Lazare Carnot

How to use this course

Lazare Carnot is remembered in France as the “organizer of victory”, the member of the Committee of Public Safety who raised, armed and directed the armies of the Republic in 1793 and 1794. He was also a serious scientist. His essay on machines of 1783 was the first theoretical treatise of engineering mechanics, and it contains a theorem on the loss of live force in collisions that still bears his name. His books on geometry helped launch the modern study of signed magnitudes and projective configurations, and his son Sadi built thermodynamics on ideas taken from his father’s mechanics.

Each technical lesson opens with an italic paragraph for the general reader, then gives precise statements in modern notation, derivations, and exercises with folded solutions. Carnot’s own notation was verbal and cumbersome; where we translate, we say so.

Timeline

Nolay and Autun, 1753–1771

Lazare Nicolas Marguerite Carnot was born on 13 May 1753 at Nolay, a small town in Burgundy between Beaune and Autun. His father, Claude Carnot, was a notary and local judge; his mother was Marguerite Pothier. The family was large, and several brothers had public careers. Lazare and his brother Claude-Marie both entered the army as engineers.

At fourteen he went to the college at Autun run by the Oratorians, a teaching order known for its attention to mathematics and the sciences. He then studied at the Sulpician seminary in Autun, where he read logic, theology and mathematics. The Church was a possible path, but his father steered both sons toward the technical corps of the army, one of the few routes by which a family of the provincial bourgeoisie could reach a commission.

To prepare for the entrance examination of the engineering school he spent time at a preparatory school in Paris. There he came to the notice of Jean le Rond d’Alembert, whose Traité de dynamique would later shape his mechanics. In February 1771 he placed third in the examination.

Mézières and the engineer corps, 1771–1789

The École royale du génie at Mézières was the best technical school in Europe. Students learned fortification, surveying, stone-cutting, hydraulics and mechanics. Its most famous teacher, Gaspard Monge, was developing descriptive geometry there, a method for representing three-dimensional objects by projections that the army kept secret for years. Carnot graduated on 1 January 1773 as a first lieutenant and was posted to garrison towns in the north and east, including Calais, Cherbourg and Arras.

Garrison life left time for study. In 1777 the Paris Academy of Sciences set a prize question on the theory of simple machines, including friction and the stiffness of ropes. Carnot submitted a memoir, revised it for a second round in 1780, and received an honourable mention; the prize went to Charles-Augustin de Coulomb for his study of friction. Carnot published the theoretical part as the Essai sur les machines en général in 1783, and a second printing under his name followed in 1786. It is the subject of Lessons 04 to 06.

In 1784 he won the prize of the Dijon Academy for a eulogy of Vauban, the great military engineer of Louis XIV, which defended walled fortresses against the fashion for mobile field armies. He also entered the Berlin Academy’s competition on the theory of the infinite in mathematics, set in 1784; his entry received an honourable mention and became, much revised, the Réflexions of 1797 (Lesson 07). He was promoted captain in December 1783 and elected to the Dijon Academy in 1787.

In Arras he joined a literary society, the Rosati, among whose members was the young lawyer Maximilien Robespierre. In 1790 he was injured in a riding accident and nursed by Sophie Dupont, from a family of Saint-Omer; they married on 17 May 1791.

The organizer of victory, 1789–1797

The Revolution opened politics to men like Carnot. In 1791 the Pas-de-Calais elected him to the Legislative Assembly, where he sat on the committee for public instruction and on military committees. In 1792 he was elected to the National Convention. In January 1793 he voted for the death of Louis XVI, the vote that later made him a proscribed regicide.

France was then at war with most of Europe. Carnot was sent on mission to the armies, and on 14 August 1793 he joined the Committee of Public Safety, the twelve-member executive of the Terror, with responsibility for war. Together with Prieur de la Côte-d’Or, also an engineer, he turned the mass levy of August 1793 into an organized army: he planned campaigns, standardized supply, promoted young officers on merit and pressed for concentrated offensives. In October 1793 he was at the front with Jean-Baptiste Jourdan for the victory of Wattignies, which relieved the siege of Maubeuge. By 1794 the Republic had more than half a million men under arms, and the invasions had been turned back. The Convention’s phrase for him, the organizer of victory, stuck.

He signed many of the Committee’s orders, including some of the Terror’s, but his work was the war, and he clashed with Robespierre and Saint-Just. He survived the fall of Robespierre in July 1794 and left the Committee in October. With Monge and others he took part in founding the École centrale des travaux publics, soon renamed the École polytechnique, which became the nursery of French science and engineering.

In November 1795 he became one of the five Directors who ruled France under the new constitution, and in 1796 he presided over the Directory. He supported the appointment of Napoleon Bonaparte to command the Army of Italy. Moderate by now, he lost the struggle with his fellow Directors, and in the coup of 18 Fructidor (4 September 1797) he was proscribed. He escaped to Switzerland and then to Germany. At the Institut, where he had sat in the mechanical arts section, he was struck off and replaced by Bonaparte.

Machines, live force and work

For the general reader. Eighteenth-century engineers wanted to know how much a water wheel, a pulley or a pump could really do. Carnot’s answer was that a machine only transmits motion; it creates nothing. What it transmits is measured by force times distance, a quantity he called the moment of activity and we call work. The best machine is the one that loses least of it.

Carnot’s definition

Carnot defined a machine as an intermediary body or system that transmits motion between bodies, the movers and the resistances, that do not act on one another directly. His theory therefore ignored the detailed construction of the machine and asked only about the relations between the motions it permits. This abstraction is the origin of the idea, later central to engineering mechanics, that machines can be compared by energy accounting alone.

Live force and moment of activity

Carnot used the live force (vis viva) \(\sum m v^2\) of Leibniz and the Bernoullis. A weight \(P=mg\) raised through height \(h\) requires a “moment of activity” \(Ph\), and a body falling freely through \(h\) gains \(v^2=2gh\). In modern form:

Proof. Dot Newton’s law \(m_i\dot{\mathbf v}_i=\mathbf F_i\) with \(\mathbf v_i\) and sum; the left side is the time derivative of the kinetic energy. Integrate in time. \(\square\)

Applied to a machine working in a steady cycle, with movers doing work \(W_{\text{in}}\) and resistances absorbing \(W_{\text{out}}\), the kinetic energy returns to its initial value, so \(W_{\text{in}}=W_{\text{out}}+W_{\text{lost}}\), with \(W_{\text{lost}}\ge0\) for friction and shocks. Gaspard-Gustave Coriolis proposed the word travail (work) for Carnot’s moment of activity in 1829, and the factor \(\tfrac12\) in kinetic energy became standard at the same time.

Geometric motions

For the general reader. A lever can only turn, a piston can only slide. Carnot called the motions that a system’s shape allows, whatever forces are acting, its geometric motions. He showed that the laws of equilibrium and of impact can be read off by testing forces against these motions. The idea is very close to the virtual displacements of Lagrange, and Carnot reached it independently in the language of an engineer.

Definition

Consider particles at positions \(\mathbf r_i\) subject to ideal constraints. A set of velocities \(\mathbf u_i\) is a geometric motion if the system could move with those velocities without violating the constraints, and if the reversed velocities \(-\mathbf u_i\) are also allowed. In his 1803 Principes fondamentaux, Carnot characterized these as motions during which the constraints do no work. In modern terms, the \(\mathbf u_i\) form the tangent space of the configuration manifold, the reversible virtual velocities.

Carnot’s own fundamental equation, for systems of hard bodies, was written with the velocities “lost” in the motion. If \(\mathbf v_i\) is the velocity a body would have had and \(\mathbf w_i\) the velocity it actually has, the lost velocity is \(\mathbf v_i-\mathbf w_i\), and

This is a version of d’Alembert’s principle: the lost motions would be in equilibrium among themselves.

Conservation laws from rigid motions

For a free system with internal constraints only, any rigid translation \(\mathbf u_i=\mathbf a\) and any rigid rotation \(\mathbf u_i=\boldsymbol\omega\times\mathbf r_i\) is a geometric motion. Inserting them in the fundamental equation gives conservation of momentum and angular momentum across impulsive interactions. Carnot obtained the conservation of moment of momentum this way.

Carnot’s theorem on shocks

For the general reader. When two lumps of clay collide and stick, momentum survives but some motion disappears into heat and deformation. Carnot found an exact formula for the loss: it equals the kinetic energy that the bodies would have if they moved with the velocities they lost. The same formula, applied to water suddenly widening in a pipe, gives the loss of head that engineers call the Borda–Carnot equation.

Proof. In a perfectly inelastic impact the bodies end up moving together in a way the constraints allow, so \(\mathbf w\) is a geometric motion. The impulsive internal forces are \(m_i(\mathbf w_i-\mathbf v_i)\), and by the principle of Lesson 05 they do no work on a geometric motion: \(\sum m_i(\mathbf v_i-\mathbf w_i)\cdot\mathbf w_i=0\). Now expand

and the middle term vanishes. \(\square\)

The identity is a Pythagorean decomposition. With the mass-weighted inner product \(\langle\mathbf a,\mathbf b\rangle=\sum m_i\mathbf a_i\cdot\mathbf b_i\), the post-impact velocity \(\mathbf w\) is the orthogonal projection of \(\mathbf v\) onto the subspace of geometric motions. So the shock always loses energy, and loses none only if the motion was already compatible with the constraints. Carnot drew the engineering conclusion that a machine should transmit motion gradually, without percussion.

The Borda–Carnot equation

When an incompressible fluid flowing at speed \(v_1\) in a pipe passes a sudden enlargement into a pipe where its speed is \(v_2<v_1\), the fast stream collides with the slower fluid. Momentum balance on the control volume, with the upstream pressure acting over the large section, gives the mechanical energy lost per unit volume,

the “kinetic energy of the lost velocity”. Jean-Charles de Borda had found this result in 1766; Carnot’s theorem explains it as an instance of a general law.

The metaphysics of the calculus

For the general reader. Eighteenth-century calculus worked, but no one could say why. Infinitely small quantities were treated as zero when convenient and as non-zero otherwise, and George Berkeley had mocked them as the “ghosts of departed quantities”. Carnot argued that the method is exact because its errors cancel: one deliberately makes two small mistakes whose effects compensate, and the final equation contains no trace of either.

Compensation of errors

The Réflexions sur la métaphysique du calcul infinitésimal (1797, much expanded 1813) grew from Carnot’s entry to the Berlin prize. Its central thesis can be put as follows.

In modern terms, if two fixed numbers \(A\) and \(B\) satisfy \(\lvert A-B\rvert<\varepsilon\) for every \(\varepsilon>0\), then \(A=B\). This is the core of the limit method that Augustin-Louis Cauchy made rigorous in the 1820s. Carnot’s contribution was to explain why the infinitesimal algorithms, applied according to their rules, never produce a false final result.

An example

Find the subtangent of the parabola \(y^2=px\) at \((x,y)\). Let \(MT\) be the tangent and take a nearby point at \((x+h,y+k)\) on the curve. Treating the chord as the tangent (an error) gives subtangent \(\approx y\,h/k\). From the curve, \(2yk+k^2=ph\), and dropping \(k^2\) (a second error) gives \(h/k\approx2y/p\). Combining, subtangent \(=2y^2/p=2x\). The two errors compensate exactly, as the limit computation confirms: \(dx/dy=2y/p\) and the subtangent is \(y\,dx/dy=2x\).

Geometry of position and signed quantities

For the general reader. A Greek geometer proving a theorem had to redraw it for every case: point inside the segment, point outside, and so on. Carnot wanted one statement that covered all positions at once. His solution was to give lengths and angles a sign that records their position, and to follow a figure as it is continuously deformed. The approach led directly to the projective geometry of Jean-Victor Poncelet and Michel Chasles.

Correlation of figures

In De la corrélation des figures de géométrie (1801) and Géométrie de position (1803), Carnot compared a “primitive” figure with “correlative” figures obtained by moving its points continuously. A relation proved for the primitive figure carries over, but some quantities may change sign when a point passes through another. Carnot himself held that negative quantities had no existence alone and only marked a reversal of position, a view that later geometers dropped in favour of signed segments \(\overline{AB}=-\overline{BA}\) on oriented lines. The method of following a theorem through all positions became Poncelet’s principle of continuity.

Proof

The central angle on chord \(BC\) is \(2A\), so \(d_a=R\cos A\), and the sign convention makes this hold also when \(A\) is obtuse (\(\cos A<0\)). Hence \(d_a+d_b+d_c=R(\cos A+\cos B+\cos C)\). The identity

finishes the proof. To prove it, write \(\cos A+\cos B=2\sin\frac C2\cos\frac{A-B}2\) and \(1-\cos C=2\sin^2\frac C2\), so the left side minus 1 is \(2\sin\frac C2\big(\cos\frac{A-B}2-\cos\frac{A+B}2\big)=4\sin\frac A2\sin\frac B2\sin\frac C2\), and \(r=4R\sin\frac A2\sin\frac B2\sin\frac C2\) is standard. \(\square\)

Transversals, conics and five points

For the general reader. A line crossing the three sides of a triangle cuts them in ratios whose product is always one; this is the ancient theorem of Menelaus. Carnot made such “transversal” theorems a subject in their own right and extended them from lines to curves. His theorem on conics is a direct ancestor of the cross-ratio methods of projective geometry.

Proof sketch. Let the conic be \(f(\mathbf x)=0\) with \(f\) a quadratic polynomial. Parametrize line \(BC\) by \(\mathbf x(t)=B+t(C-B)\); then \(g(t)=f(\mathbf x(t))\) is quadratic with roots \(t_1,t_2\) at \(A_1,A_2\), so \(g(t)=c\,(t-t_1)(t-t_2)\). Evaluating at \(t=0\) and \(t=1\):

since the two sign changes from \(\overline{A_iC}=-\overline{CA_i}\) cancel. Multiply the three such ratios: \(\frac{f(B)}{f(C)}\cdot\frac{f(C)}{f(A)}\cdot\frac{f(A)}{f(B)}=1\). (The vertices are assumed not on the conic.) \(\square\)

When the conic degenerates into the given transversal line taken twice, each pair of points coincides, and the theorem gives the square of Menelaus’ relation. The same argument works for curves of any degree \(n\), with \(n\) points on each side.

Five points in space

In his memoir of 1806 on the distances between five points, Carnot sought the relation that must hold among the ten mutual distances of five points in three-dimensional space, since only nine of them can be chosen freely. Arthur Cayley gave the relation its determinant form in 1841: if \(d_{ij}\) are the distances among points \(P_0,\dots,P_4\) in \(\mathbb R^3\), then

because this Cayley–Menger determinant is proportional to the squared volume of the 4-simplex on the five points, which is zero in three dimensions.

From Lazare to Sadi: continuity and reversibility

For the general reader. In 1824 Lazare’s son Sadi Carnot asked how much work a heat engine can produce, and founded thermodynamics. His argument copied his father’s. A water wheel is best when the water enters without shock and leaves without speed; a heat engine is best when heat passes without any useless flow between bodies at different temperatures. Both are best when the process could run backwards.

The principle of continuity

From the theorem of Lesson 06, every sudden change of velocity loses live force. Lazare concluded that a machine attains its maximum effect when motion is transmitted by insensible degrees, without percussion or turbulence, and when the fluid leaves with no remaining velocity. Engineers attributed this principle to him until the mid-nineteenth century, when it was absorbed into energy conservation.

Sadi’s own analogy compared heat falling from a high temperature to a low one with water falling from a height. The analogy was imperfect, because in his caloric theory heat was conserved, while energy conservation required that some heat be converted into work. Rudolf Clausius and William Thomson rebuilt the argument on the first and second laws around 1850, keeping Sadi’s reversibility criterion, the descendant of Lazare’s geometric motions, which can be run in either direction.

Consulate, Empire and the Hundred Days, 1800–1815

After Bonaparte seized power in November 1799, Carnot returned to France. He served as Minister of War from 2 April to 8 October 1800, organizing the campaign that ended at Marengo, then resigned. He was readmitted to the Institut and in 1802 entered the Tribunate. There he voted against the life consulship in 1802 and against the proclamation of the Empire in 1804, publicly and nearly alone. After the coronation he withdrew from politics.

The next decade was his most productive in science. In 1801 came De la corrélation des figures de géométrie, in 1803 the Géométrie de position and the Principes fondamentaux de l’équilibre et du mouvement, and in 1806 the memoir on five points with its essay on transversals. He served on many Institut commissions reviewing inventions and mathematical memoirs. In 1809 he published De la défense des places fortes, which argued for active defence by sorties and vertical fire.

In January 1814, as the allied armies entered France, Carnot offered his services and was made governor of Antwerp. He held the city through the siege and surrendered it only on orders after Napoleon’s abdication. During the Hundred Days he was Minister of the Interior from 20 March to 22 June 1815, made a count of the Empire and a peer. In that office he took up public education, including plans for elementary schooling. After Waterloo he sat on the provisional government commission that ruled until the return of Louis XVIII.

Exile, death and legacy, 1815–1823

The restored Bourbons proscribed the regicides who had served Napoleon in the Hundred Days. Carnot left France, went first to Warsaw, and from November 1816 lived at Magdeburg in Prussia. He was expelled from the Academy of Sciences in the purge of 1816. In exile he wrote verse and memoirs and followed the education of his sons. He died at Magdeburg on 2 August 1823, aged 70, and was buried there. In 1889, during the presidency of his grandson Sadi Carnot, his remains were brought to the Panthéon in Paris.

Family

His elder son, Nicolas Léonard Sadi Carnot (1796–1832), trained at the École polytechnique and published Réflexions sur la puissance motrice du feu in 1824, the founding work of thermodynamics. His younger son, Hippolyte Carnot (1801–1888), was a politician and minister of public instruction in 1848. Hippolyte’s son Marie François Sadi Carnot was President of the Republic from 1887 until his assassination in 1894.

Legacy

• Mechanics: the first general theory of machines; work as the measure of what machines transmit; geometric motions; Carnot’s theorem on shocks; the Borda–Carnot head loss. Through Coriolis, Navier and Poncelet his ideas shaped the engineering mechanics of the École polytechnique.
• Thermodynamics: the principle of continuity and the notion of a motion that can be reversed passed directly to Sadi Carnot.
• Geometry: signed magnitudes, correlation of figures, and the theory of transversals, which fed Poncelet’s projective geometry and Chasles’s historical synthesis. Carnot’s theorems on the triangle and on conics are still taught.
• Foundations: the compensation-of-errors account of the calculus, widely read and translated before Cauchy.
• Public life: the organization of the revolutionary armies, a share in founding the École polytechnique, and a reputation for integrity across five regimes.

Connections

People linked to Lazare Carnot across this collection. A name without a link has no page here yet. See Lazare Carnot in the network of all 337 lives →

Teachers and advisors

• Gaspard Monge 1771-1773

Influenced by

• Gottfried Wilhelm Leibniz
• Jean le Rond d'Alembert
• Jean-Charles de Borda

Influenced

• Augustin-Louis Cauchy
• Gaspard-Gustave Coriolis
• Jean-Victor Poncelet
• Michel Chasles

Colleagues and collaborators

• Gaspard Monge 1794
• Joseph-Louis Lagrange
• Prieur de la Côte-d'Or 1793-1794

Rivals

• Charles-Augustin de Coulomb 1777-1781

Family

• Sadi Carnot
• Hippolyte Carnot

Also mentioned in this life

• Arthur Cayley
• Rudolf Clausius
• Leonhard Euler
• William Thomson, Lord Kelvin
• Claude-Louis Navier
• Isaac Newton

Fields

• mechanics 27 lives
• geometry 32 lives
• military engineering 3 lives

On CoursesHub

• Classical Mechanics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Lazare Nicolas Marguerite Carnot, MacTutor History of Mathematics.
2. Lazare Carnot, Wikipedia; Lazare Carnot, Wikipédia (French).
3. C. C. Gillispie, Carnot, Lazare-Nicolas-Marguerite, Dictionary of Scientific Biography.
4. Carnot, “The Great”, Fondation Napoléon.
5. M. Reinhard, Le Grand Carnot, 2 vols., Hachette, 1950–52.

Science

1. L. Carnot, Essai sur les machines en général, Dijon, 1783.
2. L. Carnot, Réflexions sur la métaphysique du calcul infinitésimal, Paris, 1797; 2nd ed. 1813.
3. L. Carnot, De la corrélation des figures de géométrie, Paris, 1801.
4. L. Carnot, Géométrie de position, Paris, 1803.
5. L. Carnot, Principes fondamentaux de l’équilibre et du mouvement, Paris, 1803.
6. L. Carnot, Mémoire sur la relation qui existe entre les distances respectives de cinq points quelconques pris dans l’espace; suivi d’un essai sur la théorie des transversales, Paris, 1806.
7. C. C. Gillispie, Lazare Carnot, Savant, Princeton University Press, 1971.
8. C. C. Gillispie and R. Pisano, Lazare and Sadi Carnot: A Scientific and Filial Relationship, Springer, 2014.
9. S. Carnot, Réflexions sur la puissance motrice du feu, Paris, 1824.
10. Carnot’s theorem (inradius, circumradius) and Carnot’s theorem (conics), Wikipedia.
11. Borda–Carnot equation, Wikipedia.