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notices of the early history of Greek mathematics. Proclus and Eutocius quote it frequently. Theodosius of Tripolis is the author of a book of little merit on the geometry of the sphere. Dionysodorus of Amisus in Pontus applied the intersection of a parabola and hyperbola to the solution of a problem which Archimedes, in his Sphere and Cylinder, had left incomplete. The problem is "to cut a sphere so that its segments shall be in a given ratio."

We have now sketched the progress of geometry down to the time of Christ. Unfortunately, very little is known of the history of geometry between the time of Apollonius and the beginning of the Christian era. The names of quite a number of geometers have been mentioned, but very few of their works are now extant. It is certain, however, that there were no mathematicians of real genius from Apollonius to Ptolemy, excepting Hipparchus and perhaps Heron.


The Second Alexandrian School.


The close of the dynasty of the Lagides which ruled Egypt from the time of Ptolemy Soter, the builder of Alexandria, for 300 years; the absorption of Egypt into the Roman Empire; the closer commercial relations between peoples of the East and of the West; the gradual decline of paganism and spread of Christianity,—these events were of far-reaching influence on the progress of the sciences, which then had their home in Alexandria. Alexandria became a commercial and intellectual emporium. Traders of all nations met in her busy streets, and in her magnificent Library, museums, lecture-halls, scholars from the East mingled with those of the West; Greeks began to study older literatures and to compare them with their own. In consequence of this interchange of ideas the Greek philosophy became fused with Oriental