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In the course of preparing lessons on projective geometry I want to give an account on the historical development. It is easy to obtain an overview of the history starting with G. Desargues. And with respect to older sources is of great help. But what I am not sure about is the question, whether Desargues was more an Euclid or an Eudoxos. Was he more collecting and reproducing the knowledge of scholars and artists like Filippo Brunelleschi, Ambrogio Lorenzetti, Pietro Perugino? Or did he invent himself most of what he wrote about? And what were his relations to his contemporary Johannes Kepler who worked in the same field?

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up vote 8 down vote accepted

I guess you'll find what you need in the monograph by J.V. Field and J.J. Gray, The Geometrical Work of Girard Desargues.

They give substantial critical and exegetical commentaries as well as valuable introductory essays placing Desargues and his various works in historical context. Particularly attractive is the authors' demonstration of how Desargues' highly original ideas developed from the contemporary technical mathematical context, and their elegant exposition of the importance of the 'practical tradition of applied geometry' in stimulating his work.

Desargues had a limited range of interactions, which apparently did not include Kepler. Quoting from Gale's Science and Its Times:

Desargues spent many years in Paris with a group of mathematicians that included Descartes and Pascal as well as the Jesuit scientist Marin Mersenne (1588-1648) and Etienne Pascal (1588-1651). Desargues's work on projective geometry was printed principally for this limited readership of friends. Unfortunately, however, his views were very unorthodox and unpopular during his life — Blaise Pascal (1623-1662) was one of his few admirers. Only 50 copies of his book on projective geometry were printed, many of them later destroyed by the publisher. Desargues's work slipped into obscurity for nearly 200 years after the publication of his defining text on the subject.

One piece of trivia, on the origin of the concept of involution caught my attention:

Desargues introduced more than 70 new terms in his book, of which only one, involution, has survived. This term, which denotes quite literally the twisted state of young leaves, is used to designate the projective transformation of a line that coincides with its inverse. (Most of the terms that Desargues proposed were based on obscure botanical references.)

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Desargues certainly pioneered original mathematics. The notion of a point at infinity in projective geometry is usually attributed to him. Kepler apparently did not work in projective geometry but rather in astronomy and pioneered a number of mathematical techniques such as infinitesimals. I am not aware of any interactions between Desargues and Kepler, but Desargues did play an interesting role of attempting to resolve a dispute between his junior colleagues Fermat and Descartes.

I now see that wiki attributes the notion of the point at infinity to Kepler, citing Coxeter. This seems like a novelty to me. Kepler did talk about points at infinity, but not in the context of projective geometry as we understand it, but rather as a way of developing a unified technique for treating conic sections through a kind of a continuity principle. This is closer to calculus than projective geometry.

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