Our symbols are modern, + and - first appearing in a German work in 1489; = in Recorde's "Whetstone of Witte" in 1557; > and < in the works of Harriot (1560-1621); and x in a publication by Oughtred (1574-1660). The most noteworthy advance in geometry in modern times was made by the great French philosopher Descartes, who published a small work entitled "La Geometrie" in 1637. From this springs the modern analytic geometry, a subject that has revolutionized the methods of all mathematics. Most of the subsequent discoveries in mathematics have been in higher branches. To the great Swiss mathematician Euler (1707-1783) is due, however, one proposition that has found its way into elementary geometry, the one showing the relation between the number of edges, vertices, and faces of a polyhedron. There has of late arisen a modern elementary geometry devoted chiefly to special points and lines relating to the triangle and the circle, and many interesting propositions have been discovered. The subject is so extensive that it cannot find any place in our crowded curriculum, and must necessarily be left to the specialist.[22] Some idea of the nature of the work may be obtained from a mention of a few propositions: The medians of a triangle are concurrent in the centroid, or center of gravity of the triangle. The bisectors of the various interior and exterior angles of a triangle are concurrent by threes in the incenter or in one of the three excenters of the triangle. The common chord of two intersecting circles is a special case of their radical axis, and tangents to the circles from any point on the radical axis are equal. If _O_ is the orthocenter of the triangle _ABC_, and _X_, _Y_, _Z_ are the feet of the perpendiculars from _A_, _B_, _C_ respectively, and _P_, _Q_, _R_ are the mid-points of _a_, _b_, _c_ respectively, and _L_, _M_, _N_ are the mid-points of _OA_, _OB_, _OC_ respectively; then the points _L_, _M_, _N_; _P_, _Q_, _R_; _X_, _Y_, _Z_ all lie on a circle, the "nine points circle." In the teaching of geometry it adds a human interest to the subject to mention occasionally some of the historical facts connected with it. For this reason this brief sketch will be supplemented by many notes upon the various important propositions as they occur in the several books described in the later chapters of this work. FOOTNOTES: [16] It was published in German translation by A. Eisenlohr, "Ein mathematisches Handbuch der alten Aegypter," Leipzig, 1877, and in facsimile by the British Museum, under the title, "The Rhind Papyrus," in 1898. [17] Generally known as Rameses II. He reigned in Egypt about 1350 B.C.