JEAN DHOMBRES IS ONE PROOF ENOUGH? TRAVELS WITH A MATHEMATICIAN OF THE BAROQUE PERIOD 1 ABSTRACT. It is reasonable to assume that a subject of presumably universal appeal must rely on just one style. In spite of its universality, mathematics employs many styles. In particular, there are many styles of proof. In this paper we present and analyse a number of proofs of a property of the area under an hyperbola due to Gregory of Saint-Vincent, a mathematician of the first half of the seventeenth century. There is a baroque and prolific quality to the architecture of his proofs, and this quality points to a connection between a culture and the discovery of a mathematical theory. An historical perspective shows that, in addition to many styles, the universality of mathematics implies a variety of procedures. THE ONE-PROOF-ONLY TRADITION Among intellectuals, mathematicians are often regarded- as they have been since Antiquity - as victims of a curious affliction: they repeatedly try to prove the obvious. While everybody agrees that a formula like A = 7rR2 requires proof, far fewer insist on a formal proof of the theorem that the ratio of the areas of two circles is equal to the square of the ratio of their radii (A/A' = (R/R')2). This, however, is what is proved in Proposition 2 of Book XII of Euclid's Elements. Euclid's proof is very useful because it introduces one of the most powerful tools of ancient geometry - the method of exhaustion - so named far later by the Bruges-born mathematician Gregory of Saint-Vincent. One reason for the general indifference toward some kinds of proof is certainly the existence of easy proofs of particular cases. In our instance, a simple proof is known for two similar triangles: the ratio of the two areas is equal to the square of the scale factor. Therefore, the same is assumed to hold without further discussion for any two similar figures, say, for two circles, where the scale ratio is the ratio of the two radii. It is interesting that even meticulous geometers who copy the rather long euclidean proof in Book XII are quite casual when dealing with other similar figures - they just consider the result obvious. Surprisingly, in some textbooks dealing with the Lebesgue integral and with Lebesgue measure in the plane, the formal (and very easy) proof of the dependence of areas of similar figures on the scale factor is often left out. This general attitude was summarized in La logique ou l'Art de penser, a book first written by Arnauld and Nicole in 1662, in which the cartesian influence is very strong: the authors deplore the common weakness of mathematicians for proving the obvious. 2 Therefore, the famous Jansenist Arnauld is quite casual when discussing the area of a circle in his Geometry of 1667 (sometimes called Educational Studies in Mathematics 24: 401-419, 1993. @ 1993 Kluwer Academic Publishers. Printed in the Netherlands.