BOOKS ON ARCHITECTURE
Author(s) 
Bullant, Jean 
Title 
Petit traicte de geometrie et d’horologiographie pratique 
Imprint 
Paris, G. Cavellat, 1562 
Localisation 

Subject 
Geometry, Quadrature 
French
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 (14711556), Pierre Ramus (15151572) or Jacques Peletier
du Mans (15171583), 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.
JeanPierre 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 15301560, Paris,
Fayard, 2006, pp. 377420.
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. 161172.
P. Renouard, Imprimeurs et libraires parisiens du XVI^{e} siècle,
Fascicule Cavellat, Marnef et Cavellat, Paris, Bibliothèque nationale, 1986, pp. 5, 14, 152,
n° 176.
