# Pierre Ossian Bonnet

### Quick Info

Born
22 December 1819
Montpellier, France
Died
22 June 1892
Paris, France

Summary
Pierre Bonnet was a French mathematician who worked in the differential geometry of surfaces. He proved the Gauss-Bonnet theorem.

### Biography

Our first task is to explain Ossian Bonnet's name. His full name is Pierre Ossian Bonnet but he wrote all his papers and books under the name 'Ossian Bonnet', using 'Ossian' as his first name. His three sons, however, all sometimes used "Ossian-Bonnet" as their surname. We give more information about Bonnet's sons below.

Ossian Bonnet's parents were Pierre Bonnet (born about 1785) and Magdelaine Messac (born about 1795). In [1] Pierre Bonnet's occupation is given as 'commisbasnquier' but this word only appears in [1], and places copied from [1], and is neither in a current French dictionary nor in one with obsolete words, so I assume it is a misprint. It is certain that it is a misprint for 'commisbanquier', a 'bank clerk'. Other reliable sources, however, give Pierre Bonnet's occupation as 'négociant', a merchant or trader, see for example [5].

Bonnet attended the Collège in Montpellier, then in 1838 he entered the École Polytechnique in Paris. He also studied at the École des Ponts et des Chaussées. While still a student engineer at the Ponts-et-Chaussée he published his first paper Note sur l'intégrale $\large\int _0^\infty \Large\frac{x^{a-1}}{1+x}\normalsize dx$ (1841) in the Journal de Mathématiques Pures et Appliquées. In fact Bonnet published his first few papers in the Journal de Mathématiques Pures et Appliquées, the journal founded by Joseph Liouville in 1836 which is the second oldest mathematical journal in the world.

On graduating he was offered a post as an engineer. After some thought, however, Bonnet decided on a career in teaching and research in mathematics instead. His second paper was Note sur la convergence et la divergence des séries (1843). He introduces the paper as follows:-
M Augustus De Morgan and M Bertrand have found, each on their own, several systems of rules serving to recognise in a certain way the convergence or the divergence of series with positive terms. I have since come up with a few other rules of the same kind which are fairly simple to apply. The exposition of these new rules is the main object of this Note.
Turning down the engineering post had not been an easy decision since Bonnet was not well off financially. He had to do private tutoring so that he could afford to accept a position at the École Polytechnique in 1844. Nevertheless his position was sufficiently secure to allow him to marry and he married Aurélie Marie Louise Angélique Bruneau. They had three sons, Emile Pierre Ossian-Bonnet (1845-1901), Georges Pierre Alfred Ossian-Bonnet (1852-1912) and Gaston-Pierre Ossian-Bonnet. Let us say a little at this point about these three sons. Emile became a medical doctor and spent most of his life in Brazil and Tunisia. He became a professor at the Faculty of Medicine of Rio de Janeiro. Georges studied mathematics at the École Polytechnique from 1872 and published the paper Démonstration nouvelle de deux théorèmes de M Bertrand  (1883). He did not have a career as a mathematician, however, and became squadron leader in the artillery and became a knight of the Légion d'honneur. Gaston worked for the Ministry of the Colonies and was appointed head of office at the Colonial Central Administration in 1904.

Returning to the subject of this biography, as we saw above he had written a paper on the convergence of series with positive terms in 1843. Another paper on series in 1849 was to earn him an award from the Belgium Academy of Science. However between these two papers on series, Bonnet had begun his work on differential geometry in 1844 with the paper Sur quelques propriétés générales des surfaces et des lignes tracées sur les surfaces . This paper contains the famous Gauss-Bonnet theorem which gives a formula for the line integral of the geodesic curvature along a closed curve. Gauss had published a special case of this result and Bonnet writes in this paper:-
The illustrious geometer Gauss makes use, in the fine work 'Disquisitiones', of analytical considerations which are very interesting and very elegant, but which perhaps leave something to be desired in terms of simplicity. I proposed to ... take up the same questions using the methods of pure geometry.
You can see our list of 90 publications by Bonnet at THIS LINK.

Let us look at Bonnet's career before we return to look at his other mathematical work. His appointment to the École Polytechnique in 1844 which we mentioned above, was as a tutor in descriptive geometry. In fact Eugène Catalan, who was already working as an assistant tutor at he École Polytechnique, had been listed first for appointment to tutor in November 1844 but Bonnet was appointed to the tutor position leaving Catalan as an assistant tutor. Two admission examiners were appointed for the École Polytechnique in 1848Charles Hermite and Joseph Serret with Abel Transon (1805-1876) and Bonnet as their deputies. Again Catalan was a candidate but overlooked. There does appear to have been a certain tension between Bonnet and Catalan over the years. For example in his paper Observations sur les surfaces minima (1855) Bonnet writes:-
M Catalan does not seem to have had any knowledge of this work ...
Catalan responds in his paper Mémoire sur les surfaces dont les rayons de courbure en chaque point sont égaux et de signes contraires (1855):-
I would think I was disrespecting the Academy if I kept talking about the attacks and insinuations of M Pierre-Ossian Bonnet for longer.
Bonnet later became a tutor in analysis at the École Polytechnique. He also taught at the Sainte-Barbe College in Rue Valette in Paris. This school, founded in 1460, was designed to prepare students for sitting the entrance examinations for the École Polytechnique and the École Normale Supérieure. We learn a little about Bonnet's teaching there by examining his book Leçons de mécanique élémentaire (1858). He says he is a former pupil of the École Polytechnique and is a tutor in analysis there and states that the book is for candidates to the École Polytechnique and the École Normale supérieure. He writes in the Preface:-
This work is, with a few additions, the faithful reproduction of the lessons I have been teaching for several years at the Institution Sainte-Barbe. By publishing it, my main goal is to facilitate the work of my students.

The order I have followed and the point of view I have taken are those of the Official Programme; I have confined myself to developing the questions contained in this Programme by the most suitable and simple methods, having regard to the little mathematical instruction of my readers; thus it was necessary to proscribe the differential notations, and to disguise integration under the form of evaluations of areas. Hence the inevitable length.

Deeply convinced that poorly demonstrated truths are never well understood and often give rise to false ideas, I made myself a law of extreme rigour. All the results have been generalised with care; at the risk of appearing perhaps too thorough, I have explained in detail the meaning of each term used, and the meaning of each element considered.

I have constantly adopted the method of limits, with which the pupils are familiarised by the study of geometry. The method of infinitesimals is undoubtedly faster; but by exposing the true principles on which it rests, I feared adding new difficulties to those which are specific to mechanical engineering and to demand too much from the average intelligence of the pupils.

I thank, in closing, my friend and former student M Rouché for kindly helping me to correct the proofs.
On the title page of the book, Bonnet is given as Doctor of Science. In fact he had submitted two theses to the Faculty of Science in Paris on 15 April 1852 for his doctorate. One was a Mechanics Thesis, namely Sur le développement des Fonctions en Séries ordonnées suivant les Fonctions $X_{n}$ et $Y_{n}$ , while the second was an Astronomy Thesis, Sur la Théorie mathématique des Cartes géographiques . These were published as two consecutive papers in the Journal de Mathématiques Pures et Appliquées and also the two were put together and published as a book.

In 1853 Bonnet was appointed as a substitute professor of mathematical astronomy at the Collège de France standing in for Jacques Binet whose health had begun to deteriorate. In 1861 he was appointed as an Admissions Examiner for the École Polytechnique.

In 1860 there was a vacancy for the Academy of Sciences and four candidates were ranked in the order Joseph Serret, Ossian Bonnet, Victor Puiseux and Eugène Catalan. Serret was elected and Bonnet had to wait two years, then he was elected to the Academy in 1862 to replace Biot. He defeated Bour for this position which was slightly surprising for the following reason. In 1860 the Academy of Sciences had announced a prize competition on the problem of deformation of surfaces. The prize was:-
... to find all surfaces of a given linear element ...
Bour won the prize with his paper Théorie de la déformation des surfaces (1862), which is why we say Bonnet's election ahead of Bour was slightly surprising. Bonnet and Delfino Codazzi received honourable mentions for their submissions. Bonnet's entry was published as Mémoire sur la théorie des surfaces applicables sur une surface donnée . In it he used a special coordinate system for a surface such as isothermic and tangential coordinates. In this paper he also generalised Gauss's theorem on the sum of the angles of a geodesic triangle.

From 1868 Bonnet assisted Chasles at the École Polytechnique, and three years later he became a director of studies there. In addition to this post he also taught as a lecturer in differential calculus at the École Normale Supérieure from 1869 to 1872. Bonnet acted as a substitute professor for Chasles at the university of Paris in 1871-72 and again 1874-77.

In 1878 Bonnet succeeded Le Verrier to the chair at the Sorbonne, but he was dismissed from his director of studies position at the École Polytechnique in that year on account of false allegations about a scandal in his private life. This incident was not understood at the time so we have little chance of being able to understand it today. First we give the report in Le Nouvelliste [6]:-
Great revolution at the École Polytechnique. M Ossian Bonnet, one of the most distinguished teachers of the school has just been dismissed from his job. The cause: Cherchez la femme. In other words, M Ossian Bonnet, who is sixty years old, has a governess of about the same age as him, and the Ministry of War, which has the shyness and modesty of a young girl, could not any longer endure this scandal. At least that's the 'Temps' story.

As for the 'Siècle', it wrote gravely that M Ossian Bonnet is the victim of an infamous plot hatched by the Jesuits. These poor Jesuits, must they have a good back! They are often accused, in certain press, of being obscurantists; and every time it is a question of clarifying or rather confusing a question, they are quickly disguised willy-nilly with the role of traitor.

Anyway, it seems that the version of 'Temps' is the only true one, with this corrective which is very important: M Ossian Bonnet has never put his interests above those of his housekeeper.

Also, M Bardoux, Minister of Public Education, and all the teachers of the school have taken sides with their colleague and defend him 'tooth and nail'.

The Minister of War, with M Gambetta in reserve, sustained the shock without weakening and the battle was undecided.
Here is the report from Charles Douniol [4]:-
The sudden dismissal of Ossian Bonnet, the director of studies at the École Polytechnique, could provide a counterpart to Balzac's 'Murky Affair'. At the present time, after so many letters and articles have been exchanged, it is still impossible to know exactly what to make of this episode. The newspapers of the same shade were divided: 'Temps' against 'République française'. The 40-year-old housekeeper of the learned mathematician was thrown as a decoy into press discussions, hurled from the Capitol to the Tarpeian rock [where convicted people were flung to death in Ancient Rome], hoisted from the Tarpeian rock to the Capitol, thrown with scorn by some, offered by others for the good virtue prize. Scandal! shouted these. Admirable dedication! replied those. To the accusations of bad example those certifying good life and morals replied. Against the Minister and the Director of the School rose 'en masse' the colleagues of M Bonnet and his fellow members from the Faculty of Science. All that was needed was to involve the Jesuits in the affair. There was no lack of that: I read somewhere that M Bonnet was a supporter of the Jesuits and that is why he had been dismissed by a liberal minister; but I have read elsewhere that M Bonnet, on the contrary, is a victim of the Jesuits, whereupon public opinion has ended by no longer understanding anything; for me, that's where I started.
In 1883 Bonnet succeeded Liouville as a member of the Bureau des Longitudes.

Bonnet did important work on differential geometry in addition to Joseph Serret, Jean Frenet, Joseph Bertrand and Victor Puiseux who all worked in France on this topic. Bonnet made major contributions introducing the notion of geodesic curvature. Independently of Ferdinand Minding, Bonnet showed the invariance of the geodesic curvature under bending. Between 1844 and 1867 he published a series of papers on the differential geometry of surfaces [1]:-
In these and other papers Bonnet stressed the usefulness of special coordinate systems on a surface, such as isothermic and tangential coordinates; studied special curves, such as lines of curvature with constant geodesic curvature (1867); and investigated the conditions under which geodesic lines are the shortest connection between two points on a surface. He also paid much attention to minimal surfaces - for instance, those applicable on each other - and surfaces of constant total and constant mean curvature (1853).
Agustí Reventós Tarrida has written an excellent history of differential geometry in which he looks in detail at the work of all the mathematicians we have just mentioned, see [7]. The article is written in the Catalan language so, for those who cannot read Catalan, we have translated to English an extract of his description of Bonnet's work in this area; see THIS LINK.

Bonnet also published on cartography, algebra, rational mechanics and mathematical physics. In fact, despite most of his work being on differential geometry, as we have seen above, his two doctoral theses were on mechanics and cartography. We end this biography by giving the Introduction to Bonnet's Astronomy Theses on cartography:-
A geographical map is nothing more than a representation on a fixed surface, which is usually assumed to be a plane, of the surface of the earth or of one of its parts. This representation can be made according to any law; however, in the first maps that were constructed, it was always subject to the rules of perspective. The maps were thus simple conical or cylindrical projections, and the differences which existed between them, came from the position given to the eye and to the projection plane. We know these different maps, the most remarkable of which are due to Ptolemy. Later, some astronomers abandoned the mode of representation by perspective, which must have presented itself quite naturally to the mind, but which nothing required it to be followed, and they looked at the lines corresponding to the meridians and parallels, like any lines that one could, in each case, choose arbitrarily, according to the destination of the map. This is how the reduced nautical charts or increasing latitudes charts were constructed, in which the condition was imposed that the wind rumbs were represented by straight lines forming between them the same angles as these rumbs made in the compass rose.

Finally, Lambert considered the theory of geographical maps from an extremely important general point of view. He observed that the greatest degree of perfection of a map was to reproduce the figure of the different parts of the map, so that there was constantly similarity between any part of the earth and the corresponding part of the map; but this condition being generally impossible to fulfil, unless one supposes on the surface of the chart a particular form, Lambert proposed to determine the lines of the meridians and the parallels by the condition that the similarity take place only between the infinitely small elements. No doubt, in this way, a finite portion of the earth was distorted on the map, but the angles made on the map were always equal to the corresponding angles on the surface of the globe, an important property which had been recognised by the method of representation of Ptolemy. Lambert did not resolve in a complete manner the general problem which he had posed himself, and after him, several mathematicians, Euler, Lagrange, dealt with the question successfully, but always assuming the form of a sphere or at most a surface of revolution; it was M Gauss who, in a Memoir awarded a prize by the Academy of Copenhagen, was the first to solve the problem in all its generality and without making any assumptions about the surface of the earth and that of the map. We propose, in this Thesis, to expose the theory of geographical maps from the point of view indicated by Lambert. Our work is not essentially new. Our only goal was to simplify the solutions of the questions dealt with before us by Lagrange, Euler, and M Gauss.

### References (show)

1. P E Pilet, D J Struik, Biography in Dictionary of Scientific Biography (New York 1970-1990).
2. P Appell, Notice sur la vie et les travaux de Pierre Ossian Bonnet, Comptes rendus de l'Académie des Sciences 117 (1893), 1013-1024.
3. M Chasles, Rapport sur les progrès de la géometrie en France (Paris, 1870), 199-214.
4. C Douniol, Le Correspondant 114 (1879), 150.
5. F Huguet and B Noguès, Bonnet, Pierre Ossian, Les professeurs des facultés des lettres et des sciences en France au XIXe siècle (1808-1880) (June 2011).
http://facultes19.ish-lyon.cnrs.fr/fiche.php?indice=118
6. Mardi, Petite Revue de la Semaine, Le Nouvelliste (Sunday 8 December 1878).
7. A R Tarrida, Notes sobre els inicis històrics de la geometria diferencial, Universitat Autònoma de Barcelona.
http://www.mat.uab.cat/~agusti/hgd.pdf

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Written by J J O'Connor and E F Robertson
Last Update September 2020