triangle the squares on the sides containing the right angle are together equal to the square on the opposite side of it.  Pythagoras discovered that of all figures having the same boundary, the circle among plane figures and the sphere among solids are the most capacious.  Hippocrates treated of the duplication of the cube, and wrote elements of geometry, and knew that the area of a circle was equal to a triangle whose base is equal to its circumference and altitude equal to its radius.  The disciples of Plato invented conic sections, and discovered the geometrical foci.

It was however reserved for Euclid to make his name almost synonymous with geometry.  He was born 323 B.C., and belonged to the Platonic sect, which ever attached great importance to mathematics.  His “Elements” are still in use, as nearly perfect as any human production can be.  They consist of thirteen books.  The first four are on plane geometry; the fifth is on the theory of proportion, and applies to magnitude in general; the seventh, eighth, and ninth are on arithmetic; the tenth on the arithmetical characteristics of the division of a straight line; the eleventh and twelfth on the elements of solid geometry; the thirteenth on the regular solids.  These “Elements” soon became the universal study of geometers throughout the civilized world; they were translated into the Arabic, and through the Arabians were made known to mediaeval Europe.  There can be no doubt that this work is one of the highest triumphs of human genius, and it has 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