Mathematics for jurists: Buteo’s Opera Mathematica

Ce billet complète et prolonge l’atelier de juin 2012  « Le rôle des séries de problèmes dans la constitution de champs scientifiques » (programme) et a été rédigé par Giovanna Cifoletti

Jean Borrel, a monk and former student of the first lecteur royal ès mathématiques Oronce Fine, published in 1554 a volume under the name of Buteo: Buteonis Delphinatici Opera geometrica (Geometrical works by Buteo from Dauphiné). It is a collection of 15 booklets (works) each dealing with a single problem coming from classical tradition. For instance, the first three titles are: De Arca Noe, cuius formae, capacitatisque fuerit libellus, (Booklet on Noah’s Arch and on which shape and capacity it was); De sublicio ponte Caesaris (The Sublicius bridge);Confutatio quadraturae circuli ab Orontio Finaeo factae. (Refutation of the squaring of the circle by Oronce Fine). While the first nine problems are of miscellaneous origin and content, concerning mathematical questions from the Bible or from various classical sources (mostly in literature, but also in history and mathematics), the second part of the Opera, consisting of six problems, is devoted to questions coming from the Corpus Iuris civilis.

This collection of problems is quite different from the ones transmitted by the mathematical traditions: it does not contain either geometrical or arithmetical problems per se or related to a mathematical – liberal arts – context, but only in connection with classical literary texts. Likewise, the topics are not related to arts and crafts or commerce either: there are no commercial problems transmitted by the abacus schools, nor the problems transmitted as practical geometry, dealing with instruments or machines.

Concerning the structure of the book, there is no progression in the difficulty of problems, so it seems that there is no special technique to learn. As a consequence, there is no comparison to be made between the various problems, for they are different by all means. Rather, the structure is given by what seems to be a random order of problems. The only explicit indication about the organization of content is the distinction between the first and the second group of problems, the second stressing a more direct connection to Civil Law and bearing the title In iure civili (concerning Civil Law). What all the problems have in common is the fact of being classical problems, appearing in classical books. Another element in common is that Buteo corrects classical and non mathematical authorities on mathematical points, with some harshness at times. The first purpose here seems to be to show by instances that a deeper knowledge of mathematics is useful in reading ancient sources: this is in fact the most common justification for the need of mathematics in school training. At a deeper level, this work deals twice with Quintilian’s text: Ad locum Quintiliani geometricum explanatio and Geometriae cognitionem Iureconsulto necessariam. Quintilian’s long passage on the mathematical education necessary for jurists was a topos in the Renaissance. By working on his text Buteo emphasises this thesis, so, the book is supposed to accompany a humanist scholar in the reading of classical sources, but most particularly it is supposed to give instances of the use of mathematics in law and to accompany the jurist in his practice.

Finally, this genre of books evidences that sciences were not only conceived of as built around trees in a systematic way according to Aristotelian categories or to Ramus’s dichotomies: In fact, sixteenth-century’s way of privileging classical sources for their scientific agenda involved also the fact of taking the problems they had posed as constituting the very skeleton of new science.

Sixteenth-century professors structured teaching around questions, and most particularly questions arising from the classical tradition. Similarly, a scholar had to accept the challenges posed by classical texts. First, a student was supposed to write in notebooks and to memorize the commonplaces, that is the basic data coming from classical heritage: they are passages chosen because of their value as « rhetorical turns of phrase, dialectical arguments, factual information » (A. Blair). Second, in books the main tenets of each discipline were presented as structured in questions and answers texts. The practical technique learned at school influenced by dialectical circles was the reduction of texts into their logical question and answer structure, whatever the topic dealt with. Old problems and old solutions could be provided together with new ones.

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