Comparing now the philosophical principles on which science was
proceeding, with the principles on which ecclesiasticism rested, we see
that, while the former repudiated tradition, to the latter it was the
main support while the former insisted on the agreement of calculation
and observation, or the correspondence of reasoning and fact, the latter
leaned upon mysteries; while the former summarily rejected its own
theories, if it saw that they could not be coordinated with Nature, the
latter found merit in a faith that blindly accepted the inexplicable, a
satisfied contemplation of "things above reason." The alienation between
the two continually increased. On one side there was a sentiment of
disdain, on the other a sentiment of hatred. Impartial witnesses on all
hands perceived that science was rapidly undermining ecclesiasticism.

MATHEMATICS. Mathematics had thus become the great instrument of
scientific research, it had become the instrument of scientific
reasoning. In one respect it may be said that it reduced the operations
of the mind to a mechanical process, for its symbols often saved the
labor of thinking. The habit of mental exactness it encouraged extended
to other branches of thought, and produced an intellectual revolution.
No longer was it possible to be satisfied with miracle-proof, or the
logic that had been relied upon throughout the middle ages. Not only did
it thus influence the manner of thinking, it also changed the direction
of thought. Of this we may be satisfied by comparing the subjects
considered in the transactions of the various learned societies with the
discussions that had occupied the attention of the middle ages.

But the use of mathematics was not limited to the verification of
theories; as above indicated, it also furnished a means of predicting
what had hitherto been unobserved. In this it offered a counterpart
to the prophecies of ecclesiasticism. The discovery of Neptune is
an instance of the kind furnished by astronomy, and that of conical
refraction by the optical theory of undulations.

But, while this great instrument led to such a wonderful development in
natural science, it was itself undergoing development--improvement. Let
us in a few lines recall its progress.

The germ of algebra may be discerned in the works of Diophantus of
Alexandria, who is supposed to have lived in the second century of our
era. In that Egyptian school Euclid had formerly collected the great
tru