Géométries de Port-Royal (review)

This edition focuses on Arnauld's Nouveaux éléments de géométrie, the outcome of a friendly contest between Arnauld and Pascal regarding the best way to structure an elementary textbook on geometry. Arnauld deemed Pascal's attempts disorderly. His answer was the Nouveaux éléments (written c. 1655-56, first printed in 1677). The edition of Pascal's fragmentary Introduction àla géométrie (written c. 1655) therefore provides the immediate pre-text to the edition of Arnauld's textbook. Dominique Descotes favoured the 1667 edition of the Nouveaux éléments over both the amended but faulty 1683 one and the posthumous 1775 -83 edition. However, Arnauld's 1683 revisions to four chapters are inserted in the main text. The 1683 variants and 1775-83 additions are indicated in footnotes. For all three géométries, footnotes also include references to contemporary scholarship on the history of mathematics, as well as brief histories of a specific mathematical notion or problem, traced from classical sources (Euclid, Archimedes, Aristotle, etc.) to humanist ones (Patrizi, Telesio, Cardano, Peletier du Mans, etc), and to seventeenth- and eighteenth-century views (Gassendi, Hérigone, Lamy). Four appendices to the Nouveaux éléments provide further contextual information: Appendix I is Arnauld's early oudine (c. 1659 -60). Appendix III echoes Arnauld's discontent at the faulty 1683 edition and justifies Descotes's editorial choice. Appendix IV reproduces Arnauld's notes in his copy of Pascal's mathematical Lettres de A. Dettonville. The concluding edition of François de Nonancourt's Euclidus Logisticus (1652) in both Latin and French sheds light on a possible source of Arnauld's 1683 revision to his theory of ratio. The Nouveaux éléments follows the 'ordre naturel', that is, Cartesian method made pedagogy. The student learns from clear and distinct 'chaînes de raisons' in the vernacular, leading from the 'simple nature' of magnitude to complex geometry of solids, in fifteen chapters. Mathematical notation is minimal and initially well defined. Arnauld also elucidates 'obscurities' such as the positive product of negative magnitudes (pp. 134-36; Appendix II echoes a reader's struggle with this notion), or the incomprehensible nature of the 'raison sourde', which relates to the infinite (pp. 188, 313-16). The text illustrates architectonic relationships between mathematics, metaphysics, epistemology, and rhetoric in the seventeenth century. Descotes's magisterial edition emphasizes this breadth of scope and should appeal to the intellectual historian and the Pascal scholar alike. The Introduction situates the three géométries within the history of the reformation of Euclid's Elements. Seventeenth-century mathematicians radicalized the humanist critique (Ramus), which prompted amended editions of Euclid (Clavius), or promoted alternative classical axiomatics (Pappus). They intended various refoundations of geometry: on a topological conception of space (Pascal), on a theory of magnitudes (Arnauld), or on a theory of ratio (Nonancourt). The Introduction also specifies Jansenist views on geometry. The geometry of clear demonstrations is a good spiritual exercise (pp. 62-64; pp. 97-100 and p. 117 in the Nouveaux éléments), whereas sophisticated proofs are mere displays of vanitas. The analysis of Pascal's rejection of casuistic as a dangerous, ever-growing axiomatic in which the proof by reductio ad absurdum does not operate is a brilliant essay on Pascalian argument.