Sadi Carnot was the eldest son of Lazare Carnot and he was born in the Palais du Petit-Luxembourg. His younger brother was Hippolyte Carnot. At the time of Sadi's birth, his father was a member of the Directory, the French Revolutionary government which lasted four years from November 1795 to November 1799. Sadi was given named after a medieval Persian poet and philosopher called Sa'di of Shiraz.
Sadi Carnot was born at a time of unrest and political turmoil in France and, due the position of his father, whose fortunes changed dramatically many times, he was brought up in a totally unstable environment of interacting politics and science. His father was appointed to the high office of Napoleon's minister of war in 1799. Lazare Carnot resigned in 1807 and devoted himself to the education of his two sons. Sadi had a fine teacher in his father, who taught him mathematics and science as well as languages and music.
Under his father's tuition, Sadi Carnot showed great promise and was sent to the Lycée Charlemagne in Paris to prepare him for the examinations to the École Polytechnique in Paris. In 1812, at age 16 the minimum age possible, Carnot entered the École Polytechnique where Poisson, Ampère and Arago were among his teachers. Chasles was in the same class as Carnot and their friendship lasted throughout Carnot's life. Carnot graduated from the École Polytechnique in 1814 but, before he graduated, Carnot and other students from the École Polytechnique fought unsuccessfully with Napoleon to defend Vincennes. This skirmish against the Allies was fought just outside Paris, to the east of the city.
After graduating, Carnot went to the École du Génie at Metz to take the two year course in military engineering. In 1815 Napoleon returned from exile for his famed Hundred Days rule. Carnot's father was appointed minister of the interior and Sadi Carnot was put in a somewhat difficult position in the military academy with his father in such a prominent position. In October 1815 after the defeat of Napoleon, Carnot's father was exiled and he went to Germany never to return to France.
It is probable that a military career was not the easiest one for Carnot with his father in exile. Carnot was moved from place to place, given jobs of inspecting fortifications, drawing up plans and writing reports. However, it seems that he was not well treated and his recommendations were ignored. Unhappy at his lack of promotion and the refusal to give him a job which allowed him to make use of his training, in 1819, he sat and passed the examinations to join the recently formed General Staff Corps in Paris. Almost immediately he took leave on half pay, living in Paris in his father's former apartment, but he remained on call for army duty.
Carnot began to attend courses at various institutions in Paris, including the Sorbonne and the Collège de France. At this time he became interested in industrial problems and, in particular, began to study the theory of gases. His interests were wide :-
He made frequent visits to factories and workshops, studied the latest theories of political economy, and left in his notes detailed proposals on such current problems as tax reform. Beyond this, his activity and ability embraced mathematics and the fine arts.
Carnot visited his father in 1821 in his exiled home in Magdeburg. Hippolyte Carnot, his brother, was living with his father at this time. It is clear that there were many discussions about steam engines. The first steam engine had come to Magdeburg three years earlier and had interested Lazare Carnot, and Sadi Carnot left Magdeburg filled with enthusiasm to develop a theory for steam engines. After returning to Paris, Carnot began the work which led to the mathematical theory of heat and helped start the modern theory of thermodynamics. E Mendoza explains in  the background:-
The problem occupying Carnot was how to design good steam engines. Steam power already had many uses - draining water from mines, excavating ports and rivers, forging iron, grinding grain, and spinning and weaving cloth - but it was inefficient. The import into France of advanced engines after the war with Britain showed Carnot how far French design had fallen behind. It irked him particularly that the British had progressed so far through the genius of a few engineers who lacked formal scientific education. British engineers had also accumulated and published reliable data about the efficiency of many types of engines under actual running conditions; and they vigorously argued the merits of low- and high-pressure engines and of single-cylinder and multi-cylinder engines.
The first of Carnot's major works was a paper which he wrote in 1822-23. This paper attempted to find a mathematical expression for the work produced by one kilogram of steam. The paper is similar in its aims and also in its methods to several other papers which appeared around this time by Hachette, Navier and Petit. However :-
Carnot's work is distinguished for his careful, clear analysis of the units and concepts employed and for his use of both an adiabatic working stage and an isothermal stage in which work is consumed.
From its style and detailed descriptions the paper was clearly intended for publication but Carnot never published it and its existence was only discovered in manuscript form in 1966.
When Lazare Carnot died in August 1823, Hippolyte Carnot returned to Paris and there he helped Sadi Carnot to make the book on steam engines that he was working on at the time more understandable to the general public. In 1824 Carnot published this work, the only one he published during his lifetime, Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissanceⓉ which includes his description of the "Carnot cycle". This book only became well known after Clapeyron published an analytic reformulation of it in 1834. Carnot's ideas were later incorporated into the thermodynamic theory of Clausius and Thomson.
Carnot continued with his research after the publication of his book and although nothing of this was published, notes that Carnot made as his ideas developed have survived. In 1827, however, the General Staff Corps in Paris was reorganised and he was recalled to full time duties. He served for rather less than one year as a military engineer, being posted first to Lyon and then to Auxonne. Still unhappy with his career, Carnot retired permanently and returned to live in Paris where he aimed to continue with his research into the theory of heat.
Following his father's political views, Carnot was strongly republican so he was pleased with the direction France seemed to be going with the July 1830 Revolution. Around this time he became interested in public life, in particular he was interested in improving public education. He was suggested for a government position, but he declined and, after the monarchy was restored, he returned to his scientific work. In June 1832 he took ill and had not fully regained his strength when the cholera epidemic of 1832 hit Paris. Although only 36 years of age, he died within a day of contracting cholera.
That Carnot's important book Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissanceⓉ should have been neglected at the time of its publication is certainly not because it went unnoticed. It was published on 12 June 1824 and on 26 July of that year Pierre Girard gave a long review of it to the Académie des Sciences in Paris. Pierre Girard's review was a very positive one and was published in the Revue encyclopédique. Perhaps the problem with the review was that although it stated the theorems and the conclusions of Carnot's work fully, it did not comment on the highly original reasoning which Carnot had employed to achieve his results.
It is possible that Carnot's personality played a role in the lack of enthusiasm for his work in his lifetime. He is described in  as follows:-
Friends described him as reserved, almost taciturn, but insatiably curious about science and technical processes.
In  he is described in similar terms:-
Although sensitive and perceptive, he appeared extremely introverted, even aloof, to all but a few close friends ...
The Réflexions was an attempt by Carnot to answer two fundamental questions, firstly whether there was an upper limit to the power of heat, and secondly whether there was a better means than steam to produce this power. His attack on these questions was to produce a much more general theory than had previously been attempted based not on heat as random motion of atoms, as Hooke and Boyle had done, but by the concept of heat as a weightless fluid called caloric. He introduced the concept of the "Carnot engine", an ideal heat engine. He showed that the efficiency of the "Carnot engine" depends only on the temperature difference within the engine and not on the substance such as steam that drives the mechanism. He also introduced the important concept of reversibility, namely that motive power can be used to produce the temperature difference in the engine.
Notes which Carnot made between 1824 and 1826 show that he was moving away from the caloric theory. He set out in his notes details of experiments which he intended to carry out to study the temperature effects of friction in fluids. Some of these proposed experiments were identical with those actually performed by Joule twenty years later. It is likely that Carnot would have made many more significant scientific contributions had his life not been so tragically short.
Article by: J J O'Connor and E F Robertson