conic section, these omissions being first filled by Pappus in the third century A.D. It is shown that the normal makes equal angles with the focal radii to the point of contact, and that the latter have a constant sum for the ellipse, a constant difference for the hyperbola. This book, he says in the letter quoted above, "contains many curious theorems, most of them are pretty and new, useful for the synthesis of solid loci. . . . . . In the invention of these, I observed that Euclid had not treated synthetically the locus . . . but only a certain small portion of it, and that not happily, nor indeed was a complete treatise possible at all without my discoveries." These three books, which are indeed based largely on the earlier work of Euclid and others, contain most of the properties of conic sections discussed in modern text-books on analytic geometry. Book IV discusses the intersections of conics, treating tangency correctly as equivalent to two ordinary intersections. In Book V Apollonius even undertakes the difficult problem of determining the longest and shortest lines which can be drawn from a given point to a conic, identifying this with the problem of drawing normals from a given point. He succeeds in discovering the points for which two such normals coincide, i. e. what we call the centre of curvature. Book VI deals with equal and similar conics, reaching the problem of passing through a given cone a plane which shall cut out a given ellipse. Book VII deals with conjugate diameters and the complementary chords parallel to them. Book VIII is lost. On the whole, in this remarkable work of some 400 propositions he achieved nearly all the results which are included in our modern elementary analytic geometry, even approximating the introduction of a system of coördinates by his use of lines parallel to the principal axes. It is noteworthy that Fermat, one of the inventors of modern analytic geometry, was led to it by attempting to restore certain lost proofs of Apollonius on loci. Of his other mathematical writings little more than the titles are known. Among these are one on burning mirrors, one on stations and retrogressions of the planets, and one on the use and theory of the screw. In astronomy he is believed to have sug gested expressing the motions of the planets by combining uniform circular motions, an idea afterwards elaborated by Hipparchus and Ptolemy. How far his mathematical results were new, how far he merely compiled and coördinated the work of others, notably Euclid and Archimedes, cannot be precisely determined, but the proportion of original work is certainly very large. On the arithmetical side he obtained a closer approximation than Archimedes for the value of π, invented an abridged method of multiplication, and employed numbers of higher order in the manner of Archimedes. This last experiment if followed out to its logical conclusions might have had fundamental significance for the future development of computation. In the words of Gow: he, as well as Archimedes, lost the chance of giving to the world once for all its numerical signs. That honor was reserved by the irony of fate for a nameless Indian of an unknown time, and we know not whom to thank for an invention which has been as important as any to the general progress of intelligence. APOLLONIUS AND ARCHIMEDES. - With Apollonius and Archimedes the ancient mathematics had accomplished whatever was possible without the resources of analytic geometry and infinitesimal calculus, which, though already foreshadowed, were not fully realized until the seventeenth century. It is not only a decided preference for synthesis and a complete denial of general methods which characterize the ancient mathematics as against our newer science (modern mathematics): besides this external formal difference there is another real, more deeply seated, contrast, which arises from the different attitudes which the two assumed relative to the use of the concept of variability. For while the ancients, on account of considerations which had been transmitted to them from the philosophic school of the Eleatics, never employed the concept of motion, the spatial expression for variability, in their rigorous system, and made incidental use of it only in the treatment of phoronomically generated curves, modern geometry dates from the instant that Descartes left the purely algebraic treatment of equations and proceeded to investigate the variations which an algebraic expression undergoes when one of its variables assumes a continuous succession of values. Hankel. - In one of the most brilliant passages of his Aperçu historique Chasles remarks that, while Archimedes and Apollonius were the most able geometricians of the old world, their works are distinguished by a contrast which runs through the whole subsequent history of geometry. Archimedes, in attacking the problem of the quadrature of curvilinear areas, established the principles of the geometry which rests on measurements; this naturally gave rise to the infinitesimal calculus, and in fact the method of exhaustions as used by Archimedes does not differ in principle from the method of limits as used by Newton. Apollonius, on the other hand, in investigating the properties of conic sections by means of transversals involving the ratio of rectilineal distances and of perspective, laid the foundations of the geometry of form and position. - Ball. The works of Archimedes and Apollonius marked the most brilliant epoch of ancient geometry. They may be regarded, moreover, as the origin and foundation of two questions which have occupied geometers at all periods. The greater part of their works are connected with these and are divided by them into two classes, so that they seem to share between them the domain of geometry. The first of these two great questions is the quadrature of curvilinear figures, which gave birth to the calculus of the infinite, conceived and brought to perfection successively by Kepler, Cavalieri, Fermat, Leibnitz and Newton. The second is the theory of conic sections, for which were invented first the geometrical analysis of the ancients, afterwards the methods of perspective and of transversals. This was the prelude to the theory of geometrical curves of all degrees, and to that considerable portion of geometry which considers, in the general properties of extension, only the forms and situations of figures, and uses only the intersection of lines or surfaces and the ratios of rectilineal distances. These two great divisions of geometry, which have each its peculiar character, may be designated by the names of Geometry of Measurements and Geometry of Forms and Situations, or Geometry of Archimedes and Geometry of Apollonius. - Chasles (Gow). MEDICAL SCIENCE AT ALEXANDRIA. BEGINNINGS OF HUMAN ANATOMY.- Alexandria is famous in the history of medicine for many reasons. It was here that human, -as contrasted with comparative, anatomy was first freely studied (probably favored by the Egyptian practice of disemboweling and embalming the dead) with the result that many of the grotesque errors of the earlier Greeks, including even Aristotle, were corrected. In this connection two names, and those of rivals, have come down to us as of chief importance, Herophilus and Erasistratus. The former, himself a student at Cos, was a close follower of the teachings of Hippocrates and regarded by the ancient world as his worthy successor. Erasistratus, on the contrary, opposed the Hippocratic doctrines. Both became distinguished anatomists. It is believed that the valves of the heart were first recognized and named by Erasistratus, who also studied and described the divisions, cavities and membranes of the brain, as well as the true origin and nature of the nerves. Herophilus likewise studied the brain, the pulmonary artery and the liver, besides giving to the duodenum the name (twelve-inch) which it still bears. Physiology, meanwhile, made little or no progress, and Cicero, two centuries later, still speaks of the arteries as "air tubes." It appears also that vivisection as well as anatomy was practised at Alexandria, and probably even upon human beings. Pergamum, in Asia Minor, was for a time a rival centre of medical learning and medical education, but was eventually overshadowed by the more famous Alexandrian school. Of this last the most celebrated pupil was Galen (born 130 A.D.), the most noted medical man of the ancient Roman world. Galen was a native of Pergamum who, having first studied at home and at Smyrna, spent some years at Alexandria. He then returned to Pergamum, but soon went to Rome, where he became physician to the Emperor Commodus. Galen was an original and voluminous writer on anatomy. That his name is still constantly linked with that of Hippocrates is probably the best evidence of his importance in the history of medical science. |