MENELAUS OF ALEXANDRIA

flourished 1st century AD, , Alexandria Greek mathematician who first conceived and defined a spherical triangle (a triangle formed by three arcs of great circles on the surface of a sphere). Menelaus' most important work is Sphaerica, on the geometry of the sphere. In Book I he established the basis for a mathematical treatment of spherical triangles analogous to Euclid's treatment of plane triangles. Furthermore, he originated the use of arcs of great circles instead of arcs of parallel circles on the sphere, a major turning point in the development of spherical trigonometry. Book II covers the application of spherical geometry and trigonometry to astronomical measurements and calculations. Book III, the last, concentrates on spherical trigonometry and introduces Menelaus' theorem. The form of this theorem for plane triangles, well known to his contemporaries, was expressed as follows: if the sides of a triangle are crossed by a straight line (one of the sides is extended beyond one of its vertices), then the product of three of the nonadjacent line segments thus formed is equal to the product of the three remaining line segments of the triangle. Menelaus modified this theorem and applied it to spherical triangles. This form of the theorem became of fundamental importance in spherical trigonometry and astronomy, and the theorem has since been known by his name. Other notable works have been attributed to him, including one (in six books) on chords in a circle and one (in three books) on elements of geometry, but his only extant work is Sphaerica.