# The French Grandes Ecoles

### Mathieu Ayel

### The Teachers and Methods

A number of prominent French scientists contributed to the development of the Ecole Polytechnique's mathematical education and to the Grandes Ecoles teaching more generally. The first important works and textbooks were written in the Royal school of Engineering in the eighteenth century. Etienne Bézout was a mathematician of the Académie des Sciences who worked as an examiner for the Gardes de la Marine and the Corps d'Artillerie. Between 1770 and 1782 Bézout's

In 1794 when the Ecole Polytechnique was opened, the teaching staff recruited by the directors Monge and Carnot (himself a former student of Monge at the Ecole du Génie de Mezières) was very much a collection of the big names of French science at the time like Lagrange or De Prony. Interestingly enough many of the lecturers were former members of the defunct Académie des Sciences.

De Prony was appointed at the opening of the school as a lecturer in analysis and at a later date also taught mechanics. He pursued his teaching at the school while being nominated in 1798 as director of the Ecole des Ponts et Chaussées, and published two sets of texts from his lectures:

Monge was appointed to the chair of descriptive geometry at the opening and taught at the Ecole Polytechnique until 1809. His approach to mathematics was particularly important as for the first few years of the school he was in charge of the instruction of the future teachers. Monge gave special emphasis to geometry and geometrical reasoning and developed a range of graphical methods for construction problems.

Lagrange, one of the most famous mathematicians of the time, was the first lecturer of analysis at the Ecole Polytechnique. He was a relatively poor lecturer according to several of his students, having an accent which it was difficult to understand. He presented analysis in a very abstract manner not suitable to the wide range of mathematical background of his students. Lagrange published in two volumes his calculus lecture but his most important contribution to the Ecole Polytechnique was without a doubt the prestige he bestowed on the institution.

A short lived educational institution of importance was created by the Convention in 1794: l'Ecole Normale Supérieure was founded to train future school teachers. This school, although not a proper part of the Grandes Ecoles as such, was essential in the development of most of the next generation of lecturers of the Ecole Polytechnique. Mathematicians like Hachette or Fourier came into contact at L'Ecole Normale with famous scientists like Laplace who gave a number of lectures at the school and in particular a course on probability. Hachette taught several courses in descriptive geometry both at the Ecole Normale and from 1799 at the Ecole Polytechnique as a full professor. He heavily promoted the research done at the Ecole Polytechnique and participated in various scientific publication of the school, enhancing the school reputation for mathematical excellence. On the other hand Fourier was a student of the Ecole Normale in 1795 and was appointed in 1797 at the chair of analysis and mechanics at the Ecole Polytechnique, on the departure of Lagrange. Another name related to both the Ecole Normale and the Ecole Polytechnique is that of Sylvestre Lacroix who held a chair of mathematics in the Ecole Normale and was a professor of mathematics and in particular analysis from 1799 at the Ecole Polytechnique. The work of Lacroix was particularly important as it had a great influence in mathematics in France. His

Finally an essential contribution to the development of the teaching methods at the Ecole Polytechnique was made by Augustin Cauchy. Cauchy entered the Ecole Polytechnique in 1805, attended the lectures of Lacroix, De Prony and Hachette and graduated two years later from the school. He then entered the Ecole des Ponts et Chaussées and was noticed as an outstanding student. In 1815 he was appointed assistant professor in analysis at the Ecole Polytechnique. His mathematical work and textbooks are most remarkable for the importance given to the rigour of the proofs and definitions. In 1821 he published an analysis textbook,

*Cours complet de Mathématiques*was published in six volumes. The mathematical textbooks of Bézout are of particular interest because they remained for many years the standard textbooks for the students taking the entry exams of l'Ecole Polytechnique. The method described in these books is rather peculiar compared to the traditional rigour professed in French mathematics: geometry is treated before algebra as the reader will be more likely to grasp the geometric proofs before understanding the algebraic version. Other important textbooks were written by Charles Bossut who was professor of mathematics at the Ecole du Génie de Mezières. He had a profound influence on the design of the mathematics and mechanics courses at the school and his textbooks were widely used by the Grandes Ecoles. His*Cours complet de Mathematiques*and his*Mechanique en général*written in 1765 and 1792 respectively were his most successful works. Charles Bossut also exerted some influence in the academic career of Gaspard Monge.In 1794 when the Ecole Polytechnique was opened, the teaching staff recruited by the directors Monge and Carnot (himself a former student of Monge at the Ecole du Génie de Mezières) was very much a collection of the big names of French science at the time like Lagrange or De Prony. Interestingly enough many of the lecturers were former members of the defunct Académie des Sciences.

De Prony was appointed at the opening of the school as a lecturer in analysis and at a later date also taught mechanics. He pursued his teaching at the school while being nominated in 1798 as director of the Ecole des Ponts et Chaussées, and published two sets of texts from his lectures:

*Lecons de Mechanique Analytique*and*Sommaire des Lecons du Cours de Mechanique*.Monge was appointed to the chair of descriptive geometry at the opening and taught at the Ecole Polytechnique until 1809. His approach to mathematics was particularly important as for the first few years of the school he was in charge of the instruction of the future teachers. Monge gave special emphasis to geometry and geometrical reasoning and developed a range of graphical methods for construction problems.

Lagrange, one of the most famous mathematicians of the time, was the first lecturer of analysis at the Ecole Polytechnique. He was a relatively poor lecturer according to several of his students, having an accent which it was difficult to understand. He presented analysis in a very abstract manner not suitable to the wide range of mathematical background of his students. Lagrange published in two volumes his calculus lecture but his most important contribution to the Ecole Polytechnique was without a doubt the prestige he bestowed on the institution.

A short lived educational institution of importance was created by the Convention in 1794: l'Ecole Normale Supérieure was founded to train future school teachers. This school, although not a proper part of the Grandes Ecoles as such, was essential in the development of most of the next generation of lecturers of the Ecole Polytechnique. Mathematicians like Hachette or Fourier came into contact at L'Ecole Normale with famous scientists like Laplace who gave a number of lectures at the school and in particular a course on probability. Hachette taught several courses in descriptive geometry both at the Ecole Normale and from 1799 at the Ecole Polytechnique as a full professor. He heavily promoted the research done at the Ecole Polytechnique and participated in various scientific publication of the school, enhancing the school reputation for mathematical excellence. On the other hand Fourier was a student of the Ecole Normale in 1795 and was appointed in 1797 at the chair of analysis and mechanics at the Ecole Polytechnique, on the departure of Lagrange. Another name related to both the Ecole Normale and the Ecole Polytechnique is that of Sylvestre Lacroix who held a chair of mathematics in the Ecole Normale and was a professor of mathematics and in particular analysis from 1799 at the Ecole Polytechnique. The work of Lacroix was particularly important as it had a great influence in mathematics in France. His

*Cours de Mathématiques*, 10 volumes published between 1797 and 1799, and his*Traite de Calcul Différentiel et Intégral*, three volumes published between 1797 and 1800, were both significant in the development of mathematical teaching in France: Lacroix's approach of totally separating algebra from geometry became very much the norm in mathematical textbooks.Finally an essential contribution to the development of the teaching methods at the Ecole Polytechnique was made by Augustin Cauchy. Cauchy entered the Ecole Polytechnique in 1805, attended the lectures of Lacroix, De Prony and Hachette and graduated two years later from the school. He then entered the Ecole des Ponts et Chaussées and was noticed as an outstanding student. In 1815 he was appointed assistant professor in analysis at the Ecole Polytechnique. His mathematical work and textbooks are most remarkable for the importance given to the rigour of the proofs and definitions. In 1821 he published an analysis textbook,

*Cours d'Analyse*, intended for the teaching at the Ecole Polytechnique in which calculus was presented and developed with a maximum amount of rigour.*Lecons sur le calcul différentiel*published in 1829 presented the same rigour. Cauchy was mainly responsible for the emphasis given at l'Ecole Polytechnique to formal theory and pure mathematics and the lesser importance given to applied mathematics. Cauchy's mathematics teaching method implemented at the Grandes Ecoles despite a history of vigorous opposition starting from the reforms proposed by De Prony in the nineteenth century, continues to prevail today. The engineering schools created during the nineteenth century to satisfy the demands of the industry in specialised engineers have not modified the traditional education based on knowledge building and abstraction of the Grandes Ecoles of the Polytechnicien system.