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 suspended from them, the weights will remain at equal altitudes.  And we may further notice how, in this first step of rational mechanics, we see illustrated that truth awhile since referred to, that as magnitudes of linear extension are the only ones of which the equality is exactly ascertainable, the equalities of other magnitudes have at the outset to be determined by means of them.  For the equality of the weights which balance each other in scales, wholly depends upon the equality of the arms:  we can know that the weights are equal only by proving that the arms are equal.  And when by this means we have obtained a system of weights,—­a set of equal units of force, then does a science of mechanics become possible.  Whence, indeed, it follows, that rational mechanics could not possibly have any other starting-point than the scales.