as been valued more than any single
monument of antiquity; it is still a text-book, in various English
translations, in all our schools. Euclid also wrote various other works,
showing great mathematical talent.

Perhaps a greater even than Euclid was Archimedes, born 287 B.C. He
wrote on the sphere and cylinder, terminating in the discovery that the
solidity and surface of a sphere are two thirds respectively of the
solidity and surface of the circumscribing cylinder. He also wrote on
conoids and spheroids. "The properties of the spiral and the quadrature
of the parabola were added to ancient geometry by Archimedes, the last
being a great step in the progress of the science, since it was the
first curvilineal space legitimately squared." Modern mathematicians may
not have the patience to go through his investigations, since the
conclusions he arrived at may now be reached by shorter methods; but the
great conclusions of the old geometers were reached by only prodigious
mathematical power. Archimedes is popularly better known as the inventor
of engines of war and of various ingenious machines than as a
mathematician, great as were his attainments in this direction. His
theory of the lever was the foundation of statics till the discovery of
the composition of forces in the time of Newton, and no essential
addition was made to the principles of the equilibrium of fluids and
floating bodies till the time of Stevin, in 1608. Archimedes detected
the mixture of silver in a crown of gold which his patron, Hiero of
Syracuse, ordered to be made; and he invented a water-screw for pumping
water out of the hold of a great ship which he had built. He contrived
also the combination of pulleys, and he constructed an orrery to
represent the movement of the heavenly bodies. He had an extraordinary
inventive genius for discovering new provinces of inquiry and new points
of view for old and familiar objects. Like Newton, he had a habit of
abstraction from outward things, and would forget to take his meals. He
was killed by Roman soldiers when Syracuse was taken; and the Sicilians
so soon forgot his greatness that in the time of Cicero they did not
know where his tomb was.

Eratosthenes was another of the famous geometers of antiquity, and did
much to improve geometrical analysis. He was also a philosopher and
geographer. He gave a solution of the problem of the duplication of the
cube, and applied his geometrical knowledge to the measurem