Jean-Gaston Darboux

“ When the times are ripe, when the fundamental principles have been recognized and enunciated, nothing stops the march of ideas; the same discoveries, or discoveries almost equivalent, appear at nearly the same instant, and in places the most diverse. Without undertaking a discussion of this sort, which, besides, might appear useless or become irritating, it is, however, of importance to bring out a fundamental difference between the tendencies of the great geometers who, about 1830, gave to geometry a scope before unknown. ”

“ Poinsot, the creator of the theory of couples, developed, by a method purely geometric, 'that, said he, where one never for a moment loses from view the object of the research,' the theory of the rotation of a solid body that the researches of d'Alembert, Euler and Lagrange seemed to have exhausted: Chasles made a precious contribution to kinematic by his beautiful theorems on the displacement of a solid body, which have since been extended by other elegant methods to the case where the motion has divers degrees of freedom. ”

“ Life is short; geometers know and also practise the principle of least action. Despite these fears, which should discourage no one, it seems to us that under the first form given it by von Staudt, projective geometry must become the necessary companion of descriptive geometry, that it is called to renovate this geometry in its spirit, its procedures and its applications. ”

“ Preceded by innumerable researches on systems of straight lines, due to Poinsot, Moebius, Chasles, Dupin, Malus, Hamilton, Kummer, Transon, above all to Cayley, who first introduced the notion of the coordinates of the straight, researches originating perhaps in statics and kinematics, perhaps in geometrical optics, Pluecker's geometry of the straight line will always be regarded as the part of his work where are met the newest and most interesting ideas. ”

“ It is in its school we have learned; our successors must learn never to be blindly proud, of methods too general, to envisage the questions in themselves and to find, in the conditions particular to each problem, perhaps a direct way towards a solution, perhaps the means of applying in an appropriate manner the general procedures which every science should gather. ”

“ It is sometimes said that the principle of duality shows that the plane as well as the point may be considered as a space element. That is true; but in adding the straight line to the plane and point as possible space element, Pluecker was led to recognize that any curve, any surface, may also be considered as space element, and so was born a new geometry which already has inspired a great number of works, which will raise up still more in the future. ”

“ In geometry some simple methods of transformation of figures were already known; homology even had been employed in the plane, but without extending it to space, as did Poncelet, and especially without recognizing its power and fruitfulness. ”

“ The 'Elements' of Euclid, which have withstood the action of so many centuries, will have at least the honor before ending of arousing a long series of works admirably enchained which will contribute, in the most effective way, to the progress of mathematics, at the same time that they furnish to the philosophers the points of departure the most precise and the most solid for the study of the origin and of the formation of our cognitions. ”

“ The second characteristic which Chasles assigns to his system of geometry is the employment of imaginaries. Here, his method was really new and he illustrates it by examples of high interest. One will always admire the beautiful theories he has left us on homofocal surfaces of the second degree, where all the known properties and others new, as varied as elegant, flow from the general principle that they are inscribed in the same developable circumscribed to the circle at infinity. ”

“ Nothing bears less resemblance to a sphere than a straight line, and yet Lie had imagined a singular transformation which made a sphere correspond to a straight, and permitted, consequently, the connecting of every proposition relative to straights with a proposition relating to spheres and vice versa. ”

“ For the rest, if we were tempted to neglect it, it would soon find in the applications of mathematics, as it did once before, means to rise up again and develop itself anew. It is like the giant Anteus who recovered his strength in touching the earth. ”

“ A beautiful discovery, of which we shall speak further on, has already connected the geometry of spheres with that of straight lines and permits the introduction of the notion of coordinates of a sphere. ”

“ The properties of the polars of algebraic curves are developed by Pluecker and above all by Steiner. The study, already old, of curves of the third order is rejuvenated and enriched by a crowd of new elements. Steiner, the first, studies by pure geometry the double tangents of curves of the fourth order, and Hesse, after him, applies the methods of algebra to this beautiful question, as well as to that of points of inflection of curves of the third order. ”

“ The geometers are intent on discovering in every question some general principle, usually undemonstrable without the aid of analysis, in order to make flow from it without effort a crowd of particular consequences, solidly bound to one another and to the principle whence they are derived. ”

“ The advantages of what is sometimes called intrinsic geometry are united to those resulting from the use of the regular analysis. Besides, this analysis is by no means abandoned; the complications of calculation which it almost always carries with it, in its applications to the study of surfaces and rectilinear coordinates, usually disappear if one employs the notion on the invariants and the covariants of quadratic powers of differentials which we owe to the researches of Lipschitz and Christoffel, inspired by Riemann's studies on the non-Euclidean geometry. ”

“ A contemporary of Moebius and Pluecker, Magnus believed he had found the most general transformation which makes a point correspond to a point, but the researches of Cremona teach us that the transformation of Magnus is only the first term of a series of birational transformations which the great Italian geometer teaches us to determine methodically, at least for the figures of plane geometry. ”

“ The theory of ensembles has come to permit the following of these lines in their course on a given surface.
The solution of a problem relative to the representation of two surfaces one on the other has greatly increased the interest of discoveries of Jacobi and of Liouville relative to a particular class of surfaces of which the geodesic lines could be determined. The results concerning this particular case led to the examination of a new question: to investigate all the problems of the calculus of variations of which the solution is given by curves satisfying a given differential equation. ”

“ If we have not been able, as has been said, to determine the surfaces applicable on the sphere, other surfaces of the second degree have been attacked with more success, and, in particular, the paraboloid of revolution.
The systematic study of the deformation of general surfaces of the second degree is already entered upon; it is one of those which will give shortly the most important results.
The theory of infinitesimal deformation constitutes to-day one of the most finished chapters of geometry. ”

“ Otto Hesse, brilliant disciple of Jacobi, develops in an admirable manner that method of homogeneous coordinates to which Pluecker perhaps had not attached its full value. Boole discovers in the polars of Bobillier the first notion of a covariant; the theory of forms is created by the labors of Cayley, Sylvester, Hermite, Brioschi. Later Aronhold, Clebsch and Gordan and other geometers still living gave to it its final notation, established the fundamental theorem relative to the limitation of the number of covariant forms and so gave it all its amplitude. ”

“ Some, like Chasles and Steiner, who consecrated their entire life to research in pure geometry, opposed what they called synthesis to analysis and, adopting in the ensemble if not in detail the tendencies of Poncelet, proposed to constitute an independent doctrine, rival of Descartes' analysis. ”

“ The notion of class introduced by Gergonne, the study of a paradox in part elucidated, by Poncelet and relative to the respective degrees of two curves reciprocal polars one of the other, give birth to the researches of Pluecker relative to the singularities called ordinary of algebraic plane curves. The celebrated formulas to which Pluecker is thus conducted are later extended by Cayley and by other geometers to algebraic skew curves, by Cayley again and by Salmon to algebraic surfaces.
The singularities of higher order are in their turn taken up by the geometers ”

“ Some differential equations more or less complicated to integrate, some chapters to add to the integral calculus, and one seemed about to touch the very outmost bounds of science.
Laplace had achieved the explanation of the system of the world and laid the foundations of molecular physics. ”

“ The new ideas which came to them in the fecund period of youth and which they introduced into the common domain have given them all they could have expected; they have fulfilled their task and feel the need of turning their mental activity towards wholly new subjects. This need, as we recognize, manifested itself with particular force at the epoch of Lagrange. At this moment, in fact, the program of researches opened to geometers by the discovery of the infinitesimal calculus appeared very nearly finished up. ”

“ Moreover, beside the 'Géometrie descriptive' we must not forget to place that other master-piece, the 'Application de l'analyse a la geometrie'; nor should we forget that to Monge are due the notion of lines of curvature and the elegant integration of the differential equation of these lines for the case of the ellipsoid, which, it is said, Lagrange envied him. To be stressed is this character of unity of the work of Monge. ”

“ If it happens often that, employed without discernment, transformations multiply uselessly the number of theorems, it must be recognized that they often aid us to better understand the nature of the propositions even to which they have been applied. ”

“ However, they may be characterized by saying that they rest upon the introduction of those elementary geometric forms which Desargues had already considered, on the development he was able to give to Bobillier's theory of polars, and finally on the construction of curves and surfaces of higher degrees by the aid of sheaves or nets of curves of lower orders. In default of recent researches, analysis would suffice to show that the field thus embraced has just the extent of that into which the analysis of Descartes introduces us without effort. ”