# André Sainte-Laguë

### Quick Info

Saint Martin de Curton, Lot-et-Garonne, Aquitaine, France

Paris, France

**André Sainte-Laguë**was French mathematician who is famed for his work in inventing a proportional voting method and for writing a thesis which may be thought of as the first ever textbook on graph theory. He promoted new methods of education and was a leader of many associations.

### Biography

**André Sainte-Laguë**was born in the village of Saint Martin de Curton in southwest France, between Bordeaux and Toulouse. He was the son of Jean Abel Sainte-Laguë (1857-1924) and Claire Desgans (1860-). Abel Sainte-Laguë, the son of a farmer, was a school teacher in Casteljaloux who married Claire Desgans, the daughter of a sailor, in Gujan-Mestras, Gironde, Aquitaine on 11 December 1880. They had two sons, André Sainte-Laguë, the subject of this biography born in 1882 and Pierre René Sainte-Laguë, born on 16 November 1883 in Gujan-Mestras, Gironde, Aquitaine. Pierre René Sainte-Laguë died in Morocco in 1914; we say a little more about this below.

André's family moved to Haiti where he spent his early years. He was sent back to France to become a boarder at a boys' high school in Agen, Lot-et-Garonne. He was awarded his baccalaureate in 1899 and then spent two years from 1900 to 1902 in the special mathematics class at Lycée Montaigne in Bordeaux, Gironde, preparing to take the entry examinations for the Grandes Écoles. He took the examinations in 1902 and was offered a place in both the École Polytechnique and the École Normale Supérieure on Rue d'Ulm. He chose the École Normale Supérieure but did a year of military service before beginning his studies in 1903. He passed the Aggregation of mathematics in 1906 being ranked sixth.

After graduating from the École Normale Supérieure, Sainte-Laguë began his teaching career. He was employed first as a provisional substitute for the professor of mathematics at the high school of Évreux, the capital of the department of Eure, Normandy. He spent the year 1906-07 in this post, and while he was employed there he married Adélaïde Menu on 24 January 1907 in Paris. Adélaïde Geneviève Georgette Menu (6 February 1878 - 18 May 1960) had been born in Paris, the daughter of a military doctor. André and Adélaïde Sainte-Laguë had three children: Madeleine Christiane Sainte-Laguë (1910-1998); Jean Bertrand Sainte-Laguë (1911-1993); and Fernande Arlette Sainte-Laguë (1918-2004). Madeleine was awarded a degree in engineering from the École Centrale Paris, became an aeronautical civil engineer and married the engineer Henri Eugène Adolphe Mazuel (1906-1950); Jean Bertrand was also awarded a degree in engineering from the École Centrale Paris and became an engineer; and Fernande married the engineer Paul Jean Joseph Burty (1910-2002).

In 1908 Sainte-Laguë was appointed as professor of special mathematics at the high school of Douai, Nord. It was during his time at this school that he published the paper for which he is probably best known today, namely

*La représentation proportionelle et la méthode des moindres carrés*Ⓣ [44]:-

In 1882 Professor Victor d'Hondt, of the University of Ghent in Belgium, proposed the system of party proportionality that bears his name, which is widely used in Europe today. ... D'Hondt proposed that one should allocate the seats to the parties one at a time. At each stage, a party $P_{i}$ with $v_{i}$ votes and (currently) $s_{i}$ seats is assigned the quotient $\Large\frac{v_i}{s_i+1}$, and the next seat is given to the party with the largest quotient. ... The Frenchman A Sainte-Laguë devised his arithmetic-mean system (but in the form 1, 3, 5, 7 ... ) in 1910, in order to correct the tendency of d'Hondt's rule to favour the large parties.Michael Gallagher writes in [9] that the Sainte-Laguë method:-

... at the theoretical level is probably the soundest of all the measures.Today the Sainte-Laguë method is used in many countries, for example Germany, New Zealand and Sweden.

Sainte-Laguë taught at Douai until 1912 when he moved to the Lycée de Besançon in Doubs. In July 1914 he was appointed to the Lycée - Cité Scolaire Pasteur in Neuilly-sur-Seine, Paris. He had only just taken up this position, however, when war broke out in Europe and on 2 August 1914 Sainte-Laguë was called up for military service. Germany declared war on France on the following day and Sainte-Laguë began training as a sergeant. In January 1915 he was sent to the front as a second lieutenant and soon was promoted to lieutenant. Twice wounded, he was able to continue fighting until 24 June 1916 when he was wounded for the third time. His thigh bone was shattered by shrapnel [7]:-

... while going to ensure the restoration of a telephone line himself, under a violent bombardment.He had five operations on his wounded thigh, and was declared 'unfit for active service'. In 1916, for his courageous action, he was awarded the Croix de Guerre avec Palmes, and made Chevalier de la Légion d'Honneur Militaires. His time in military hospitals was not wasted for there he had the opportunity to continue his research on graph theory that he had begun while a student at the École Normale Supérieure. When he was recovered sufficiently to leave hospital, he still had to undertake war service. Not fit for active duty, in 1917 he was sent to study long range shells at the Inventions Department of the École Normale Supérieure. He had an interest in aviation since his brother Pierre Sainte-Laguë had trained at the Blériot military aviation centre in Pau and later died in an accident on 4 May 1914 in Lalla Itto in Morocco when returning from a reconnaissance mission.

After completing his war work, Sainte-Laguë was able to resume his appointment at the Lycée Pasteur which had been made just before the outbreak of war. He was given leave of absence from 25 April to 30 September 1919 before starting teaching there. He also taught at the Lycée Carnot which was restarting its normal school activities after part had been turned into a Franco-Belgian hospital during the war. Many teachers had lost their lives during World War I so those who had survived were in great demand. At the start of the 1920 school year, Sainte-Laguë took up an appointment at the prestigious Lycée Janson de Sailly where Élie Cartan had been a pupil thirty years earlier. At the same time he began teaching at the connected Centrale assisting Salomon Bloch (1858-1926) who was teaching large numbers of students preparing for the Grandes écoles entrance examination. Bloch was a highly successful teacher who published several books including the two volume work

*Cours de géométrie descriptive à l'usage des candidats à l'École polytechnique, à l'École normale supérieure, aux-écoles centrale, des arts et manufactures, des ponts et chaussées et des mines de Paris et de Saint-Étienne*Ⓣ (1921). Bloch was allowed to retire on 1 April 1922 "at his request and for seniority of age and services" and Sainte-Laguë took over his teaching in addition to his own. Sainte-Laguë was also an admissions examiner for the École Supérieure d'Aéronautique from 1922 to 1929.

In parallel with these teaching commitments, Sainte-Laguë was working on undertaking research for a doctoral thesis. He submitted two theses,

*Les réseaux*Ⓣ and

*Surfaces minima*Ⓣ to the Faculty of Science in Paris on 6 June 1924. His main thesis

*Les réseaux*Ⓣ is a remarkable piece of work on what today we would call graph theory. He begins by giving the definition of a graph (which he calls a network) which would not look out of place in a 21st century graph theory text [25]:-

We will call a network a set of points or crossroads which will consist of vertices, joined together by lines or sides which will be the paths of the network. The shape of these paths is of no interest and the only thing that matters for two given points A and B is whether or not they are joined by one or more paths. Two networks will be said to be homeomorphic if one can establish between their vertices on the one hand, as well as between their paths on the other hand, a reciprocal and unambiguous correspondence. Two homeomorphic networks will be considered identical. It follows that the shape of the diagrams, planes or curves, by which it may be convenient to represent networks, has no theoretical importance. The location of the vertices can be arbitrary as well as the shape of the paths joining two given points. Care must be taken not to confuse the graphic crossing of two paths drawn on paper with a crossroads.He ends this thesis with the following conclusion:-

The study of networks can be pursued in many different ways and each of the definitions posed at the beginning makes it possible to initiate new research. We simply wanted to study, as best we could, two of the simplest cases one could think of. Simple as they are, they show, we believe, the complexity of the questions raised and the diversity of the methods which it is essential to employ. The subject, limited as it may seem at first glance, is in fact .. very vast and seems quite difficult. We have systematically left aside, so as not to lengthen the thesis, not only all the research that we have already started on spherical networks, but also all the practical applications of networks. Many questions of higher arithmetic, positional geometry, game theory, for example the research of Lucas, is immediately related to network theory. Other applications of more immediate use could also be considered, such as certain questions of graphic statics or especially of stereochemistry. We couldn't think of addressing them here.The examining committee for the thesis consisted of Émile Picard, as president, and Émile Borel and Paul Montel as examiners.

The significance of this thesis is clearly explained in the Foreword to [10] written by Michel Habib. The Foreword begins:-

The appearance in 1926 of 'Les Réseaux (ou Graphes)' was an important milestone in the historical development of graphs and their applications, from its first introduction by Euler (1735) with his famous problem on the bridges of Königsberg, to some early applications in chemical theory with the graphs of organic molecules. Most of the research in graph theory in the beginning of the last century still came from mathematical games and puzzles, but in his introduction, Sainte-Laguë seemed to be convinced of the great potential of applications of graph theory. Sainte-Laguë's book was the first textbook devoted to the study of graphs, containing most of what was known at his time, as for example, the four colour problem. In many aspects his book is very modern, by its subject, its presentation, and with many examples. It is interesting that many important notions had already been defined: trees, centres, chains, cycles, Eulerian cycles, Hamiltonian cycles, etc., although some of the notations are not the ones we use today.For more information about

*Les Réseaux (ou Graphes)*, see THIS LINK.

In 1927 Sainte-Laguë was appointed as a lecturer in mathematics at the National Conservatory of Arts and Crafts. The professor of mathematics was Raoul Bricard (1870-1943) who had published on Hilbert's Third Problem in 1896 (before Hilbert included it in his list in 1900) and had investigated flexible polyhedra. He had been awarded the Poncelet Prize in mathematics from the Paris Academy of Sciences in 1932 for his work in geometry. When Bricard retired in 1938, Sainte-Laguë was appointed to the chair of mathematics, a position he retained until his death in 1950. He was very successful as a lecturer as Jérôme Chastenet de Géry relates in [5]:-

His classes enjoyed considerable success, unequalled up to that time, and one of his lectures had around 2500 listeners, requiring him to give it three times in the large 900-seat Paul Painlevé amphitheatre at the Conservatoire National des Arts et Métiers. His warm, loud voice filled the lecture theatre, and his lectures were lively, fast, and clear. When teaching mathematics to practitioners, he insisted on the necessity of theoretical, numerical, and graphical exercises, and also on the need to distinguish between rigorous reasoning and that which is not rigorous, as well as the importance of using correct language. "We must not respond to the students' desire to know only recipes," he said. A forerunner in what may now be called 'new educational technologies', from 1928 onwards, he also used films for his geometry lessons.Sainte-Laguë was not only an outstanding teacher, but he also played major roles in organisations. The Société des agrégés was founded in 1914 with the aim of ensuring fair and demanding education for all regardless of their status in society. Sainte-Laguë was elected president serving from 1917 to 1919. He was one of the founders of the Association Amicale des Anciens Combattants de l'enseignement supérieur et de l'enseignement secondaire publics, and is listed as one of two Vice-Presidents for 1924-25. This Associations aimed to:-

... participate in the preparation of school youth, and to study and to bring about all the reforms concerning education.He was also a founding member of the Confédération des Travailleurs Intellectuels in March 1920 and remained a staunch supporter all his life. He was its president from 1929 until his death. In January 1932 the Confederation proposed a programme with three main aims: improved wages, availability of work, and reduction in working hours. France, like so much of the world, was suffering during the Great Depression at this time and the Confederation stated [6]:-

We are not afraid to call the current crisis a crisis of our economic regime and consider participating in a study of ways to adapt the general economy to new conditions. We feel that the world is in the process of being renovated and that we can no longer, as immediately after the war, wait for others to think for us, intellectual workers. The new regime must be with our support, otherwise it will be against us.Sainte-Laguë was interviewed in his role as president of the Confederation in June 1932. He said (see for example [6] or [7]):-

Capitalism, considered here simply as an economic phenomenon and a financial system, seems to have had its day. It has fulfilled its role and it has done great things just as feudalism once did, but in more recent times it has not been able to foresee anything, or to organise anything.The Exposition Internationale des Arts et Techniques dans la Vie Moderne was held from 25 May to 25 November 1937 in Paris. Le Palais de la Découverte was to have mathematics rooms and these were under the direction of Émile Borel and Paul Montel. They asked Sainte-Laguë to organise the mathematics displays and, beginning the work in 1933, he wrote the book

*Avec des Nombres et des Lignes: Récréations Mathématiques*Ⓣ (see [32]). Martin Charles Golumbic writes in [10]:-

In the 1937 brochure of the Section des Mathématiques, André Sainte-Laguë is listed as the secretary and organiser of the exhibition. The other committee members were Émile Borel (president), Paul Montel (vice-president), Raoul Bricard and Georges Darmois. In Salle 31 (Room 31), we have the following: "Three albums of curves by A Sainte-Laguë, present 50 curves each to the visitors: algebraic, transcendental, and various others."The official volume on the Palais de la Découverte from the 1937 Paris Exposition reports:

For those visitors who are fascinated by the research carried out by engineers, and who rave at the wonders of modern industry, the organisers presented some of the properties of the strength of materials in the simple and easily assimilated form of a film. The film, 'De la similitude des longueurs et des vitesses' Ⓣ was written by André Sainte-Laguë.On 1 September 1939 Germany invaded Poland and, two days later, France declared war on Germany. On 10 May 1940 Germany invaded Holland and two days later the Panzer Corps had crossed into France and had reached the Meuse. Sainte-Laguë immediately urged resistance against German occupation. At the general assembly of the Confédération des Travailleurs Intellectuels in the spring of 1940 he supported the setting up of a Solidarity Centre for intellectual workers. He said [7]:-

Even more in times of war than in times of peace, 'trade unionism' must be synonymous with 'solidarity'.He was one of the founders of the Organisation Civile et Militaire. Jérôme Chastenet de Géry writes in [5]:-

He participated in the underground resistance in September 1940 and, known to be of a liberal mind, he was arrested by the Germans at his home in early October 1941 and imprisoned. But he knew how to remain discreet, and after many interrogations, the Germans, not finding evidence of his participation in clandestine activities, released him. When he resumed his class, he simply said to his pupils, "Gentlemen, I have been able to reflect for a long time, where I was, on the properties of unicursal curves ... ."Another of his wartime activities is related by Laurent Schwartz in [40]:-

It was not until the Liberation that they learned that he had been a leader of the Organisation Civile et Militaire. He was decorated with the medal of the Resistance, and was appointed a member of the Provisional Consultative Assembly.

A cousin of my mother's, Guy Iliovici, a professor of mathematics in the Lycée Saint-Louis, also remained in Paris and wore a yellow star. Because he was Jewish, he lost his job, but his colleague and close friend Sainte-Laguë gave him half of his salary. He was deported with his wife Suzanne and his daughter Janine; they never returned.We have mentioned some of Sainte-Laguë's publications above but these are only a very few from the large number covering a wide range of topics. We list a few in references: [13], [14], [17]-[19], [22]-[38].

John McMasters comments in [20] about the claim that an engineer "proved that bumblebees should not be able to fly." It appears that Sainte-Laguë was involved in this "proof." August Magnan writes in [16]:-

At first prompted by what is done in aviation, I applied the laws of air resistance to insects, and I arrived with A Sainte-Laguë at this conclusion that their flight is impossible.The Bumblebees Conservation Trust explains the apparent paradox as follows:-

This myth stems from a well-known story of some engineers who proved that bumblebees shouldn't be able to fly because their wings are too small for the size of their bodies. In reality, bumblebees fly in quite a complicated way with their four wings, they don't just flap them up and down which probably would make it impossible for them to fly. In fact, they flap their wings front to back and simultaneously rotate them, like a figure-8, to create enough lift!Sainte-Laguë died suddenly at the age of 67. He had continued his heavy teaching load up to the time before his death. He had been a member of the committee of the Société des Amis de l'Institut Métapsychique since 1934, was its vice-president in 1949 and, just before his death, had accepted becoming president in 1950. His granddaughter Dominique Sainte-Laguë (born 18 August 1946) gave many documents concerning her grandfather to the Conservatoire National des Arts et Métiers in 2016 which allowed historians to better understand Sainte-Laguë as a pioneer of new educational technologies and of graph theory [41]:-

Dominique Sainte-Laguë, granddaughter of André Sainte-Laguë (1882-1950), professor of 'General Mathematics with a View to Applications' at the National Conservatory of Arts and Crafts from 1938 to 1950, has just donated to the central library of the Conservatoire National des Arts et Métiers (CNAM) a set documents related to her grandfather's activities. This gift, made up of printed matter, books and off-prints and manuscripts, as well as a set of photographs and glass plates that served as conference materials, provides a better understanding of the popularising and teaching activity of this pioneer of new educational technologies and graph theory.

...

This fund now enriches the collections of the CNAM, which did not keep all the publications of André Sainte-Laguë, and allows us to better understand his activity as a lecturer and writer, through numerous off-prints, as well as a set of articles published in the press and the manuscripts of his works such as that of 'La Machine humaine' Ⓣ, published by Fayard under the title 'De l'homme au robot' Ⓣ in 1953. The knowledge of the mathematician and the teacher, as well as that of his methods will no doubt benefit from the study of his manuscript notebooks.

### References (show)

- A-S Aguilar, Les artistes et le syndicalisme intellectuel dans la France de l'entre-deux-guerres: Le cas de la Confédération des travailleurs intellectuels, in
*Jean-Philippe Garric (ed.), Les dimensions relationnelles de l'art Processus créatifs, mise en valeur, action politique*(Éditions de la Sorbonne, Paris, 2018), 173-195. - André Sainte-Laguë Papers,
*International Institute of Social History*.

https://archive.wikiwix.com/cache/index2.php?url=http%3A%2F%2Fhdl.handle.net%2F10622%2FARCH02477#federation=archive.wikiwix.com&tab=url - R C Archibald, Review: Notions de Mathématiques, by André Sainte-Laguë,
*Bulletin of the American Mathematical Society***19**(1913), 421-422. - R Brasseur, André Sainte-Laguë,
*Dictionnaire des professeurs de mathématiques spéciales 1852-1914*(7 June 2012).

https://archive.wikiwix.com/cache/index2.php?url=https%3A%2F%2Fsites.google.com%2Fsite%2Frolandbrasseur%2F5---dictionnaire-des-professeurs-de-mathematiques-speciales#federation=archive.wikiwix.com&tab=url - J Chastenet de Gery, Sainte-Laguë, André (1882-1950): Professeur de Mathématiques générales en vue des applications (1938-1950), in
*Claudine Fontanon and André Grelon (eds.), Les Professeurs du Conservatoire National des Arts et Métiers, Histoire Biographique de l'Enseignement*(INRP, Paris, 1994). - A Chatriot, La lutte contre le « chômage intellectuel » : l'action de la Confédération des Travailleurs Intellectuels (CTI) face à la crise des années trente,
*Le Mouvement Social***214**(1) (2006), 77-91. - A Dalançon, Sainte-Laguë Jean [André],
*Le Maitron*(17 November 2021).

https://maitron.fr/spip.php?article195138 - V Dančišin, Nedocenený André Sainte-Laguë,
*Annales Scientia Politica***4**(1) (2015), 56-61. - M Gallagher, Proportionality, disproportionality and electoral systems, Electoral studies 10 (1) (1991), 33-51.
- M C Golumbic and André Sainte-Laguë,
*The zeroth book of graph theory - an annotated translation of Les Réseaux (ou Graphes) - André Sainte-Laguë (1926), Lecture Notes in Math.***2261**(Springer, Cham, 2021). - H Gropp, On configurations and the book of Sainte-Laguë,
*Discrete Math.***191**(1-3) (1998), 91-99. - M Habib, Foreword, in
*Martin Charles Golumbic and André Sainte-Laguë, The zeroth book of graph theory - an annotated translation of Les Réseaux (ou Graphes) - André Sainte-Laguë (1926), Lecture Notes in Math.***2261**(Springer, Cham, 2021), vii. - G Iliovici and André Sainte-Laguë, Mathématiques Appliquées à l'Usage des Ingénieurs, des Élèves-ingénieurs et des Étudiants des Facultés des Sciences (Eyrolles, Paris, 1933).
- G Iliovici and André Sainte-Laguë, Cours d'Algèbre et d'Analyse à l'Usage des Élèves de Mathématiques Spéciales (2 Volumes) (Eyrolles, Paris, 1933).
- A Lijphart and Robert W Gibberd, Thresholds and payoffs in list systems of proportional representation,
*European Journal of Political Research***5**(1977), 219-244. - A Magnan,
*Le vol des insectes*(Hermann and Cle, Paris, 1934). - A Magnan and André Sainte-Laguë, Étude des trajectoires et des qualités aérodynamiques d'un avion par l'emploi d'un appareil cinématographique de bord (Gauthier-Villars, Paris, 1932).
- A Magnan and André Sainte-Laguë, Le vol au point fixe, Actualités Scientifiques et Industrielles 60 (Hermann, Paris, 1933).
- A Magnan and André Sainte-Laguë, De Quelques Méthodes en Morphologie (Masson, Paris, 1933).
- J McMasters, The Boeing company, Seattle, Washington.

https://web.archive.org/web/20071012143442/http://www.math.niu.edu/~rusin/known-math/98/bees - A Quirós Gracián, D'Hondt does not take away earned seats, Sainte-Laguë maybe does (Spanish), Gac. R. Soc. Mat. Esp. 22 (1) (2019), 60.
- A Sainte-Laguë, La représentation proportionelle et la méthode des moindres carrés, Académie des Sciences: Comptes Rendus Hebdomadaires 151 (1910), 377-378.
- A Sainte-Laguë, La représentation proportionelle et la méthode des moindres carrés, Annales scientifiques de l'É.N.S. (3) 27 (1910), 529-542.
- A Sainte-Laguë, Notions de Mathématiques (Hermann, Paris, 1913).
- A Sainte-Laguë, Les Réseaux (thesis Privat à Toulouse) (Hermann, Paris, 1924).
- A Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques 18 (Gauthier-Villars, Paris, 1926).
- A Sainte-Laguë, Notions de Géométrie Vectorielle (Vuibert, Paris, 1927).
- A Sainte-Laguë, Géométrie de Situation et Jeux, Mémorial des Sciences Mathématiques 41 (Gauthier-Villars, Paris, 1929).

http://eudml.org/doc/192569 - A Sainte-Laguë, Probabilités et Morphologie, Actualités Scientifiques et Industrielles 36 (Hermann, Paris, 1932).
- A Sainte-Laguë, Les Outils du Mathématicien (Eyrolles, Paris, 1933).
- A Sainte-Laguë, La Règle à Calcul (Eyrolles, Paris, 1934).
- A Sainte-Laguë, Avec des Nombres et des Lignes: Récréations Mathématiques (Vuibert, Paris, 1937).
- A Sainte-Laguë, Du Connu à l'Inconnu (Gallimard, Paris, 1941).
- A Sainte-Laguë, Le Monde des Formes (Fayard, Paris, 1948).
- A Sainte-Laguë, Géométrie Descriptive et Géométrie Cotée (Eyrolles, Paris, 1948).
- A Sainte-Laguë, Algèbre, Analyse et Géométrie Analytique (Eyrolles, Paris, 1948).
- A Sainte-Laguë, Dessinons un Carré (Vuibert, Paris, 1950).
- A Sainte-Laguë, De l'Homme au Robot (Hermann, Paris, 1953).
- L Schwartz,
*Un Mathématicien aux Prises avec le Siècle*(Odile Jacob, Paris (1997), 215. - L Schwartz,
*A Mathematician Grappling with His Century*(Birkhäuser, Boston, 2001). - Un don André Sainte-Laguë rejoint les collections de la bibliothèque centrale du CNAM,
*Conservatoire national des arts et métiers*(December 2016).

https://listes.services.cnrs.fr/wws/arc/athena/2016-12/msg00039/Communiqu___Andr___Sainte-Lagu__.pdf - Y Verneuil, Valeurs et combats de la Société des agrégés depuis 1914,
*Revue d'histoire***77**(1) (2003), 69-84.

https://www.cairn.info/revue-vingtieme-siecle-revue-d-histoire-2003-1-page-69.htm - R Wilson, J J Watkins, D J Parks,
*Graph Theory in America: The First Hundred Years*(Princeton University Press, 2023). - D R Woodall, How Proportional is Proportional Representation?,
*The Mathematical Intelligencer***8**(4) (1986), 36-46.

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Written by J J O'Connor and E F Robertson

Last Update September 2023

Last Update September 2023