indeed, we may see exemplified the truth, which the subsequent
history of science frequently illustrates, that before any more
abstract division makes a further advance, some more concrete division
must suggest the necessity for that advance--must present the new order
of questions to be solved. Before astronomy presented Hipparchus with
the problem of solar tables, there was nothing to raise the question of
the relations between lines and angles; the subject-matter of
trigonometry had not been conceived. And as there must be subject-matter
before there can be investigation, it follows that the progress of the
concrete divisions is as necessary to that of the abstract, as the
progress of the abstract to that of the concrete.

Just incidentally noticing the circumstance that the epoch we are
describing witnessed the evolution of algebra, a comparatively abstract
division of mathematics, by the union of its less abstract divisions,
geometry and arithmetic--a fact proved by the earliest extant samples of
algebra, which are half algebraic, half geometric--we go on to observe
that during the era in which mathematics and astronomy were thus
advancing, rational mechanics made its second step; and something was
done towards giving a quantitative form to hydrostatics, optics, and
harmonics. In each case we shall see, as before, how the idea of
equality underlies all quantitative prevision; and in what simple forms
this idea is first applied.

As already shown, the first theorem established in mechanics was, that
equal weights suspended from a lever with equal arms would remain in
equilibrium. Archimedes discovered that a lever with unequal arms was in
equilibrium when one weight was to its arm as the other arm to its
weight; that is--when the numerical relation between one weight and its
arm was _equal_ to the numerical relation between the other arm and its
weight.

The first advance made in hydrostatics, which we also owe to Archimedes,
was the discovery that fluids press _equally_ in all directions; and
from this followed the solution of the problem of floating bodies:
namely, that they are in equilibrium when the upward and downward
pressures are _equal_.

In optics, again, the Greeks found that the angle of incidence is
_equal_ to the angle of reflection; and their knowledge reached no
further than to such simple deductions from this as their geometry
sufficed for. In harmonics they ascertained the fact that three stri