# The Jubilee of Gaston Darboux

On Sunday 21 January 1912 the Jubilee of Gaston Darboux was held in the Council Room of the University of Paris. The proceedings of the Jubilee were published in 'G Lippmann, H Poincaré, P Appell, E Lavisse, V Volterra, L Belugou, É Picard, L Lévy, C Guichard and G Darboux, Le jubilé de M Gaston Darboux, Revue internationale de l'enseignement 63 (1912), 97-125'. We present below an English translation of some parts of the Jubilee. In particular we give a version of
(i) Henri Poincaré's speech,
(ii) Vito Volterra's speech,
(iii) Émile Picard's speech,
(iv) an extract from Gaston Darboux's reply.

1. Henri Poincaré's speech

It is to Geometry that you have spent the most time and work; not only did this science naturally attract you, perhaps for the same reason which ensured it the predilection of the Greeks, because it easily leads to finished results, satisfying both for the mind and for the aesthetic imagination, but the duties of your teaching kept bringing you back to it and compelling you to deepen it. However, it is your work of pure Analysis that I will first recall, because the precious qualities of your mind, elegance, clarity, the quest for simplicity, stand out even better in a field where they meet more rarely.

I will first cite your memoir on the functions of very large numbers. Certain expressions, which depend on an integer, get complicated quickly when this integer increases, but can be replaced with sufficient approximation by very simple functions when this integer becomes very large. In a host of questions, it is precisely the cases that exclusively concern us; this is especially true in applications; the physicist, in the theory of gases for example, only has averages for very large numbers, he does Statistical Mechanics; also those who cultivate Celestial Mechanics know the important role played by the high order terms of the perturbing function; finally, the pure mathematician faces the same difficulties whenever he deals with questions of convergence. The general method you have created for solving these problems is elegantly simple and easy to use, since it is only a question of forming a Taylor series and studying the singularities of the function it represented.

Second order partial differential equations are one of the objects that most resist the efforts of analysts; there are, however, cases where integration can be performed without partial quadrature; only one was known, thanks to Monge's work; you are the one who introduced us to all the others; you have shown us how they are linked together and how a regular sequence of operations can surely lead us to the result if this result is possible.

A seemingly simpler problem, the algebraic integration of first order and first degree differential equations, also occupied your attention; you showed us how the integrability cases are classified and what role the singular points and certain exponents attached to them play. No one can doubt that it is by the way that you have started that one day we will be able to come to recognise for sure if a given equation can be integrated algebraically, and that it is still by this way that we will be able to approach the systematic study of integrals in cases where they are transcendent.

We thought for a long time that all differential equations had singular solutions: we thought we had established it by specious, but somewhat sketchy reasoning. You have shown how wrong we are; what was believed to be the rule was only the exception, what was believed to be the exception was the rule; this is a kind of adventure to which mathematicians would often be exposed, if the sagacity of the masters did not warn them of the trap.

When approaching the works that have especially exhibited your glory, your geometric research, I realise that I have already greatly abused the attention of the audience and yours and that I have very little time left. Fortunately your discoveries are in everyone's memory, all geometers have read the volumes of your Surface Theory, your treatise on orthogonal systems and curvilinear coordinates.

The geometers seem to be divided into two schools; some regard analysis as an intruder, which Descartes has unduly introduced into a field which did not belong to him; they would like to give back to the science that they love the purity that it had in Euclid's time; the others hardly see geometry as being other than a branch of analysis, which one could do without making figures. Fortunately, you have moved between these two opposing tendencies; you know that nothing can be done today without analysis, but you also know how precious is what is called geometric meaning; you have shown us that we can keep it as safe and as fine as it was with the ancient Greeks and yet handle the calculation with skill.

Analytical geometry is sometimes purely algebraic, it then studies surfaces and curves of finite and well determined degree, and it studies them as a whole; but often it also uses infinitesimal calculus, so it takes a microscope to show us in detail what is happening in the vicinity of each point on a surface. Without neglecting the first point of view, as your beautiful studies of cyclides show, on the surface of Kummer, on the surface of the wave, you especially focused on the second. Triple orthogonal systems owe their importance to the use that can be made of them to define curvilinear coordinates; they depend, as you know, on a third-order equation that Bonnet had discovered and which you found by another means; this is a subject that seems inexhaustible and to which you have often returned, each time fruitfully. I would say the same about the deformation of surfaces, an extremely difficult problem, which is not about to be solved in general; the day it is, we will not forget what you did to prepare the solution.

Geometry, as you understand it, led you naturally to Mechanics, and by two routes, on the one hand infinitesimal Geometry is intimately linked to Kinematics; on the other hand, the problem of geodesic lines is basically a problem of analytical dynamics.

It was easy to obtain beautiful and many partial results, you knew how to embrace them from an overview, and summarise them in a masterful work which you made one of the classics of Geometry.

Allow me to stop, because your research is too abundant for me to think of being complete; your colleagues, of whom I have been an imperfect interpreter, are pleased with this opportunity to show you both their friendship and their admiration.

2. Vito Volterra's speech

The Committee set up for your scientific jubilee had entrusted Hermann Schwarz, Dean of the Academy's Correspondents for the Geometry Section, with the honourable mission of expressing the common sentiments of the foreign scholars who associated themselves with its celebration.

To his great regret, he was unable to attend this moving ceremony today; this is why, in the opinion of the Committee, it is to me, happy to be in Paris on this occasion, that the great honour is given to present to you the feelings of admiration and deep respect that mathematicians of all the parts of the world nourish for you, Monsieur Darboux, for your genius, and for your noble character.

I could not do my job better than by appropriating the beautiful and cordial expressions that our Dean, Hermann Schwarz, had prepared for you. Here they are:

Your merits, M Darboux, are too great and too numerous for it to be possible to appreciate them at their true value in a few words. All mathematicians agree that the services you have rendered in the field of applications from Analysis to Geometry, as well as in pure analysis, in mechanics and in other branches of mathematics, are too considerable so that their action was not felt outside the borders of your country. Not only in France, but everywhere where mathematics is studied, there are enthusiastic students who consider you as their master. They declare highly, that in your scientific works, in your teaching books, in your Bulletin, not only have you been able to present with marvellous clarity the results of your research and that of your compatriots, but that you have put into full light the importance of the work of your fellow foreign mathematicians.

Your example clearly shows that the emulation, which exists between scientists of all countries, does not make them enemies; but brings them together all the more intimately since the treasures discovered by each of them immediately become the good of all.

Every mathematician fulfils a duty by advising young foreign scholars and students from all countries to come to Paris to study, at its source, the results of the research of French mathematicians. To many of these foreigners you have kindly favoured the support and assistance necessary to reach their goal, and you have facilitated their relations with the master mathematicians of France. They all keep a respectfully grateful memory of your friendly help.

The most famous Academies and Learned Societies of Europe and America have been happy and honoured to call you among their members. Several foreign universities have appointed you professor and honorary doctor.

Most recently, the Prussian Academy of Sciences awarded you the Steiner Prize for 1910 in recognition of the progress you have made in geometry. I cannot fail to add that at the last International Congress of Mathematicians in Rome, you were the object of testimonies of the keenest affection, the highest esteem, the admiration of all those who had come from the different parts of the world to attend this scientific assembly.

Today, on the occasion of your jubilee, all foreign mathematicians are united in spirit to French mathematicians to thank you and to praise with all their admiration your remarkable scientific works which have advanced not only geometry, but also mathematics in general.

Foreign mathematicians join their sincere wishes with those of your compatriots. May you be granted to add to your life, already shining with scientific work, still several jewels, and may you enjoy, admired by all, for a large number of years, the fruits of a life of work totally devoted to science.

This is what we wish you from the bottom of our hearts.

3. Émile Picard's speech

We are sure it will be a pleasure to you by recalling in this ceremony your title of President of the Society of the Friends of Science. You like action as much as thought, and we have just been told what activity you spend at the Academy and the services you rendered to the Faculty of Science. But your desire to be useful never tires; you still wanted to give some of your time to more discreet works that required real dedication. Nowhere more than at the Society of Friends of Science have you put the old saying into better practice, that noise does not do good and that good does not make noise.

The Society founded in 1857 by Baron Thénard has a particularly high goal: it is a Support Society, but where the securities to be invoked are services rendered to the pure and applied sciences, industry and agriculture. Those who collaborate with you in this work know how carefully you strive to respect the thought of its founder. When it comes to your dear Society of the Friends of Science, you spare neither your time nor your pain, soliciting the opinions of the most varied skills, and going, if necessary, to take your own information to inform our decisions. You dream of a great work of scientific solidarity, where those, and they are legion, who profit from the progress and discoveries of science, would all come to help researchers only concerned with their work, carefree of the future for them and those around them. Your emotional calls were often heard, and generous hands were extended to us. But, unfortunately, the miseries we should be rescuing are increasing faster than our resources, and we lack many actions on which we would be entitled to count; it's one of your sorrows that science, on which so many eloquent speeches are made, still garners too much ingratitude.

You work, my dear President, with all your energy to relieve noble and sometimes glorious misfortunes, and you thus show that your heart is at the peak of your intelligence. May you remain at our head for a long time and see the work to which you are piously attached continue to grow.

When, almost a year, friends, devoted students thought of celebrating the fiftieth anniversary of my entry into teaching and my academic silver wedding anniversary, I had some fear about the result of their efforts. During my career, I had the opportunity to organise the jubilee of Charles Hermite, that of Joseph Bertrand and of Berthelot; but I am far from comparing myself to these illustrious masters. The success that their high scientific position had naturally acquired it for them, but it is your benevolence, your affection, which give it to me today. Let me start by thanking you from the bottom of my heart.

My dear Poincaré,

The praise you give to my work bears the mark of your natural benevolence; they fill me with joy as coming from the one I consider the greatest living mathematician. I will always remember the lovely relationships I had with you as the dean. You were always willing to be of service to a colleague, to punctually perform the sometimes thankless tasks entrusted to you. With men like you, the Faculty ran on its own. There is more, when the consideration of the good of the service determined me to ask you to change teaching, you did it without hesitation, the first time to take the chair of mathematical physics, the second time to pass to that of celestial mechanics. And so today I have the joy and the pride to think that I could advance the moment when, at the same time as great mathematician, you were proclaimed by all great physicists and great astronomers. Why does the Faculty also not have a chair in scientific philosophy? I could also have asked you to occupy it.

Senator Vito Volterra,

Called by the Council of our University, you have come to exhibit here, in the Sorbonne, your original discoveries which have opened a new path in our modern mathematics. You belong to a great nation that has distinguished itself in science, no less than in letters and the arts. Your mathematicians excel in all branches of our science. I am pleased to take this opportunity to thank them for the welcome they have always given to my work. It is undoubtedly in Italy that this work has aroused the most interest and sparked the most research; I'm very honoured to belong to five of your Academies. You have kindly added your personal congratulations, and instructed you to read the address prepared by M Schwarz, the dean of the correspondents of our Geometry Section.

I have been used to counting him as a friend for a long time. His speech today confirms me with this pleasant thought. But his praise touches me particularly, because they come from a mathematician whose high and rare qualities I particularly appreciate. Peculiar thing! At the moment when our Cauchy displayed so much genius and invention in his works, without however succeeding in giving them a precise and definitive form, the great German mathematicians Gauss, Jacobi, and Dirichlet showed us in all their writings qualities that we would be tempted to call good French ones: clarity, rigour, perfection of form, finish of presentation. Pauca, sed matura [Few, but ripe] was their motto. At a time when this motto seems a little neglected, M Schwarz wanted to keep it for himself. His works, like those of the illustrious masters to whom he is naturally attached, assure him of a lasting and fruitful influence on the development of the beautiful science to which, the one and the other, have devoted our lives.

Last Updated July 2020