all his innovations is one that seems
much less striking. It is the answer to the question, What is the
relation in bulk between a sphere and its circumscribing cylinder?
Archimedes finds that the ratio is simply two to three. We are not
informed as to how he reached his conclusion, but an obvious method
would be to immerse a ball in a cylindrical cup. The experiment is one
which any one can make for himself, with approximate accuracy, with the
aid of a tumbler and a solid rubber ball or a billiard-ball of just the
right size. Another geometrical problem which Archimedes solved was the
problem as to the size of a triangle which has equal area with a circle;
the answer being, a triangle having for its base the circumference of
the circle and for its altitude the radius. Archimedes solved also
the problem of the relation of the diameter of the circle to its
circumference; his answer being a close approximation to the familiar
3.1416, which every tyro in geometry will recall as the equivalent of
pi.

Numerous other of the studies of Archimedes having reference to conic
sections, properties of curves and spirals, and the like, are too
technical to be detailed here. The extent of his mathematical knowledge,
however, is suggested by the fact that he computed in great detail the
number of grains of sand that would be required to cover the sphere of
the sun's orbit, making certain hypothetical assumptions as to the size
of the earth and the distance of the sun for the purposes of argument.
Mathematicians find his computation peculiarly interesting because it
evidences a crude conception of the idea of logarithms. From our present
stand-point, the paper in which this calculation is contained has
considerable interest because of its assumptions as to celestial
mechanics. Thus Archimedes starts out with the preliminary assumption
that the circumference of the earth is less than three million stadia.
It must be understood that this assumption is purely for the sake of
argument. Archimedes expressly states that he takes this number because
it is "ten times as large as the earth has been supposed to be by
certain investigators." Here, perhaps, the reference is to Eratosthenes,
whose measurement of the earth we shall have occasion to revert to in a
moment. Continuing, Archimedes asserts that the sun is larger than the
earth, and the earth larger than the moon. In this assumption, he says,
he is following the opinion of the majority o