Lazare Carnot graduated from the School of Engineering in Mézières in 1773. In 1778 he wrote Essai sur les machines en général to submit for a prize in a competition. He revised it in 1781 and it was eventually published in 1783. It deals with mechanics and areas of engineering. The following year he declined an invitation to enter the Prussian service and, in the same year he was promoted to captain.
From 1787 he became a member of the Dijon Academy while he was elected to the Legislative Assembly in 1791 and to the National Convention in 1792.
He directed the Army of the North after April 1793 becoming in that year a leading member of the Committee of General Defence and a member of the Committee of Public Safety.
In 1794, under direction from Carnot and Monge, a 'grande école' was set up called 'École centrale des travaux publics' but its name was changed to 'École Polytechnique' in the following year. Two years later his son Sadi Carnot was born.
The year 1797 was an eventful one for Carnot. In this year he published his famous text Réflexions sur la métaphysique du calcul infinitésimal Ⓣ. The book is introduced with the words:-
As however everything indicates that there will be a new turn in the culture of mathematics, the author deems it apposite to publish this monograph.
Carnot's approach to mathematics shows strongly his engineering background. Thiele writes in  that he
accepted mathematical expressions only insofar as the quantities contained in them were real and the operations involved held meaning. ... to Carnot negative quantities are impossible, and zero, just like infinity, is a limit. ... infinitely small quantities are real objects, being representable as differences between limits...
In the same year, 1797, the political situation in France became such that he could no longer remain with his strong republican views, and he fled to Switzerland going on to Nuremberg in Germany.
The following year Carnot returned to France when Napoleon became First Consul. He became Napoleon Bonaparte's minister of war for a period of five months and was promoted further to the rank of lieutenant-general.
Carnot is best known as a geometer. In 1801 he published De la correlation des figures de géométrie Ⓣ in which he tried to put pure geometry into a universal setting. He showed that several of the theorems of Euclid's Elements can be established from a single theorem.
In 1803 he published Géométrie de position Ⓣ in which sensed magnitudes were first used systematically in geometry. This work greatly extended his work of 1801 and in it Carnot again shows what quantities mean to him writing:-
Every quantity is a real object such that the mind can grasp it or at least its representation in calculation.
Carnot's military masterpiece De la défense des places fortes Ⓣ was published in 1809. He later served as military governor of Antwerp but after Napoleon's final defeat at Waterloo he went into exile. He fled to Magdeburg, after going first to Warsaw, arriving in Magdeburg in November 1816.
Carnot's interests turned toward the steam engine with the first steam engine coming to Magdeburg in 1818. His son Sadi Carnot visited him in Magdeburg in 1821 and it is clear that Lazare Carnot influenced his son. Sadi Carnot published his masterpiece on the thermodynamics of the steam engine three years later.
Article by: J J O'Connor and E F Robertson