Evolving geometric proofs in the seventeenth century
from icons to symbols
pp. 237-251
in: Gila Hanna, Hans N. Jahnke, Helmut Pulte (eds), Explanation and proof in mathematics, Berlin, Springer, 2010Abstract
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.The purpose of this paper is to understand the meaning of the expression "arithmetization of geometry" in the seventeenth century in the context of a new meaning for proof designed to enlighten and not just to convince. For this purpose, we compare five proofs of Thales' proposition on geometrical proportion. Two of these proofs are from Euclid's Elements (one for magnitudes, one for numbers), one in New elements of geometry (1667) by Antoine Arnauld and two in Elements of geometry (2nd edn. 1695 and 5th edn. 1731) by Bernard Lamy. In each case we use Peirce's classification of signs to examine the use of magnitudes and their roles in reasoning.
