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

TRANSFORMATION 163
stated. To envisage the matter in a large way, conceive
two immense canvases suspended parallel to each other
each of them bisecting the universe of space. Imagine
yourself comfortably seated between them; fancy that
on the face of one of them are marked and drawn the
points of a plane, its point loci,-curves and sets thereof
limitless in number and variety,-and that on the face of
the other canvas are recorded the real number pairs, pair
systems-(x, y)-equations and sets thereof, more numerous than the sands of the sea; choose a unit of length
and in the former face a pair of axes. What happens?
You behold the phenomena on either sheet transformed
into those of the other-and this infinitely multitudinous
transformation is the method of plane analytical geometry.
The like is true of analytical geometry of three or more
dimensions. You see that instead of transfomation
being a chapter in analytical geometry, the latter is itself
only a huge chapter in the infinitely more embracing
theory of Transformation.
And now do you ask what transformations are good for?
That is very much like asking what Thinking is good for;
for without transformations, thinking could not go on.
We have just seen that analytical geometry is born of
transformation and does its work thereby; we have seen
that the Olympian concept of Infinity owes its birth to
transformation; we have seen that, except for transformation, we could not even count the cattle in a field; we have
seen that transformation pervades the practical thinking
of the workaday world; in previous lectures, as you will
recall, we saw that certain simple transformations,the inversion transformation and others,-enable us to
establish divers verifications of the doctrinal functions of
Hilbert, doctrireç being thus derived from doctrines and