and that "such harmony is in immortal souls," it leaves to poet and
philosopher. The timbre, loudness, pitch, of musical tones, is a
concern of science; but for this a Beethoven symphony is no better
than the latest ragtime air from the music halls. In brief, science
deals only with _phenomena_, and its gift to man is power over his
material environment.


MATHEMATICS

The gift of pure mathematics, on the other hand, is primarily to the
mind and spirit: the fact that man uses it to get himself out of his
physical predicaments is more or less by the way. Consider for a
moment this paradox. Mathematics, the very thing common sense swears
by and dotes on, contradicts common sense at every turn. Common
sense balks at the idea of _less than nothing_; yet the _minus_
quantity, which in one sense is less than nothing in that something
must be added to it to make it equal to nothing, is a concept
without which algebra would have to come to a full stop. Again, the
science of quaternions, or more generally, a vector analysis in
which the progress of electrical science is essentially involved,
embraces (explicitly or implicitly) the extensive use of _imaginary_
or _impossible_ quantities of the earlier algebraists. The very
words "imaginary" and "impossible" are eloquent of the defeat of
common sense in dealing with concepts with which it cannot
practically dispense, for even the negative or imaginary solutions
of imaginary quantities almost invariably have some physical
significance. A similar statement might also be made with regard to
_transcendental_ functions.