838 THE REVIVAL OF PROJECTIVE GEOMETRY Figure 35.1 120° Figure 35.2 spheres with the plane are the foci of the conic section,and the intersections of the plane with the planes of the circles along which the spheres touch the cone are the directrices of the conic. Another interesting theme pursued in the nineteenth century was the solution of maximum and minimum problems by purely geometric methods, that is,without relying upon the calculus of variations.Of the several theorems Jacob Steiner proved by using synthetic methods,the most famous result is the isoperimetric theorem:Of all plane figures with a given perimeter the circle bounds the greatest area.Steiner gave various proofs. Unfortunately,Steiner assumed that there exists a curve that does have maximum area.Dirichlet tried several times to persuade him that his proofs were incomplete on that account but Steiner insisted that this was self-evident.Once,however,he did write(in the first of the 1842 papers): and the proof is readily made if one assumes that there is a largest figure." The proof of the existence of a maximizing curve baffled mathe- maticians for a number of years until Weierstrass in his lectures of the 1870s at Berlin resorted to the calculus of variations.Later Constantin Caratheo- dory (1873-1950)and Study7 in a joint paper rigorized Steiner's proofs without employing that calculus.Their proofs(there were two)were direct rather than indirect as in Steiner's method.Hermann Amandus Schwarz, who did great work in partial differential equations and analysis and who 4.Jour.fur Math.,18,1838,281-96;and24,1842,83-l62,189-250;the1842 papers are in his Ges.Werke,2,177-308. 5.Gs.Werk,2,197. 6.Wrk,7,257-64,301-2. 7.Math.Ann.,68,1909,133-40 Caratheodory,Ges.math.Schriften,2,3-11
SYNTHETIC EUCLIDEAN GEOMETRY 839 Figure 35.3 served as a professor at several universities including Gottingen and Berlin, gave a rigorous proof for the isoperimetric problem in three dimensions.8 Steiner also proved (in the first of the 1842 papers)that of all triangles with a given perimeter,the equilateral has the greatest area.Another of his results states that if A,B,and C are three given points (Fig.35.2)and if each of the angles of triangle ABCis less than 120,then the point P for which PA PB PC is a minimum is such that each of the angles at P is 120.If, however,one angle of the triangle,say angle A,is equal to or larger than 120,then P coincides with A.This result had been proven much earlier by Cavalieri (Exercitationes Geometricae Sex,1647)but it was undoubtedly unknown to Steiner.Steiner also extended the result to n points. Schwarz solved the following problem:Given an acute-angled triangle, consider all triangles such that each has its vertices on the three sides of the original triangle;the problem is to find the triangle that has least perimeter. Schwarz proved synthetically0 that the vertices of this triangle of minimum perimeter are the feet of the altitudes of the given triangle(Fig.35.3) A theorem novel for Euclidean geometry was discovered in 1899 by Frank Morley,professor of mathematics at Johns Hopkins University,and proofs were subsequently published by many men.12 The theorem states that if the angle trisectors are drawn at each vertex of a triangle,adjacent trisectors meet at the vertices of an equilateral triangle (Fig.35.4).The novelty lies in the involvement of angle trisectors.Up to the middle of the nineteenth century no mathematician would have considered such lines because only those elements and figures that are constructible were regarded as having legitimacy in Euclidean geometry.Constructibility guaranteed 8.Nachrichten Konig.Ges.der Wiss.zu Gott.,1884,1-13 =Ges.math.Abh.,2,327-40. 9.Monatsber.Berliner Akad,1837,144 Ges.Werke,2,93 and 729-31. 10.Paper unpublished,Ges.Math.Abh.,2,344-45. 11.Schwarz's proof can be found in Richard Courant and Herhert Robbins,What Is Mathematies?,Oxford University Press,1941,pp.346-49.A proof using the calculus was given by J.F.de'Toschi di Fagnano (1715-97)in the Acta Eruditorum,1775,p.297. There were less elegant geometrical proofs before Schwarz's. 12.For a proof and references to published proofs see H.S.M.Coxeter,Introduction to Geometry,John Wiley and Sons,1961,pp.23-25
840 THE REVIVAL OF PROJECTIVE GEOMETRY Figure 35.4 existence.However,the conception of what established existence changed as we shall see more clearly when we examine the work on the foundations of Euclidean geometry. A number of efforts were directed toward reducing the use of straight- edge and compass along the lines initiated by Mohr and Mascheroni(Chap. 12,sec.2).In his Traite of 1822 Poncelet showed that all constructions possible with straightedge and compass(except the construction of circular arcs)are possible with straightedge alone provided that we are given a fixed circle and its center.Steiner re-proved the same result more elegantly in asmall book,Die geometrischen Constructionen ausgefuhrt mittelst der geraden Linie und eines festen Kreises (The Geometrical Constructions Executed by Means of the Straight Line and a Fixed Circle).13 Though Steiner intended the book for pedagogical purposes,he does claim in the preface that he will establish the conjecture which a French mathematician had expressed. The briefsampling above of Euclidean theorems established by synthetic methods should not leave the reader with the impression that analytic geometric methods were not also used.In fact,Gergonne gave analytic proofs of many geometric theorems which he published in the journal he founded,the Annales de Mathematiques. 3.The Revival of Synthetic Projective Geometry The major area to which Monge and his pupils turned was projective oreehaemthe rise of analytic geometry,the calculus,and analysis.As we have already noted,Desargues's major work of 1639 was lost sight of until 1845 and Pascal's major essay on conics(1639)was never recovered.Only La Hire's books,which used some of Desargues's results,were available.What the nineteenth-century men learned from La Hire's books they often incorrectly credited to him.On the whole,however,these geometers were ignorant of Desargues's and Pascal's work and had to re-create it. 13.Published 1833=Werke,1,461-522
THE REVIVAL OF SYNTHETIC PROJECTIVE GEOMETRY 845 6 Figure 35.5 The revival of projcctive geometry was initiated by Lazare N.M.Carnot (1753-1823),a pupil of Monge and father of the distinguished physicist Sadi Carnot.His major work was Gometrie de position(1803)and he also contributed the Essai sur la theorie des transversales (1806).Monge had espoused the joint use of analysis and pure geometry,but Carnot refused to use analytic methods and started the championship of pure geometry.Many of the ideas we shall shortly discuss more fully are at least suggested in Carnot's work. Thus the principle that Monge called contingent relations and which became known also as the principle of correlativity and more commonly as the principle of continuity is to be found there.To avoid separate figures for various sizes of angles and directions of lines Carnot did not use negative numbers,which he regarded as conradictory,but introduced a complicated scheme called"correspondence of signs.' Among the early nineteenth-century workers in projective geometry we shall just mention Frangois-Joseph Servois and Charles-Julien Brianchon (1785-1864),both of whom made applications of their work to military problems.Though they aided in reconstructing,systematizing,and extending old results,the only new theorem of consequence is Brianchon's famous result,4 which he proved while still a student at the Ecole.The theorem states that if there are six tangents to a conic (Fig.35.5),thus forming a circumscribed hexagon,the three lines,each of which joins two opposite vertices,pass through one point.Brianchon derived this theorem by using the pole-polar relationship. The revival of projective geometry received its main impetus from Poncelet(1788-1867).Poncelet was a pupil of Monge and he also learned much from Carnot.While serving as an officer in Napoleon's campaign against Russia,he was captured and spent the year 1813-14 in a Russian prison at Saratoff.There Poncelet reconstructed without the aid of any books what he had learned from Monge and Carnot and then proceeded to create new results.He later expanded and revised this work and published 14.Jour de 'Ecole Po4,6,1806,297-311
842 THE REVIVAL OF PROJECTIVE GEOMETRY it as the Traite des propridtes projectives des figures (1822).This work was his chief contribution to projective geometry and to the erection of a new discipline.In his later life he was obliged to devote a great deal of time to government service,though he did hold professorships for limited periods. Poncelet became the most ardent champion of synthetic geometry and even attacked the analysts.He had been friendly with the analyst Joseph- Diez Gergonne (1771-1859)and had published papers in Gergonne's Annales de Mathdmatiques,but his attacks were soon directed to Gergonne also. Poncelet was convinced of the autonomy and importance of pure geometry. Though he admitted the power of analysis he believed that one could give the same power to synthetic geometry.In a paper of 1818,published in Gergonne'snalshe said that the power of analytic methods lay not in the use of algebra but in its generality and this advantage resulted from the fact that the metric properties discovered for a typical figure remain ap- plicable,other than for a possible change of sign,toall related figures which spring from the typical or basic one.This generality could be secured in synthetic geometry by the principle of continuity (which we shall examine shortly). Poncelet was the first mathematician to appreciate fully that projective geometry was a new branch of mathematics with methods and goals of its own.Whereas the seventeenth-century projective geometers had dealt with specific problems,Poncelet entertained the general problem of seeking all properties of geometrical figures that were common to all sections of any projection of a figure,that is,remain unaltered by projection and section. This is the theme that he and his successors took up.Because distances and angles are altered by projection and section Poncelet selected and developed the theory of involution and of harmonic sets of points but not the concept of cross ratio.Monge had used parallel projection in his work;like Desargues, Pascal,Newton,and Lambert,Poncelet used central projection,that is, projection from a point.This concept Poncelet elevated into a method of approach to geometric problems.Poncelet also considered projective transf ormation from one space figure to another,of course in purely geometric form.Here he seemed to have lost interest in projective properties and was more concerned with the use of the method in bas-relief and stage design His work centers about three ideas.The first is that of homologous figures.Two figures are homologous if one can be derived from the other by one projection and a section,which is called a perspectivity,or by a sequenc of projections and sections,that is,a projectivity.In working with homol- ogous figures his plan was to find for a given figure a simpler homologous figure and by studying it find properties which are invariant under projection 15.Ann.de Math.,8,1817/18,141-55.This paper is reprinted in Poncelet's Applications d'analyse et de gdomdtrie (1862-64),2,466-76