# The diameter and the Traversale : inside Girard Desargues' workshop

## TL;DR

This paper explores Girard Desargues' detailed development of the concept of traversale in his Brouillon Project, revealing his early understanding of conic polarity and involution, predating modern notions.

## Contribution

It uncovers Desargues' original ideas on conic polarity and involution, showing how he developed these concepts through detailed analysis and notes, predating 19th-century formalizations.

## Key findings

- Desargues' notes reveal an early theory of conic polarity.
- He understood that a conic induces an involution on each line.
- The paper clarifies Desargues' conceptual approach to conic sections.

## Abstract

In his \textit{Brouillon Project} on conic sections, Girard Desargues studies the notion of \textit{traversale}, which generalizes that of diameter introduced by Apollonius. One often reads that it is equivalent to the notion of \textit{polar,} a concept that emerged in the beginning of 19th century. In this article we shall study in great detail the developments around that notion in the middle part of the \textit{Brouillon project.} We shall in particular show, using the notes added by Desargues after the first draft was written, how here arises a new \textit{theory,} that of the \textit{polarity} associated to a conic section. We shall show that besides a natural correspondance between points and lines, Desargues has understood that a conic induces on every line in its plane an involution, and how he progressively handles the considerable conceptual difficulties he has to deal with when trying to explain clearly his novative ideas.