been gained by man through his study of these heavenly diagrams, marked
out by the sun and the moon, by the planets and the comets, upon the
tablets of the sky? Yet, without the ellipse, without the conic
sections of Plato and Apollonius, astronomy would have been to this day
a sealed science, and the labors of Hipparchus, Ptolemy, Tycho, and
Copernicus would have waited in vain for the genius of Kepler and of
Newton to educe divine order from the seeming chaos of motions.

But the obligations of man to the ellipse do not end here. The
eighteenth and nineteenth centuries also owe it a debt of gratitude.
Even where the knowledge of conic sections does not enter as a direct
component of that analytical power which was the glory of a Lagrange, a
Laplace, and a Gauss, and which is the glory of a Leverrier, a Peirce,
and their companions in science, it serves as a part of the necessary
scaffolding by which that skill is attained,--of the necessary
discipline by which their power was exercised and made available for
the solution of the great problems of astronomy, optics, and
thermotics, which have been solved in our century.

There is another curve, generated by a simple law from a circle, which
has played an important part at various epochs in the intellectual
history of our race. A spot on the tire of a wheel running on a
straight, level road, will describe in the air a series of peculiar
arches, called the cycloid. The law of its formation is simple; the law
of its curvature is also simple. The path in which the spot moves
curves exactly in proportion to its nearness to the lowest point of the
wheel. By the simplicity of its law, it ought, according to the canon,
to be a beautiful curve. Now, although artists have not shown any
admiration for the cycloid, as they have for the ellipse, yet the
mathematicians have gazed upon it with great eagerness, and found it