CURTIUS, ERNST (korlse-os), a German archaeologist and historian; born in Liibeck, Sept. 2, 1814. His studies were all directed toward Grecian antiq uity, and he visited Greece repeatedly on scientific missions. "Peloponnesus" curve of double curvature or a skew, tuous or twisted curve. Ordinary curves can be defined as geometrical loci, by a prescribed kinematic movement of a point or a line, according to the methods of an alytic geometry, by an equation between co-ordinates, as the intersection of a plane by an irregular surface. The ellipse for example can be represented in all four of these methods: as the geometrical locus of all points for which the sum of the distances of two given points—the foci—is constant. Kinematically by an ellipsograph or oval; by an equation of the second rank, and by the section of a cone by a plane.
The consideration of curves as geomet rical loci is based on the principles of the geometry of Euclid and is the most an cient method of studying curves and dis covering new kinds. Far more fruitful and speedy in their results are the meth ods of analytical geometry, the science of which was established by Descartes in 1637, especially through the use of the differential and integral calculus. In this way the peculiarities of curves may be investigated on purely mathematical methods, and on the other hand the ana lytical geometry of the theory of func tions offers a means of establishing the functions as curves and thereby giving a clear image of their course. According to the nature of the equation on which they are based, curves are called alge braic, containing powers of x and y, or transcendental, where they involve loga rithms. Algebraic curves are distin guished according to the rank or order of the equation. Thus, we have curves of the 2d rank or conic sections, of the 3d rank or cubic curves, of which there are many varieties, including Newton's foli ate or 41st species, and the 4th rank or quartic, and so on. The analytic investi
gation of a curve is especially directed toward the characteristics of its tangents and normals, toward its point of oscula tion as well as toward its asymptotes and its peculiar points or singularities. Curves can be likewise defined according as one prescribes their tangents or nor mals or the characteristics of their cur vation from which the equation of the curve is deduced. A frequently recur rent condition of curves is that they are regarded as inclusive of their tangents whereby, for example, the caustic curves, the trajectories and tractories are found. Also through investigation of the nadir curves and the evolutes arise many forms of curves and relations among well known kinds. The number of points in which a curve of any order in general is drawn is called its rank; the number of tangents which in general may be drawn from any given point to a curve is called its class. Between rank, class, and the number of their distinguished points and tangents, double points, return points, double tangents, periodic tangents, come a series of continuously valid relations, the Pliicker's Formulas. For example, every curve of the 3d rank without double point is of the 6th class, with double point is of the 4th class, with return point of the 3d class. Besides the an alytical methods for the investigation of curves there are the synthetic methods devised especially by Poncelet, Steiner, and Staudt. Projection geometry has proved of great use in the investigation of cones. For description and illustra tion of the principal curves, see their respective titles.