away and the universal alone remains. He
then seeks to combine the universals which he has disengaged from sense,
not perceiving that the correlation of them has no other basis but the
common use of language. He never understands that abstractions, as Hegel
says, are 'mere abstractions'--of use when employed in the arrangement
of facts, but adding nothing to the sum of knowledge when pursued apart
from them, or with reference to an imaginary idea of good. Still the
exercise of the faculty of abstraction apart from facts has enlarged the
mind, and played a great part in the education of the human race.
Plato appreciated the value of this faculty, and saw that it might be
quickened by the study of number and relation. All things in which
there is opposition or proportion are suggestive of reflection. The
mere impression of sense evokes no power of thought or of mind, but when
sensible objects ask to be compared and distinguished, then philosophy
begins. The science of arithmetic first suggests such distinctions. The
follow in order the other sciences of plain and solid geometry, and of
solids in motion, one branch of which is astronomy or the harmony of
the spheres,--to this is appended the sister science of the harmony
of sounds. Plato seems also to hint at the possibility of other
applications of arithmetical or mathematical proportions, such as we
employ in chemistry and natural philosophy, such as the Pythagoreans and
even Aristotle make use of in Ethics and Politics, e.g. his distinction
between arithmetical and geometrical proportion in the Ethics (Book V),
or between numerical and proportional equality in the Politics.

The modern mathematician will readily sympathise with Plato's delight
in the properties of pure mathematics. He will not be disinclined to say
with him:--Let alone the heavens, and study the beauties of number and
figure in themselves. He too will be apt to depreciate their application
to the arts. He will observe that Plato has a conception of geometry,
in which figures are to be dispensed with; thus in a distant and
shadowy way seeming to anticipate the possibility of working geometrical
problems by a more general mode of analysis. He will remark with
interest on the backward state of solid geometry, which, alas! was not
encouraged by the aid of the State in the age of Plato; and he will
recognize the grasp of Plato's mind in his ability to conceive of
one science of solids in motion including the