priori element must be treated first, and must be carefully
distinguished from that which is supplied from other sources. Any
other method can only lead to irremediable confusion.

Mathematics and physics are the two theoretical sciences which
have to determine their objects a priori. The former is purely a
priori, the latter is partially so, but is also dependent on other
sources of cognition.

In the earliest times of which history affords us any record,
mathematics had already entered on the sure course of science, among
that wonderful nation, the Greeks. Still it is not to be supposed that
it was as easy for this science to strike into, or rather to construct
for itself, that royal road, as it was for logic, in which reason
has only to deal with itself. On the contrary, I believe that it
must have remained long--chiefly among the Egyptians--in the stage
of blind groping after its true aims and destination, and that it
was revolutionized by the happy idea of one man, who struck out and
determined for all time the path which this science must follow, and
which admits of an indefinite advancement. The history of this
intellectual revolution--much more important in its results than the
discovery of the passage round the celebrated Cape of Good Hope--and
of its author, has not been preserved. But Diogenes Laertius, in
naming the supposed discoverer of some of the simplest elements of
geometrical demonstration--elements which, according to the ordinary
opinion, do not even require to be proved--makes it apparent that
the change introduced by the first indication of this new path, must
have seemed of the utmost importance to the mathematicians of that
age, and it has thus been secured against the chance of oblivion. A
new light must have flashed on the mind of the first man (Thales, or