Gaspard Monge

French mathematician

• Born: May 10, 1746
• Birthplace: Beaune, France
• Died: July 28, 1818
• Place of death: Paris, France

Monge founded modern descriptive geometry and revitalized analytic geometry. An enthusiastic supporter of the French Revolution, he helped establish the metric system and the École Polytechnique, an important engineering school.

Early Life

Gaspard Monge (gah-spahr mohnzh) was born on May 10, 1746, in Beaune, a small town 166 miles southeast of Paris. He was the eldest son of Jacques Monge, an itinerant peddler and knife-grinder, and Jeanne Rousseaux, a woman of humble Burgundian origin. Jacques deeply respected education and sent his three sons to the local school run by the Oratorian religious order and to their Collège de la Trinité in Lyons. Although all three brothers eventually made successful careers in mathematics, Gaspard was clearly the genius. He was the golden boy of the Oratorians, and he regularly won academic prizes and became, at the age of sixteen, a physics teacher at Lyons.

In the summer of 1764, during a vacation to Beaune, Monge used surveying instruments of his own invention and construction to make a detailed map of the town. A military officer who later saw the map was so impressed by the boy’s ability that he recommended Monge to the commandant of the military school at Mézières. Created in 1748, the École Royale du Génie had become a prestigious institution for the training of officers, who derived mostly from the nobility. Upon his arrival at Mézières in 1765, Monge learned that he would not study with the officers but would be trained as a draftsman to do the routine work of military surveying. Within a year, however, he had an opportunity to show that his mathematical skills were vastly superior to those of the officers. He was assigned the problem of computing the best places to locate guns in a proposed fortress at Metz. At the time, the calculation of positions shielded from enemy firepower in intricate fortifications was a long and arduous arithmetic procedure, but Monge developed a geometric method that obtained results so quickly that the commandant was initially skeptical. Upon detailed inspection by skilled officers, the advantages of Monge’s invention, which formed the basis of what later came to be known as descriptive geometry, became evident. In fact, Monge’s method was so highly valued that it was preserved as a military secret for twenty-five years.

Life’s Work

Monge spent the first fifteen years of his career at the military academy of Mézières, where he was répétiteur (assistant) to the professor of mathematics, then a teacher of mathematics, and finally a royal professor of mathematics and physics. Through his excellent teaching, he was able to improve the French engineering corps and influence several students who went on to brilliant military careers. His lectures on descriptive geometry (then called stereotomy) allowed him to develop his ideas about perspective, the properties of surfaces, and the theory of machines. Descriptive geometry is basically a way to represent three-dimensional figures on a plane. Albrecht Dürer, the German painter and engraver, had used the idea of orthogonal projection of the human figure on mutually perpendicular planes in the early sixteenth century, and in 1738, A. F. Frézier had suggested a method of representing solid objects on plane diagrams, but Monge developed descriptive geometry into a special branch of mathematics. He systematized its principles, developed its basic theorems, and applied this knowledge to problems of military engineering, mechanical drawing, and architecture.

Documents from his Mézières period reveal that Monge did extensive research in several areas of mathematics. He wrote memoirs on various curves and studied their radii of curvature. He analyzed evolutes (the loci of centers of curvature for a given curve) and systematically applied the calculus, in particular partial differential equations, to his investigations of the curvature of surfaces. In 1775, he presented to the French Academy of Sciences in Paris a paper on a developable surface, that is, a surface that can be flattened on a plane without distortion, a subject of great interest to mapmakers.

During the middle 1770’s, Monge’s interests began to switch from mathematics to physics and chemistry. In physics, he helped develop the material theory of heat (he called the heat substance “caloric”). This theory was useful to physicists and chemists in the eighteenth century (it was replaced by the kinetic theory of heat in the nineteenth century). Working alone at Mézières and with Antoine-Laurent Lavoisier on his trips to Paris, Monge carried out experiments on the expansion, solution, and reaction of various gases. To enable him to better carry out his research in chemistry and physics, Monge established a well-equipped laboratory in the late 1770’s at the École Royale du Génie.

In 1777, Monge married a twenty-year-old widow, Catherine (Huart) Horbon, for whose honor he had earlier tried to fight a duel with one of her rejected suitors. The couple had three daughters. Since she owned a forge, Catherine had an indirect influence on her husband’s interest in metallurgy. Supervising its operation led Monge to study the smelting and properties of metals. His outstanding work in mathematics and the physical sciences led to his election to the Academy of Sciences in 1780. This honor forced him to divide his time between Paris and Mézières. During his stays in Paris, he taught hydraulics (the science and technology of fluids) at the Louvre, and during his time at Mézières he taught engineering to the military officers and prepared memoirs on physics, chemistry, and mathematics for presentation at the Academy of Sciences.

Monge’s researches in chemistry consumed so much of his time that he arranged for a substitute to deliver many of his lectures at Mézières. In the summer of 1783, he carried out his famous experiments on the synthesis of water from its component elements. Monge mixed hydrogen and oxygen gases (then called inflammable air and dephlogisticated air) in a closed glass vessel and ignited the explosive reaction between the gases by an electric spark from a voltaic battery. He found that the weight of the pure water he obtained was very nearly equal to the weights of the two gases. These studies became part of the so-called water controversy over the first discoverer of the compound nature of water. Monge deserves credit for showing quantitatively that water is composed of two elemental gases, but he did not formally publish until 1786. Henry Cavendish, the English physicist who published his results in 1784, showed that water is produced when inflammable air is burned in dephlogisticated air, but he interpreted his experiments in terms of the confusing phlogiston theory. Lavoisier correctly interpreted the reaction as the oxidation of hydrogen.

In the fall of 1783, Monge accepted yet another responsibility in Paris—the examiner of naval candidates. For a while, he tried to continue his professorship at Mézières along with this new position, but this proved impossible; in December, 1784, he resigned from the school where he had spent nearly twenty years of his life. His post as examiner also required him to make tours of inspection of naval schools outside Paris, and this enabled him to reform the teaching of science and technology in the provinces. His time in Paris was spent participating in the activities of the academy and conducting research in chemistry, physics, and mathematics. During the 1780’s, he did important work on the composition of nitrous acid, the liquefaction of sulfur dioxide, the nature of different types of iron and steel, and the action of electricity on carbon dioxide gas. In these chemical researches, he interpreted his results through Lavoisier’s new oxygen theory rather than the outdated phlogiston theory. In physics, he did research on the double refraction of Iceland spar (a transparent calcite); he also studied capillary action. In mathematics, he continued his work on curved surfaces and partial differential equations.

When the French Revolution began in 1789, Monge was one of its most ardent supporters. His humble birth and his negative experiences with aristocrats gave him firsthand knowledge of the poverty of the masses and the corruption of the ancien régime. In 1791, he served on the committee that established the metric system. In 1792, he became minister of the navy and played a significant role in organizing the defense of France against the counterrevolutionary armies. In 1793, he voted in favor of the death of King Louis XVI. After all this, he was still bitterly attacked for not being revolutionary enough. These attacks forced him to resign from his ministerial post. Nevertheless, he continued to support the republic, and as a member of the committee on arms and munitions he worked hard to improve the extraction and purification of saltpeter and the construction and operation of powder-works in Paris and in the provinces. He also became involved in establishing a new system of scientific and technical education. It was during his teaching at the short-lived École Normale that his work on descriptive geometry was finally published.

Monge was very much concerned about preserving the nation’s cultural and intellectual heritage during this time of revolutionary turmoil. He was convinced of the value of a national school for training civil and military engineers. As an influential member of the commission of public works, he helped institute the École Polytechnique in 1794. Monge became an important administrator and popular teacher at this school. His textbook on analytic geometry, which appeared in 1795, was used in the course he taught on the application of algebra to geometry. In pursuing the correspondence between algebraic analysis and geometry, Monge recognized that families of surfaces could be described both geometrically and analytically. He founded a school of geometers at the École Polytechnique, who would exert a powerful influence on the development of mathematics in the nineteenth century.

The last stage of Monge’s career began in 1796 and was dominated by his fascination with—some have called it his mesmerization by—Napoleon I. Monge had actually met Napoleon earlier, when he cordially welcomed the young artillery officer from Corsica to the military school at Mézières. Though Monge had forgotten this meeting, Napoleon remembered and called Monge to Italy as a member of the committee supervising the selection of the paintings, sculptures, and other valuables that the victorious army was to bring back to France. Although this looting disturbed Monge’s conscience, he accepted it as a way to finance Napoleon’s military campaigns. Monge’s duties took him to many cities throughout Italy and gave him the opportunity to become Napoleon’s confidant and friend.

In the fall of 1797, Monge returned to Paris to begin his new post as director of the École Polytechnique, but his stay was brief, for Napoleon called him back to Rome to conduct a political inquiry. Monge also participated in the creation of the Republic of Rome and in the preparations for Napoleon’s Egyptian adventure. Monge arrived in Cairo on July 21, 1798, the day after Napoleon’s victory at the Battle of the Pyramids. Napoleon made Monge president of the Institut d’Égypte (Egyptian Institute), modeled on the Institut de France, the revolutionary organization intended to replace the royal academies. Monge was heavily involved in many of the projects of the Institut d’Égypte. He was also a companion of Napoleon on a trip to the Suez region, on his disastrous Syrian expedition, and on his secret voyage back to France in 1799.

Upon his return to Paris, Napoleon rewarded Monge for his services. These favors continued throughout the period of the consulate as well as during Napoleon’s reign as emperor. Monge was given more powerful administrative responsibilities along with extensive land grants. Napoleon created Monge count of Péluse, an honor he accepted gratefully, although he had once voted to abolish all titles. Napoleon also named Monge a senator, and by accepting the position Monge became publicly and irrevocably tied to Napoleon. A representation of Monge at this time depicts him in a powdered wig, looking slightly uncomfortable in the trappings of nobility. His strong and stocky build seems awkwardly confined by the expensive clothes, but his piercing eyes radiate intelligence and confidence, showing him to be a man ready to meet any challenge.

During these Napoleonic years, Monge divided his time among his duties in the senate, at the École Polytechnique, and in the Academy of Sciences. At the École Polytechnique, Monge influenced many young French mathematicians in various kinds of geometry—synthetic, analytic, and infinitesimal. Monge’s great contribution to synthetic geometry was his Géométrie descriptive (1798; An Elementary Treatise on Descriptive Geometry, 1851), a summation of his life’s work in descriptive geometry and a book that proved useful not only to mathematicians but also to artists, architects, military engineers, carpenters, and stonecutters. In 1801, Monge published a book on analytic geometry that revealed how useful geometry could be for algebra, and vice versa. In the same year, he published an expanded version of his lectures on infinitesimal geometry, his favorite subject, in which he used ordinary and partial differential equations to study complex surfaces and solids.

Monge’s health began to decline in 1809, when he stopped teaching at the École Polytechnique. His health worsened during the autumn of 1812, when Napoleon’s army suffered great losses on its retreat from Moscow. Monge deliberately fled Paris and did not participate in the senate session of 1814 that dethroned Napoleon. Seeing Napoleon as the standard-bearer of the revolutionary ideals of liberty, equality, and fraternity, Monge refused to condemn him, and during the so-called Hundred Days in 1815, when Napoleon tried to recover his throne, Monge pledged his allegiance to the emperor, remaining loyal even after Napoleon’s defeat at Waterloo and his abdication. When the Bourbons were restored to the French monarchy in 1815, Monge, who refused to modify his anti-Royalism, was deprived of all of his honors and positions, even his membership in the Academy of Sciences. The last years of his life were filled with further humiliations and greater physical sufferings. Following a stroke, he died on July 28, 1818. Many students at the École Polytechnique asked to attend his funeral, but the king refused permission. Although they observed the king’s refusal, the next day the students marched en masse to the cemetery and laid a wreath on the grave of their beloved teacher.

Significance

Gaspard Monge’s reputation derives from his work in geometry, and there is no doubt that he was responsible for the revival of interest in geometry that occurred in the late eighteenth and early nineteenth centuries. As a result of his inspiration, a golden age of modern geometry began, and his methods flourished first in France and later throughout Europe, blazing the way for such nineteenth century mathematicians as Carl Friedrich Gauss and Georg Friedrich Bernhard Riemann. Though Monge was not strictly the inventor of descriptive geometry, he was the first to elaborate its principles and methods and to detail its applications in mathematics and technology. He also made valuable contributions to analytic and infinitesimal (or differential) geometry.

Despite his reputation as a geometer, Monge’s accomplishments were actually much broader. Besides his exceptional sense of spatial relations, he was also an insightful analyst who could transform geometric problems into algebraic relations. For Monge, geometry and analysis supported each other, and in every problem he emphasized the close connection between the mathematical and practical aspects. His treatment of partial differential equations has a geometrical flavor, and he believed that problems involving differential equations could be solved more readily when visualized geometrically. On the other hand, some problems involving complex surfaces led to interesting differential equations. Many historians of mathematics ascribe to Monge the revival of the alliance between algebra and geometry. René Descartes may have created analytic geometry, but it was Monge and his students who made it a vital field.

Monge was a Renaissance man in the Age of the Enlightenment. He possessed a broad combination of talents: He was a creative mathematician, an excellent chemist, and a talented physicist and engineer. Furthermore, he was an adroit politician, a capable administrator, and an inspiring teacher. His skill as a teacher can be seen in his distinguished pupils, some who continued on paths he had opened and others who created new paths. Charles Dupin applied Monge’s methods to the theory of surfaces. Victor Poncelet, the most original of Monge’s students, became the founder of projective geometry. Jean Hachette and Jean Baptiste Biot developed the analytic geometry of conics and quadrics.

Throughout his career, Monge was interested in the practical consequences of his work in science and mathematics. At Mézières and at the École Polytechnique, he was interested in the structure and functioning of machines. He took his work on such practical problems as windmill vane design as seriously as the highly abstract problems of differential geometry. He believed that technical progress helped to augment human happiness, and, since technical progress depended on the development of science and mathematics, he supported France’s efforts to improve education in these basic fields. In his unified view, science freed the intellect with the truth about the world, and this was the only valid way to social progress.

Bibliography

Bell, Eric T. Men of Mathematics. New York: Simon & Schuster, 1937. Bell, who spent most of his teaching career at the California Institute of Technology, was skilled in unraveling the mysteries of mathematics for the general reader. In this book, he uses the lives of the men who created modern mathematics to explain some of the most important ideas animating mathematics today. Monge and Joseph Fourier are treated together in the chapter “Friends of an Emperor.”

Boyer, Carl B. History of Analytic Geometry. New York: Scripta Mathematica, 1956. Reprint. Princeton Junction, N.J.: Scholar’s Bookshelf, 1988. An important study of the development of analytic geometry from ancient times to the nineteenth century. Boyer’s approach is conceptual rather than biographical, but Monge’s work on analytic geometry is extensively analyzed. Boyer’s emphasis is on the history of mathematical ideas, and some knowledge of algebra and geometry is assumed.

‗‗‗‗‗‗‗. A History of Mathematics. New York: John Wiley & Sons, 1968. A textbook for college students at the junior or senior level. Though he assumes an understanding of calculus and analytic geometry, much of the material is accessible to readers with weaker mathematical backgrounds. Boyer analyzes both Monge’s contributions to mathematics and his involvement in politics. Extensive chapter bibliographies and a good general bibliography.

Crabbs, Robert Alan. “Gaspard Monge and the Monge Point of the Tetrahedron.” Mathematics Magazine 76, no. 3 (June, 2003): 193. Explains the Monge Point of the Tetrahedron and discusses some of Monge’s other geometric concepts. Includes a biographical profile of Monge, summarizing his career and his contributions to the development of applied and higher mathematics.

Kemp, Martin. “Monge’s Maths, Hummel’s Highlights.” Nature 395, no. 6703 (October 15, 1998): 649. A brief, scientific discussion of the isometric perspective and how Monge developed its geometric properties.

Kline, Morris. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, 1972. Monge’s contributions to descriptive geometry and partial differential equations are extensively discussed. Kline’s book is aimed at professional and prospective mathematicians, and a knowledge of advanced mathematics is necessary to understand his analysis of Monge’s contributions.

Partington, J. R. A History of Chemistry. Vol. 3. London: Macmillan, 1962. This volume, dealing with the seventeenth, eighteenth, and early nineteenth centuries, contains an excellent analysis of chemistry in France during the time of Monge’s career. Discusses Monge’s contributions to chemistry in depth, with extensive references to original documents, some of which are translated into English. Accessible to the general reader.

Taton, René, ed. The Beginnings of Modern Science. Translated by A. J. Pomerans. New York: Basic Books, 1964. This work, the third volume in Taton’s History of Science series, covers the period from 1450 to 1800. Monge’s contributions to geometry and chemistry are discussed in their historical contexts, but this book is best used as a reference rather than for narrative reading.