After submitting his doctoral thesis Quelques applications de la théorie des formes algébriques à la géométrie Ⓣ, Le Paige was given an oral examination and graduated on 28 July 1875. After graduation, he was appointed as a lecturer at the University of Liège on 10 October 1876. It is a clear indication of his remarkable productivity that nine of his papers were published in 1876 and ten were published in the following year. Some of these papers were on topics he had worked on before he settled on geometry as his main interest, for example there are papers on continued fractions, differential equations, the difference calculus, and Bernoulli numbers. Beginning in the year he was appointed to the University, he taught courses on 'Elements of the theory of determinants', which is not surprising since this was Catalan's speciality, and he also taught a 'Higher analysis' course. He taught the course on the theory of determinants throughout his 46-year career. He began teaching geometry courses in 1879, taking over courses that had previously been taught by his research advisor François Folie. He was appointed as an extraordinary professor at the University of Liège in 1882, promoted to full professor three years later, and remained there for the whole of his career, retiring in 1922. He was Rector of the University from 1895 to 1898, and administrateur-inspecteur from 1905 to 1922. He was Director of the Institute Astrophysique de Cointe-Sclession, annexed to the University of Liège, from 1897 to 1922. After Catalan retired in 1884, Le Paige took over teaching his probability theory course. Around the same time François Folie left Liège to become the director of the Royal Observatory, Uccle, Brussels, leading to a considerable increase in Le Paige's teaching and administrative duties. During the 1890s he began teaching courses on 'Celestial mechanics and analytical mechanics', the 'Elements of astronomy and geodesy', the 'History of mathematical and physical science', and 'Physical astronomy'. He continuing to teach all these courses until he retired. Another of his colleagues at the University of Liège was Joseph Neuberg who, although about twelve years older than Le Paige, had been appointed as a professor at Liège within a couple of years of Le Paige. Neuberg retired in 1910 and again Le Paige took over certain additional responsibilities.
Le Paige married Marie Joséphine Ernst. They had a son Ubric Le Paige who married Teresa De Walque. Ubric and Teresa had eleven sons, including Father Gustave Le Paige De Walque (1903-1980) who was Director of the Museo Arqueológico de la Universidad del Norte in San Pedro de Atacama. We mentioned above that Le Paige was administrateur-inspecteur of the University of Liège from 1905 to 1922. This period, of course, contained the extremely difficult years of World War I when, between 1914 and 1918, much of Belgium was occupied by German troops. Although Belgium declared itself neutral in 1914, after it refused passage to German troops to cross its lands to attack France, the country was invaded in August 1914. The Germans took control of Liège and made demands of the University that Le Paige was largely able to resist. After the end of the war his dedication to duty helped the university to return quickly to its former activities.
There was a tradition of geometry at the University of Liège which created the background to Le Paige's early research. Germinal Dandelin was professor of mining engineering at Liège from 1825 to 1830 and during that time he taught analytic geometry. It was Jean-Baptiste Brasseur who learnt geometry from Dandelin at this time and he, in turn, taught course on Higher Geometry in which he outlined the theory of algebraic curves and surfaces. François Folie was Brasseur's student and he took over teaching the Higher Geometry course in 1876. Of course, as we have indicated above, Le Paige was Folie's student and he took over teaching the course when Folie left Liège. Folie's approach to geometry was a purely geometrical one, but Le Paige approached the topic in a much more algebraic way, following the fashion that was beginning to take over the subject. He worked on the theory of algebraic forms, a topic whose study was initiated by George Boole in 1841 and then developed by Arthur Cayley, James Joseph Sylvester, Charles Hermite, Alfred Clebsch and Siegfried Aronhold. In particular Le Paige studied the geometry of algebraic curves and surfaces, building on this earlier work. H L L Busard writes :-
Le Paige's investigations touched mainly upon the geometry of algebraic curves and surfaces, and the theory of invariants and involutions. He coordinated and generalized the extensions which at that time had been tried. His best-known achievement was the construction of a cubic surface given by nineteen points. Starting from the construction of a cubic surface given by a straight line, three groups of three points on a line, and six other points, Le Paige comes to the construction of a cubic surface given by three lines and seven points. From this he proceeds to the construction of a cubic surface given by a line, three points on a line, and twelve other points, and by means of the construction of a cubic surface given by three points on a line and sixteen other points, he arrives at a surface given by nineteen points.Le Paige studied the generation of plane cubic and quartic curves, developing further Chasles's work on plane algebraic curves and Steiner's results on the intersection of two projective pencils. The history of mathematics was another topic which interested Le Paige. His most significant work in this area was his publication of de Sluze's correspondence with Pascal, Huygens, Oldenburg and Wallis. Other topics in this area which interested him were the history of mathematical notation, and the history of astronomy, particularly Belgium astronomy. Here is a list of the main historical papers he wrote: Correspondance de René-François de Sluse, publiée pour la première fois et précédée d'une introduction Ⓣ (1884); Un géomètre belge du XVIIe siècle: René-François de Sluse Ⓣ (1887); Lettre de M le Paige à M G Longchamps (relative au géomètre montois J-F Lepoivre) Ⓣ (1887); Un astronome belge du XVIIe siècle: Godefroid Wendelin Ⓣ (1891); Notes pour servir à l'Histoire des Mathématiques dans l'ancien Pays de Liège Ⓣ (1890); Sur l'origine de certains signes d'opération Ⓣ (1891-1892); Sur les notations algébriques avant Descartes Ⓣ (1895-1896); Discours sur l'astronomie des Grecs Ⓣ (1895); and Discours sur l'astronomie au temps de Kepler Ⓣ (1896).
After Le Paige was appointed director of the Astrophysical Institute, he wrote a number of astronomical texts. For example, in the year of his appointment, 1897, he published: De l'action du Soleil sur les plaques photographiques Ⓣ; Sur la photographie de l'atmosphère (suite à une note de M De Heen) Ⓣ; Sur la photographie du Soleil Ⓣ; and Discours sur l'astronomie moderne Ⓣ. Let us note some of his students who went on to outstanding academic careers: Jacques Deruyts, François Deruyts, Marcel Dehalu, and Henry Janne.
Le Paige received many honours. He was elected a corresponding member of the Royal Belgium Academy of Sciences on 15 December 1885, becoming a full member on to the 15 December 1890. In 1907 he became Director of the Sciences Class of the Academy. He was elected to the Royal Society of Sciences of Liège in 1878, becoming its Secretary General in 1886. He was elected to the Royal Society of Bohemia in 1881, and in the same year became a foreign corresponding member of the Accademia Pontificia dei Nuovi Lincei in Rome. He was elected to the Royal Academy of Sciences of Lisbon in 1883 and, in the same year, to the Leopoldino-Carolina Naturae Curiosorum Academia in Halle. He became an honorary member of the Mathematical Society of Amsterdam in 1886.
Finally, we note Le Paige's love of collecting old books. In  the sale of Le Paige's collection by Joseph Baer and Co. of Frankfurt-am-Main on 26 May 1933, four years after his death, was announced. The firm:-
... will sell the collection of the well-known bibliophile, le Chevalier Constantin Le Paige, of Liège, consisting of incunabula, antique manuscripts, valuable illustrated books, French literature, as, for example, first editions of Balzac, Bossuet, Corneille, Pascal, Rousseau, Voltaire; also mathematical and astronomical rarities, the commentary of Albertus de Brudzewo of 1494, printed in Milan, of which only two examples are known, the "Opus calculationum" of Suiseth (Padua of about 1477) and the first edition of Boethius "Arithmetica." The chief attraction is Dante's manuscript of 1493, which resembles the books written in Naples for King Matthias Corvinus. A great rarity is the French translation of the "Hortus sanitatis," which appeared in Paris in 1500. The "Divisie Chronijk" of 1517 is celebrated for its wood-cuts by Lucas van Leyden, and a "Livre d'Heures" of 1513 is printed on parchment and illustrated with coloured wood engravings. The sale contains also rare books on art and science, music books, geographical works, etc.
Article by: J J O'Connor and E F Robertson