BIBLIOGRAPHY 833 Numbers and its Recent Development,"Amer.Math.Soc.Bull.,29,1923, 445-63. Reichardt,Hans,ed.:C.F.Gauss,Leben und Werk,Haude und Spenersche Verlags buchhandlung,60,Pp.1;also B.G.Teubner, 1057 Scott,J.F.:AHistoryMathematics,Taylor and Francis,18,Chap.15. Smith,David E.:A Source Book in Mathematics,Dover (reprint),1959,Vol.1, 107-48. Smith,H.J.S.:Collected Mathematical Papers,2 vols.(1890-94),Chelsea (reprint),165.Vol.I contains Smith's Report on f Numbers, which is also published separately by Chelsea,1965. Vandiver,H.S.:"Fermat's Last Theorem,"Amer.Math.Monthly,53,1946,555-78
35 The Revival of Projective Geometry The doctrines of pure geometry often,and in many questions, give a simple and natural way to penetrate to the origin of truths,to lay bare the mysterious chain which unites them, and to make them known individually,luminously and completely. MICHEL CHASLES 1.The Renewal of Interest in Geometry For over one hundred years after the introduction of analytic geometry by Descartes and Fermat,algebraic and analytic methods dominated geometry almost to the exclusion of synthetic methods.During this period some men for example the English mathematicians who persisted in trying to found the calculus rigorously on geometry,produced new results synthetically. Geometric methods,elegant and intuitively clear,always captivated some minds.Maclaurin,especially,preferred synthetic geometry to analysis.Pure geometry,then,retained some life even ifit was not at the heart of the most vital developments of the seventeenth and eighteenth centuries.In the early nineteenth century several great mathematicians decided that synthetic geometry had been unfairly and unwisely neglected and made a positive effort to revive and extend that approach. One of the new champions of synthetic methods,Jean-Victor Poncelet did concede the limitations of the older pure geometry.He says,"While analytic geometry offers by its characteristic method general and uniform means of proceeding to the solution of questions which present themselves. while it arrives at results whosc gencrality is without bound,the other [synthetic geometry]proceeds by chance;its way depends completely on the sagacity of those who employ it and its results are almost always limited to the particular figure which one considers."But Poncelet did not believe that synthetic methods were necessarily so limited and he proposed to create new ones which would rival the power of analytic geometry. Michel Chasles(1793-1880)was another great proponent of geometrical 834
THE RENEWAL OF INTEREST IN GEOMETRY 835 methods.In his Apercu historique sur Porigine et le diveloppement des mithodes en geomdtrie(1837),a historical study in which Chasles admitted that he ignored the German writers because he did not know the language,he states that the mathematicians of his time and earlier had declared geometry a dead language which in the future would be of no use and influence.Not ony does Chasles deny this but he cites Lagrange,who was entirely an analyst,as aaoEemeiwamyneaR of celestial mechanics, old geometric methods,which one usually,but improperly,calls synthesis, there are nevertheless problems in which the latter appear more advanta geous,partly because of their intrinsic clarity and partly because of the elegance and ease of their solutions.There are even some problems for which the algebraic analysis in some measure does not suffice and which,it appears, the synthetic methods alone can master."'Lagrange cites as an example the very difficult problem of the attraction of an ellipsoid of revolution exerted on a point (unit mass)on its surface or inside.This problem had been solved purely Chasles also gives an extract from a letter he received from Lamb Adolphe Quetelet (1796-1874),the Belgian astronomer and statistician. Quetelet says,"It is not proper that most of our young mathematicians value pure geometry so lightly."The young men complain of lack of generality of method,continues Quetelet,but is this the fault of geometry or of those who cultivate geometry,he asks.To counter this lack of generality, Chasles gives two rules to prospective geometers.They should generalize special theorems to obtain the most general result,which should at the same time be simple and natural.Second,they should not be satisfied with the proof of a result if it is not part of a general method or doctrine on which it is naturally dependent.To know when one really has the true basis for a theorem,he says,there is always a principal truth which one will recognize because other theorems will result from a simple transformation or as a ready consequence.The great truths,which are the foundations ofknowledge, always have the characteristics of simplicity and intuitiveness Other mathematicians attacked analytic methods in harsher language. Carnot wished"to free geometry from the hieroglyphics of analysis."Later in the century Eduard Study (1862-1922)referred to the machine-like process of coordinate geometry as the"clatter of the coordinate mill." The objections to analytic methods in geometry were based on more than a personal preference or taste.There was,first of all,a genuine question of whether analytic geometry was really geometry since algebra was the essence of the method and results,and the geometric significance of both were hidden.Moreover,as Chasles pointed out,analysis through its formal 1.Nouv.Mim.de I'Acad.de Berlin,1773,121-48,pub.1775 =Euvres,3,617-58
836 THE REVIVAL OF PROIECTIVE GEOMETRY processes neglects all the small steps which geometry continually makes.The quick and perhaps penetrating steps of analysis do not reveal the sense of what is accomplished.The connection between the starting point and the final result is not clear.Chasles asks,"Is it then sufficient in a philosophic and basic study of a science to know that something is true if one does not know why it is so and what place it should take in the series of truths to which it belongs?"The geometric method,on the other hand,permits simple and intuitively evident proofs and conclusions There was another argument which,first voiced by Descartes,still appealed in the nineteenth century.Geometry was regarded as the truth about space and the real world.Algebra and analysis were not in themselves significant truths even about numbers and functions.They were merely methods of arriving at truths,and artificial at that.This view of algebra and analysis was gradually disappearing.Nevertheless the criticism was still vigorous in the early nineteenth century because the methods of analysis were incomplete and even logically unsound.The geometers rightly ques- tioned the validity of the analytic proofs and credited them with merely suggesting results.The analysts could retort only that the geometric proofs were clumsy and inelegant. The upshot of the controversy is that the pure geometers reasserted their role in mathematics.As if to revenge themselves on Descartes because his creation ofanalyticgeometry had caused the abandonment ofpure geometry, the early nineteenth-century geometers made it their objective to beat Descartes at the game of geometry.The rivalry between analysts and geometers grew so bitter that Steiner,who was a pure geometer,threatened to quit writing for Crelle's Journal fur Mathematik if Crelle continued to pub- lish the analytical papers of Plucker The stimulus to revive synthetic geometry came primarily from one man, Gaspard Monge.We have already discussed his valuable contributions to analytic and differential geometry and his inspiring lectures at the Ecole Polytechnique during the years 1795 to 1809.Monge himself did not intend to do more than bring geometry back into the fold of mathematics as a suggestive approach to and an interpretation of analytic results.He sought only to stress both modes of thought.However,his own work in geometry and his enthusiasm for it inspired in his pupils,Charles Dupin,Frangois- Toseph Servois,Charles-Tulien Brianchon,Jean-Baptiste Biot (1774-1862), Lazare-Nicholas-Marguerite Carnot,and Jean-Victor Poncelet,the urge to revitalize pure geometry. Monge's contribution to pure geometry was his Traite de gdomdtrie descriptive (1799).This subject shows how to project orthogonally a three- dimensional object on two planes (a horizontal and a vertical one)so that from this representation one can deduce mathematical properties of the object.The scheme is useful in architecture,the design of fortifications,per-
SYNTHETIC EUCLIDEAN GEOMETRY 837 spective,carpentry,and stonecutting and was the first to treat the projection of a three-dimensional figure into two two-dimensional ones.The ideas and method of descriptive geometry did not prove to be the avenue to subse- quent developments in geometry or,for that matter,to any other part of mathematics 2.Synthetic Euclidean Geometry Though the geometers who reacted to Monge's inspiration turned to pro- jective geometry,we shall pause to note some new results in synthetic Euclidean geometry.These results,perhaps minor in significance,never theless exhibit new themes and the almost inexhaustible richness of this old subject.Actually hundreds of new theorems were produced,of which we can give just a few examples. Associated with every triangle ABC are nine particular points,the midpoints of the sides,the feet of the three altitudes,and the midpoints of the segments which join the vertices to the point of intersection of the altitudes All nine points lie on one circle,called the nine-point circle.This theorem was first published by Gergonne and Poncelet.2 It is often credited to Karl Wilhelm Feuerbach (1800-34),a high-school teacher,who published his proof in Eigenschaften einiger merkwurdigen Punkte des geradlinigen Dreiecks (Properties of Some Distinctive Points of the Rectilinear Triangle,1822).In this book Feuerbach added another fact about the nine-point circle.An excircle (escribed circle)is one which is tangent to one of the sides and to the extensions of the other two sides.(The center of an escribed circle lies on the bisectors of two exterior angles and the remote interior angle.) Feuerbach's theorem states that the nine-point circle is tangent to the inscribed circle and the three excircles In a small book published in 1816,Uber einige Eigenschaften des ebenen Crle omepomeanle the lines joining P to the vertices of the triangle and the sides of the triangle make equal angles.That is,3 in Figure 35.1.There is also a point P'different from P such that寸P'AC=文P'CB=寸P'BA. The conic sections,we know,were treated definitively by Apollonius as sections of a cone and then introduced as plane loci in the seventeenth century.In 1822 Germinal Dandelin (1794-1847)proved a very interesting theorem about the conic sections in relation to the cone.3 His theorem states that if two spheres are inscribed in a circular cone so that they are tangent to a given plane cutting the cone in a conic section,the points of contact of the 2.Ann de Math,11,1820/2l,205-20. 3.Nouv Mim.de PAcad.Roy.des Sci.,Bruxelles,2,1822,169-202