he years, months, and days between eclipses.
Consequently there must have been a mode of registering numbers;
probably even a system of numerals. The earliest numerical records, if
we may judge by the practices of the less civilised races now existing,
were probably kept by notches cut on sticks, or strokes marked on walls;
much as public-house scores are kept now. And there seems reason to
believe that the first numerals used were simply groups of straight
strokes, as some of the still-extant Roman ones are; leading us to
suspect that these groups of strokes were used to represent groups of
fingers, as the groups of fingers had been used to represent groups of
objects--a supposition quite in conformity with the aboriginal system of
picture writing and its subsequent modifications. Be this so or not,
however, it is manifest that before the Chaldeans discovered their
_Saros_, there must have been both a set of written symbols serving for
an extensive numeration, and a familiarity with the simpler rules of
arithmetic.

Not only must abstract mathematics have made some progress, but concrete
mathematics also. It is scarcely possible that the buildings belonging
to this era should have been laid out and erected without any knowledge
of geometry. At any rate, there must have existed that elementary
geometry which deals with direct measurement--with the apposition of
lines; and it seems that only after the discovery of those simple
proceedings, by which right angles are drawn, and relative positions
fixed, could so regular an architecture be executed. In the case of the
other division of concrete mathematics--mechanics, we have definite
evidence of progress. We know that the lever and the inclined plane were
employed during this period: implying that there was a qualitative
prevision of their effects, though not a quantitative one. But we know
more. We read of weights in the earliest records; and we find weights in
ruins of the highest antiquity. Weights imply scales, of which we have
also mention; and scales involve the primary theorem of mechanics in its
least complicated form--involve not a qualitative but a quantitative
prevision of mechanical effects. And here we may notice how mechanics,
in common with the other exact sciences, took its rise from the simplest
application of the idea of _equality_. For the mechanical proposition
which the scales involve, is, that if a lever with _equal_ arms, have
_equal_ weights suspend