Every day during these years of hard work he descended from the highlands of Condroz to the foot of the rocks bordering the picturesque Hoyoux valley. With the enthusiasm of youth, and his happy enthusiastic nature, he went through the woods of Sandron, of Barse and of Bailli, whose invigorating scent cleansed him and prepared him for the hard daily exercises and the intellectual duals that he wanted to win. And in the evening, when he retook the same road ... he meditated amid the splendid nature, preparing to revise, to work on, in the family retreat, the lessons that with sustained effort, the result of his iron will, he had been able to assimilate on the benches of the school.Mansion was particularly grateful to three teachers at the College for the positive influence they had on him, namely J Poumay, who taught him French and German, G Smiet, who taught him mathematics, and J Kunders , who taught him Latin and Greek. He left the College in Hoy and entered the École Normale des Sciences, attached to the University of Ghent, on 15 October 1862. He graduated with his first degree on 3 July 1865 and continued to undertake research for his doctorate. In November 1865 he began lecturing on advanced algebra, analytic geometry and descriptive geometry at the School of Civil Engineering, attached to the University of Ghent. He taught these courses for two years while he undertook research guided by Félix Dauge and Mathias Schaar. He submitted his thesis Théorie de la multiplication et de la transformation des fonctions elliptiques Ⓣ and defended it before a jury of professors of the University of Ghent and the University of Brussels on 3 August 1867. Lucien Godeaux writes that the thesis :-
... studied, by a particularly simple original method, the multiplication and transformation of elliptic functions.Mansion was awarded his doctorate in mathematical and physical sciences with distinction.
Let us say a little about Félix Dauge (1829-1899) and Mathias Schaar (1817-1867) who had such a strong influence on Mansion and on the direction of his research throughout his life. We quote from Mansion himself (see ):-
Dauge, whose student I had the good fortune to be, was an eminent professor. He taught advanced algebra, analytic geometry, astronomy and mathematical methodology. His methodology course was especially remarkable. ... [Schaar] raised the level of studies at the University of Ghent by introducing into the doctoral courses the first principles of the general theory of functions and the theory of elliptic integrals. [His work on] Stirling's formula and the law of quadratic reciprocity gives his name a lasting place in the history of mathematics.Schaar was not quite fifty years old when he died in April 1867 leaving the Chair of Differential and Integral Calculus and Higher Analysis vacant. By October 1867 Mansion, who had only been awarded his doctorate two months earlier, had been appointed to teach Schaar's advanced mathematics courses. He was appointed as an extraordinary professor of mathematics at the University of Ghent in 1870 and became a full professor four years later. After his appointment as an extraordinary professor, Mansion married to Marie-Cécile Belpaire (1846-1924), the daughter of Alphonse Belpaire (1817-1854), a mathematician, poet and musician, and Elizabeth Teichmann (1821-1900). Paul and Marie-Cécile Mansion had several children: Alphonse Mansion 1872-1873, Marie Mansion 1874-1912, Cécile Mansion 1875-1894, Joseph Mansion 1877-1937, and Victor Mansion 1879-1902. In 1892 Mansion succeeded Emmanuel-Joseph Boudin when he was appointed to the Chair of the Calculus of Probabilities at Ghent.
According to J Pelseneer , Mansion:-
... held an eminent position in the scientific world of Belgium despite his extreme narrow-mindedness.Alphonse Demoulin writes :-
Author of many works on mathematical analysis, the calculus of probabilities, non-Euclidean geometry, the history and philosophy of science, [Mansion] held a prominent place in the Belgian scientific world.In fact Mansion's first publication, appearing before he submitted his doctoral thesis, was a 13-page paper on probability Sur le problème des partis Ⓣ which appeared in the Mémoires of the Royal Belgium Academy of Science in 1868. This was the first of his many publications on probability. We give a few examples of works on this topic published near the end of his career: Démonstration du théorème de Bernoulli Ⓣ. Sur une intégrale considérée en calcul des probabilités Ⓣ. Sur une intégrale considérée par Poisson en calcul des probabilités Ⓣ (1902), Sur la portée objective du calcul des probabilités Ⓣ (1903), Sur une intégrale considérée en calcul des probabilités Ⓣ (1904), and Exceptions apparentes au théorème de Jacques Bernoulli en calcul des probabilités Ⓣ (1913).
The Royal Belgium Academy of Science proposed for its prize competition for 1871 the task "to summarise and simplify the theory of partial differential equations of the first two orders". This was a vast and difficult undertaking and Mansion decided to enter but to restrict himself to the theory of first order partial differential equations. The competition was extended by two years setting 1873 as the new date for submissions. Mansion submitted the 289-page memoir Mémoire sur la théorie des équations aux dérivées partielles du premier ordre which was judged the winning entry. Joseph de Tilly, one of the commissioners judging the entries, is quoted in the report as stating that:-
Mansion has produced a considerable body of work in which he has shown deep analytical knowledge and vast erudition.In 1875 Mansion's winning entry was published in the Mémoires of the Royal Belgium Academy of Science. An extended German translation of this work was published as Theorie der partiellen Differentialgleichungen erster Ordnung in 1892. The scale of the additions are clear when we note that this German publication ran to 489 pages.
Of course living in Ghent made Mansion particularly aware of the famous mathematician Adolphe Quetelet who was born in that city some 50 years before Mansion was born, and who had studied at the university there. Quetelet and Jean-Guillaume Garnier (1766-1840), the professor of astronomy and higher mathematics at Ghent, had edited the Belgium publication Correspondance mathématique et physique and in 1874 Mansion, together with Eugène Catalan and Joseph Neuberg, founded the journal Nouvelle correspondance mathématique named to honour the earlier Correspondance mathématique et physique and to follow the naming pattern set by the French journal Nouvelles Annales de Mathématiques which followed the Annales de Mathématiques. The journal which Mansion, Catalan and Neuberg founded was published between 1874 and 1880. During these years Mansion published 15 papers in the journal, for example: Démonstration d'un théorème de Liouville Ⓣ (1874); Théorie analytique des transformations linéaires Ⓣ (1874, 1876, 1887, 1888); Sur les carrés magiques Ⓣ (1876); Remarques sur les théorèmes arithmétiques de Fermât Ⓣ (1879); and Dérivées des fonctions élémentaires d'une variable imaginaire Ⓣ (1880). After this journal ceased publication, Catalan encouraged Mansion and Neuberg to collaborate in publishing a new journal and, indeed, they did precisely this, publishing Mathesis from 1881 onwards. Mansion became director of Mathesis and continued with this project until he retired in 1910. He published over 80 papers and notes in Mathesis up to 1915, the time that it temporarily stopped publication because of the difficulties caused by World War I.
After he retired in 1910, Mansion was made professor emeritus on 9 November of that year. He continued to publish numerous article on all the topics that had interested him throughout his career, for example Quadrilatère asymptotique lobatchefskien Ⓣ (1910), Sur un principe de calcul des probabilités Ⓣ (1911), Sur les figures isopérimètres en géométrie plane Ⓣ (1913), and Sur la constante spatiale en géométrie non euclidienne Ⓣ (1914). In 1916 he published the book Leçons de calcul des probabilités Ⓣ which consisted of Emmanuel-Joseph Boudin's lectures at the University of Ghent given between 1846 and 1890, together with numerous notes and additions by Mansion. In the Preface to the book, Mansion dedicated the work to his father-in-law, writing:-
We dedicate this book to the memory of the eminent man, Alphonse Belpaire (1817-1854), who was the Boudin's advisor at the beginning of his career and to whom we are closely related, as he is the grandfather of our children.Mansion translated into French mathematical works by Riemann, Plücker and Clebsch. He did not restrict himself to translating mathematical texts, however, for he translated works by other famous authors such as Dante into French, publishing La Divine Comédie Ⓣ in 1887. He wrote on the history of Greek mathematics and on many mathematicians including: Hermite, Abel, de la Vallée Poussin, Saccheri, Lobachevsky, de Tilly, Poincaré, Copernicus, Galileo, Kepler, Descartes, Huygens, Leibniz, Newton, d'Alembert, Euler, Laplace, Ampère, Faraday, Quetelet, Lord Kelvin, and Helmholtz. He also wrote on the history of physics and on Greek astronomy in Note sur le caractère géométrique de l'ancienne astronomie Ⓣ (1899). Being a narrow-minded man with highly orthodox Roman Catholic views, his evaluation of the work of authors such as Copernicus and Galileo was somewhat biased and indeed many would consider his writings on these authors as intended to justify the views of the Roman Catholic Church.
Despite the rather critical tone of these comments we should point out that, as the reader will have already observed, Mansion was highly productive. When Alphonse Demoulin wrote the obituary of Mansion  (published in 1929) he included a list of 349 of his works although in fact the list contains many papers with the same title published in parts over many years which each appear as a single item. Nor were these works merely of local interest, for many were considered important enough to be translated into German or to be republished in foreign publications. Demoulin, who was a student of Mansion's and wrote a doctoral dissertation Sur l'application d'une méthode vectorielle à l'étude de divers systèmes de droites (complexes, congruences, surfaces réglées) Ⓣ (1890) under his supervision, is full of praise for his teacher :-
Mansion was an outstanding teacher. His lessons, clear and modern, illustrated with historical references which he filled effortlessly with the treasures of his erudition, were a real treat for his listeners.Among the honours which Mansion received was election to the Royal Belgium Academy of Science. Elected as a corresponding member in 1882, he became a full member in 1887. He was further honoured by being made president of the Academy in 1903, giving a particularly influential presidential address on the theory of probability. He was also a member of many other societies and academies such as the Royal Society of Sciences of Liège, the Mathematical Society of Amsterdam, and the Accademia Pontificia dei Nuovi Lincei in Rome. He was a founder member of the Brussels Scientific Society.
Article by: J J O'Connor and E F Robertson
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