The convents of Port-Royal (one in Paris, one in Magny-les-Hameaux) played a curious role in 17th century French culture. Port-Royal-des-Champs became the site of a group of schools known as the “petites écoles,” which became famous for the quality of the education they provided. The convent tended to favor the Jansenists in their controversy with the Jesuits (and eventually with the Catholic hierarchy as a whole). Several important French intellectuals of the time, most notably Blaise Pascal, were sympathetic to the Jansenists and so found themselves spending time at Port-Royal. The result was “a kind of miracle of culture.”
The most famous texts associated with Port-Royal are the Provincial Letters of Pascal and the “Port-Royal Logic,” written by Antoine Arnauld and Pierre Nicole (the actual title was La Logique, ou L’art de Bien Penser). But Port-Royal is also associated with an important geometry textbook, Arnauld’s Nouveaux Éléments de Géométrie. This seems to have originated from an encounter between Pascal and Arnauld at Port-Royal, around 1655.
Pascal, it turns out, had written a text called Introduction à la Géométrie in which he set out a new approach to the subject. This text is now lost to us except for some notes found in Liebniz’s papers. These concern only the very first bit of the text, in which the foundations are laid. They are reproduced here with extensive annotations, on pages 85–90 of this book. Short of some sensational manuscript discovery, this is all we will ever know about Pascal’s geometry.
It is difficult to reconstruct from this fragment exactly what Pascal had in mind, but it is clear that his approach is a conscious and radical departure from the Euclidean tradition. The fragment begins with a few definitions and basic notions, including a description of space as three-dimensional, infinite, and immobile. Then follows a list of “theorems known naturally” which will serve, it seems, as axioms. The reference to what is “known naturally” reflects an interest in the epistemology of geometry and the role of postulates and theorems that was shared by all the Port-Royal authors. There is also a clear interest in improving on Euclid. For example, Pascal includes a “theorem known naturally” that guarantees that the two circles in Euclid Proposition 1 must have a point in common, a well-known missing postulate in Euclid.
Pascal’s Introduction was not meant as a textbook. A discussion between Pascal and Antoine Arnauld of how one might use this approach to write an actual textbook seems to have led to a “friendly challenge” to go ahead and do that. The result was Arnauld’s Nouveauxc Éléments, written probably in 1655–56, but published only in 1667. The book was successful enough to receive a second edition in 1683.
Between writing the first and second editions, Arnauld seems to have come into contact with another geometric work that issued from Jansenist circles: Euclides Logisticus, written by François de Nonancourt around 1652. This is a discussion of Euclid’s theory of ratios (ratio in Greek is logos, hence the title) in which Nonancourt, probably influenced by Gregory of St. Vincent, argued that ratios should be considered a kind of magnitude, and hence that it made sense to speak of ratios of ratios.
The goal of all of these books was to improve on Euclid by, in particular, setting out the theory in its “proper” order. Euclid’s Elements was marred, they argued, by the sudden detour in book five from geometry to an abstract theory of ratios, only to return to geometry in book six. It would be more reasonable to start with a general theory of magnitudes (not to be found in Euclid), then a theory of ratios of magnitudes (presumably along the lines of Euclid V), and then to delve into the specific magnitudes and ratios that occur in geometry. (The genus should be treated before looking at any of the species.) This is exactly Arnauld’s procedure.
Another concern was about the role and function of proofs. These were not smply to guarantee the truth of a proposition, but rather to increase one’s understanding. This led them to object, for example, to proofs that merely showed that something is, without showing us why it is. Arnauld felt that many proofs by contradiction suffered from this fault, and therefore tried to use them as sparingly as possible.
All of these texts are presented here in a new critical edition. Pascal’s Introduction (more properly, the fragment found among Leibniz’s papers) comes first, and fills only a few pages. Then comes the Nouveaux Éléments, which fills the majority of the book. When there are significant differences between the first and second edition, Descombes includes both texts. Finally, Nonancourt’s Euclides Logisticus is presented both in the original Latin and in a French translation.
Descombes provides a fascinating introduction that argues, inter alia, for the importance of these texts. I was particularly fascinated by the discussion of the reasons given by these authors for the study of geometry. Their major concerns, after all, were theological and moral, and they insisted that not one truth of geometry could make a difference in anyone’s eternal salvation. Why spend time on it, then? Their answer was that studying geometry helped make “honest men” who knew the difference between true and false argument and were prepared to submit themselves to the truth. This strikes me as a very nice point: one of the lessons of mathematics is that one must be ruthless in searching out errors in one’s own arguments.
This is a well-produced and valuable volume. Descombes has rendered us a significant service by putting it together. Anyone who is interested in the history of mathematics and/or the history of mathematics education will find in it much to think about.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.