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knowledge. His proof of the principal theorem in this is obtained by the “method of exhaustion,” which had been invented by Eudoxus but was greatly developed by Archimedes. This method contains in itself the germ of that most powerful instrument of mathematical analysis in the modern time, the calculus.

      Another very important work was “The Sphere and the Cylinder.” This was more appreciated in his own time, and as a consequence, after his death the figure of a sphere inscribed in a cylinder was cut on his tomb in commemoration of his favorite theorem, that the volume of the sphere is two-thirds that of the cylinder and its surface is four times that of the base of the cylinder. It was by searching for this symbol, famous in antiquity, that Cicero was enabled to find his tomb according to the story that I have already related.

      Within the last few years the reputation of Archimedes in pure mathematics has been greatly enhanced by the discovery by Professor Heiberg of a lost work of the great Alexandrian professor in Constantinople. Archimedes himself stated in a dedication of the work to Eratosthenes the method employed in this. He says: “I have thought it well to analyze and lay down for you in this same book a peculiar method by means of which it will be possible for you to derive instruction as to how certain mathematical questions may be investigated by means of mechanics. And I am convinced that this is equally profitable in demonstrating a proposition itself, for much that was made evident to me through the medium of mechanics was later proved by means of geometry, because the treatment by the former method had not yet been established by way of a demonstration. For of course it is easier to establish a proof, if one has in this way previously obtained a conception of the questions, than for him to seek it without such a preliminary notion. . . . Indeed, I assume that some one among the investigators of to-day or in the future, will discover by the method here set forth still other propositions which have not yet occurred to me.” On this Professor Smith comments: “Perhaps in all the history of mathematics no such prophetic truth was ever put into words. It would almost seem as if Archimedes must have seen as in a vision the methods of Galileo, Cavalieri, Pascal, Newton, and many other great makers of the mathematics of the Renaissance and the present time.”

      Many other distinguished professors of mathematics have, since this declaration of Archimedes came under their notice, declared that he must have had almost a prophetic vision of certain developments of mathematics and especially applied mathematics and mechanics and their relation to one another, that were only to come in much later and indeed comparatively modern times. Undoubtedly Archimedes’ works proved the germ of magnificent development not only immediately after his own time but in the long-after time of the Renaissance, when their translation awakened minds to mathematical problems and their solutions that would not otherwise have come.

      We know much less of the life of the third of the great trio of teachers and students of Alexandria, Apollonius of Perga. Perhaps it should be enough for us to know that his contemporaries spoke of him as “the great geometer,” though they were familiar with Euclid’s book and with Archimedes’ mighty work. Apollonius was surely a student of Alexandria for many years and he was probably also a professor of mathematics there. He developed especially what we know now as conic sections. His book on the subject contains practically all of the theorems to be found in our text-books of analytical geometry or conic sections of the present time. It was developed with rigorous mathematical logic and Euclidean conclusiveness. These three men show us beyond all doubt how finely the mathematical side of the university developed.

      After Archimedes the greatest mechanical genius of the University of Alexandria was Heron. To him we owe a series of inventions and discoveries in hydrostatics and the construction of various mechanical toys that have been used in the laboratories since. There is even a little engine run by steam—the aeolipile—invented by him, which shows how close the old Greeks were to the underlying principles of discoveries that were destined to come only after the development of industries created a demand for them in the after time. Heron’s engine is a globe of copper mounted on pivots, containing water, which on being heated produces steam that finds its way out through tubes bent so as to open in opposite directions on each side of the globe. The impact of the escaping steam on the air sets the globe revolving, and the principle of the turbine engine at work is clear. We have used steam for nearly 200 years always with a reciprocating type of movement, so that to apply energy in one direction the engine has had to move its parts backwards and forwards, but here was a direct-motion turbine engine in the long ago. Our great steamboats, the Lusitania and the Mauretania, now cross the ocean by the use of this principle and not by the reciprocating engine, and it is evident that it is along these lines the future developments of the application of steam are to take place.

      Another extremely interesting invention made by Heron is the famous fountain called by his name, and which still is used to illustrate principles in pneumatics in our classrooms and laboratories. By means of condensed air water is made to spring from a jet in a continuous stream and seems paradoxically to rise higher than its source. Probably his best work in the domain of physics is that on pneumatics in which are given not only a series of discussions, but of experiments and demonstrations on the elasticity of air and of steam. These experiments could only have been conducted in what we now call a physical laboratory. Indeed these inventions of his are still used in laboratories for demonstration purposes. While we may think, then, that the foundation of laboratories was reserved to our day, there is abundant evidence for their existence at the University of Alexandria. We shall return to this subject a little later, when the evidence from other departments has been presented, and then it will be clear, I think, that the laboratory methods were favorite modes of teaching at the University of Alexandria and were in use in nearly all departments of science both for research and for demonstration purposes.

      The work of the other great teacher at Alexandria which was to influence mankind next to that of Euclid, was not destined to withstand the critical study of succeeding generations, though it served for some 1,500 years as the basis of their thinking in astronomy. This was the work of Ptolemy, the great professor of astronomy at Alexandria of the first century after Christ. It is easy for us now to see the absurdity of Ptolemy’s system. It is even hard for us to understand how men could have accepted it. It must not be forgotten, however, that it solved all the astronomical problems of fifteen centuries and that it even enabled men, by its application, to foretell events in the heavens, and scientific prophecy is sometimes claimed to be the highest test of the truth of a system of scientific thought. Even so late as 1620 Francis Bacon refused to accept Copernicanism, already before the world for more than a century, because it did not, as it seemed to him, solve all the difficulties, while Ptolemy’s system did. As great an astronomer as Tycho Brahe living in the century after Copernicus still clung to Ptolemy’s teaching. It must not be forgotten that when Galileo restated Copernicanism, the reason for the rejection of his teaching by all the astronomers of Europe almost without exception, was that his reasons were not conclusive. They preferred to hold on to the old which had been so satisfying than to accept the new which seemed dubious. Their wisdom in this will be best appreciated from the fact that none of Galileo’s reasons maintained themselves.

      Though his system has been rejected, still Ptolemy must be looked up to as one of the great teachers of mankind and his work the “Almagest” as one of the great contributions to human knowledge. The fact that he represented a climax of astronomical development at Alexandria some four centuries after the foundation of that university, serves to show how much that first modern university occupied itself for all the centuries of its highest prestige, with physical science as well as with mathematics. Astronomy, physics, especially hydrostatics and mechanics, were all wonderfully developed. Generations of professors had given themselves to research and to the publication of important works quite as in the modern time, and Alexandria may well claim the right to be placed beside any university for what it accomplished in physical science, and rank high if not highest in the list of great research institutions adding new knowledge to old, leading men across the borderland of the unknown in science and furnishing that precious incentive to growing youth to occupy itself with the scientific problems of the world around it.

      The most important part of the scientific work of the University of Alexandria to my mind remains to be spoken of, and that is the medical department. It is a well-known law in the history of medicine that, whenever medical schools are attached to universities in such a way that

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