eason is that the latter is
assured to us by experiment, and is not contrasted with the certainty
of ideal or mathematical knowledge. But the ancient philosopher never
experimented: in the Timaeus Plato seems to have thought that there
would be impiety in making the attempt; he, for example, who tried
experiments in colours would 'forget the difference of the human and
divine natures.' Their indefiniteness is probably the reason why he
singles them out, as especially incapable of being tested by experiment.
(Compare the saying of Anaxagoras--Sext. Pyrrh.--that since snow is made
of water and water is black, snow ought to be black.)

The greatest 'divination' of the ancients was the supremacy which they
assigned to mathematics in all the realms of nature; for in all of them
there is a foundation of mechanics. Even physiology partakes of figure
and number; and Plato is not wrong in attributing them to the human
frame, but in the omission to observe how little could be explained by
them. Thus we may remark in passing that the most fanciful of ancient
philosophies is also the most nearly verified in fact. The fortunate
guess that the world is a sum of numbers and figures has been the most
fruitful of anticipations. The 'diatonic' scale of the Pythagoreans
and Plato suggested to Kepler that the secret of the distances of the
planets from one another was to be found in mathematical proportions.
The doctrine that the heavenly bodies all move in a circle is known by
us to be erroneous; but without such an error how could the human mind
have comprehended the heavens? Astronomy, even in modern times, has
made far greater progress by the high a priori road than could have been
attained by any other. Yet, strictly speaking--and the remark applies
to ancient physics generally--this high a priori road was based upon a
posteriori grounds. For there were no facts of which the ancients were
so well assured by experience as facts of number. Having observed that
they held good in a few instances, they applied them everywhere; and in
the complexity, of which they were capable, found the explanation of the
equally complex phenomena of the universe. They seemed to see them in
the least things as well as in the greatest; in atoms, as well as in
suns and stars; in the human body as well as in external nature. And
now a favourite speculation of modern chemistry is the explanation of
qualitative difference by quantitative, which is at present ver