III - REAL PROJECTIVE GEOMETRY: POLARITIES, CONICS AND QUADRICS

Summary

Two-dimensional projectivities. The history of conies begins about 430 B.C., when Hippocrates of Chios expressed the “duplication of the cube” as a problem which his followers could solve by means of intersecting curves. Some seventy years later, Menaechmus showed that these curves can be defined as sections of a right circular cone by a plane perpendicular to a generator. Their metrical properties (such as the theorem regarding the ratio of the distances to focus and directrix) were described in great detail by Aristaeus, Euclid, and Apollonius. Apollonius introduced the names ellipse, parabola, and hyperbola, and discovered the harmonic property of pole and polar. But the earliest genuinely non-metrical property is the theorem of Pascal (1623-1662), who obtained it at the age of sixteen. (See 3.35.) A hundred years later, Maclaurin used similar ideas in one of his constructions for the conic through five given points. The first systematic account of projective properties is due to Steiner (1796-1863). But his definition in terms of related pencils (3.34) lacks symmetry, as it specializes two points on the conic (the centres of the pencils); moreover, several steps have to be taken before the self-duality of a conic becomes apparent. Von Staudt (1798-1867) made the important discovery that the relation which a conic establishes between poles and polars is really more fundamental than the conic itself, and can be set up independently (§3.2).