**Jean-Baptiste Bélanger** was a French engineer and mathematician. His ideas influenced strongly the teaching of statics and dynamics, and mechanics in France and Europe.

- Jean-Baptiste was baptised by the local Roman Catholic priest on the day he was born, which was standard practice at this time.
- In 1808 Bélanger sat the entrance examination for the École Polytechnique, as did Gaspard-Gustave de Coriolis who became his classmate and friend.
- On graduating Bélanger was ranked second in his class and, in 1812, he entered the École Nationale Supérieure des Ponts et Chaussées in Paris to train to become an engineer.
- Bélanger qualified as an engineer in 1816 and went to undertake work at La Réole, a town on the river Garonne about 50 km south east of Bordeaux.
- However, work on this does not appear to have begun until some years after Bélanger left the town.
- Another project, with a link to Bélanger, was to bridge the river Garonne at La Réole.
- Work on a suspension bridge was begun in 1833 by Gustave Eiffel, who had been a student of Bélanger.
- In 1821 Bélanger left La Réole and moved to work on the Somme navigation canal project.
- Bélanger's contributions were similar to those on the Somme canal.
- While working on these canals from 1821 onwards, Bélanger began to apply high powered mathematical techniques to study the flow of water in these channels.
- In his new theory, Bélanger supposed a slow variation of the section of the channel and slight differences between the surface and bottom slopes.
- Bélanger's first work on this was published in a report that appeared in the Journal des Mines in 1823 but he was unhappy with this and felt that it required considerable improvements, particularly needing a stronger theoretical foundation.
- The application of the momentum principle to the hydraulic jump is commonly called the Bélanger equation, but few know that his original treatise was focused on the study of gradually varied open channel flows (Bélanger 1828).
- The originality of Bélanger's 1828 essay was the successful development of the backwater equation for steady, one-dimensional gradually-varied flows in an open channel, together with the introduction of the step method, distance calculated from depth, and the concept of critical flow conditions.
- In 1828, Jean-Baptiste Bélanger understood the rapidly varied nature of the jump flow, but he applied incorrectly the Bernoulli principle to the hydraulic jump.
- Bélanger gave some specific examples in the paper to show the applicability of his equation.
- Bélanger also gave a few examples of calculations of the backwaters before a weir, with the navigability of rivers in view.
- When Bélanger wrote this work he was employed on the Ardennes canal project.
- Bélanger had a teaching career in addition to his engineering career.
- Bélanger produced an important publication Notes sur l'Hydraulique Ⓣ(Notes on water resources) in 1841 being his lecture notes for session 1841-42 at the École des Ponts et Chaussées.
- When we gave an indication of Bélanger's 1828 work we mentioned that he applied incorrectly the Bernoulli principle to the hydraulic jump.
- We learn from Bélanger's lecture notes that he realised his error in 1838 and a version with the momentum principle correctly applied is presented in Notes sur l'Hydraulique Ⓣ(Notes on water resources) (1841).
- This 1841 work contained the hydraulic jump equation now known as the Bélanger equation.
- Bélanger gave at Centrale, the Ponts et Chaussées, and the Polytechnique in the 1840s and 1850s integrated ...
- In addition to these lecture notes, Bélanger published a number of books.
- In 1858 Bélanger published the treatise Théorie de la résistance de la torsion et de la flexion plane des solides Ⓣ(Theory of the resistance of the torsion and bending of a planar solid); a second edition was published in 1862.
- We remarked above that Gustave Eiffel (1832-1923) was a student of Bélanger's at the École Centrale des Arts et Manufactures.
- Eiffel, of course, went on to design the Eiffel Tower and Bélanger is one of the 72 scientists whose names are engraved on the facades of the Tower.
- Jacques Antoine Charles Bresse (1822-1883), mentioned in this quote, was greatly influenced by Bélanger and succeeded him as a professor at the École Nationale Supérieure des Ponts et Chaussées in 1855.
- His name, like that of Bélanger, in engraved on the Eiffel Tower.

Born 4 April 1790, Valenciennes, France. Died 8 May 1874, Neuilly-sur-Seine, France.

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**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive