HERO OF ALEXANDRIA), whose personal history, like that of Euclid, is practically unknown, and to whom it is difficult to assign a date even within a century. His most interesting contri bution to elementary geometry is the formula for the area of a triangle in terms of the sides, A`=-"Vs(s—a) (s—b) (s— c).
Possibly contemporary with Hero lived Menelaus, whose theorem, known in the Mid dle Ages as the Regula sex quantitatutn, has made his name well known. His most im portant discovery, however, was the projective property of the anharmonic ratio. By this time the age of discovery in geometry had passed in Greece, and the efforts of the Neopythagoreans at Alexandria were productive of little that is remembered. Pappus (c. 300 A.D.), an Alexan drian mathematician and geographer, may be called the last of the Greek geometricians who showed any originality. He suggested the the ory of involution of points, restated the pro jective property discovered by Menelaus, and discovered the theorem (which also bears the name of Guldin) concerning the volume gen erated by a plane figure revolving about an axis.
The Orientals contributed but little to ele mentary geometry, their interests being rather directed to algebra (q.y.) and trigonometry (q.v.), with astronomy as the leading applica tion for their advanced mathematics. Brahma gupta, a Hindu, born in 598, generalized the Hero formula, showing that the area of an in scribed quadrilateral is expressed by A=V(s—a)(s--b) (s—c) (s—d), but aside from problems in mensuration, geom etry played but little part in India. The Bagdad school of c. 800 was chiefly inter ested in geometry only as it concerned trig onometry, and its greatest contribution to the science Consisted in the preservation of the works of the Greeks. Euclid, for example, was first made known to Christian Europe in the Middle Ages through a translation from the Arabic possibly by Adelhard of Bath, c. 1120.
Among the first books on mathematics to be printed was Euclid's 'Elements' (1482), a fact which made this famous work again a standard. The appearance of this classic had the same effect as in the later centuries of Greek culture, to encourage commentators rather than investi gators. In the way of original work, only such
minor efforts as the study of stellar polygons and the geometry of a single opening of the compasses characterized the closing decades of the Middle Ages and the opening years of the Renaissance. Not until Kepler (q.v.) suggested the principle of continuity (1604), and Cava Beni set forth the method of indivisibles (1629: published in 1635), and Desaroues began the theory of modern geometry (1639), was there any material advance in the subject. When, however, this advance was undertaken it was so vigorous as to lead from elementary geometry to higher fields. In the latter part of the 19th century there was a renaissance of investigation in the elementary domain, leading to an inter esting but not very productive study of the geometry of the circle and the triangle, notably in the work of Lemoinc and Brocard. The 19th century also saw an exhaustive study of non Euclidean geometries (q.v.), those based on other postulates than those of Euclid. This ,study began with the works of Lobachevsky and Bolyai (qq.v.), and has led to very interesting results, hardly to be ranked, however, in the domain of elementary geometry.
Bibliography.— Allman,