Author(s) Bullant, Jean
Title Petit traicte de geometrie et d’horologiographie pratique
Imprint Paris, G. Cavellat, 1562
Subject Geometry, Quadrature
Consult in image mode


     On January 2, 1562, the Parisian bookseller Guillaume Cavellat signed a publisher's agreement with Jean Bullant. He took over the royal privilege which had been granted to the architect on January 14, 1561 "pour ung petit traicté de geometrye et orlogiographie"– it appears in the Recueil d’horlogiographie printed in 1561 by Jean Bridier for Vincent Sertenas – and bought from him "six cens dudit livre de orologiographie que ledit vendeur avoit fait imprimer suyvant ledit privilege avec les figures qu’il avoit dicelluy, le tout pour cent livres tournois" (Archives nationales, Minutier central, LXXIII, 26). Guillaume Cavellat also had at his disposal a little treatise on geometry that Bullant had just drafted. Of these two treatises, he made a book which he published in 1562 and then in 1564.
In his word to the reader Bullant indicates that it is a matter of rendering the rules of geometry "familières aux artisans, comme elles sont aux gens doctes et plus curieux" and that horlogiography and more generally all the liberal arts depend "du premier degré des belles disciplines et noble science de géométrie". However, in this treatise on geometry one will look in vain for remarks useful for engraving sundials or references to constructions of the treatise on geometry in the treatise on gnomonics. The book is in French, not a novelty for the period in the area of scientific texts, but sufficiently rare to be pointed out. Thus Bullant put himself within the reach of artisans who, like him, did not understand Latin. Above all, he became part of a long line of scholars who, like Oronce Fine (1494 -1555), Charles de Bovelles (1471-1556), Pierre Ramus (1515-1572) or Jacques Peletier du Mans (1517-1583), tried to "replanter en notre plaine française", to repeat du Bellay's expression, a little of their mathematical knowledge.
Oronce Fine's influence is particularly strong in Bullant's geometry. Fine taught mathematics for a long time at the royal college. He is generally considered as the restorer of mathematics in France, even if he was more a popularizer than an innovating mathematician. Thus the rondeau on the fourth page of Bullant's treatise is not his own, but was written by Oronce Fine for the Géométrie pratique by Charles de Bovelles published in 1551, at the shop of Simon de Colines. This influence is also perceptible in the choice of the problems treated by Bullant and most particularly the one relative to squaring the circle, already well (badly) handled by Fine.
Aristotle had expressed the opinion that with a ruler and a compass, it was possible in a finite number of steps, to draw a square with the same area as that determined by a given circle. Although, as early as 1544, the mathematician Michael Stifel spread the idea that squaring the circle was impossible, numerous geometers, and not the least important, proposed solutions, obviously mistaken ones. Humble Jean Bullant also attacked this problem but only proposed to "réduire la superficie ronde à la superficie carrée, au plus juste que m’a été possible". If he had a lot in common with the circle squarers, let us recognize that he claimed only to be an approximate one, which should gain him indulgence on the part of the severest geometers .
First Bullant deals with squaring the rectangle (in an original and perfectly accurate way), without proofs, indeed, but well exemplified, then with squaring various triangles (isoceles or equilateral, an isoceles rectangle and lastly scalene) and with that of a diamond. Next he solves the inverse problem consisting of constructing several rectangles with the same area as that of a given square. Having done this, Bullant attempts to construct, as accurately as possible, a segment of line with the same length as the circumference of a circle or of an arc of a circle (the rectification of the circle is a problem related to squaring the circle), and inversely, to construct an arc whose length is given by a segment.
These constructions are original, simple and clearly explained thanks to the accompanying illustrations. In performing the calculations, one finds very acceptable values approaching П, even though this was not Bullant's focus as he remained within the strict framework of the geometry of the ruler and the compass. These constructions determine segments of different lengths for the rectification of a same arc. One can thus assume that Bullant was aware that no construction is exact. Next he rectifies and squares the oval (a geometric figure formed by four circles whose centers form two adjacent equilateral triangles) and proposes two solutions for the rectification and the squaring of the circle, but his approximations are less close than the preceding ones. Alas, finally he misuses a squaring of the circle, a procedure recommended by Oronce Fine, to propose a cubature of the sphere which is frankly wrong.
The last four pages of the treatise are more disparate. The geometer becomes an architect again. He puts a church portal and the osteau above it in good proportion, then becomes a land surveyor, measuring one length on a parcel with the help of a mason's square. Without falling into plagiarism, Bullant describes a process inspired by the ones Fine exhibits in his 1556 work, La composition et usage du quarré géométrique.
This treatise on geometry is entirely in keeping with the "gens doctes et curieux" of that century who deepened the study of mathematics by bringing their small touch of originality, as much through personal inclination as through professional necessity. Thus one is dealing with a "maçon cultivé", to take up Yves Pauwel's expression, but also with one of those cultural intermediaries who would change the social status of knowledge by popularizing it and using it as a means of promotion. Writing his book in French and using an effective pedagogy (in particular the use of illustrations), Bullant spread his knowledge and rendered it accessible to new social strata. But he also used it to increase the standing of his trade and to place it as high as possible on the scale of the liberal arts. In order to do this he does not insist on the directly utilitarian aspect of mathematics for the mason or the manufacturer of sundials. On the contrary, while apologizing hypocritically "de son débile et petit entendement", he does not hesitate to approach questions debated by the most expert mathematicians of his period. Unfortunately among those questions, squaring the circle was not within the reach of Renaissance geometers. The modern reader is therefore somewhat disinterested in this naïve geometry, lacking precision, in which certain results are inaccurate. Yet it played an important role; it is so true that even in this domaine, errors are indispensable stages on the way to truth. With this treatise Jean Bullant made a humble addition to this journey.
The Recueil d'horlogiographie dated 1561 is bound at the end of the treatise on geometry. In fact Guillaume Cavellat had bought six hundred copies from Bullant of his treatise on gnomonics.

Jean-Pierre Manceau (Tours) – 2009

Critical bibliography

J. d’Hombres, "La mise à jour des mathématiques par les professeurs royaux", A. Thuilier (ed.), Histoire du Collège de France. La création 1530-1560, Paris, Fayard, 2006, pp. 377-420.

J.-P. Manceau, "La place des mathématiques dans les écrits de Jean Bullant et Philibert De l’Orme", Journal de la Renaissance, 6, 2008, pp. 161-172.

P. Renouard, Imprimeurs et libraires parisiens du XVIe siècle, Fascicule Cavellat, Marnef et Cavellat, Paris, Bibliothèque nationale, 1986, pp. 5, 14, 152, n° 176.