On the Historiography of Mathematics

The opening session of the 1900 International Congress of Mathematicians took place on Monday 6 August 1900 in the Palais de Congrès in Paris. In addition to other business, there were two plenary talks given during that session. The first one was the lecture Sur l'Historiographie des Mathématiques by Moritz Cantor (Heidelberg). We note that, being in French, his name is given as Maurice Cantor on the published talk. We give an English version of this talk below.

On the Historiography of Mathematics.

By Moritz Cantor (Heidelberg).

When in June 1899 the Organizing Committee did me the honour of asking for a lecture in one of the two general sessions of the International Congress of Mathematicians in Paris, the first question, before which I hesitated, was that of the language in which I would speak. The international nature of the Congress and the regulations adopted in Zurich in August 1897 allowed me to use my own language, on the other hand, and without prejudice to my esteem for the profound erudition of my audience to which no language can be foreign in theory, I told myself that in practice I would be better understood by a certain number of people by speaking French.

With this in mind, I so decided without consulting with the other speakers chosen to speak in the general sessions. The four of us will use the language of the country in the capital of which we are gathered today, which is that of the vast majority of the members present.

The second question I had to focus on was that of the subject about which I would try to talk to you. Mathematics in the century at the end of which we find ourselves has had an almost incredible momentum.

It has lost its previous unity, so to speak, to multiply itself. The mathematician escaped through a hatch previously invisible in the podium of the Theatre of Sciences, and from another hatch we saw the emergence of geometers, analysts, algebraists, arithmologists, astronomers, theoretical physicists, and even historians. The latter, if you will allow me to speak on their behalf, have never pretended to advance the Mathematical Sciences in any of these branches just mentioned. We neither advanced towards the Arctic pole of the calculation of functions, nor towards the Antarctic pole of algebra further than our predecessors, we did not discover the properties of the more or less steep slopes of geometry, we have not gone to the bottom in the abyss of differential equations. It is you, and I could easily cite names and point the finger at the bearers of these immortalized names, to whom this merit is vested. But we, the historiographers of Mathematics, what have we done?

We have composed Travellers Guides. We have described that at such a time such a river was made navigable, such a canal, such a highway, such a railway was built, abandoned today in some cases, used more than ever in others, deserving of being repeated in yet another. We have shown that these means of communication while travelling from one place to another did not fail to miss some remarkable places, worthy of stopping there and not yet as well known as they may deserve to be. We have claimed each path as much as possible for the one who created it, an act of recognition whose justice should be most liberally confessed by those who themselves are entitled to similar recognition.

It is well-understood that charity that begins at home. Why should recognition not imitate it? Let me do this and make a brief sketch of the history of the historiography of mathematics.

To start at the beginning, we should take in a very high point. Aristotle's successors, the Peripateticists, loved to seek and find a personal source of human knowledge, and they rejoiced all the more if it were possible for them to discover this source in one of the distant countries subjugated by Alexander, the disciple of their master. It was from this inclination that the first essays in the History of Science were born. Eudemus of Rhodes composed a History of Mathematics at a time very close to 300 years before the beginning of the Christian era. This History of Mathematics, we do not have. Only certain fragments have been preserved in one way or another and these make us deeply regret what we have lost. Eudemus, judging by these fragments, liked to draw with a bold pen the scientific profile of the mathematicians whose names he kept for us. He seems to have given a history of their ideas rather than of their lives, who perhaps in his day were still too well known for him to have to do this.

You will not blame me if I jump more than two thousand years in a single jump in order to arrive in the middle of the 18th century. Not that there was in all this interval no writing about the History of Mathematics. Far from it! I could quote you writers formerly famous in France, in Italy, in Holland, in Germany, who did their best, but unfortunately their best was hardly worthy of comparison with what we have done since and what we ask today. Petrus Ramus, Bernardino Baldi, Geradrus Vossius, and Jakob Heilbrunner generally contented themselves with saying that such and such an author had written on such and such a work, but if one wanted to know what was new in these works or what was the scientific work of an author, this remained unanswered. I therefore limit myself to this short remark to motivate my somersault from Eudemus to Montucla.

Who does not know Jean-Étienne Montucla, of Lyon, whose life lasted from 1725 to 1799? Having acquired unusual knowledge of foreign languages and mathematics in the Jesuit college in his hometown, he was equipped with the two main tools he needed to accomplish his scientific task before he even knew what it would be. Indeed, he began by studying law in Toulouse and it was only in the second place that he came to Paris to do general studies. He became a regular at the hospitable house of the publisher-bookseller Charles-Antoine Jombert, and it was there that he met the other intimate friends of the house, among whom I will only mention Jean Paul Da Gua de Malves, Jean d'Alembert, and Jérôme de Lalande. It is perhaps Da Gua de Malves, whose algebraic memoirs of 1741, beginning with very extensive historical introductions, gave Montucla the idea of attacking a mathematical problem himself and following its historical development. The Histoire des recherches sur la quadrature du cercle appeared in 1754, a conscientious work if ever there was one, teeming with details long forgotten that had to be dug up in authors themselves less known than they deserved to be. The small volume was a complete success, and it was not much later in 1758 that Montucla followed it with his Histoire des Mathématique. Today's readers generally know only the second editions of the works that we have just cited, the edition of Recherches sur la quadrature of 1831 by Lacroix, and the edition of the Histoire des Mathématique of 1799 by de Lalande. In the latter, the two volumes emerging from the pen of the same author Montucla are followed by two other volumes composed by de Lalande. Judged by themselves the latter volumes would not be bad, if only the proximity of the first two volumes did not harm them. Unintentionally we start to compare, we find an enormous distance in value from the first two volumes to the last two, and instead of praising Montucla we are content to blame Lalande.

For a while, it was even fashionable to disparage Montucla too. He ignored the existence of such a monument, he did not recognize the significance of such a manuscript, he badly translated such a sentence. All of this is perfectly true, but can we reasonably ask from a single author what twelve or more others did later by their combined forces? Can I, if I am allowed to reproduce a sentence that I used in the past, criticize a cartographer of the year 1850 for having left the central part of Africa blank before it was explored? No, and a thousand times no! Certainly, Montucla is no longer today the almost infallible author who must be consulted on any question of the history of mathematics, he has repeatedly fallen into inevitable errors, but he is still and will remain perhaps always a model that every historiographer of sciences must follow, a model also insofar as he tested his strengths in a monograph before starting his all-embracing work.

The century was nearing its end, when two books appeared, which asked us to cross the borders of France and to go first to Germany, then to Italy.

Abraham Gotthelf Kaestner, born in Leipzig in 1719, died in Göttingen in 1800, published, in the last four years of his life, as many volumes of the History of Mathematics. Kaestner was praised during his lifetime, it is written on his bust: "Kaestner the only one", and, later, we made fun of him; he was called "the best poet among mathematicians, the best mathematician among poets of his time", and it is Gauss who is the author of this horrible epigram. It would not be difficult to demonstrate that there was exaggeration on both sides in these evaluations as opposite as possible from the same author, but I stick and I must stick to his Histoire des Mathématique. I will not give weight to Kaestner's age during this publication. Being in your 80s is an excuse not to do anything, but not to throw bad books out to the public. Do not believe, however, that this is the quality that I attribute to his History. It is a work which is far from presenting to us what its title promises, but which, in spite of this, in the unanimous opinion of the experts, one cannot do without among historical works. Kaestner must have had a very rich library, and in his Histoire des Mathématique, he gave a conscientious and detailed description of it by bringing together, in chronological order, the works with similar content. This is how, through Kaestner, we know, and even know quite well, a number of authors whose very names would be unknown to us without him, and a number of volumes not found today.

The Italian work that I announced earlier is of a very different quality. It is the critical history of the origin of algebra, its arrival in Italy and the progress that it made there, by Pietro Cossali, printed in two volumes, from 1797 to 1799, in Parma where the author was professor of astronomy and associated sciences. In 1806 Cossali moved to Padua where he died in 1815, aged sixty-seven and a half. It has been said, with a great deal of truth, that the Cossali treatise made superfluous any other work on algebra in Italy in the era limited by the years 1200 and 1600, and, with the Italians, in this era, who walked at the head of algebra, we only need research on some algebraists from other countries who, like Jordanus Nemorarius, like Nicolas Chuquet, like Michael Stifel, like François Viète, wrote their names on missing pages and must be missing from Cossali, but are absolutely necessary in a general history of algebra. Cossali's merit is all the greater since, for his research on Leonardo of Pisa, an author whom he has, so to speak, discovered, he could only have recourse to manuscripts which were sometimes difficult to find, and even more difficult to decipher. It was Cossali who put Prince Boncompagni in the footsteps of this thirteenth-century merchant, friend of a prince, himself a prince among the friends of the mathematical sciences. And it was Cossali also who knew how to enter the mind of Girolamo Cardano, who brought out the truths discovered by this man of genius by translating them from a barely comprehensible language into a language within the reach of any mathematician. We blamed this translation. It has been said that to change an author's way of speaking is to distort an author. I do not share this opinion. No one more than I warns against the inclination to lend to an author ideas after his century, but if it is only a change of expressions, I believe that we must be grateful to those who save us from having to use a dictionary at all times, and Cossali, in my opinion, did nothing more.

I take you back to France where Charles Bossut, born in 1730 in Tartaras, died in 1814 in Paris, as an examiner at the École Polytechnique, published in 1810, when also an octogenarian like Kaestner, two small in-octavo volumes bearing the title of Histoire générale des Mathématiques depuis leur origine jusqu'à l'année 1808. In these two volumes, you will not find a geometric figure, nor an equation. It is enough to tell you that Bossut did not want to go into details about the science, of which he only signals the very general development. These are overviews, quick glances taken from certain points of view, reasoning full of attractions for those who know, of little use for those who need to learn. As for the errors found in Bossut, and in Cossali, it is the same as with the errors of Montucla. They should not be imputed to them; these are the defects of their time and not of their person.

Let's go back over thirty years to come up with authors that the oldest of us could have known personally when we were young. Yes, there must be among you people who, as well as I, remember the small, affable and spiritual figure of Michel Chasles, who believe they still hear the kind words by which he was pleased to give encouragement to those of his young emulators that had been terrified by a first failure, to spur on those that a first success could have made vain and lazy. I always see him before me, speaking to me for the first time in the corridor which leads to the hall of the sessions of the Academy of Sciences, giving me the warmest welcome, treating me as a colleague, me a young man who had published only a poor little memoir that I had dared to send him. I see him receiving me in a country house in Saint-Germain, in the company of Iréneé-Jules Bienaimé and Joseph Bertrand whom, in this way, he introduced me to. I see him visiting me in a small room on the fifth of rue Saint-Lazare and sitting next to the bed where I was kept being ill.

Forgive me, if I speak to you about these personal memories before speaking of the Aperçu historique sur l'origine et le développement des méthodes en Géométrie of 1837. It is like a photograph of the author put at the head of his work and enhancing the interest shown to him. Of course, the Aperçu historique does not need such an advertisement, a work which had the honour of being translated into German as early as 1839 and reprinted in 1875.

In the Aperçu historique, a distinction must be made between the text and the notes. The text gives a history of geometry, especially synthetic geometry, fairly brief and condensed, calling on the reader's faith for the theses which are formulated there. These are in the explanatory notes at the bottom of the pages, and especially the notes with the size of real memoirs and which serve as an appendix to the volume, where we find discussed in detail the documents that the deep knowledge of the author has been able to unearth. This is the model part of the book, on which more than one historiographer of mathematics has been trained. In these notes, Chasles exceeded the goal announced in the title. It is not only the methods of geometry in question, numeration and calculation, algebra, and mechanics, are treated there in turn with a masterful erudition. Chasles again made in 1871 a Rapport sur les progrès de la Géométrie full of interest, but perhaps dealing with things and people too close to be able to pass a historical judgment on them. Let us add that in the Rapport the ignorance of the German language, less harmful in the Aperçu historique, which generally only deals with works written or translated into French, Latin, Italian, rarely in English, is made to feel unpleasant, despite the care Chasles took to have it translated for his use and even writing on German works which he did not understand and which he wanted to know.

Guillaume Libri lived close to Michel Chasles in Paris. Count Libri Carucci della Sommaja was born in Florence in 1803. Professor of Mathematics in Pisa, he was guilty of opinions if not actions deemed, at that time, to be subversive, and had to take refuge in France in 1830. He was naturalized there in 1833, became Professor of Analysis at the Sorbonne, a member of the Academy of Sciences, and general inspector of libraries. We know the abuse he made of this last position. In 1848, he left the soil of France, which had become too hot for him. He went to England, later to Italy, where he died at Villa Fiesole, in 1869. In the years 1837 to 1841, Libri published his Histoire des Mathématique en Italie, leading, in four volumes, to the death of Galileo, that is to say until the middle of the 17th century.

Can we properly write the History of Mathematics of any country? I doubt it very much, and here's why. If there is an international science par excellence, it is mathematics. Law, theology, philosophy, easy literature, history, can and generally do carry, in fact, a national stamp which I neither want to praise nor to blame, but only to point out. In mathematics, it is quite different. Since the earliest times, the influence of one people on another when it is a question of mathematical knowledge has not been hidden for a moment. One cannot understand the development of mathematics in Greece without knowing the state of this science in Egypt and Babylonia. Roman mathematics arises from Greek mathematics, which they presuppose. For the mathematics of the Arabs, the situation is similar, impossible to orient yourself without having studied the mathematics of the Egyptians, the Greeks, and the Hindus. We come to the times after the invention of the printing press; it is something else entirely. As long as the Latin language was the language of scholars of any country, there was no border for books, and later, when we got used to writing each in his own idiom, the barely established borders were blurred again for those who, besides science, also possessed several modern languages. How then to discern, in the current of the common river, the sources of each small stream which contributed to it; how reciprocally to describe a source while stopping there or to pass beside it while neglecting it, before knowing if it was intended to produce, in another country, a river, or to dry up in the sand? I will be told that, however, all people had their time when they walked at the head of one science or another. It is perfectly true; but, because it is true for all people, this proves all the more the difficulty of writing the history of this science for a single people, if not for the time during which these people advanced this science. You will remember that in other words, I already said about the same thing when speaking of Cossali's works.

The intervals during which other people had to accomplish the same scientific mission will remain gaps, crevasses, so to speak, that one cannot neither jump nor go round without setting foot in foreign ground. How did Libri get out of this dilemma in his Histoire des Mathématique en Italie? He just evaded it. For him and for the reader who lets himself be carried away by an admirable style, by the accents of a warm but badly placed patriotism, it is only the Italians and a few French who have made progress in mathematics. He stops near some, he names the others in passing and that's all. Authors like Michael Stifel do not exist for him, and if he finds in an Italian, say in Tartaglia, inventions of this inconvenient predecessor, it is Tartaglia to whom we must be indebted and who is the author. You see by this little example that one can only follow Libri with great care. These precautions taken, it is indisputable that Libri rendered enormous services to the historiography of mathematics. He has studied a number of manuscripts, of which he gives extracts which are for the most part very exact, and, as I said already, he uses language with an art completely offline. His Histoire des Mathématique en Italie reads like a novel, even in the parts where it is not one.

A year after the fourth volume of Libri, the friends of the History of Mathematics were able to salute in Germany the appearance of a masterpiece worthy of being put alongside the Aperçu historique of Chasles. I am talking about Nesselmann's Histoire de l'Algèbre, to which I know only one fault, that of stopping after the first volume.

Not that time ran out for Nesselmann to continue the work of which he published the first volume at the age of 31, in 1842. Nesselmann lived until 1881, but, for reasons that I could not manage to clarify, he abandoned the path he had started with so much success, and, from 1845, he published only works on philology. We therefore have from him, like the History of Algebra, only the history of Greek algebra.

The chronological sequence of publications leads me to C-J Gerhardt, who died in May 1899, aged 83. His first occupation with a historical subject even goes back to 1837, when he wrote a Memoir: Sur les principes du calcul différentiel, awarded a prize by the University of Berlin who had asked this question as the subject of a university prize, and, since this time, Gerhardt did not cease devoting himself to the study of Leibniz, his rivals and his followers. He searched several times, in the library of Hanover, the literary trade and the manuscripts left by Leibniz, he found there documents of a date sometimes certain, sometimes probable, which he published, and which allowed him to make the true history of the invention of infinitesimal calculus. It is there and in the writing of Leibniz's Complete Works of Mathematics and Philosophy, entrusted to Gerhardt by the Berlin Academy, that his great merit rests which one should not dispute by relying on small errors which have slipped here and there into his works as in those of any other.

A reproach that one can rightly address to him is that of a certain contempt for contemporary work, which perhaps he did not even read, which he certainly did not use as he should have. It was because he was imbued with the conviction of being alone in reviving the studies of the History of Mathematics. But pride never fails to be harmful. Another task was assigned to Gerhardt, that of writing a History of Mathematics in Germany, and the small volume by which he discharged it, bears the date of 1877. Speaking of Libri, I said what I thought of the History of Mathematics in one country. The volume of 306 pages published by Gerhardt is no better for taking us from the year 700 approximately until 1850. Traveling, so to speak, in a balloon, it only stops on the tops of the mountains and still forgets quite a bit.

Adolphe Quételet, director of the Brussels observatory, who in his long life, from 1796 until 1874, deserved credit for the most varied sciences, published two volumes, one in 1864, the other in 1866 , on the History of Mathematical and Physical Sciences in Belgium. I quote them to admit that they seem to me to be much preferable to the work of which I have just said a few words.

I go back a few years to name the Histoire des Mathématiques pures of Arthur Arneth from 1852. The author was for several years my colleague in Heidelberg, where he was born in 1802 and died in 1858. Arneth wanted to tell the History Mathematics as part of the history of the development of the human mind, and, by marking this goal, there was posed a problem worthy of being resolved, but which cannot be dealt with in 291 pages.

The Introduction and a very general first Part teeming with remarks as spiritual as they are deep fill 67 pages; 73 pages are devoted to Greek Mathematics, 45 to those of the Hindus, 49 to those of the Arabs, the Romans and the Middle Ages, leading up to the middle of the 16th century. Do the calculations and you will find that there are just 58 pages left for the era starting in about 1550 and ending in 1800. That says it all. Arneth could have made an excellent, albeit laconic, superlative book if he had taken care to have the space agreed with the publisher. He neglected this first rule of an author (and let me add, in brackets, of a speaker too) and his book suffered.

The last work, which I want to talk about, had a very different fate from those I have named so far. All were published by their authors monitoring the publication and therefore making themselves responsible for it. If Montucla could not complete his second edition, at least he saw the first two volumes printed. Hermann Hankel died in 1873 at the age of only 34, before a page of his work on the history of mathematics was printed. He left only a chest, but a chest of such beauty that he would have been sorry not to bring it to light. There are inequalities, slight flaws that the artist's chisel would no doubt have removed if he had had the time, but the general beauty is only minimally diminished.

Hankel was, as you all know, a very talented mathematician, and he likewise possessed all the knowledge and all the skills to be desired for the historiographer of mathematics. The fragment published in 1874 after his death shows us this and makes us regret what we would have expected if he had lived.

I have taken you very briefly through a review of a fairly large number of works on the history of mathematics, and I could easily have doubled if not tripled the number. I have omitted all the works whose authors are still alive and of which I do not want to hurt either their modesty or the opinion, perhaps too favourable, that they have made of their work. I have omitted some which were written by authors just as dead as their books; let us beware of resuscitating them. I have omitted the authors who, like Maclaurin, Lagrangs, and Gauss, should have been mentioned for the admirable historical Introductions with which they have enriched their famous productions. I have also omitted to speak to you of a large number of writers who have composed only Historical Memories, either dispersed or united in a volume, but which do not constitute a coherent work. I think of Reimer, of de Morgan, of Biot, of Giesel, of Ofterdinger, of Bierens de Haen, which I certainly would not have been silent if I had not feared exceeding by much the time allowed.

But should I attract this reproach, I cannot pass in silence Prince Baldassare Boncompagni, this Roman patron who had devoted his time, his work, and his fortune to the worship of the History of Mathematics. His research on Gherard of Cremona, on Plato of Tivoli, and on Leonardo of Pisa is recognized as of great value, but it is especially with the edition made at his expense of the works of Leonardo of Pisa and various other authors, and by publication of the twenty volumes of his Bulletino distributed with a generous hand which he gave a lively and persistent impulse to the science he cherished. Boncompagni had assembled an unparalleled mathematical library containing more than six hundred manuscripts alongside thousands of printed books. It is to be regretted that no State or no Academy has had the desire or the means to acquire all these riches, which, subject to public sale, were dispersed without the possibility of return.

Perhaps there are people in my audience who are quite surprised to see how large is the cohort of authors who are entitled to the title of historian of mathematics. Indeed this branch has never been as neglected as we thought. However, it has never been as successful as it has been in the last twenty years or so. Science is feminine, and the feminine gender likes to get into fashion that always changes and comes back after a while. In science it is this or that subject that was formerly in vogue, which we hardly talk about today, which will come back in a new form. Now that Scientific Historiography has managed to get in favour, let me finish with a few words on the future way of writing the History of Mathematics.

Do I need to explain myself more fully on what I have in mind? I am in no way thinking of these parts of the History of Mathematics, which others have, of which I myself have tried to give a detailed account. I am thinking of the History of Mathematics since 1759, a time less fortuitous than it seems, since it is the year in which Lagrange published his first Memoir, this illustrious mathematician that Italy possessed from his birth in 1734 until 1766, then Germany until 1786, finally France until his death in 1813, the creator so to speak of modern mathematics. My excellent friend, M Paul Tannery, maintained, and many others agreed, that from this time it became impossible to treat the subject in question in the same way as in previous times. The sections which one is obliged to introduce into any history book in order to be able to orient oneself there would have multiplied from the year 1600 approximately by treating more and more shortened intervals, on the other hand the chapters being based on the matters treated would have increased disproportionately, where would we come to at the end? We ask for a new distribution consistent with the distribution of mathematical work that was discussed at the beginning of my talk. We now want, and certainly we are not entirely wrong, that the general history of mathematics give way to histories of geometry, analysis, algebra, etc., or even parts much more special. I say that we are not entirely wrong or, which amounts to the same thing, that we are not quite right, because I believe that what we ask for is not enough in itself.

Let me use an image. I was talking about fashions earlier. I saw earlier a ladies hairstyle of the end of century or new century, as you like, because I do not like the discussions on the tastes, the colours and the beginning of the century, I saw, I say, a hairstyle consisting of a multitude of intertwined locks which formed plaits each of which touched the others in an artistically combined manner. I tried to follow the contours of a single lock. Nothing easier. You just have to remove it from the hairstyle. Yes, but the hairstyle itself will be destroyed.

You will tell me that by basing a demonstration on images we can easily prove anything we want. I agree provided you agree that in my image there is truth and here is the consequence that I draw from it. I admit that the history of modern mathematics will need to be studied in various volumes each intended for the detailed history of a special branch.

But these partial stories, once written as essential preparations, will require a final volume summarizing everything, bringing out the great ideas of the century, no matter what problem that will appear to be created, because in the end the word that is credited to Jacobi, who would have started a university course with him, is very true. Mathematics is a science of which one cannot understand a part without knowing all the others. This last volume, the Histoire des Idées as I have allowed myself to call it, will be very difficult to compose, much more difficult than the volumes which will precede, but it will be essential. This is how I believe I understand the task of our successors.

JOC/EFR January 2020