Monge attended the Oratorian College in Beaune. This school was intended for young nobles and was run by priests. The school offered a more liberal education than other religious schools, providing instruction not only in the humanities but also in history, mathematics, and the natural sciences. It was at this school that Monge first showed his brilliance. In 1762, at the age of 16, Monge went to Lyons 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.
Bossut was elected to the Académie des Sciences in 1768 and he left the École in Mézières to become professor of hydrodynamics at the Louvre. On 22 January 1769 Monge wrote to Bossut explaining that he was writing a work on the evolutes of curves of double curvature. He asked Bossut to give an opinion on the originality and usefulness of the work. 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. This paper, in which Monge generalised the results obtained by Huygens on space curves (as part of Huygens's investigation of the pendulum) and added many important new discoveries, is described in detail in . The completed work was submitted to the Académie des Sciences in Paris in October 1770 and read before the Académie in August 1771 (although it was not published by the Académie until 1785).
When Bossut left the École Royale du Génie at Mézières, Monge was appointed to succeed him in January 1769. In 1770 he received an additional post at the École Royale du Génie when he was appointed as instructor in experimental physics. 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. Over the next few years he submitted a series of important papers to the Académie on partial differential equations which he studied from a geometrical point of view. His interest in subjects other than mathematics began to grow and he became interested in problems in both physics and chemistry.
In 1777 Monge married Cathérine Huart and, since his wife had a forge, he became interested in metallurgy in addition to his wide range of mathematical and scientific interests. Still deeply involved in teaching at the École Royale du Génie at Mézières he organised the setting up of a chemistry laboratory there. From 1780, however, he devoted less time to his work at the École at Mézières since in that year he was elected as adjoint géomètre at the Académie des Sciences in Paris. From that time he spent long periods in Paris, teaching a course in hydrodynamics as a substitute for Bossut as well as participating in projects undertaken by the Académie in mathematics, physics and chemistry. It was not possible to do all this and to teach all his courses at Mézières but he kept his posts there and received his full salary out of which he paid others to teach some courses in his place.
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. Over the next five years, despite heavy duties as an examiner, Monge undertook research in a wide range of scientific subjects presenting papers to the Académie on :-
...the composition of nitrous acid, the generation of curved surfaces, finite difference equations, partial differential equations (1785); double refraction and the structure of Iceland spar, the composition of iron, steel, and cast iron, and the action of electricity sparks on carbon dioxide gas (1786); capillary phenomena (1787); and the causes of certain meteorological phenomena (1788); and a study in physiological optics (1789).Of course 1789 was an eventful year in French history with the storming of the Bastille on 14 July 1789 marking the start of the French Revolution. This was to completely change the course of Monge's life. At the onset of the Revolution he was one of the leading scientists in Paris with an outstanding research record in a wide variety of sciences, experience as an examiner and experience in school reforms which he had undertaken in 1786 as part of his duties as an examiner. 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. By this time he was on the major Académie Commission on Weights and Measures.
Louis XVI attempted to flee the country on 20 June 1791, but was stopped at Varennes and brought back to Paris, and this put an end to attempts to share government between the king and an assembly. Relations with Europe deteriorated when the National Assembly declared that a people had the right of self-determination. France declared war on Austria and Prussia on 20 April 1792. French defeats led to unrest in France and, on 10 August 1792, there was further revolutions by the people with nobles and clergy murdered during September. On 21 September the monarchy was abolished in France and a republic was declared. 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. Although he tried hard in difficult circumstances, he survived only eight months in the post before he gave up the incessant battle with those around him, and he submitted his resignation on 10 April 1793. 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. He wrote papers on the topics and also gave courses on these military topics. He continued to serve on the Commission on Weights and Measures which survived despite ending the Académie des Sciences. He also proposed educational reforms to the National Convention but, despite being accepted on 15 September 1793, it was rejected on the following day. Such was the volatile nature of decisions at this unstable time.
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). Not only was he a major influence in setting up the École using his experience at Mézières to good effect, but he was appointed as an instructor in descriptive geometry on 9 November 1794. His first task as instructor was to train future teachers of the school which began to operate from June 1795. 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. He was also a strong believer in the Académie des Sciences and worked hard to see it reinstated as the Institut National. The National Convention approved the new body on 26 October 1795. 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. Of particular significance was the fact that he became friendly with Napoleon Bonaparte during his time in Italy. Napoleon had defeated Austria and signed the Treaty of Campo Formio on 17 October 1797 which was an exceptionally good treaty for France, preserving most of the French conquests. 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  the author describes these events using letters which Monge sent to his wife from Rome at that time. 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. Malta was occupied on 10 June 1798, Alexandria taken by storm on 1 July, and the delta of the Nile quickly taken. However, on 1 August 1798 the French fleet was completely destroyed by Nelson's fleet in the Battle of the Nile, so that Napoleon found himself confined to the land that he was occupying. 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.
Napoleon abandoned his army and returned to Paris in 1799, he soon held absolute power in France. Monge was back in Paris on 16 October 1799 and took up his role as director of the École Polytechnique. He discovered that his memoir Géométrie descriptive Ⓣ had been published earlier in 1799. This had been done at his wife's request and had been put together by Hachette from Monge's lectures at the École Normale. On 9 November 1799 Napoleon and two others seized power in a coup and a new government, the Consulate, was set up. 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. The truth must be that Monge was :-
... dazzled by Napoleon ... and accepted all the honours and gifts the emperor bestowed upon him: grand officer of the Legion of Honour in 1804, president of the Senate in 1806, Count of Péluse in 1808, among others.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. Slowly he became less involved in mathematical research, then from 1809 he gave up his teaching at the École Polytechnique as his health began to fail.
In June 1812 Napoleon assembled his Grande Armée of about 453,000 men, including men from Prussia and from Austria who were forced to serve, and marched on Russia. The campaign was a disaster but by September Napoleon's army had entered a deserted Moscow. Napoleon withdrew, the Prussians and Austrians deserted the Grande Armée and in there were attempts at a coup against Napoleon in Paris. Monge was dismayed at the situation and his health suddenly collapsed. Slowly his health returned after Napoleon left the remains of his army and returned to Paris to assert his authority. After Napoleon had some military success in 1813, the allied armies against him strengthened. 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. Napoleon escaped from Elba, where he had been banished, and by 20 March 1815 he was back in Paris. 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. Two days after his return he was expelled from the Institut de France and from then on his life was desperately difficult as he was harassed politically and his life was continually threatened. On his death the students of the École Polytechnique paid tribute to him despite the insistence of the French Government that no tributes should be paid.
In  Monge's political career is treated kindly but G Jorland, in a review of that paper, takes a harder view:-
[Monge's] tenure at the Ministry of the Navy was a complete failure and he presided over the cultural pillage of Italy and Egypt. If Napoleon actually said that Monge loved him like a mistress, it proves that the utmost mathematical clarity can go hand in hand with political blindness.We have commented quite frequently regarding Monge's scientific work above. He 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-dimensional space. He developed a general method of applying geometry to problems of construction. He also introduced two planes of projection at right angles to each other for graphical description of solid objects. These techniques were generalised into a system called géométrie descriptive, which is now known as orthographic projection, the graphical method used in modern mechanical drawing.
The basic philosophy behind Monge's approach to mathematics is discussed in  where the author states that Monge's aims were the:-
... geometrisation of mathematics based on:Monge regarded analysis as being :-
(a) the analogy or correspondence of operations in analysis with geometric transformations;
(b) the genetic classification and parametrisation of surfaces through analysis of the movement of generating lines.
... not a self-contained language but merely the 'script' of the 'moving geometrical spectacle' that constitutes reality.
... [His] new approach addressed itself to the most profound, intimate and universal relations in space and their transformations, putting him in a position to interconnect geometry and analysis in a fertile, previously unheard-of fashion. 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 ...
Article by: J J O'Connor and E F Robertson