# Gaston Darboux plenary talk ICM 1908

At the International Congress of Mathematicians held in Rome in 1908, Gaston Darboux was a plenary speaker, delivering the lecture

*The Origins, Methods and Problems of Infinitesimal Geometry*. He gave this lecture at 3:30 on Tuesday 7 April 1908, the session being chaired by Simon Newcomb. We give an English version of Darboux's lecture below.**The Origins, Methods and Problems of Infinitesimal Geometry by Gaston Darboux**

**0.**

**Introduction.**

Over the past thirty years, a profound change has taken place before our eyes in the direction of mathematical studies. The theory of functions, since it must be called by its name, attracts today, with irresistible force, the research of the youngest, the most active, the most inventive among us. A new spirit drives them: it has already earned us beautiful and deeply original discoveries, and promises us even more for the future. If we compare the beginnings of the 20th century with the beginnings of the one that has just ended, we are forced to recognise that there has been a radical transformation in the methods and theories over the past 100 years, in the points of view and in guiding principles. The 19th century left us with memorable writings and far-reaching works; but, at least in its first half, it was content to complete the task which had been bequeathed to it by the great scientific geniuses of the two centuries which had preceded it. On the contrary, what opens up before us offers truly new perspectives, presenting us with entirely unexplored fields of research. Nothing stops ardent and curious minds; they have no fear of attacking the very foundations of the building that so many works, so long pursued, had seemed to establish on unshakable foundations. And not content to cultivate our science, to pursue directions that they consider the best, they have the pretension, to which I applaud for my part, to make the most original and precise contributions to this essential branch of Philosophy which has as its own object the origin, nature and scope of our knowledge.

By entrusting a man who follows this movement with sympathy, but who has not taken part directly in its writings, to take care of one of the lectures in the programme, the organisers of our Congress, who he would like to thank warmly, have done him a great honour because he feels the full honour. But they also wanted to show, I imagine, that they do not intend to neglect any of the branches of mathematical knowledge. In science as elsewhere, perhaps more than elsewhere, because researchers are always passionate, there are fashions and practices which should not be abandoned without reserve. Despite the growing number of followers of mathematical science, the old disciplines run the risk of being somewhat neglected. It is characteristic of meetings like this to operate a task of coordination, to draw attention in particular to the questions which call for research, so that our science works, with an equal and firm step, without interruptions or jolts, towards its essential goal, which is the search for the truth.

Those who are called upon to hold your attention for some time probably have the first duty of choosing only subjects on which they can speak with competence and, if I dare say, first hand experience. By conforming to this so natural rule, I am happy to think that the questions which I will tell you about are among those which more particularly attracted the attention of the geometers of this country, and thus to pay homage to the beautiful genius of Italy, where Geometry, in all its forms, was cultivated, is cultivated even today, by so many eminent masters. Some of them, Bellavitis, Brioschi, Cremona, Casorati were my guides or my friends. Allow me, before getting into the subject, to remember one of the greatest and kindest, Eugenio Beltrami, with whom I had the most affectionate relations, and who left us so many writings where the deepest science is combined with the elegance of form and clarity of exposition.

**I.**

**The theory of maps and the work of Gauss.**

Like many other branches of human knowledge, infinitesimal geometry was born in the study of a problem posed by practice. The ancients were already concerned with obtaining planar representations of the different regions of the Earth, and they had adopted the natural idea of projecting the surface of our globe onto a plane. For a very long time, we focused exclusively on these projection methods, by simply limiting ourselves to studying the best means of choosing, in each case, the point of view and the projection plane. It is the most penetrating geometer, Lambert, the highly esteemed colleague of Lagrange at the Berlin Academy, who, for the first time pointed out a property common to Mercator maps, called reduced maps, and to those provided by the projection stereographic, first considered the theory of maps from a truly general point of view, and proposed, with all its breadth, the problem of the representation of the earth's surface on the plane with similarity of infinitely small elements. This beautiful question, which gave rise to research by Lambert himself, by Euler and two very important Memoirs by Lagrange, was dealt with for the first time by Gauss in all its generality. The work that the great mathematician composed on this subject, and which was crowned in 1822 by the Royal Society of Sciences in Copenhagen, was followed, in 1827, by the famous

*Disquisitiones generales circa superficies curvas*, in which we find the first elements of the a theory which was to receive great developments. Among the essential notions introduced by Gauss, we should note the systematic use of curvilinear coordinates on a surface, the idea of considering a surface as a flexible and inextensible fabric, which led the great mathematician to his famous theorem on invariability of the total curvature, the beautiful properties of geodesic lines and their orthogonal trajectories, the generalisation of Albert Girard's theorem on the area of the spherical triangle, all these concrete and definitive truths which, like many other results due to the genius of the great mathematician, are intended to preserve, through the ages, the name and the memory of the one who first revealed them.

Driven by so many other works, Gauss did not develop the consequences of the principles he had introduced. For example, the first-order partial differential equation he gave for families of orthogonal trajectories of geodesic lines contained in seed the fundamental discoveries that Jacobi developed later on Analytical Mechanics. I have also shown, in my

*Lessons*, that by very simply combining three linear second order equations which are found in the

*Disquisitiones*and which satisfy the rectangular coordinates of a point on the surface, one could form with all possible generality the partial differential equation of surfaces which admit a given linear element, and thus obtain the complete solution of a question which was to be proposed, only 35 years later, as a subject of prize by the Academy of Sciences of Paris. But Gauss had in mind from his earliest works, and he expressly says so in the preface to the Memoir crowned by the Academy of Copenhagen, only the construction of geographical maps and the applications of a general theory to Geodesy. He returned to these subjects in two Memoirs which bear the dates of 1843 and 1846, and which can appear without disadvantage alongside the

*Disquisitiones*. Taking up, with more precision, a method which Prony had published in 1808 to replace our elliptical globe by a suitably chosen sphere, he makes known, for the construction of the geographical maps of a large country, simple and practical methods which have been faithfully followed in Germany and which certainly deserve to be adopted in other countries. The beautiful publication of Gauss's

*Reliquia*that we owe to Felix Klein has taught us that the great mathematician had not stopped continuing his studies on surfaces in general, and that he had notably obtained the very essential notion of what we call today the geodesic curvature. We also found a short fragment relating to the determination of the applicable surfaces on a given surface; it seems, however, that Gauss never paid his attention, in a well-followed manner, to the problem that his first works seemed to suggest: the determination of the surfaces which admit a given linear element.

**II.**

**The first works of French geometers Monge, Dupin, Lamé, Jacobi.**

At the time when Gauss published in Germany the research we have just mentioned, France was, so to speak, in full geometric bloom. Through his original designs on the generation of surfaces and his creation of descriptive geometry, through the renovation which he had also been able to imprint on analytical geometry, Monge had exerted an influence which Lagrange noted with some regret. We know the famous words that escaped the illustrious geometer, at the end of a Monge lesson at the École Polytechnique: "With his descriptive geometry and his generations of surfaces, this devil of a man will make himself immortal." Monge was not content to make discoveries; he also had pupils, which is sometimes better. One of them, perhaps the most gifted, Charles Dupin, knew how to complete his master's work by following him in all directions, like him combining analytical geometry with the infinitesimal method; by introducing, alongside the notion of the lines of curvature due to Monge, those of the indicator, conjugate tangents and consequently asymptotic lines, which are all projective as we say today, and thereby acquired a role preponderant in the most recent research. Lamé, for his part, this apostle of natural philosophy, gradually rose to the notion of curvilinear coordinates of space, to that of differential parameters, which had been suggested to him by his studies in Mathematical Physics and that he developed in a masterful way in a Memoir inserted in 1833 in the Journal of the École Polytechnique. Soon the beautiful discovery of Jacobi on the geodesic lines of the ellipsoid with three unequal axes came to bring a new food to the research of geometry. Chasles, recalling one of his most elegant theorems according to which two homofocal surfaces of the second degree seem to intersect at right angles, from whatever point in the space we are looking from, showed that this theorem could have given him everything at least, a prime integral of the second order differential equation on which the determination of the geodesic lines of the ellipsoid depends. Thanks to a fertility of means which has never been exceeded, he managed, with only resources of Geometry, to integrate, not only the differential equation of geodesics, but also these systems of two differential equations to which Jacobi had given the name of abelian equations. A host of mathematicians, including Liouville, Frenet, Puiseux, Joseph Bertrand, Alfred Serret, Minding, Joachimsthal, Bouquet, Paul Serret, and more particularly Ossian Bonnet, to whom his research almost assigns the role of a creator, took part with ardour in this work of development, which was to set up one of the most interesting and elegant branches of modern geometry. All these masters had to prepare for themselves worthy successors. I would like to analyse here their main discoveries; but it is fitting that I remember that time is measured for all of us; I must not abuse your attention and arrive as soon as possible at the researches which are occupied people at present; I can therefore only refer those of you who would like more details on the somewhat old period which I must abandon, to the lecture I had the honour of giving before the International Congress of Saint Louis in the United States of America, under the chairmanship of the illustrious astronomer who came to take part in our work.

In the rest of this talk, I will therefore limit myself to following the example given to us, 8 years ago, by David Hilbert at our Paris Congress, and I will indicate, in a quick review of the main branches of infinitesimal geometry, what are the problems whose solution matters most to us, and which we can hope to solve in the more or less near future.

**III.**

**Methods in infinitesimal geometry. Role of the analytico-geometric method.**

A word first on the methods that should be followed in the study of problems of infinitesimal geometry. Here, as in the geometry of finite elements, we tried to build a body of doctrine entirely independent of Analysis. Jacobi had set an example by reviving the infinitely small, proscribed by Lagrange; Joseph Bertrand and Ossian Bonnet followed him in this direction. Today there seems to be little time to follow two different methods to study the same questions; and we have attached ourselves almost exclusively, in infinitesimal geometry, to the analytical methods which employ the coordinate axes; for my part, I am far from finding it bad, with the proviso, however, that research will be constantly invigorated and inspired by the geometric spirit, which must never cease to be present. To explain what I just said, let me share with you one of my oldest and most specific memories. More than 30 years ago, a most distinguished analyst brought me work, which he had just completed, on the developable surface circumscribed at the same time to a sphere and to a surface of the second degree. His formulas were elegant, symmetrical and expertly deduced; they had led him to this conclusion, which had surprised him a lot, that the cusp of its evolute was algebraically rectifiable. I had no trouble showing him, without calculation, that the discovery of which he was so proud and which, moreover, was not without interest, was geometrically evident, and could even be extended to any evolue circumscribed to a sphere; because it had its origin in this property that has the edge of a cusp to be one of the evolutes of the contact curve of the developable surface and the sphere. This is what my analyst friend would have recognised by himself, if he had not been betrayed and abandoned, for a short time I hope, by the god of Geometry.

So let's follow the analytical methods, but don't follow them blindly. Like the main roads, they often provide us with the safest ways; but the side roads have their charm, and they shed much more light on the real connections between places and things.

Analytical methods are sometimes criticised for their length and obscurity; it has even been argued that the incessant use of coordinates is artificial. The use of the movable trihedron and the method of relative movements, combined with a judicious choice of curvilinear coordinates, most often escapes this objection. It highlights the close and fundamental relationships that link Geometry to the science of movement and provides the mechanical meaning, so essential to know, of the geometric elements that arise in research.

There is yet another advantage that Analysis presents in these studies. Whatever some people say, more or less foreign to our science, the mathematician is by no means a machine to deduce or calculate. His work brings into play all the faculties of his mind. Finesse, inventiveness, imagination, are perhaps more necessary to him than order and correctness in reasoning. But by creating himself, as it often happens in Geometry, the very purpose of his studies, by introducing new notions and elements, he must guard himself above all from these pointless speculations, whose least disadvantage is to hold a place that could be better occupied. It is here that Analysis comes into play with its own virtue. She has no signs, according to Fourier, for confused notions; nor does it have methods for idle and hopeless searches. On the other hand, it puts in our hands, when we have found the right path, all the elements that we need for the development of our concepts.

**IV.**

**Need to openly and completely introduce imaginaries into Geometry. Supporting examples.**

I will add, and on this point I was pleased to collect more than once the assent and the support of Sophus Lie, that the mathematician must never back down before the frank and complete use of imaginary elements. With Cauchy, the imaginary imposed its victorious imprint on all parts of Analysis. Their place, in the same way and even more than in Analysis, is marked in Geometry. It happened to me in my youth that I met some retarded teachers who raised their arms to the sky when one ventured to speak to them about the imaginary circle of the infinite.

These times have passed and no one disputes the importance of the notions of the imaginary introduced into the geometry of finite elements; they have even become so commonplace that one no longer dreams of citing their first inventor. I remember that my friend Laguerre, having started one of his Memoirs with a sentence such as this one: "We know that all the circles of the plane can be considered as passing through two points at infinity", was genuinely picked up by Poncelet. "We know, yes we know," said the illustrious mathematician, "but you should have said who taught you that." In infinitesimal geometry, as in Poncelet geometry, the use of imaginaries will be no less fruitful than in Analysis. But it took much longer to win. Why? I do not know. Is it because, according to some, Geometry must always have in view real objects and lead to constructions? Perhaps. In any case there is much progress to be made in this regard. Monge had nevertheless set an example for us; on this point as on several others, he was ahead of his time, since he did not hesitate to devote a whole chapter of his work to the imaginary surface whose radii of curvature are equal and of the same sign. These imaginaries, before which he did not back down, allowed his successors to interpret and invigorate this integration of minimal surfaces, which is one of his most beautiful discoveries, and thus to constitute the most attractive and perfect chapter of infinitesimal geometry. Is it not also thanks to the imaginaries that we were able to deduce from two famous theorems of Julius Weingarten, which would not spoil the great Memoir of Gauss, the complete determination of the applicable surfaces on the paraboloid of revolution?

I would like to point out another example in which the imaginaries reveal to us the true origin of certain properties, unexpected at first sight. You know that, by very varied methods, we were able to complete certain Monge researches, and to determine all the surfaces in which the lines of curvature of one of the systems, or of the two systems, are plane or spherical. For the sake of clarity, consider the surfaces with plane lines of curvature in one system. We observe singular relationships between the planar lines of curvature of the same surface. If only one of them is algebraic, all the others are algebraic. If one of them is a circle, all the others are necessarily circles. It is by bringing in the imaginary that we go back to the source of these curious relationships. Imagine that we deform the developable surface enveloping planes of lines of curvature, assuming that each tangent plane of this developable carries with it the line of curvature contained therein. It is easy to establish that, among all the deformations thus considered, there will be one, necessarily imaginary, in which all the lines of curvature will be placed on an isotropic developable, that is to say circumscribed to the circle of infinity. Consequently, if one of the lines of curvature of the surface is known to us, all the others will necessarily be equal to plane sections suitably chosen from the isotropic developable which contains this first curve. This remark obviously gives the key to the relations that we had pointed out. At the same time, it provides us with the simplest construction of surfaces with plane lines of curvature. We will take two developables (D), (J) the second of which is isotropic, then we will assume that (D) deforms, its tangent planes carrying with them the sections they have determined in the developable (J). The new positions of these sections will generate the area sought.

**V.**

**Some problems of infinitesimal geometry. Constant twist curves.**

Lets go from methods to problems. I will start with the following:

Constant torsion curves are found in a large number of studies. According to a curious theorem of Enneper, asymptotic lines have curves with constant torsion for constant curvature. Likewise, the surfaces applicable to the paraboloid of revolution, which we have already mentioned, can be derived, by an elegant geometric construction, from imaginary curves whose torsion is constant. For the surface to be algebraic, it is necessary and sufficient that the curve with constant torsion from which it derives is algebraic. It is also on the determination of the algebraic curves with constant torsion that the construction of the minimum algebraic surfaces circumscribed to a sphere depends.

There would therefore be a real interest in obtaining, among these curves with constant torsion that J.-A. Serret taught us to determine by three quadratures, those which are algebraic or, only, unicursal. Even limited to unicursal curves, the research takes the form of a problem of higher Algebra, well worthy of tempting a disciple of Lagrange and Cauchy. Unfortunately, despite efforts that have given us partial results, which are quite extensive, the complete and general solution of this beautiful and difficult question still remains to be found.

**VI.**

**Constantly curved surfaces and their transformations.**

Among the non-developable surfaces of the second degree, the paraboloid of revolution and two other imaginary paraboloids tangent to the circle of infinity are the only ones from which we know how to obtain the most general deformation. During the competition that the Academy of Sciences of Paris instituted in 1860 on the deformation of surfaces, and which brought together competitors such as Bour, Ossian Bonnet, Delfino Codazzi and another Italian mathematician whose name Beltrami entrusted to me, Bour had announced that he had been able, by applying the famous method of variation of the constants due to Lagrange, to determine all the surfaces applicable on any surface of revolution and, in particular, on the sphere. Bour died shortly after, without being able to confirm the fine result he had obtained; and his Memoir, kept by Joseph Bertrand, perished in the fires of the Commune. It was in vain that, taking up the path opened by Bour, we tried to reconstruct his method. But efforts in this direction have been far from fruitless. They made us aware in particular of transformation processes which make it possible to derive, from a surface applicable on the sphere or, which is the same thing, from a surface with constant curvature, an unlimited sequence of surfaces of the same definition. These methods of transforming surfaces with constant curvature have a simplicity and elegance that deserves us to stop for a few moments.

All the countries took part in their progressive elaboration, from Italy, which started, to the extreme north of Europe which gave the crowning. We managed to derive from any surface with constant curvature a new surface of the same constant curvature, which depends on two parameters; so that the repetition of the method constantly supplies new surfaces, which contain in their equation parameters in as large a number as may be desired.

...

**VII.**

**The notion of the general integral, as given by Cauchy.**

In this talk, which I continue in a somewhat irregular manner, I was led to speak to you about the general integral as Ampère had been led to conceive it. Cauchy, who also endeavoured to introduce precision, and sometimes even too great precision, in the study of the different problems of Integral Calculus, has lifted all the difficulties relating to this subject, by subjecting each integral to fulfil certain conditions which complete its determination, and in a way make it possible to distinguish it from all the others. This really new way of posing the problems of Computation of Integrals has been the source of immense progress. The initial conditions chosen by Cauchy were perhaps a little restrictive; but, by simple changes of variables, it was easy to widen them, and I proposed, to designate under the name of the

*Cauchy problem*, which was unanimously adopted, that which consists in determining and characterising the integral that one seeks by initial conditions chosen as broadly as possible. For example, if it is a partial differential equation with two independent variables, the integral surface will be subject to passing through any given curve, when the equation is first order; if the equation is of the second order, the integral surface must also be written along the curve at a given developable; and so on for higher orders. This new point of view was to lead, in the most precise way and in all cases, to the notion of characteristics, which had remained so vague in Monge's work: characteristics are the assemblages for which the Cauchy problem becomes undetermined.

**VIII.**

**Application to minimal surfaces. The Plateau problem.**

The Cauchy problem, as we have just formulated it, was solved for minimal surfaces, using elegant formulas which bear the name of their author and are universally used.

We must be careful not to confuse it with the research of a completely different nature and which has as its object the determination of a minimum continuous surface passing through a closed contour. On this point, it must be recognised, we are less advanced than the physicists. To obtain the desired surface, Plateau contented himself with physically making the outline he had in view, and then immersing it in a solution of the glyceric liquid. The mathematicians have not yet been able to compete with these experimental achievements. Their ingenious and skilful efforts brought them a host of partial successes. The progress made in the study of the calculus of variations, the improvements made to modern analysis and to the fruitful method of successive approximations, will soon enable them, I have the firm hope, to resolve in all its generality the beautiful problem bequeathed to them by Plateau, and thus to record to their credit a new and brilliant success.

**IX.**

**Progress and problems of the theory of maps. Geodetic representation. Chebyshev problem.**

Like the study of minimum areas, the theory of maps has given rise to many interesting problems. It has been proposed, for example, to investigate whether the correspondence established by parallel tangent planes between any two surfaces can give a conformal representation of one of the surfaces on the other; this research has been extended to the correspondence which is established, between any two surfaces, by the points of contact of a sphere which must be tangent to one and to the other. By studying a property of the central projection, as Lambert had studied a property of the stereographic projection, Beltrami was led to deviate from the point of view which, in the hands of Lagrange and Gauss, had given such perfect results, and to search if it is possible to make a map of a given surface, subject to this unique condition that the geodesics of the surface are represented by the lines of the plane. His elegant analysis showed him that this mode of geodesic representation could only intervene in the case of surfaces with constant curvature. In turn, Beltrami's studies and results have been extended to correspondence between any two surfaces and have given us new properties of this form of the linear element which bears the names of Liouville and Jacobi.

A Russian mathematician, Chebyshev, who occupies a special place in the development of modern Mathematics, proposed to us a new and curious subject of research which we can present as follows, by generalising it somewhat. Imagine a net such as those in which the ladies encase their hair, or as worn by people who go to the market. Whatever their shapes and dimensions, these nets are always composed of two series of threads attached at their meeting points, so that, for each stitch, the angles only of the sides can vary, but not their lengths. Imagine that we put such a net on a surface or, what amounts to the same thing, that we introduce a solid body, terminated by any surface, inside the net, and we propose to determine the form that will take the net.

...

He had not noticed that, in the case of the sphere, its complete and general solution can be linked to the determination of surfaces with constant negative curvature; in the sense that each surface with negative constant curvature gives, by the spherical representation of its asymptotic lines, a solution of the problem of Chebyshev for the sphere, and vice versa.

**X.**

**Negative constant curvature surfaces and quadratic forms of differentials. Non-Euclidean geometry.**

You see, my dear colleagues, it is in vain that we would like to escape from these surfaces with constant curvature, which seem, at first sight, so particular. This is the case of repeating the poet's verse:

I avoid him everywhere, everywhere he pursues me.They present themselves to us, in all the researches which we undertake, by virtue undoubtedly of these properties of invariance which are highlighted by the theorem of Gauss. But it should also be noted, their study borrows an exceptional interest from the relationships so close that it presents with this research on the principles of Geometry which were inaugurated by the works of Lobachevsky, Gauss, Bolyai and Riemann. This beautiful and vast subject, to which modern mathematicians are passionately attached, merits a special study in itself. I will limit myself, for the moment, to noting that Riemann, by attaching it to the consideration of a quadratic form of differentials, has introduced by this very fact, into Infinitesimal Geometry, an element of study and coordination called to play the most essential role in all research.

Riemann and Beltrami had opened the way: one by defining, the other by studying, in a detailed way, the notion of spaces with constant curvature, which is, in a way, adequate for non-Euclidean geometry. Since then, Lipschitz and Christoffel have considered quadratic forms of differentials in all their generality; and their scholarly research, welcomed everywhere with great favour, has been continued with great success in this country. This is to be welcomed, because the quadratic forms of differentials occur almost everywhere. We know the key role which has been assigned to them in analytical mechanics by the immortal discoveries of Lagrange; to confine oneself to Geometry, one obtains them, not only by forming the linear elements of surfaces and spaces with any number of dimensions, but also by calculating the moment of two infinitely neighbouring lines, the angle of two infinitely neighbouring spheres, etc. Their study has been much advanced as regards the differential and invariant part; we have even attached to their consideration a new kind of calculation which forms a curious extension of differential calculus. We have also begun to study in all its generality the first and most elementary of the problems of this theory: I mean the search for the conditions which are necessary and sufficient for two forms to be applicable to one another. In its delicate and difficult part, this research is obviously reduced to that of quadratic forms which can be transformed into themselves by a continuous series of substitutions. Here, the masterful discoveries of Lie on group theory have found, and will no doubt still find, a beautiful field of application.

**XI.**

**Reducing problems to each other. Recurrence processes. Isothermal surfaces. Deformation of second degree surfaces.**

Recent advances in Analysis have led us to consider the integration of differential and partial differential equations from a completely new point of view. We know today that we must most often give up integrating them in the old sense of the word; that it will be necessary to study them directly, to classify them, to bring them back to one another, to seek, if necessary, the transformations which reproduce them. This study is just beginning, and will require long efforts; but, as of now, it is permissible to affirm that infinitesimal geometry will offer to Analysis, when the time comes, the most interesting applications, and above all the most precious examples.

We have already mentioned the methods of transforming surfaces with constant curvature. You could almost say that these recurrence processes are the rule in Geometry, because they are found everywhere.

Let us consider, for example, isothermal surfaces, that is to say surfaces whose lines of curvature form an isothermal system. Their determination depends on a fourth order partial differential equation, which we have managed to write in the most symmetrical forms, but whose integration constitutes perhaps the most difficult problem encountered in Geometry. We know, it is true, many particular integrals: surfaces of the second degree, the most general cyclids, surfaces of revolution, those whose mean curvature is constant, certain surfaces with plane lines of curvature in a system, etc. ., are isothermal surfaces; but, in comparison to the general integral which, according to the old terminology, must include four arbitrary functions, these particular solutions do not count so to speak. Here, however, as in the theory of surfaces with constant curvature, some of the simplest geometric constructions allow us to attach, by simple quadratures, to each isothermal surface, a whole series of new isothermal surfaces, in the definition of which will appear a number of bigger and bigger constants. One could therefore be tempted to take up the efforts of Lie, to deduce from this set of solutions the general integral itself, by relying, no longer on Ampère's conceptions, but on the correct definition of the general integral, as given by Cauchy. Let me defer these kinds of considerations, and move on to another example of a very current interest, in which it is not a matter of transforming an equation into itself, but of the reduction of one problem to another, different and simpler problem.

It seems that, in its development, infinitesimal Geometry is called to follow the same paths as the analysis of Descartes. The deformation of the sphere had first occupied mathematicians; the natural course of their studies led them to deal with the deformation of general surfaces of the second degree. Two different paths have been followed for the study of this beautiful question. Some have become attached, and have succeeded, in constituting recurrence processes, transformation methods analogous to those which had been given for surfaces with constant curvature. The others tried to relate the new problem which was posed to them as the problem, simpler in its statement, of the deformation of the sphere, and they succeeded in a fairly large number of cases: for the paraboloid whatever, for all quadrics of revolution, and even for the more general surface subject to the unique condition of being tangent at a single point to the circle of infinity. Can this result extend further to the most general quadric surface? The answer to this question, whether affirmative or negative, will be equally interesting in both cases. I believe it will soon be given; but for the moment I am reduced to saying, speaking Latin in the Eternal City,

*Adhuc sub judice lis est.*

The argument is still under dispute.

**XII.**

**Linear partial differential equations and their role in Infinitesimal Geometry.**

It seems indeed, from what precedes, that Infinitesimal Geometry is destined to become somehow an illustration of this part of infinitesimal calculus which deals with partial differential equations. And yet I haven't told you anything yet about the close, almost surprising, relationships it presents with linear partial differential equations with two independent variables. These equations, which also occur, but in a different form, in Mathematical Physics, had already been considered by Laplace, who had given a truly original method to integrate them. Their analytical study was taken up by Riemann and Théodore-Florentin Moutard. Their theory has been perfected, a special theory of invariants made up for them. In particular, those whose invariants are equal form a group of exceptional interest. Moutard had indicated a method to integrate them when possible which allowed objections, but which was found to be correct. So that geometrical research has again contributed to perfecting analytical theories.

They even posed problems for us which Analysis would not have thought of on its own. Thus, among these linear equations which have proved to be particularly worthy of being formed and studied are those which have one or more groups of particular solutions linked by a quadratic equation.

For example, the search for surfaces with constant curvature comes down to that of equations of the form (D) which have three solutions whose sum of squares is equal to unity. That of isothermal surfaces is also reduced to those of equations of the same form having five solutions for which the sum of the squares is zero, etc.

Among the questions whose solution depends exclusively on the integration of a linear equation, two must be more particularly highlighted: the problem of the spherical representation and the problem of the infinitely small deformation. They can be reduced to one another, and their solution can today be considered as completely finished.

In particular, the theory of infinitely small deformations constitutes the first somewhat extensive application of a method which seems destined for a great future.

**XIII.**

**The auxiliary system in Geometry. Finite deformation and infinitely small deformation.**

Given a system of differential or partial differential equations which contains a certain number of unknown functions, it is necessary to add to it a system which we have called auxiliary and which determines the solutions of the system infinitely close to any system, given in advance of solutions. The auxiliary system is obviously linear. Its relatively easy study provides the most precious information on the primitive system itself, and on the possibility of obtaining its integration. For example, with regard to the deformation of a surface, the application of the Laplace criteria to the auxiliary equation will make it possible to recognise whether it is possible to integrate the partial differential equation on which the complete and general solution of the problem depends.

The consideration of the infinitely small deformation still makes it possible to apply to the problem of the finite deformation a theorem of Cauchy which, even today, is hardly known, but which excited the admiration of Liouville, this judge so severe and so difficult, especially for the work of Cauchy.

The important mathematician has shown that one can extend to partial differential equations a theorem long demonstrated by Lagrange for linear equations with ordinary derivatives and integrate, by simple quadratures, any nonhomogeneous linear system of partial differential equations, when we obtained the general integral of this same system, made homogeneous by the suppression of the second terms of the equations.

According to this, if we want to find the surfaces which result from the deformation of a given surface, we can, as I have shown in my

*Lessons*, solve with an approximation as large as we want, and by simple quadratures, the problem of finite deformation, whenever we have integrated the linear equation of the form (D) on which depends the complete solution of the problem of infinitely small deformation.

**XIV.**

**Expansion of Geometry frameworks. Need for general and uniform methods allowing the necessary simplifications.**

I hope, my dear colleagues, to have shown you that, even if we limit ourselves to the old and already explored parts, we find before us many questions which can tempt both the geometers and the analysts. But what about this almost intact part of the Geometry that has been bequeathed to us by the work of Plücker, Sophus Lie and their successors. It was not enough for them to introduce the plane and to consider it, like the point, as an element of space. The reciprocal polar theory had broken the levees. To the point and to the plane came successively to add the straight line, the sphere, the circle, then the curves and the surfaces accompanied by their tangent planes; we have even brought together these already complex beings in assemblies, which we have taken pleasure in considering themselves as primordial elements. The study of infinitesimal relationships has barely begun for this new Geometry. At the very most, we have outlined the theory of the straight line and the sphere, linked to each other by the admirable transformation of Sophus Lie. The circle and the helix gave rise to some research; we have, in a way, isolated and studied certain classes of rectilinear congruences, among which we must point out the isotropic congruences of Ribaucour, which play such an elegant role in the theory of minimum surfaces; the theory of curvilinear coordinate systems and, in particular, triple systems of orthogonal surfaces has been perfected at several points. But subsequent progress will depend exclusively on the order, uniformity and generality that we manage to introduce into the methods. It is only at this price that we will be able to achieve, in this branch, the simplifications without which any progress would become impossible. I trust that here as elsewhere, the extreme generality of the problems will generate the simplicity of the solutions.

Last Updated July 2020