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 measurement of the magnitude of the earth,—­being one of the first who brought mathematical methods to the aid of astronomy, which in our day is almost exclusively the province of the mathematician.

Apollonius of Perga, probably about forty years younger than Archimedes, and his equal in mathematical genius, was the most fertile and profound writer among the ancients who treated of geometry.  He was called the Great Geometer.  His most important work is a treatise on conic sections, which was regarded with unbounded admiration by contemporaries, and in some respects is unsurpassed by any thing produced by modern mathematicians.  He however made use of the labors of his predecessors, so that it is difficult to tell how far he is original.  But all men of science must necessarily be indebted to those who have preceded them.  Even Homer, in the field of poetry, made use of the bards who had sung for a thousand years before him; and in the realms of philosophy the great men of all ages have built up new systems on the foundations which others have established.  If Plato or Aristotle had been contemporaries with Thales, would they have matured so wonderful a system of dialectics?  Yet if Thales had been contemporaneous with Plato, he might have added to the great Athenian’s sublime science even more than did Aristotle.  So of the great mathematicians of antiquity; they were all wonderful men, and worthy to be classed with the Newtons and Keplers of our times.  Considering their means and the state of science, they made as great though not as fortunate discoveries,—­discoveries which show patience, genius, and power of calculation.  Apollonius was one of these,—­one of the master intellects of antiquity, like Euclid and Archimedes; one of the master intellects of all ages, like Newton himself.  I might mention the subjects of his various works, but they would not be understood except by those familiar with mathematics.