A History of Mathematics/Recent Times/Synthetic Geometry
SYNTHETIC GEOMETRY.
The conflict between geometry and analysis which arose near the close of the last century and the beginning of the present has now come to an end. Neither side has come out victorious. The greatest strength is found to lie, not in the suppression of either, but in the friendly rivalry between the two, and in the stimulating influence of the one upon the other. Lagrange prided himself that in his Mecanique Analytique he had succeeded in avoiding all figures; but since his time mechanics has received much help from geometry.
Modern synthetic geometry was created by several investigators about the same time. It seemed to be the outgrowth of a desire for general methods which should serve as threads of Ariadne to guide the student through the labyrinth of theorems, corollaries, porisms, and problems. Synthetic geometry was first cultivated by Monge, Carnot, and Poncelet in France; it then bore rich fruits at the hands of Möbius and Steiner in Germany and Switzerland, and was finally developed to still higher perfection by Chasles in France, von Staudt in Germany, and Cremona in Italy.
Augustus Ferdinand Möbius (1790–1868) was a native of Schulpforta in Prussia. He studied at Göttingen under Gauss, also at Leipzig and Halle. In Leipzig he became, in 1815, privat-docent, the next year extraordinary professor of astronomy, and in 1844 ordinary professor. This position he held till his death. The most important of his researches are on geometry. They appeared in Crelle's Journal, and in his celebrated work entitled Der Barycentrische Calcul, Leipzig, 1827. As the name indicates, this calculus is based upon properties of the centre of gravity.[58] Thus, that the point is the centre of gravity of weights placed at the points respectively, is expressed by the equation
.
His calculus is the beginning of a quadruple algebra, and contains the germs of Grassmann's marvellous system. In designating segments of lines we find throughout this work for the first time consistency in the distinction of positive and negative by the order of letters , . Similarly for triangles and tetrahedra. The remark that it is always possible to give three points such weights that any fourth point in their plane will become a centre of mass, led Möbius to a new system of co-ordinates in which the position of a point was indicated by an equation, and that of a line by co-ordinates. By this algorithm he found by algebra many geometric theorems expressing mainly invariantal properties,—for example, the theorems on the anharmonic relation. Möbius wrote also on statics and astronomy. He generalised spherical trigonometry by letting the sides or angles of triangles exceed 180°.
Jacob Steiner (1796–1863), "the greatest geometrician since the time of Euclid," was born in Utzendorf in the Canton of Bern. He did not learn to write till he was fourteen. At eighteen he became a pupil of Pestalozzi. Later he studied at Heidelberg and Berlin. When Crelle started, in 1826, the celebrated mathematical journal bearing his name, Steiner and Abel became leading contributors. In 1832 Steiner published his Systematische Entwickelung der Abhängigkeit geometrischer Gestalten von einander, "in which is uncovered the organism by which the most diverse phenomena (Erscheinungen) in the world of space are united to each other." Through the influence of Jacobi and others, the chair of geometry was founded for him at Berlin in 1834. This position he occupied until his death, which occurred after years of bad health. In his Systematische Entwickelungen, for the first time, is the principle of duality introduced at the outset. This book and von Staudt's lay the foundation on which synthetic geometry in its present form rests. Not only did he fairly complete the theory of curves and surfaces of the second degree, but he made great advances in the theory of those of higher degrees. In his hands synthetic geometry made prodigious progress. New discoveries followed each other so rapidly that he often did not take time to record their demonstrations. In an article in Crelle's Journal on Allgemeine Eigenschaften Algebraischer Curven he gives without proof theorems which were declared by Hesse to be "like Fermat's theorems, riddles to the present and future generations." Analytical proofs of some of them have been given since by others, but Cremona finally proved them all by a synthetic method. Steiner discovered synthetically the two prominent properties of a surface of the third order; viz. that it contains twenty-seven straight lines and a pentahedron which has the double points for its vertices and the lines of the Hessian of the given surface for its edges.[55] The first property was discovered analytically somewhat earlier in England by Cayley and Salmon, and the second by Sylvester. Steiner's work on this subject was the starting-point of important researches by H. Schröter, F. August, L. Cremona, and R. Sturm. Steiner made investigations by synthetic methods on maxima and minima, and arrived at the solution of problems which at that time altogether surpassed the analytic power of the calculus of variations. He generalised the hexagrammum mysticum and also Malfatti's problem.[59] Malfatti, in 1803, proposed the problem, to cut three cylindrical holes out of a three-sided prism in such a way that the cylinders and the prism have the same altitude and that the volume of the cylinders be a maximum. This problem was reduced to another, now generally known as Malfatti's problem: to inscribe three circles in a triangle that each circle will be tangent to two sides of a triangle and to the other two circles. Malfatti gave an analytical solution, but Steiner gave without proof a construction, remarked that there were thirty-two solutions, generalised the problem by replacing the three lines by three circles, and solved the analogous problem for three dimensions. This general problem was solved analytically by C. H. Schellbach (1809–1892) and Cayley, and by Clebsch with the aid of the addition theorem of elliptic functions.[60]
Steiner's researches are confined to synthetic geometry. He hated analysis as thoroughly as Lagrange disliked geometry. Steiner's Gesammelte Werke were published in Berlin in 1881 and 1882.
Michel Chasles (1793–1880) was born at Epernon, entered the Polytechnic School of Paris in 1812, engaged afterwards in business, which he later gave up that he might devote all his time to scientific pursuits. In 1841 he became professor of geodesy and mechanics at the Polytechnic School; later, "Professeur de Géométrie supérieure à la Faculté des Sciences de Paris." He was a voluminous writer on geometrical subjects. In 1837 he published his admirable Aperçu historique sur l'origine et le développement des méthodes en géométrie, containing a history of geometry and, as an appendix, a treatise "sur deux principes généraux de la Science." The Aperçu historique is still a standard historical work; the appendix contains the general theory of Homography (Collineation) and of duality (Reciprocity). The name duality is due to Joseph Diaz Gergonne (1771–1859). Chasles introduced the term anharmonic ratio, corresponding to the German Doppelverhältniss and to Clifford's cross-ratio. Chasles and Steiner elaborated independently the modern synthetic or projective geometry. Numerous original memoirs of Chasles were published later in the Journal de l'École Polytechnique. He gave a reduction of cubics, different from Newton's in this, that the five curves from which all others can be projected are symmetrical with respect to a centre. In 1864 he began the publication, in the Comptes rendus, of articles in which he solves by his "method of characteristics" and the "principle of correspondence" an immense number of problems. He determined, for instance, the number of intersections of two curves in a plane. The method of characteristics contains the basis of enumerative geometry. The application of the principle of correspondence was extended by Cayley, A. Brill, H. G. Zeuthen, H. A. Schwarz, G. H. Halphen (1844–1889), and others. The full value of these principles of Chasles was not brought out until the appearance, in 1879, of the Kalkül der Abzählenden Geometrie by Hermann Schubert of Hamburg. This work contains a masterly discussion of the problem of enumerative geometry, viz. to determine how many geometric figures of given definition satisfy a sufficient number of conditions. Schubert extended his enumerative geometry to -dimensional space.[55]
To Chasles we owe the introduction into projective geometry of non-projective properties of figures by means of the infinitely distant imaginary sphero-circle.[61] Remarkable is his complete solution, in 1846, by synthetic geometry, of the difficult question of the attraction of an ellipsoid on an external point. This was accomplished analytically by Poisson in 1835. The labours of Chasles and Steiner raised synthetic geometry to an honoured and respected position by the side of analysis.
Karl Georg Christian von Staudt (1798–1867) was born in Rothenburg on the Tauber, and, at his death, was professor in Erlangen. His great works are the Geometrie der Lage, Nürnberg, 1847, and his Beiträge zur Geometrie der Lage, 1856–1860. The author cut loose from algebraic formulæ and from metrical relations, particularly the anharmonic ratio of Steiner and Chasles, and then created a geometry of position, which is a complete science in itself, independent of all measurements. He shows that projective properties of figures have no dependence whatever on measurements, and can be established without any mention of them. In his theory of what he calls "Würfe," he even gives a geometrical definition of a number in its relation to geometry as determining the position of a point. The Beiträge contains the first complete and general theory of imaginary points, lines, and planes in projective geometry. Representation of an imaginary point is sought in the combination of an involution with a determinate direction, both on the real line through the point. While purely projective, von Staudt's method is intimately related to the problem of representing by actual points and lines the imaginaries of analytical geometry. This was systematically undertaken by C. F. Maximilien Marie, who worked, however, on entirely different lines. An independent attempt has been made recently (1893) by F. H. Loud of Colorado College. Von Staudt's geometry of position. was for a long time disregarded, mainly, no doubt, because his book is extremely condensed. An impulse to the study of this subject was given by Culmann, who rests his graphical statics upon the work of von Staudt. An interpreter of von Staudt was at last found in Theodor Reye of Strassburg, who wrote a Geometrie der Lage in 1868.
Synthetic geometry has been studied with much success by Luigi Cremona, professor in the University of Rome. In his Introduzione ad una teoria geometrica delle curve piane he developed by a uniform method many new results and proved synthetically all important results reached before that time by analysis. His writings have been translated into German by M. Curtze, professor at the gymnasium in Thorn. The theory of the transformation of curves and of the correspondence of points on curves was extended by him to three dimensions. Ruled surfaces, surfaces of the second order, space-curves of the third order, and the general theory of surfaces have received much attention at his hands.
Karl Culmann, professor at the Polytechnicum in Zürich, published an epoch-making work on Die graphische Statik, Zürich, 1864, which has rendered graphical statics a great rival of analytical statics. Before Culmann, B. E. Cousinery had turned his attention to the graphical calculus, but he made use of perspective, and not of modern geometry.[62] Culmann is the first to undertake to present the graphical calculus as a symmetrical whole, holding the same relation to the new geometry that analytical mechanics does to higher analysis. He makes use of the polar theory of reciprocal figures as expressing the relation between the force and the funicular polygons. He deduces this relation without leaving the plane of the two figures. But if the polygons be regarded as projections of lines in space, these lines may be treated as reciprocal elements of a "Nullsystem." This was done by Clerk Maxwell in 1864, and elaborated further by Cremona.[63] The graphical calculus has been applied by 0. Mohr of Dresden to the elastic line for continuous spans. Henry T. Eddy, of the Rose Polytechnic Institute, gives graphical solutions of problems on the maximum stresses in bridges under concentrated loads, with aid of what he calls "reaction polygons." A standard work, La Statique graphique, 1874, was issued by Maurice Levy of Paris.
Descriptive geometry (reduced to a science by Monge in France, and elaborated further by his successors, Hachette, Dupin, Olivier, J, de la Gournerie) was soon studied also in other countries. The French directed their attention mainly to the theory of surfaces and their curvature; the Germans and Swiss, through Schreiber, Pohlke, Schlessinger, and particularly Fiedler, interwove projective and descriptive geometry. Bellavitis in Italy worked along the same line. The theory of shades and shadows was first investigated by the French writers just quoted, and in Germany treated most exhaustively by Burmester.[62]
During the present century very remarkable generalisations have been made, which reach to the very root of two of the oldest branches of mathematics,—elementary algebra and geometry. In algebra the laws of operation have been extended; in geometry the axioms have been searched to the bottom, and the conclusion has been reached that the space defined by Euclid's axioms is not the only possible non-contradictory space. Euclid proved (I. 27) that "if a straight line falling on two other straight lines make the alternate angles equal to one another, the two straight lines shall be parallel to one another." Being unable to prove that in every other case the two lines are not parallel, he assumed this to be true in what is generally called the 12th "axiom," by some the 11th "axiom." But this so-called axiom is far from axiomatic. After centuries of desperate but fruitless attempts to prove Euclid's assumption, the bold idea dawned upon the minds of several mathematicians that a geometry might be built up without assuming the parallel-axiom. While Legendre still endeavoured to establish the axiom by rigid proof, Lobatchewsky brought out a publication which assumed the contradictory of that axiom, and which was the first of a series of articles destined to clear up obscurities in the fundamental concepts, and to greatly extend the field of geometry.
Nicholaus Ivanovitch Lobatchewsky (1793–1856) was born at Makarief, in Nischni-Nowgorod, Russia, studied at Kasan, and from 1827 to 1846 was professor and rector of the University of Kasan. His views on the foundation of geometry were first made public in a discourse before the physical and mathematical faculty at Kasan, and first printed in the Kasan Messenger for 1829, and then in the Gelehrte Schriften der Universität Kasan, 1836–1838, under the title, "New Elements of Geometry, with a complete theory of Parallels." Being in the Russian language, the work remained unknown to foreigners, but even at home it attracted no notice. In 1840 he published a brief statement of his researches in Berlin. Lobatchewsky constructed an "imaginary geometry," as he called it, which has been described by Clifford as "quite simple, merely Euclid without the vicious assumption." A remarkable part of this geometry is this, that through a point an indefinite number of lines can be drawn in a plane, none of which cut a given line in the same plane. A similar system of geometry was deduced independently by the Bolyais in Hungary, who called it "absolute geometry."
Wolfgang Bolyai de Bolya (1775–1856) was born in Szekler-Land, Transylvania. After studying at Jena, he went to Göttingen, where he became intimate with Gauss, then nineteen years old. Gauss used to say that Bolyai was the only man who fully understood his views on the metaphysics of mathematics. Bolyai became professor at the Reformed College of Maros-Vásárhely, where for forty-seven years he had for his pupils most of the present professors of Transylvania. The first publications of this remarkable genius were dramas and poetry. Clad in old-time planter's garb, he was truly original in his private life as well as in his mode of thinking. He was extremely modest. No monument, said he, should stand over his grave, only an apple-tree, in memory of the three apples; the two of Eve and Paris, which made hell out of earth, and that of Newton, which elevated the earth again into the circle of heavenly bodies.[64] His son, Johann Bolyai (1802–1860), was educated for the army, and distinguished himself as a profound mathematician, an impassioned violin-player, and an expert fencer. He once accepted the challenge of thirteen officers on condition that after each duel he might play a piece on his violin, and he vanquished them all.
The chief mathematical work of Wolfgang Bolyai appeared in two volumes, 1832–1833, entitled Tentamen juventutem studiosam in elementa matheseos puræ…introducendi. It is followed by an appendix composed by his son Johann on The Science Absolute of Space. Its twenty-six pages make the name of Johann Bolyai immortal. He published nothing else, but he left behind one thousand pages of manuscript which have never been read by a competent mathematician! His father seems to have been the only person in Hungary who really appreciated the merits of his son's work. For thirty-five years this appendix, as also Lobatchewsky's researches, remained in almost entire oblivion. Finally Richard Baltzer of the University of Giessen, in 1867, called attention to the wonderful researches. Johann Bolyai's Science Absolute of Space and Lobatchewsky's Geometrical Researches on the Theory of Parallels (1840) were rendered easily accessible to American readers by translations into English made in 1891 by George Bruce Halsted of the University of Texas.
The Russian and Hungarian mathematicians were not the only ones to whom pangeometry suggested itself. A copy of the Tentamen reached Gauss, the elder Bolyai's former room-mate at Göttingen, and this Nestor of German mathematicians was surprised to discover in it worked out what he himself had begun long before, only to leave it after him in his papers. As early as 1792 he had started on researches of that character. His letters show that in 1799 he was trying to prove a priori the reality of Euclid's system; but some time within the next thirty years he arrived at the conclusion reached by Lobatchewsky and Bolyai. In 1829 he wrote to Bessel, stating that his "conviction that we cannot found geometry completely a priori has become, if possible, still firmer," and that "if number is merely a product of our mind, space has also a reality beyond our mind of which we cannot fully foreordain the laws a priori." The term non-Euclidean geometry is due to Gauss. It has recently been brought to notice that Geronimo Saccheri, a Jesuit father of Milan, in 1733 anticipated Lobatchewsky's doctrine of the parallel angle. Moreover, G. B. Halsted has pointed out that in 1766 Lambert wrote a paper "Zur Theorie der Parallellinien," published in the Leipziger Magazin für reine und angewandte Mathematik, 1786, in which: (1) The failure of the parallel-axiom in surface-spherics gives a geometry with angle-sum > 2 right angles; (2) In order to make intuitive a geometry with angle-sum < 2 right angles we need the aid of an "imaginary sphere" (pseudo-sphere); (3) In a space with the angle-sum differing from 2 right angles, there is an absolute measure (Bolyai's natural unit for length).
In 1854, nearly twenty years later, Gauss heard from his pupil, Riemann, a marvellous dissertation carrying the discussion one step further by developing the notion of -ply extended magnitude, and the measure-relations of which a manifoldness of dimensions is capable, on the assumption that every line may be measured by every other. Riemann applied his ideas to space. He taught us to distinguish between "unboundedness" and "infinite extent." According to him we have in our mind a more general notion of space, i.e. a notion of non-Euclidean space; but we learn by experience that our physical space is, if not exactly, at least to high degree of approximation, Euclidean space. Riemann's profound dissertation was not published until 1867, when it appeared in the Göttingen Abhandlungen Before this the idea of dimensions had suggested itself under various aspects to Lagrange, Plücker, and H. Grassmann. About the same time with Riemann's paper, others were published from the pens of Helmholtz and Beltrami. These contributed powerfully to the victory of logic over excessive empiricism. This period marks the beginning of lively discussions upon this subject. Some writers—Bellavitis, for example—were able to see in non-Euclidean geometry and -dimensional space nothing but huge caricatures, or diseased outgrowths of mathematics. Helmholtz's article was entitled Thatsachen, welche der Geometrie zu Grunde liegen, 1868, and contained many of the ideas of Riemann. Helmholtz popularised the subject in lectures, and in articles for various magazines.
Eugenio Beltrami, born at Cremona, Italy, in 1835, and now professor at Rome, wrote the classical paper Saggio di interpretazione della geometria non-euclidea (Giorn. di Matem., 6), which is analytical (and, like several other papers, should be mentioned elsewhere were we to adhere to a strict separation between synthesis and analysis). He reached the brilliant and surprising conclusion that the theorems of non-Euclidean geometry find their realisation upon surfaces of constant negative curvature. He studied, also, surfaces of constant positive curvature, and ended with the interesting theorem that the space of constant positive curvature is contained in the space of constant negative curvature. These researches of Beltrami, Helmholtz, and Riemann culminated in the conclusion that on surfaces of constant curvature we may have three geometries,—the non-Euclidean on a surface of constant negative curvature, the spherical on a surface of constant positive curvature, and the Euclidean geometry on a surface of zero curvature. The three geometries do not contradict each other, but are members of a system,—a geometrical trinity. The ideas of hyper-space were brilliantly expounded and popularised in England by Clifford.
William Kingdon Clifford (1845–1879) was born at Exeter, educated at Trinity College, Cambridge, and from 1871 until his death professor of applied mathematics in University College, London. His premature death left incomplete several brilliant researches which he had entered upon. Among these are his paper On Classification of Loci and his Theory of Graphs. He wrote articles On the Canonical Form and Dissection of a Riemann's Surface, on Biquaternions, and an incomplete work on the Elements of Dynamic. The theory of polars of curves and surfaces was generalised by him and by Reye. His classification of loci, 1878, being a general study of curves, was an introduction to the study of -dimensional space in a direction mainly projective. This study has been continued since chiefly by G. Veronese of Padua, C. Segre of Turin, E. Bertini, F. Aschieri, P. Del Pezzo of Naples.
Beltrami's researches on non-Euclidean geometry were followed, in 1871, by important investigations of Felix Klein, resting upon Cayley's Sixth Memoir on Quantics, 1859. The question whether it is not possible to so express the metrical properties of figures that they will not vary by projection (or linear transformation) had been solved for special projections by Chasles, Poncelet, and E. Laguerre (1834–1886) of Paris, but it remained for Cayley to give a general solution by defining the distance between two points as an arbitrary constant multiplied by the logarithm of the anharmonic ratio in which the line joining the two points is divided by the fundamental quadric. Enlarging upon this notion, Klein showed the independence of projective geometry from the parallel-axiom, and by properly choosing the law of the measurement of distance deduced from projective geometry the spherical, Euclidean, and pseudospherical geometries, named by him respectively the elliptic, parabolic, and hyperbolic geometries. This suggestive investigation was followed up by numerous writers, particularly by G. Battaglini of Naples, E. d' Ovidio of Turin, R. de Paolis of Pisa, F. Aschieri, A. Cayley, F. Lindemann of Munich, E. Schering of Göttingen, W. Story of Clark University, H. Stahl of Tübingen, A. Voss of Würzburg, Homersham Cox, A. Buchheim.[55] The geometry of dimensions was studied along a line mainly metrical by a host of writers, among whom may be mentioned Simon Newcomb of the Johns Hopkins University, L. Schläfli of Bern, W. I. Stringham of the University of California, W. Killing of Münster, T. Craig of the Johns Hopkins, R. Lipschitz of Bonn. R. S. Heath and Killing investigated the kinematics and mechanics of such a space. Regular solids in -dimensional space were studied by Stringham, Ellery W. Davis of the University of Nebraska, R. Hoppe of Berlin, and others. Stringham gave pictures of projections upon our space of regular solids in four dimensions, and Schlegel at Hagen constructed models of such projections. These are among the most curious of a series of models published by L. Brill in Darmstadt. It has been pointed out that if a fourth dimension existed, certain motions could take place which we hold to be impossible. Thus Newcomb showed the possibility of turning a closed material shell inside out by simple flexure without either stretching or tearing; Klein pointed out that knots could not be tied; Veronese showed that a body could be removed from a closed room without breaking the walls; C. S. Peirce proved that a body in four-fold space either rotates about two axes at once, or cannot rotate without losing one of its dimensions.