Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

186 MATHEMATICAL PHILOSOPHY
mathematical invariants, which are formulated precisely;
yet they evidently belong, like the latter, to the type of
invariantive matter and will more and more approximate
or even attain precision in course of the progress of
analysis and definition.
Let us now have another little dip in the boundless sea
of strictly mathematical invariants. In Lecture IV we
were introduced to the pole-polar transformation of a
plane with respect to a circle; we saw that it converts a
point (as pole) into a line (as polar), and a line (as polar)
into a point (as pole); it is, you see, a one-to-one reciprocal
transformation-a twofold affair composed of a transformation (I) of points into lines and the converse transformation (2) oflines into points. Let us first think of (I) alone;
we readily detect certain variants and certain invariants
under it; the property of being a point is not preserved,
since the transform of a point is a line; a range of points
loses the range property, since the transform of a range is
a pencil (of lines); distance is lost, since the distance
between two points has for transform the angle between
two lines (the transforms of the points); now, as you
know, a curve has two aspects (called dual aspects), one
as the locus of its points, the other as the envelope of its
(tangent) lines; the property of being a curve is invariant
(preserved), for under (I) the transform of a curve is a
curve; but the locus aspect is lost, its transform being
the envelope aspect; as we saw in Lecture IV, the relation
of order and the relation of congruence are exceedingly
important invariants under (I); by (I) the ordinary
geometry Di of the plane was transformed into the
geometry D2 of lines and pathopencils; and, as D1 and D2
are the same in respect to form, we see that under (I)
doctrinal form, or logical structure, is invariant. I