Evelyne Barbin

Integrating history: research perspectives

The question of judging the effectiveness of integrating historical resources into mathematics teaching may not be susceptible to the research techniques of the quantitative experimental scientist. It is better handled through qualitative research paradigms such as those developed by anthropologists.

History of Teaching Geometry

Since the beginning of the transmission of geometric knowledge, two aspects of geometry have been present: the abstract “speculative” and the practical. These two aspects correspond to an essential dialectic in geometry teaching, between a deductive/rational science and a practical/intuitive one. In the nineteenth and twentieth centuries, the stress on the second aspect led to experimental geometry. After the work of Descartes, another tension concerned the solution of problems: between “pure” methods and methods coming from algebra or analysis. Attempts to find a new language for school geometry reached their apex when geometry was substituted by linear algebra in the 1960s.

Mathematical Physics in the Style of Gabriel Lamé and the Treatise of Emile Mathieu

The Treatise of Mathematical Physics of Emile Mathieu, published from 1873 to 1890, provided an exposition of the specific French “Mathematical Physics” inherited from Lame, himself an heir of Poisson, Fourier, and Laplace. The works of all these authors had significant differences, but they were pursuing the same goal, described here with its relation to Theoretical Physics.

Evolving Geometric Proofs in the Seventeenth Century: From Icons to Symbols

My purpose is to understand what “arithmetization of geometry” meant in the seventeenth century. I compare five proofs of the main proposition on geometrical proportion: two proofs in Euclid’s Elements (one for magnitudes, one for numbers), one proof in Antoine Arnauld’s New Elements of Geometry (1667) and two proofs in Bernard Lamy’s Elements of Geometry (2nd edn, 1695, 5th edn 1731). For each of these proofs, I examine the signs used both for magnitudes and for reasoning, using Peirce’s classification of signs. This examination clearly shows that in the seventeenth century geometry had undergone a process of arithmetization through the use of symbolization, and that the outcome of this process of arithmetization had a strong influence on proofs in mathematics.