But I must not attempt to give you a lecture on the _Elements_ of Euclid, of which some of you may have evil recollections. For it is the misfortune, as well as the glory, of a great work not only to be repeated for centuries, but to be parroted and travestied by those who merely accept its greatness from the voice of ages, and who come to think that the words of inspiration only require blind repetition to instruct men. So if Euclid has become in many classical schools a sort of amulet or fetish (which must for common decency be put in the programme but which may be learned by committing the proofs to memory without any intelligence) such a misfortune is not the fault of Euclid, but the most pathetic tribute to his genius. Let me also add, for the benefit of those of you who have never seen more than six books of the _Elements_, and who probably thought six more than enough, that these are but the introduction to the discussion of higher and more complex questions, which show the large advance made by the Greeks in this science, and which explain also how in other arts, such as architecture, there is no defect for want of scientific accuracy. Books VII-X are not on geometry, but on higher arithmetic, and even treat, as in Book X, of incommensurable or irrational quantities. With XI he begins to teach solid geometry, the measurement of pyramids, cones, spheres, and the like, ending (XIII) with the discussion of the five regular polyhedra, of which Plato had long since spoken. From the great sequence of discoverers and teachers of pure mathematics, I need only here pick out three immortal names: Apollonius of Perga, living about 200 B.C., whose geometrical treatment of conic sections is, I am informed, a splendid monument of genius, which would still be the basis of modern study had not the treatment of these figures by analysis entirely superseded the geometrical method. Then there is Pappus in the second century A.D., who gives us in eight books a review of all the previous masters, with important additions of his own. The third name is Diophantus, who lived much later, perhaps in the fourth century, and whose work is considered the first great step toward the science of Algebra. All these speculations were developed in the direction of mathematical physics by Archimedes, Heron, and other great men of the Alexandrian school. The triumphs of Archimedes in mechanics astonished the Romans, who, in the defence of Syracuse against their attack, found him equal to a host. But how little Archimedes confined himself to practical problems is shown by his famous method of determining the area of a circle by approximation, by inscribing and circumscribing polygons of a great number of sides, which can of course be treated and measured as a complex of triangles. This is still, I am told, the proof admitted by modern mathematicians as the best.