In 1765 Condorcet published Essai sur le calcul intégral . He was elected to the Académie des Sciences in 1769. During this period he produced several important works, including one in 1772 on the integral calculus which was described by Lagrange as:-
filled with sublime and fruitful ideas which could have furnished material for several works.Soon after the publication of his 1772 work, Condorcet met Turgot, a French economist who became an administrator under Louis XV. Turgot became Controller General of Finance in 1774 under Louis XVI and he had Condorcet appointed Inspector General of the Mint.
Turgot was dismissed from his post in 1776 and Condorcet tended his resignation. However Condorcet's resignation was refused and he continued to fill this post until 1791.
In 1777 Condorcet was appointed Secretary of the Académie des Sciences. He had been advised by Voltaire and by d'Alembert to become an expert in writing obituaries in order to improve his chances of getting the post. It certainly was good advice but it severely curtailed his mathematical output.
His most important work was on probability and the philosophy of mathematics. His most important treatise was Essay on the Application of Analysis to the Probability of Majority Decisions (1785). This is an extremely important work in the development of the theory of probability.
He is known for the Condorcet Paradox which points out that it is possible that a majority prefers option A over option B, a majority prefers option B over option C, and yet a majority prefers option C over option A. (Thus, "majority prefers" is not transitive.)
Condorcet published Vie de M Turgot (1786) and Vie de Voltaire (1789). In these biographies he showed that he favoured Turgot's economic theories and agreed with Voltaire in his opposition to the Church. Also in 1786 he again worked on his ideas for the differential and integral calculus, giving a new treatment of infinitesimals. However his treatise was never printed.
When the French Revolution broke out Condorcet championed the liberal cause. He was elected as the Paris representative in the Legislative Assembly and he became the secretary of the Assembly. He drew up plans for a state education system which were adopted.
By 1792 Condorcet had become one of the leaders of the Republican cause. He joined the moderate Girondists and argued strongly that the King's life should be spared.
When the Girondists fell from favour and the Jacobins, a more radical political group led by Robespierre, took over, Condorcet argued strongly against the new, hurriedly written, constitution which was drawn up to replace the one which he himself had been chiefly responsible for drawing up. This showed a lack of sense and he paid for it when a warrant was issued for his arrest.
Condorcet went into hiding and wrote a very interesting philosophical work Esquisse d'un tableau historique des progrès de l'esprit humain (1795). In March 1794 he thought that the house in which he was hiding in Paris was being watched by his enemies and he no longer felt safe. He fled from Paris and after three days he was arrested and imprisoned on 27 March 1794. Two days later he was found dead in his prison cell and it is not known if he died from natural causes or whether he was murdered or took his own life.
J Herival described Condorcet as follows:-
... Condorcet was no politician. His uncompromising directness of manner and inability to suffer illogical windbags in silence made him many enemies and few friends. His weak voice, lack of oratorical powers, and tendency to bore the Convention by the excessive height of his arguments was one of the tragedies of the Revolution.His life is summed up by H B Acton in  as follows:-
Wholly a man of the Enlightenment, an advocate of economic freedom, religious toleration, legal and educational reform, and the abolition of slavery, Condorcet sought to extend the empire of reason to social affairs. Rather than elucidate human behaviour, as had been done thus far, by recourse to either the moral or physical sciences, he sought to explain it by a merger of the two sciences that eventually became transmuted into the discipline of sociology.
Article by: J J O'Connor and E F Robertson