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

154 MATHEMATICAL PHILOSOPHY
how to solve; in the chapter was a section headed, Transformation of Equations; in it the x of our equations was
replaced by something else; so we got new equations;
these were managed so that, after a few "stunts," we
had the roots sought; just what part of the proceeding
was dubbed transformation and what not, we were left to
conjecture, but, in those days, even conjecture,-guessing,
-was a kind of sin, for mathematics was "the exact science," it was just "pure reasoning." I wonder if you,
who are of a later generation, were more fortunate.
Well, my second introduction to the term in question occurred in analytical geometry, for the book contained,
over toward the middle of it, an arid little chapter entitled Transformation of Coordinates-a meagre, dull,
stupid, stupefying parched little desert discussion, of
which no use was made there and very little in the subsequent part of the course. In neither of the "introductions" was there an illuminating word by text-book or
by instructor to signalise the significance of the matter in
hand-no insight, no outlook, apparently no sense of
being in the presence of a great matter, at once a powerful instrument and a subject of first-rate importance.
What I have said of the term "transformation" might
be said with equal truth of other great terms,-of Function, for example, which is at length happily winning its
way to due recognition in elementary instruction,-of
Invariant and Group, of which I hope to speak at a later
stage-and especially of Relation, which, though long
current in mathematical literature as a convenient term
used in a sense semi-scientific (or semi-technical) and
semi-literary, is at length coming to be recognized, owing
to recent work in the logical foundations of mathematics,
as denoting better than any other term the ultimate tissue