From this time on information on Arbogast becomes more plentiful. While at the Collège de Colmar Arbogast entered a mathematical competition which was run by the St Petersburg Academy. His entry was to bring him fame and an important place in the history of the development of the calculus. Perhaps a little background will make the reason for the question that was posed for this competition more meaningful.
Euler and d'Alembert had fallen out in the 1750s over a number of different matters, some mathematical and others more to do with their personalities. The particular mathematical dispute which prompted the question set by the St Petersburg Academy in 1787, however, concerned the arbitrary functions which appeared when a differential equation was integrated. d'Alembert claimed that these arbitrary functions were required to be continuous and must always be expressed in terms of algebraic or transcendental equations. Euler argued that more general functions could be introduced when differential equations were integrated.
The actual question the St Petersburg Academy posed was:-
Do the arbitrary functions introduced when differential equations are integrated belong to any curves or surfaces either algebraic, transcendental, or mechanical, either discontinuous or produced by a simple movement of the hand? Or should they legitimately be applied only to continuous curves susceptible of being expressed by algebraic or transcendental equations?Arbogast submitted an essay to the St Petersburg Academy in which he came down firmly on the side of Euler. In fact he went much further than Euler in the type of arbitrary functions introduced by integrating, claiming that not only could the functions be discontinuous in the limited sense that Euler claimed, but discontinuous in a more general sense that he defined which allowed the function to consist of portions of different curves. Arbogast won the prize with his essay and his notion of discontinuous function became important in Cauchy's more rigorous approach to analysis.
In 1789 Arbogast moved from Colmar to Strasbourg where he taught mathematics at the École d'Artillerie. Also in 1789 he submitted a major report on the differential and integral calculus to the Académie des Sciences in Paris which was never published. In the Preface of a later work he described the ideas which prompted him to write the major report of 1789. Essentially he realised that there was no rigorous methods to deal with the convergence of series. This fact caused him to:-
... reflect on fundamental principles. ... I then foresaw the birth of the first inkling of the ideas and methods which, when developed and extended, formed the substance of the calculus of derivatives.Arbogast's career in Strasbourg reached new heights. In addition to his mathematics post, he was appointed as professor of physics at the Collège Royal in Strasbourg and from April 1791 he served as its rector until October 1791 when he was appointed rector of the University of Strasbourg. By this time, however, Arbogast was involved in politics and in 1791 he was elected to the Legislative Assembly. As a member he introduced plans for reforms in the schools at all levels. Then in 1792 he was returned as member for Haguenau to the Convention Nationale.
In 1794 he was appointed Professor of Calculus at the École Centrale (soon to become the École Polytechnique) but he taught at the École Préparatoire. In July 1795 he was put in charge of planning the École Centrale du Bas-Rhin and, once set up, he became the professor of mathematics there in 1796, holding the post until 1802.
His contributions to mathematics show him as a philosophical thinker somewhat ahead of his time. As well as introducing discontinuous functions, as we discussed above, he conceived the calculus as operational symbols. The formal algebraic manipulation of series investigated by Lagrange and Laplace in the 1770s was put in the form of operator equalities by Arbogast in 1800 in Calcul des dérivations. In  Ljusternik and Petrova explain that in this work:-
... operators and functions of operators were separated from the functions on which they act.Arbogast is clear about what he is doing writing in the Preface explaining that his method separates the scale of operations:-
This method is generally thought of as separating from the functions of variables when possible, the operational signs which affect this function. Then, of treating the expression formed by these signs applied to any quantity whatsoever, an expression which I have called a scale of operation, to treat it, I say, nevertheless as if the operational signs which compose it were quantities, then to multiply the result by the function.Arbogast was interested in the history of mathematics and classified Mersenne's papers and collected manuscript copies of memoirs and letters of Fermat, Descartes, Johann Bernoulli, Varignon, de L'Hôpital and others. This is an extremely important collection, part of which is now in Paris and part in Florence.
Arbogast was friendly with François Français and together they worked on the calculus of derivations and the operational calculus. After Arbogast died in 1803, François Français inherited his collection of manuscripts, and also his mathematical papers. He continued Arbogast's work on the operational calculus and presented a memoir on this topic, in particular applying the methods to study projectiles in a resistant medium, to the Académie des Sciences in 1804. This memoir was very highly praised by Biot in a report of 22 April 1805, but the work was not published.
The historical manuscripts which went to François Français on Arbogast's death were bought by Libri from a bookseller in Metz in 1839.
We should mention one other important contribution made by Arbogast. He :-
... was responsible for the law introducing the decimal metric system in the whole of the French Republic.Arbogast was elected to the Académie des Sciences in 1792 and the mathematics section of the Institut National in 1796.
Article by: J J O'Connor and E F Robertson