CENTRES. A point such that every radius vec tor (see CO6RDINATES ) drawn from it to a point on the curve is matched by another vector of the same length in the opposite direction is called the centre of a curve. See also CIRCLE and the paragraph on Carrot are above.
'hen a plane fignre moves in any manner in its own plane, the instantaneous centre of rota tion is the intersection of two lines drawn through two points perpendicular to the direc tions in which the points are moving.
The number of kinds of curves that might be drawn is infinite. A large number are known by specific names, and are objects of great in terest on account of their beauty. their remark able properties, or their relation to physical problems. Among those discussed under sepa rate titles are the conic sections, cissoid, eon choid, lemniscate, cycloid, trochoid, witch, car dioid, cartesians. Cassinian ovals, caustic curve, tractrix, curve of pursuit, catenary. curves of circular functions (e.g. curves of sines), loga rithmic curves, and spirals.
Though the history of curves is inseparable from that of geometry, it may roughly be divided into four periods: (l ) The synthetical geometry of the Greeks, in which the conic sections (q.v.) play an important role; (2) the birth of analytic geometry, in which the synthetic geometry of Guldin, Kepler, and Roberval merged into the coordinate geometry of Descartes and Fermat; (3) the period 1650 to 1800, character ized by the application of the calculus to geome try and including the names of Newton, Leib nitz, the Bernoullis, Clairaut, Maclaurin, Euler, and Lagrange; (4) the nineteenth century, the renaissance of pure geometry, characterized by the descriptive geometry of Monge, the modern synthetic geometry of Poncelet, Steiner, von Staudt. Cremona, and Plticker. Descartes's con tributions were confined to plane curves, but led to the discovery of many general properties. The scientific foundations of the theory of plane curves may be ascribed to Euler (1748) and Cramer (1750). Euler distinguished algebraic from transcendental curves, and Cramer found ed the theory of singularities. Clairaut (1731)
attacked the problem of double curvature; Monge introduced the u-se of differential equations. IINIfibiu (1852) summed up the classification of the cubic curve, Zeuthen (1874) did the same for the quartics, and Bobillier (1827) first used trilinear coOrdinates (q.v.). In 1828 Pliicker published the first volume of his nalyt isch-gco etrische Ent wiekelungen, which introduced abridged notation and marked a new era in analytic geometry. To him is due (1833) the general treatment of foci, a complete classifica tion of cubits (1835), and his celebrated 'MY equations' (1842). Hesse (1844) gave a com plete theory of inflections, and introduced the so-called Hessian curve as the first instance of a covariant of a ternary form. To Chasles (q.v.) is due the method of characteristics developed by 1Talphen (1875) and Schubert (1879). and the general theory of correspondence (1864), completed by Caylcy (1866) and Brill (1873). Cayley's influence was also very great. He ad vanced the work of Pliicker, investigated bi tangents and osculating conies, extended the properties of cnvariants and invariants, as well as Salmon's theory of reciprocal surfaces and the theory of double curvature. Mention should also be made of the labors of Jean-Claude Bou quet (1819-85) and Charles - Auguste - Albert Briot (1817-82), two of Cauchy's most eminent pupils, whose labors in the field of geometry and theory of functions are well kno‘xn. Their Lccons de g('ometrie analytique (Paris, 1847) has been translated into English (Chicago, 1890) and forms an excellent introduction to the sub ject. Besides the works of those mentioned in connection with the development of curves, con sult, for theory: Salmon, Treatise on. the Higher Plane Curres (Dublin, 1852) ; Clebsch, Vorlesun gen fiber Geometric (Leipzig, vol. i., 1875-76; vol. ii., 1891) ; and for history, Merriman and Woodward, Higher Mathematics, chap. xi. (New York, 1896) ; Brocard, :Votes de bibliographic des courbcs geomOriques (Bar-le-Due, 1S97-99).