Gaspard Monge

How to use this course

Gaspard Monge thought in pictures. He turned the drawings of stonemasons and fortification engineers into a mathematical method, descriptive geometry, which taught generations of engineers to reason about three-dimensional objects on paper. He used the same spatial intuition to found the differential geometry of surfaces, to give the first geometric account of partial differential equations, and to pose the problem of moving earth at least cost, now the theory of optimal transport.

He was also a central figure of the French Revolution and the Empire: Minister of the Navy when Louis XVI was deposed, organiser of arms production during the war of 1793, founder and director of the École Polytechnique, president of the Institut d’Égypte, and a senator and count under Napoleon. His teaching formed the next generation of French geometers.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements in modern language, derivations and exercises with folded solutions. Statements are given in their modern form; the text says where Monge’s own version was narrower.

Timeline

Beaune, Lyon and Mézières, 1746–1768

Gaspard Monge was born in Beaune, Burgundy, on 9 May 1746. His father, Jacques Monge, was a merchant originally from Haute-Savoie; his mother, Jeanne Rousseaux, came from Burgundy. Jacques Monge valued education and sent his sons to the Oratorian college in Beaune. Gaspard’s younger brother Louis also became a mathematician.

Gaspard was the best pupil at the college. In 1762 he went to the Collège de la Trinité in Lyon, also run by the Oratorians, and at 17 was put in charge of teaching physics there. On a visit home in 1764 he drew a large-scale plan of Beaune, making his own surveying instruments. An officer of engineers saw the plan and recommended him to the military engineering school at Mézières.

Mézières

The École Royale du Génie at Mézières, in the Ardennes, trained officers of the engineering corps. Officers were nobles; Monge, a commoner, was taken on in 1765 as a draughtsman in the drawing office, where non-commissioned men prepared plans and models. His chance came with a standard exercise in fortification called défilement: design a fortress so that no part of it is exposed to fire from the surrounding high ground. The accepted method was a long arithmetical computation. Monge solved the problem with a geometric construction, so quickly that his superior at first refused to look at the result. Once checked, the method was adopted. The techniques behind it became descriptive geometry (Lesson 04), which the army treated for years as a military secret.

Professor at Mézières, 1769–1792

Monge’s gifts were soon recognised. He began teaching at Mézières, and in January 1769 he succeeded Charles Bossut as professor of mathematics. In 1770 he also took over the teaching of experimental physics. He was 23.

Research followed teaching. His first paper, on the evolutes of curves in space, appeared in 1769. In 1771 he read to the Académie des Sciences in Paris a series of memoirs on curves, developable surfaces and partial differential equations, and he established contact with Jean le Rond d’Alembert and Nicolas de Condorcet. Monge’s method was always to read an analytic equation as a statement about a surface, and a surface as a family of curves. In 1777 he married Catherine Huart, a widow who owned a forge; Monge took an interest in its metallurgy.

In 1780 he was elected to the Académie des Sciences as an adjoint géomètre and began to divide his time between Mézières and Paris. His memoir of 1781 on cuts and fills (déblais et remblais) posed the transport problem and introduced the lines of curvature of a surface (Lessons 06 and 09). He also worked in chemistry and physics: in 1783 he synthesised water from hydrogen and oxygen, independently of Antoine Lavoisier and Henry Cavendish, and he studied the composition of iron and steel with Claude Louis Berthollet and Alexandre-Théophile Vandermonde.

In 1783, on the death of Étienne Bézout, Monge became examiner of naval cadets, a post that took him on long tours of the ports. He resigned his Mézières posts in December 1784. Among his students at Mézières was Jean-Baptiste Meusnier, whose theorem on the curvature of curves on a surface (1776) grew directly from Monge’s teaching.

Revolution and the École Polytechnique, 1792–1795

Monge welcomed the Revolution. After the overthrow of the monarchy on 10 August 1792 he was named Minister of the Navy in the provisional executive council, a post he held until 10 April 1793. The navy had lost many of its royalist officers, the war with Britain began in February 1793, and Monge was attacked from all sides. He resigned after eight months.

He then threw himself into the war effort. With Berthollet and other scientists he organised the production of saltpetre for gunpowder and the casting and boring of cannon for the armies of the Republic, and in 1794 he wrote a manual on the manufacture of cannon. The Académie des Sciences was abolished in August 1793.

A school for engineers

The Republic needed engineers, and the old corps schools such as Mézières had been closed or disrupted. On 11 March 1794 Monge was appointed to the commission that created the École Centrale des Travaux Publics, together with Lazare Carnot, Jacques-Élie Lamblardie and others. Monge drew up much of its programme, built around descriptive geometry and analysis, and taught descriptive geometry himself from November 1794. In 1795 the school was renamed the École Polytechnique. Monge was its most admired teacher and later its director.

Early in 1795 he also lectured on descriptive geometry at the short-lived École Normale. Stenographers recorded the lectures, and they were published, first in the school’s proceedings and in 1799 as the book Géométrie descriptive, edited by his colleague Jean Nicolas Pierre Hachette. At Polytechnique he also lectured on the application of analysis to geometry; the lecture sheets were printed as Feuilles d’analyse appliquée à la géométrie (1795). His students at the school included Étienne-Louis Malus, Charles Dupin, Charles Julien Brianchon and Jean-Victor Poncelet. Sylvestre François Lacroix, an earlier protégé, became one of the most influential textbook writers of the period.

Descriptive geometry

For the general reader. How do you draw a three-dimensional object on flat paper so precisely that a mason can cut stone from the drawing? Monge’s answer was to draw two views, one from above and one from the front, on the same sheet, linked by lines that tie each point in one view to the same point in the other. From these two views every length, angle and intersection can be constructed with ruler and compass.

Plan and elevation

Take a horizontal plane \(H\) and a vertical plane \(V\) meeting in the ground line \(g\). A point \(P\) in space has a horizontal projection \(P'\) on \(H\) (its plan) and a vertical projection \(P''\) on \(V\) (its elevation). Rotate \(V\) about \(g\) until it lies in \(H\). The sheet now carries both projections.

A line is given by its two projections; a plane by its traces, the lines in which it meets \(H\) and \(V\). Every metric question reduces to a construction in the sheet. Monge’s systematic contribution was to show that a small set of operations suffices: projecting, rotating a figure about a line parallel to a projection plane, and introducing an auxiliary projection plane.

True length

A segment \(AB\) appears foreshortened in both views. If its plan has length \(\ell'\) and the heights of its endpoints differ by \(\Delta z\), then its true length is

and its angle with the horizontal is \(\arctan(\Delta z/\ell')\). Graphically: rotate the segment about a vertical axis through \(A\) until it is parallel to \(V\); its new elevation shows it in true length. This is the operation masons had long used informally in stereotomy, the art of cutting stone for vaults and arches.

Intersections

To find where two surfaces meet, cut both with a family of auxiliary planes (or spheres) chosen so that each cut is simple. Each auxiliary plane meets each surface in a line or circle, and the points where these meet lie on the intersection curve. Shadows are intersections too: the shadow of an object is the intersection of a cylinder (or cone) of light rays with the receiving surface. Monge’s course treated tangent planes, shadows and the intersections of quadrics this way.

Space curves and developable surfaces

For the general reader. Some surfaces can be unrolled flat without stretching or tearing: a cylinder, a cone, the surface swept out by the tangent lines of a twisted curve. These developable surfaces matter to anyone who shapes sheet metal or cuts cloth. Monge found the equation they satisfy and described them as families of straight lines along which the surface does not twist.

Evolutes

The evolute of a plane curve is the locus of its centres of curvature, and the curve can be recovered by unwinding a string from the evolute. Monge’s first paper (1769) extended this to curves in space. At each point of a space curve the normal plane contains a whole line of points equidistant to first order from three neighbouring points of the curve, the polar line. The polar lines sweep out a developable surface, the polar developable. Monge showed that a space curve has infinitely many evolutes, and that they are geodesics of the polar developable.

Monge’s notation

Write a surface as a graph \(z=f(x,y)\), now called the Monge form, with

Monge used these letters throughout, and they remain standard in the classical theory of partial differential equations. The Gaussian curvature of the graph is

Sketch. If \(rt=s^2\), the Jacobian of \((x,y)\mapsto(p,q)\) vanishes, so \(p\) and \(q\) are functionally dependent, say \(q=\phi(p)\) locally. The tangent plane \(z=px+qy-w\), where \(w=xp+yq-f\), then depends on one parameter \(p\), with \(w=\psi(p)\) also a function of \(p\). A one-parameter family of planes envelopes a developable surface, and the surface is the union of the lines \(z=px+\phi(p)y-\psi(p)\), \(0=x+\phi'(p)y-\psi'(p)\). Leonhard Euler studied developable surfaces at the same time, in a paper of 1772; Monge’s 1771 memoir was published in 1785.

Lines of curvature

For the general reader. At each point of a curved surface there are two directions in which it bends most and least. Following these directions from point to point traces two families of curves crossing at right angles, the lines of curvature. Monge introduced them in 1781 and drew them on the ellipsoid, where they produce a striking net of curves around four special points.

Principal directions

Let \(N\) be the unit normal of a surface \(S\). The shape operator \(dN\) is a self-adjoint map of each tangent plane. Its eigenvalues are \(-\kappa_1,-\kappa_2\), where \(\kappa_1,\kappa_2\) are the principal curvatures, and its eigenvectors are the principal directions. Euler (1760) had found that normal curvatures at a point range between two extremes in perpendicular directions. A line of curvature is a curve on \(S\) tangent at each point to a principal direction.

Monge found lines of curvature by asking, in the transport problem of Lesson 09, when the normals to a surface along a curve can meet consecutively. The equivalence of (2) and (3): the ruled surface \(\alpha(s)+vN(s)\) is developable exactly when \(\alpha'\), \(N\) and \(N'\) are coplanar, which, since \(N'\perp N\), means \(N'\parallel\alpha'\). Away from umbilics (where \(\kappa_1=\kappa_2\)) the two families of lines of curvature form an orthogonal net.

Charles Dupin was Monge’s student. The confocal quadrics \(\frac{x^2}{a-\lambda}+\frac{y^2}{b-\lambda}+\frac{z^2}{c-\lambda}=1\) form a triply orthogonal system, so the lines of curvature of an ellipsoid are its intersections with the confocal hyperboloids of one and of two sheets. Monge had computed them directly in the 1790s. The ellipsoid with distinct axes has exactly four umbilics, in the plane of its longest and shortest axes.

First-order equations and characteristics

For the general reader. A partial differential equation of first order restricts the tangent plane of an unknown surface at each point. Monge saw this as a geometric condition: at each point of space, the allowed tangent planes envelope a small cone. Solution surfaces are woven from curves that follow these cones, the characteristics. The idea underlies the modern theory of wave propagation and shocks.

Quasi-linear equations

Consider \(a(x,y,u)\,u_x+b(x,y,u)\,u_y=c(x,y,u)\). A solution surface \(z=u(x,y)\) has normal \((u_x,u_y,-1)\), and the equation says this normal is perpendicular to the vector field \(\mathbf v=(a,b,c)\). So the surface is tangent to \(\mathbf v\) everywhere and is a union of integral curves of

These are the characteristic curves. Joseph-Louis Lagrange solved such equations analytically in the 1770s and 1780s; Monge supplied the geometric picture.

The Monge cone

For a general equation \(F(x,y,z,p,q)=0\), at a point \((x,y,z)\) the admissible tangent planes \(Z-z=p(X-x)+q(Y-y)\) form a one-parameter family. Their envelope is a cone with vertex at the point, the Monge cone. A solution surface touches the Monge cone at each point along a generator, and following these generators gives the characteristic curves. Together with the planes along them they give the characteristic strips, which satisfy Charpit’s equations

Second-order equations and Monge–Ampère

For the general reader. Monge also studied equations involving second derivatives, where the unknown surface’s curvature is constrained. The most important of these prescribes the product of the principal curvatures, or the determinant of the Hessian. Monge met such equations in his work on developables and on transport; André-Marie Ampère treated them systematically around 1820. Today they are central in geometry and in optimal transport.

Characteristics of second-order equations

For \(R\,r+S\,s+T\,t=V\) (with \(R,S,T,V\) depending on \(x,y,z,p,q\)), Monge looked for curves along which knowledge of \(z,p,q\) does not determine the second derivatives. Along a curve with direction \((dx,dy)\), \(dp=r\,dx+s\,dy\) and \(dq=s\,dx+t\,dy\). Solving these with the equation for \(r,s,t\) fails exactly when

The equation is hyperbolic if \(S^2-4RT>0\) (two real families of characteristics), parabolic if \(S^2=4RT\), and elliptic if \(S^2-4RT<0\) (no real characteristics). Monge developed a method of integration using these characteristic directions, now called Monge’s method of intermediate integrals.

The Monge–Ampère equation

By Lesson 05, prescribing Gaussian curvature \(K(x,y)\) for a graph is the equation \(rt-s^2=K\,(1+p^2+q^2)^2\). The case \(f=0\) is Monge’s equation for developables. Hermann Minkowski’s problem, to find a closed convex surface with prescribed Gaussian curvature as a function of the normal, reduces to a Monge–Ampère equation on the sphere. In optimal transport (Lesson 09), Brenier’s map \(T=\nabla u\) pushing a density \(f\) to a density \(g\) satisfies

Cuts and fills: optimal transport

For the general reader. Builders of roads and fortifications dig earth from some places (the cuts) and pile it in others (the fills). Which load should go where, to make the total effort as small as possible? Monge posed this question in 1781. It lay largely dormant until Leonid Kantorovich reformulated it in 1942. Since the 1990s optimal transport has become a major field, linking geometry, probability, economics and machine learning; Cédric Villani (2010) and Alessio Figalli (2018) received Fields Medals in part for work in it.

The problems

Let \(\mu,\nu\) be probability measures on \(\mathbb R^n\) and \(c(x,y)\ge0\) a cost.

Monge’s problem is hard because the constraint \(T_\#\mu=\nu\) is nonlinear in \(T\), and an optimal map may not exist (a point mass cannot be split by a map). Kantorovich allowed mass to split.

Monge’s observations

Monge noticed two facts. First, for \(c=|x-y|\), the transport paths of an optimal plan cannot cross. Second, the paths are straight segments that, in good cases, are normal to a family of surfaces; studying when normals of a surface form such a family led him to lines of curvature (Lesson 06).

For Monge’s own cost \(|x-y|\) the optimal plan is not unique, and the existence of an optimal map for absolutely continuous \(\mu\) was established only around 2000, after work of Vladimir Sudakov, Lawrence Evans and Wilfrid Gangbo, and others.

Elementary geometry: circles and tetrahedra

For the general reader. Several theorems of classical geometry carry Monge’s name. They share his habit of looking at a plane figure as a shadow of a figure in space, which often makes a hard plane theorem obvious. His students Brianchon and Poncelet made this habit the basis of projective geometry.

Proof by going into space. Replace each circle by the sphere with the same centre and radius. For two spheres, the external tangent cone has its apex at the external centre of similitude. Consider the two planes tangent to all three spheres (one above and one below the plane of the centres). Each of these planes is tangent to all three cones, so it contains all three apexes. The apexes lie in the plane of the circles too. So they lie on the line where a tangent plane meets the original plane.

Italy, Egypt and the Empire, 1796–1815

In May 1796 the Directory sent Monge to Italy, with Berthollet and others, on a commission to select works of art and science to be taken to Paris under the terms imposed on the Italian states by the victorious army. There he came to know the young general Napoleon Bonaparte, and the two formed a lasting attachment. Monge returned to Paris in October 1797 carrying the Treaty of Campo Formio. Early in 1798 he was sent back to Italy on the mission that set up the short-lived Roman Republic.

Egypt

In May 1798 Monge joined the expedition to Egypt, and he helped recruit its scientists, among them Joseph Fourier. In Cairo Bonaparte founded the Institut d’Égypte, and in August 1798 Monge became its president, with Fourier as secretary. He studied the antiquities and the natural phenomena of the country. His best-known paper from Egypt explained the mirage seen by the troops in the desert: air heated by the ground has a lower refractive index, and rays from the low sky are bent back upward to the observer, so the sky appears as a sheet of water on the ground. In October 1799 Monge was one of the few who returned to France with Bonaparte.

Under Napoleon

After the coup of November 1799 Monge was named senator for life. He became president of the Senate in 1806 and was created comte de Péluse (Pelusium, a site in the Nile delta) in 1808. He remained director of the École Polytechnique and continued to teach there until 1809, when his health declined. The Application de l’analyse à la géométrie, the expanded version of his Feuilles, appeared in 1807. He used his influence to protect the school and its students, who adored him.

Fall, death and legacy

When Napoleon returned from Elba in 1815, Monge rallied to him during the Hundred Days. After Waterloo his position was untenable. He left France in October 1815 and returned in March 1816. The restored monarchy reorganised the Institut de France and excluded him, along with other prominent Bonapartists. He lost his honours and his place at the École Polytechnique, which was itself reorganised. His health and spirits failed, and he died in Paris on 28 July 1818, aged 72. He was buried at Père Lachaise. In 1989, for the bicentenary of the Revolution, his remains were transferred to the Panthéon.

Legacy

• Engineering education: descriptive geometry was taught to engineers and architects across Europe and America for more than a century and is the ancestor of technical drawing and computer-aided design. The École Polytechnique became the model for engineering schools in many countries.
• Differential geometry: lines of curvature, developable surfaces and the Monge form of a surface. His students Meusnier and Dupin, and later Gauss’s intrinsic theory, built on this foundation.
• Partial differential equations: the Monge cone, characteristics, and the Monge–Ampère equation, now central in geometric analysis.
• Projective geometry: Poncelet’s Traité des propriétés projectives des figures (1822) and Brianchon’s theorem came from Monge’s school.
• Optimal transport: the 1781 memoir is the founding document of a field that now reaches from Riemannian geometry to economics and machine learning.
• Honours: his name is one of the 72 engraved on the Eiffel Tower.

Monge’s teaching mattered as much as his theorems. He showed a generation that geometry could be both rigorous and useful, and that seeing a figure in space was itself a form of proof.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Gaspard Monge, MacTutor History of Mathematics.
2. Gaspard Monge, Wikipedia; Gaspard Monge, Wikipédia (French).
3. F. Arago, Biographie de Gaspard Monge, in Œuvres complètes, vol. 2, Paris, 1854.
4. R. Taton, L’œuvre scientifique de Monge, Presses Universitaires de France, 1951.
5. F. Pairault, Gaspard Monge, le fondateur de Polytechnique, Tallandier, 2000.

Mathematics

1. G. Monge, Mémoire sur la théorie des déblais et des remblais, Histoire de l’Académie Royale des Sciences, année 1781, Paris, 1784.
2. G. Monge, Géométrie descriptive, Paris, an VII (1799).
3. G. Monge, Application de l’analyse à la géométrie, Paris, 1807; 5th edition with notes by J. Liouville, 1850.
4. L. Kantorovich, On the translocation of masses, Doklady Akad. Nauk SSSR 37 (1942).
5. Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991).
6. C. Villani, Optimal Transport, Old and New, Springer, 2009.
7. M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, 1976.
8. L. C. Evans, Partial Differential Equations, AMS, 2nd ed. 2010 (characteristics, Chapter 3).
9. A. Figalli, The Monge–Ampère Equation and Its Applications, EMS, 2017.