**Gaspard Monge** is considered the father of differential geometry because of his work _Application de l'analyse à la géométrie _ where he introduced the concept of lines of curvature of a surface in 3-space.

- Around the time that Gaspard was born Beaune, after a period of decline, was becoming prosperous again due to the success of the wine trade.
- Monge attended the Oratorian College in Beaune.
- It was at this school that Monge first showed his brilliance.
- In 1762, at the age of 16, Monge went to Lyon where he continued his education at the Collège de la Trinité.
- Despite being only 17 years of age at the time, Monge was put in charge of teaching a course in physics.
- Completing his education there in 1764, Monge returned to Beaune where he drew up a plan of the city.
- The plan of Beaune that Monge constructed was to have a major influence in the direction that his career took, for the plan was seen by a member of staff at the École Royale du Génie at Mézières.
- He was very impressed by Monge's work and, in 1765, Monge was appointed to the École Royale du Génie as a draftsman.
- Of course, in this post Monge was undertaking tasks that were not entirely to his liking, for he aspired to a position in life which made far more use of his mathematical talents.
- However the École Royale du Génie brought Monge into contact with Charles Bossut who was the professor of mathematics there.
- At first Monge's post did not require him to use his mathematical talents, but Monge worked in his own time developing his own ideas of geometry.
- About a year after becoming a draftsman, Monge was given a task which allowed him to use his mathematical skill to attack the task he was given.
- Asked to draw up a fortification plan which prevented an enemy from either seeing or firing at a military position no matter what the position of the enemy, Monge devised his own graphical method to construct such a fortification rather than use the complicated methods then available.
- This method made full use of the geometrical techniques which Monge was developing in his own time.
- His mathematical abilities were now recognised at the École Royale du Génie and it was realised that Monge was someone with exceptional abilities in both theoretical and practical subjects.
- On 22 January 1769 Monge wrote to Bossut explaining that he was writing a work on the evolutes of curves of double curvature.
- Bossut must have replied in a very positive fashion for in June a publication in the Journal Encyclopédique by Monge (his first publication) appeared giving a summary of the results which he had obtained.
- When Bossut left the École Royale du Génie at Mézières, Monge was appointed to succeed him in January 1769.
- Although this was a large step forward for Monge's career, he was more interested in making his name as a mathematician in the highest circles.
- Realising that he had to obtain advice from the leading mathematicians, Monge approached d'Alembert and Condorcet early in 1771.
- Condorcet must have been impressed with the depth of the mathematics that Monge showed him, for he recommended that he present memoirs to the Académie des Sciences in each of the four areas of mathematics in which he was undertaking research.
- The four memoirs that Monge submitted to the Académie were on a generalisation of the calculus of variations, infinitesimal geometry, the theory of partial differential equations, and combinatorics.
- After three years of dividing his time between Paris and Mézières, Monge was offered yet another post, namely to replace Bézout as examiner of naval cadets.
- Monge would have liked to keep all these positions, but after attempting to organise an impossible schedule for about a year, he decided that he would have to resign his posts in Mézières, which he did in December 1784.
- This was to completely change the course of Monge's life.
- Politically Monge was a strong supporter of the Revolution, and his first actions were to show his support by joining various societies supporting the Revolution, but he continued his normal duties as an examiner of naval cadets, and as a major figure in the work of the Académie.
- Monge was offered the post of Minister of the Navy in the government by the National Convention.
- Without disrespect to Monge, it was impossible to satisfy the quite extreme views of many people, and Monge's period as Minister of the Navy cannot be viewed as a success.
- For a few months Monge returned to his work with the Académie des Sciences but this did not last long for, on 8 August 1793, the Académie des Sciences was abolished by the National Convention.
- Still a strong republican and supporter of the Revolution, Monge worked on various military projects relating to arms and explosives.
- Monge was appointed by the National Convention on 11 March 1794 to the body that was put in place to establish the École Centrale des Travaux Publics (soon to become the École Polytechnique).
- Monge's lectures on infinitesimal geometry were to form the basis of his book Application de l'analyse à la géométrie.
- Another educational establishment, the École Normale, was set up to train secondary school teachers and Monge gave a course on descriptive geometry.
- However from May 1796 to October 1797, Monge was in Italy on a commission to select the best art treasures for the conquerors and bring them to France.
- Monge returned to Paris bringing the text of the Treaty of Campo Formio with him.
- Back in Paris Monge slotted back into his previous roles and was appointed to the prestigious new one of Director of the École Polytechnique.
- By February 1798 Monge was back in Rome, involved with the setting up of the Republic of Rome.
- In particular Monge proposed a project for advanced schools in the Republic of Rome.
- Napoleon Bonaparte now asked Monge to join him on his Egyptian expedition and, somewhat reluctantly, Monge agreed.
- Monge left Italy on 26 May 1798 and joined Napoleon's expeditionary force.
- The expedition, which included the mathematicians Fourier and Malus as well as Monge, was at first a great success.
- Monge was appointed president of the Institut d'Egypte in Cairo on 21 August.
- The Institut had twelve members of the mathematics division, including Fourier, Monge, Malus and Napoleon Bonaparte.
- During difficult times with Napoleon in Egypt and Syria, Monge continued to work on perfecting his treatise Application de l'analyse à la géométrie.
- Monge was back in Paris on 16 October 1799 and took up his role as director of the École Polytechnique.
- This had been done at his wife's request and had been put together by Hachette from Monge's lectures at the École Normale.
- Napoleon named Monge a senator on the Consulate for life.
- Monge accepted with pleasure, although his republican views should have meant that he was opposed to the military dictatorship imposed by Napoleon on France.
- Over the next few years Monge continued a whole range of activities, undertaking his role as a senator while maintaining an interest in research in mathematics but mostly his mathematical work involved teaching and writing texts for the students at the École Polytechnique.
- Monge was dismayed at the situation and his health suddenly collapsed.
- Monge was sent to Liège to organise the defence of the town against an attack.
- The allied armies began to move against France and Monge fled.
- When Napoleon abdicated on 6 April 1814, Monge was not in Paris, but soon after he did return and tried to pick up his life again.
- Monge immediately rallied to Napoleon and gave him his full support.
- After Napoleon was defeated at Waterloo, Monge continued to see him until he was put on board a ship on 15 July.
- By October Monge feared for his life and fled from France.
- Monge returned to Paris in March 1816.
- We have commented quite frequently regarding Monge's scientific work above.
- Practical concerns induced Monge to perceive the object and function of mathematics in a new way, in violation of the formalistic (linguistic) standards set by the approved patrons of mathematics ...

Born 9 May 1746, Beaune, Bourgogne, France. Died 28 July 1818, Paris, France.

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Algebra, Geometry, Group Theory

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