# A History of Mathematics/Recent Times/Analytic Geometry

ANALYTIC GEOMETRY.
In the preceding chapter we endeavoured to give a flash-light view of the rapid advance of synthetic geometry. In connection with hyperspace we also mentioned analytical treatises. Modern synthetic and modern analytical geometry have much in common, and may be grouped together under the common name "projective geometry." Each has advantages over the other. The continual direct viewing of figures as existing in space adds exceptional charm to the study of the former, but the latter has the advantage in this, that a well-established routine in a certain degree may outrun thought itself, and thereby aid original research. While in Germany Steiner and von Staudt developed synthetic geometry, Plücker laid the foundation of modern analytic geometry.
Julius Plücker (1801–1868) was born at Elberfeld, in Prussia. After studying at Bonn, Berlin, and Heidelberg, he spent a short time in Paris attending lectures of Monge and his pupils. Between 1826 and 1836 he held positions successively at Bonn, Berlin, and Halle. He then became professor of ​physics at Bonn. Until 1846 his original researches were on geometry. In 1828 and in 1831 he published his Analytisch-Geometrische Untersuchungen in two volumes. Therein he adopted the abbreviated notation (used before him in a more restricted way by Bobillier), and avoided the tedious process of algebraic elimination by a geometric consideration. In the second volume the principle of duality is formulated analytically. With him duality and homogeneity found expression already in his system of co-ordinates. The homogenous or tri-linear system used by him is much the same as the co-ordinates of Möbius. In the identity of analytical operation and geometric construction Plücker looked for the source of his proofs. The System der Analytischen Geometrie, 1835, contains a complete classification of plane curves of the third order, based on the nature of the points at infinity. The Theorie der Algebraischen Curven, 1839, contains, besides an enumeration of curves of the fourth order, the analytic relations between the ordinary singularities of plane curves known as "Plücker's equations," by which he was able to explain "Poncelet's paradox." The discovery of these relations is, says Cayley, "the most important one beyond all comparison in the entire subject of modern geometry." But in Germany Plücker's researches met with no favour. His method was declared to be unproductive as compared with the synthetic method of Steiner and Poncelet! His relations with Jacobi were not altogether friendly. Steiner once declared that he would stop writing for Crelle's Journal if Plücker continued to contribute to it. The result was that many of Plücker's researches were published in foreign journals, and that his work came to be better known in France and England than in his native country. The charge was also brought against Plücker that, though occupying the chair of physics, he was no physicist. This induced him to ​relinquish mathematics, and for nearly twenty years to devote his energies to physics. Important discoveries on Fresnel's wave-surface, magnetism, spectrum-analysis were made by him. But towards the close of his life he returned to his first love,—mathematics,—and enriched it with new discoveries. By considering space as made up of lines he created a "new geometry of space." Regarding a right line as a curve involving four arbitrary parameters, one has the whole system of lines in space. By connecting them by a single relation, he got a "complex" of lines; by connecting them with a twofold relation, he got a "congruency" of lines. His first researches on this subject were laid before the Royal Society in 1865. His further investigations thereon appeared in 1868 in a posthumous work entitled Neue Geometrie des Raumes gegründet auf die Betractung der geraden Linie als Raumelement, edited by Felix Klein. Plücker's analysis lacks the elegance found in Lagrange, Jacobi, Hesse, and Clebsch. For many years he had not kept up with the progress of geometry, so that many investigations in his last work had already received more general treatment on the part of others. The work contained, nevertheless, much that was fresh and original. The theory of complexes of the second degree, left unfinished by Plücker, was continued by Felix Klein, who greatly extended and supplemented the ideas of his master.
Ludwig Otto Hesse (1811–1874) was born at Königsberg, and studied at the university of his native place under Bessel, Jacobi, Richelot, and F. Neumann. Having taken the doctor's degree in 1840, he became docent at Königsberg, and in 1845 extraordinary professor there. Among his pupils at that time were Durège, Carl Neumann, Clebsch, Kirchhoff. The Königsberg period was one of great activity for Hesse. Every new discovery increased his zeal for still greater achievement. His earliest researches were on surfaces of the second order, ​and were partly synthetic. He solved the problem to construct any tenth point of such a surface when nine points are given. The analogous problem for a conic had been solved by Pascal by means of the hexagram. A difficult problem confronting mathematicians of this time was that of elimination. Plücker had seen that the main advantage of his special method in analytic geometry lay in the avoidance of algebraic elimination. Hesse, however, showed how by determinants to make algebraic elimination easy. In his earlier results he was anticipated by Sylvester, who published his dialytic method of elimination in 1840. These advances in algebra Hesse applied to the analytic study of curves of the third order. By linear substitutions, he reduced a form of the third degree in three variables to one of only four terms, and was led to an important determinant involving the second differential coefficient of a form of the third degree, called the "Hessian." The "Hessian" plays a leading part in the theory of invariants, a subject first studied by Cayley. Hesse showed that his determinant gives for every curve another curve, such that the double points of the first are points on the second, or "Hessian." Similarly for surfaces (Crelle, 1844). Many of the most important theorems on curves of the third order are due to Hesse. He determined the curve of the 14th order, which passes through the 56 points of contact of the 28 bi-tangents of a curve of the fourth order. His great memoir on this subject (Crelle, 1855) was published at the same time as was a paper by Steiner treating of the same subject.
Hesse's income at Königsberg had not kept pace with his growing reputation. Hardly was he able to support himself and family. In 1855 he accepted a more lucrative position at Halle, and in 1856 one at Heidelberg. Here he remained until 1868, when he accepted a position at a technic school in Munich. At Heidelberg he revised and enlarged upon his ​previous researches, and published in 1861 his Vorlesungen über die Analytische Geometrie des Raumes, insbesondere über Flächen 2. Ordnung. More elementary works soon followed. While in Heidelberg he elaborated a principle, his "Uebertragungsprincip." According to this, there corresponds to every point in a plane a pair of points in a line, and the projective geometry of the plane can be carried back to the geometry of points in a line.
The researches of Plücker and Hesse were continued in England by Cayley, Salmon, and Sylvester. It may be premised here that among the early writers on analytical geometry in England was James Booth (1806–1878), whose chief results are embodied in his Treatise on Some New Geometrical Methods; and James MacCullagh (1809–1846), who was professor of natural philosophy at Dublin, and made some valuable discoveries on the theory of quadrics. The influence of these men on the progress of geometry was insignificant, for the interchange of scientific results between different nations was not so complete at that time as might have been desired. In further illustration of this, we mention that Chasles in France elaborated subjects which had previously been disposed of by Steiner in Germany, and Steiner published researches which had been given by Cayley, Sylvester, and Salmon nearly five years earlier. Cayley and Salmon in 1849 determined the straight lines in a cubic surface, and studied its principal properties, while Sylvester in 1851 discovered the pentahedron of such a surface. Cayley extended Plücker's equations to curves of higher singularities. Cayley's own investigations, and those of M. Nöther of Erlangen, G. H. Halphen (1844–1889) of the Polytechnic School in Paris, De La Gournérie of Paris, A. Brill of Tübingen, lead to the conclusion that each higher singularity of a curve is equivalent to a certain number of simple singularities,—the node, the ordinary cusp, the double tangent, ​and the inflection. Sylvester studied the "twisted Cartesian," a curve of the fourth order. Salmon helped powerfully towards the spreading of a knowledge of the new algebraic and geometric methods by the publication of an excellent series of text-books (Conic Sections, Modern Higher Algebra, Higher Plane Curves, Geometry of Three Dimensions), which have been placed within easy reach of German readers by a free translation, with additions, made by Wilhelm Fiedler of the Polytechnicum in Zürich. The next great worker in the field of analytic geometry was Clebsch.
Rudolf Friedrich Alfred Clebsch (1833–1872) was born at Königsberg in Prussia, studied at the university of that place under Hesse, Richelot, F. Neumann. From 1858 to 1863 he held the chair of theoretical mechanics at the Polytechnicum in Carlsruhe. The study of Salmon's works led him into algebra and geometry. In 1863 he accepted a position at the University of Giesen, where he worked in conjunction with Paul Gordan (now of Erlangen). In 1868 Clebsch went to Göttingen, and remained there until his death. He worked successively at the following subjects: Mathematical physics, the calculus of variations and partial differential equations of the first order, the general theory of curves and surfaces, Abelian functions and their use in geometry, the theory of invariants, and "Flächenabbildung." He proved theorems on the pentahedron enunciated by Sylvester and Steiner; he made systematic use of "deficiency" (Geschlecht) as a fundamental principle in the classification of algebraic curves. The notion of deficiency was known before him to Abel and Riemann. At the beginning of his career, Clebsch had shown how elliptic functions could be advantageously applied to Malfatti's problem. The idea involved therein, viz. the use of higher transcendentals in the study of geometry, led him to his greatest discoveries. Not only did he apply Abelian ​functions to geometry, but conversely, he drew geometry into the service of Abelian functions.
Clebsch made liberal use of determinants. His study of curves and surfaces began with the determination of the points of contact of lines which meet a surface in four consecutive points. Salmon had proved that these points lie on the intersection of the surface with a derived surface of the degree $\scriptstyle {11n-24}$, but his solution was given in inconvenient form. Clebsch's investigation thereon is a most beautiful piece of analysis.
The representation of one surface upon another (Flächenabbildung), so that they have a (1,1) correspondence, was thoroughly studied for the first time by Clebsch. The representation of a sphere on a plane is an old problem which drew the attention of Ptolemæus, Gerard Mercator, Lambert, Gauss, Lagrange. Its importance in the construction of maps is obvious. Gauss was the first to represent a surface upon another with a view of more easily arriving at its properties. Plücker, Chasles, Cayley, thus represented on a plane the geometry of quadric surfaces; Clebsch and Cremona, that of cubic surfaces. Other surfaces have been studied in the same way by recent writers, particularly M. Nöther of Erlangen, Armenante, Felix Klein, Korndörfer, Caporali, H. G. Zeuthen of Copenhagen. A fundamental question which has as yet received only a partial answer is this: What surfaces can be represented by a (1,1) correspondence upon a given surface? This and the analogous question for curves was studied by Clebsch. Higher correspondences between surfaces have been investigated by Cayley and Nöther. The theory of surfaces has been studied also by Joseph Alfred Serret (1819–1885), professor at the Sorbonne in Paris, Jean Gaston Darboux of Paris, John Casey of Dublin (died 1891), W, R. W. Roberts of Dublin, H. Schröter (1829–1892) of Breslau. Surfaces of the ​fourth order were investigated by Kummer, and Fresnel's wave-surface, studied by Hamilton, is a particular case of Kummer's quartic surface, with sixteen canonical points and sixteen singular tangent planes.
The infinitesimal calculus was first applied to the determination of the measure of curvature of surfaces by Lagrange, Euler, and Meunier (1754–1793) of Paris. Then followed the researches of Monge and Dupin, but they were eclipsed by the work of Gauss, who disposed of this difficult subject in a way that opened new vistas to geometricians. His treatment is embodied in the Disquisitiones generales circa superficies curvas (1827) and Untersuchungen über gegenstände der höheren Geodäsie of 1843 and 1846. He defined the measure of curvature at a point to be the reciprocal of the product of the two principal radii of curvature at that point. From this flows the theorem of Johann August Grunert (1797–1872; professor in Greifswald), that the arithmetical mean of the radii of curvature of all normal sections through a point is the radius of a sphere which has the same measure of curvature as has the surface at that point. Gauss's deduction of the formula of curvature was simplified through the use of determinants by Heinrich Richard Baltzer (1818–1887) of Giessen. Gauss obtained an interesting theorem that if one surface be developed (abgewickelt) upon another, the measure of curvature remains unaltered at each point. The question whether two surfaces having the same curvature in corresponding points can be unwound, one upon the other, was answered by F. Minding in the affirmative only when the curvature is constant. The case of variable curvature is difficult, and was studied by Minding, J. Liouville (1806–1882) of the Polytechnic School in Paris, Ossian Bonnet of Paris (died 1892). Gauss's measure of curvature, expressed as a function of curvilinear co-ordinates, gave an impetus to the study of ​differential-invariants, or differential-parameters, which have been investigated by Jacobi, C. Neumann, Sir James Cockle, Halphen, and elaborated into a general theory by Beltrami, S. Lie, and others. Beltrami showed also the connection between the measure of curvature and the geometric axioms.
Various researches have been brought under the head of "analysis situs." The subject was first investigated by Leibniz, and was later treated by Gauss, whose theory of knots (Verschlingungen) has been employed recently by J. B. Listing, 0. Simony, F. Dingeldey, and others in their "topologic studies." Tait was led to the study of knots by Sir William Thomson's theory of vortex atoms. In the hands of Riemann the analysis situs had for its object the determination of what remains unchanged under transformations brought about by a combination of infinitesimal distortions. In continuation of his work, Walter Dyck of Munich wrote on the analysis situs of three-dimensional spaces.
Of geometrical text-books not yet mentioned, reference should be made to Alfred Clebsch's Vorlesungen über Geometrie, edited by Ferdinand Lindemann, now of Munich; Frost's Solid Geometry; Durege's Ebene Curven dritter Ordnung.