France's Ecole Polytechnique Becomes the Most Influential Mathematics Institution of Its Time

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France's Ecole Polytechnique Becomes the Most Influential Mathematics Institution of Its Time

Overview

The creation of France's Ecole Polytechnique in 1794 was one of the brighter consequences of the chaos of the early days of the French Revolution, and the effect of the Ecole Polytechnique on mathematics is brighter still. With its mathematical curriculum shaped at its inception by French mathematician and administrator Gaspard Monge (1746-1818), among others, the school was originally intended as a training-ground for civil engineers. In addition to engineering, though, the Ecole Polytechnique under the guidance of Monge and other distinguished mathematicians and teachers became a hotbed of mathematical (and, later in its history, revolutionary) activity, as students and professors formalized the teaching of mathematics. That process of formalization—along with the lectures, textbooks, exercises, and syllabi that accompanied it—reinvigorated mathematics as both an academic and theoretical discipline: an entire generation of Ecole Polytechnique graduates and instructors extended both the institution's influence on the teaching of mathematics and the range of pure mathematics itself. Descriptive and analytic geometry, for example, were first formally taught at the Ecole Polytechnique, rescuing geometry from a decline that some felt stretched back as far as Plato. Founded in revolution, the Ecole Polytechnique revolutionized the teaching of and the status of higher mathematics not only in France, but by example throughout the world.

Background

In the 1780s France poised precariously on the brink of chaos. Food shortages, an economic crisis, the heavy financial burden of the nobility and tens of thousands of pensioned military officers, additional financial demands by the Church and by debts incurred by three consecutive kings helped fuel a widespread social dissatisfaction. Intellectuals felt there was too much control of thought and expression; the middle class felt too heavy a tax burden, and the peasants felt themselves still imprisoned by feudal social structures. Public dissatisfaction reached its boiling point in May 1789, when King Louis XVI convened a legislative body, the Estates-General, to Versailles to approve a new tax code.

The convention quickly became something else as frustration and anger gathered force. By mid-June the Estates-General had proclaimed itself the National Assembly, and began re-shaping France. By July 14 the zeal for reform had become outright revolutionary fervor as the citizens of Paris stormed the Bastille. In August the National Assembly abolished the last vestiges of feudalism and serfdom, and shortly thereafter issued the Declaration of the Rights of Man. Within two years a new Constitution had been adopted; within four Louis XVI had been sentenced to the guillotine. Moderate reformers gave way to fanatic revolutionaries, and between fall 1793 and summer 1794 a Reign of Terror resulted in the deaths of between 20,000 and 40,000 people. As the tide of terror receded and calmer (un-guillotined!) heads prevailed, France had ceased to be a monarchy and become a republic.

It was not all terror. Among the revolutionary impulses was a call for educational reform on a grand scale. France's archaic university system was overhauled, with new institutions created to provide more specialized training than the universities offered. Already in pre-Revolutionary days some "Grandes Ecoles" had been created: in 1748 the Ecole du Génie Militaire (Army Corps of Engineers School) and the Ecoles de Constructeurs de Vasseaux Royaux (Royal Shipbuilding School) in 1765.

The Revolution gave added impetus to the Ecole movement, and in 1794 the Conservatoire National des Arts et Métiers (National Conservatory for Arts and Crafts) and the Ecole Centrale des Travaux Publiques (Central School for Public Works) were established. The School for Public Works would include a curriculum stretched far beyond civil engineering and would soon be renamed the Ecole Polytechnique, under which name it would become France's, and the world's, leading center for mathematical thought and instruction.

The prospects for the new institution were greatly helped by the presence of Gaspard Monge on the committee setting out the charter for the school. A gifted mathematician and teacher, Monge also had a great deal of administrative and bureaucratic experience. He had been an important member of the French Académie des Sciences, a member of the Académie Commission on Weights and Measures, and was well known for applying his knowledge of geometry to practical ends such as the design of fortifications.

As a mathematician Monge was startlingly original and devoted to extending the range of geometry and related mathematical disciplines. He published numerous papers on infinitesimal and descriptive geometry, as well as work in a variety of other scientific fields.

A supporter of the moderate forces of revolution, Monge was offered, in 1792, the post of Minister of the Navy, which he accepted. For once his administrative skills failed him, and his tenure as Minister was undistinguished and brief. He continued to be involved in the Académie, lobbying for educational reforms, but to little avail: the National Commission abolished the Académie in 1793. The next year the Ecole was founded, and Gaspard Monge named an instructor there, charged with teaching descriptive geometry and training those who would later become teachers at the Ecole. Also teaching geometry at the Ecole Normale (Normal School) for the training of teachers, Monge began assembling his lectures into a book, Application de l'analyse a la géométrie, on which he would work for much of his life, and which housed his insights into the use of analysis as a geometric tool.

Another book, Géométrie Descriptive (1795), adapted graphic drafting techniques into theoretical geometric reasoning. This book's popularity led to its adoption throughout Europe as the standard text on its subject. Monge was by all accounts a superb lecturer, and the Ecole quickly became a fount of exuberant students, many of whom went on, in turn, to write their own geometry texts. The Polytechnicians, as they came to be known, poured out books on analytical and other geometries, elevating geometry from its position in the shadow of the calculus to its preeminence in modern mathematics. The modern age of pure geometry had begun, and the Ecole Polytechnique was in many ways its starting point.

By 1797, after performing various administrative and semi-diplomatic services for Napoleon, whom he admired and knew well, Monge was appointed Director of the Ecole. He would not serve long: once Napoleon took control of the French government, Monge was appointed a senator for life, and he resigned his Ecole directorship.

But Monge was far from the only distinguished mathematician on the faculty of the Ecole Polytechnique (or, for that matter, the Ecole Normale; many professors taught at both institutions.) Among Monge's influential colleagues at the Ecole Polytechnique was Joseph-Louis Lagrange (1736-1813), who lectured on algebra and the calculus, particularly differential calculus. Named professor of analysis at the Ecole, Lagrange had a wide-ranging mathematical mind and had made many contributions, notably to the place of variables in the calculus, the nature of orbital positions in celestial mechanics, and the study of four-dimensional spaces, which would prove crucial to extending math to the study of the physical mechanics of solids and fluids. In 1793, as president of the weights and measures commission, Lagrange insisted that France adopt a 10-place, or decimal, rather than 12-place system; he is considered in many ways to be the father of modern metrics.

Lagrange's great contribution to education, to the mathematical discourse, had begun to take shape before the creation of the Ecole. In 1788 he published his book Mécanique analytique, in which he discussed the mathematics of physical mechanics and did so without recourse to a single drawing or diagram. Such an approach had never been taken: since the Greeks, all mathematical treatises had been leavened with illustrations. But the language of Lagrange's entire book was algebra, pure and unillustrated, and it would exert an enormous influence on future mathematical texts and treatises.

Also at the Ecole Polytechnique (and particularly at the Ecole Normale), Pierre-Simon Laplace (1749-1827) had built a substantial mathematical reputation on the strength of his work in probability studies and especially his work with celestial mechanics. While his studies of celestial mechanics are perhaps his greatest contribution to science (his theory that the planets coalesced out of clouds of gas is known as the nebular hypothesis), it was his awareness of the practical applications of probability to matters such as mortality rates, games theory, and even judicial decisions that exerted a strong and lasting effect on probability studies.

By the early nineteenth century the Ecole Polytechnique was well established as a mathematics center, with teachers, mathematical theorems, and books of both instruction and theory emerging from its halls at a steady pace. Geometry remained at the center of many of these endeavors, and the discipline's and the institution's debt to Monge was large.

Some things had changed. With the fall of Napoleon, Monge was disgraced, and in 1816 he was cast out of the French scientific community. Heartbroken, Monge fled France, but returned to die in Paris barely two years later. The French Government decreed that no notice be paid to honor the mathematician. The students at the Ecole, an institution born of revolution, paid no attention to the decree, and gathered to honor the passing of Gaspard Monge.

Impact

While it is an overstatement to attribute the explosion of nineteenth-century mathematics to an institution—Monge, Laplace, Lagrange, and others were doing serious work before the Ecole Polytechnique was founded, and would doubtless have made major contributions at other institutions—the Ecole's role as a central focus, an epicenter for the geometric and mathematical flowering, cannot be doubted. It was a place where ideas came together and were tested against each other by some of the finest mathematical minds of the time. That in itself helped generate excitement and refine emergent theories.

But because the Ecole's faculty also taught mathematicians who would go on to become professors themselves as well as writers of textbooks, its influence was enormous and ongoing. Primary among those influences was Monge's shepherding of geometry out of eclipse and into prominence as a mathematical discipline. Monge's textbook and the lectures it was based upon inspired even better geometry tests: between 1798 and 1802 no fewer than four major geometries appeared, inspired by Monge and the Ecole, and authored by leading mathematicians Lacroix, Puissant, Lefrançois, and Biot. Other texts followed, with analytic geometry blazing forth in ways it had not when presented earlier by Pierre de Fermat (1601-1665) and René Descartes (1596-1650).

Likewise, Lagrange's bold decision to write in purely mathematical terms without illustration was, although not a product of the Ecole Polytechnique per se, important to the teaching of mathematics. It placed the focus directly on the math, on pure mathematical analysis, on establishing properties and deducing them from the smallest number of principles.

The influence of the Ecole Polytechnique on mathematics would be felt throughout the nineteenth century. Its influence as a revolutionary body remained strong as well, with its students taking up swords and leaping to the barricades in the revolution of 1830 and the chaotic, near-revolutionary fervor of the June Days of 1848. In the best republican revolutionary tradition, these students of the Ecole pledged themselves to stand against oppression as workers of the mind, just as the "workers of the fist" stood.

Victor Hugo wrote: "Whoever wants to vanquish France must first destroy three institutions: the Institut, the Legion d'Honneur, and the Ecole de Polytechnique." Few institutions of learning have received higher praise, or done more for the cause of pure mathematics.

KEITH FERRELL