angle) as'; for c log s— c log zi+ c log al—a log z =c log a — c log z' identically. Also c log s/s 0, whence c to siz' =— c log s'/z con formably to the like property of the ordinary notion of measure. The displacement proper ty, too, is preserved; for the scale-generating transformation zk=az converts distance (an gle) sr sp into distance (angle) a* Az p. These magnitudes are plainly equal, for the former is c log zr—c log Sp while the latter is c log A c log c log A — c to sp.
Geometric Meaning of the New Concept of Distance (Angle).— Let z=z1 and s=s, be any two points (lines) of the range (pencil). The anharmonic ratio (0 z, m si) of the pair (z., isi,) to the pair (0, oo ) of fixed elements is (Y-s.) — ri/z2 Hence the distance (angle) of two points (lines) is an arbitrary constant c times the logarithm of the anharmonic ratio of the two points (lines) to the two fixed —absolute— points (lines) of the range (pencil).