Buteo, Johannes

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Buteo, Johannes

(b. Charpey, Dauphine, France, ca. 1492; d. Romans-sur-Isère, Dauphine, ca. 1564–1572)


Buteo’s father, François, seigneur d’Espenel, is said to have had twenty children. Because he did not wish to be a burden to his parents, Buteo entered the Abbaye de St.-Antoine about 1508. He had so much feeling for languages and mathematics, we are told, that he soon could comprehend Euclid in the original Greek. In 1522 he was sent to Paris, where he studied under Oronce Fine. By 1528 he longed for his monastic life and returned to St.-Antoine; he was abbot during two of his years there. In 1562, during the first of the Wars of Religion, he had to leave the monastery and take refuse with one of his brothers in Romans-sur-Isère. He died there of grief and boredom. His original French name was Jean Borrel (bourreau means “executioner”, but is also a popular name for the buzzard, and in this last sense is translated as Buteo). There were such variants as Boteo, Butèon, and Bateon.

Buteo published his works only after he was sixty years old. The Opera goemetrica contains fifteen articles on different subjects, the last six showing his interest in law through treatment of such mathematical aspects of jurisprudence as division of land and inheritances. The first nine articles treat mechanical, arithmetical, and geometrical problems. The most original is Ad problema cubi duplicande, in which he refutes Michael Stifel’s claim of an exact solution to this problem and gives an approximate one.

This is also the main theme of De quadratura circuli, in which Buteo refutes the pretensions of those who claimed to have found the solution of the quadrature, most notably those of his master, Oronce Fine. By contrast, he discusses appreciatively the approximations found by Bryson, Archimedes, and Ptolemy. He also mentions two approximate values for π 3–17/120 (from Ptolemy) and (Indian, although he believed it to be Arab).

In the second part of this work, Buteo criticizes errors of many of his contemporaries, particularly in terminological questions. An interesting point is his proof that the author of the proofs of Euclid’s Elements was not Theon, as was the current opinion, but Euclid himself. Here, too, are the beginnings of the famous dispute involving Peletier, Clavius, and many others on the angle of contact. In the Apologia (1562) Buteo pursued his refutation of Peletier’s theories.

Buteo’s most important work, the Logistica, was divided into five books, of which the first two deal with arithmetic, the third deals with algebra, and the last two present many problems in both fields. Terms such as “million” and “zero” and symbols such as p and m for + and – show Italian influence. There is a good treatment of simultaneous linear equations, with notations borrowed from Stifel; and there are approximations to a a influenced by Chucquet through Estinne de la Roche. The work was not practical enough to be reprinted, however.

Buteo’s fame rests only on his books. He has been a solitary figure in his love of mathematics and mechanics, and he wanted to be so. As far as we know, he had no pupils; and his criticism, often excessively sharp, must have estranged other mathematicians.


I. Original Works. Buteo’s works are Opera geometrica (Lyons, 1554; reissued 1559. See British Museum, General Catalogue of Printed Books for information on reprinted articles.); Logistica, quae et arithmetica vulgo dicitur in libros quinque digesta… eiusdem ad locum Vitruvij corruptum restitutio (Lyons, 1559, 1560); Ad locum Vitruvij corruptum restitutio was reprinted in J. Polenus, Exercitationes Vitruvianae primae (Padua, 17390, and M. Vitruvius, Architecture, IV, part 2 (Utini, 1825–1830), 37–43; De quadratura circuli libri duo… Eiusdem annotationum opuscula in errores Campani, Zamberti, Orontij, Peletarij, Io. Penaeinterpretum Euclidis (Lyons, 1559); and Apologia adversus epistolam Jacobi Peletarii depravatoris Elementorum Euclidis (Lyons, 1562).

II. Secondary Literature. There is no biography of Buteo. The best sources for information on his life are J.A. de Thou, Histoire universelle… depuis 1543 jusqu’en 1610 (The Hague, 1740), III, 493; and L. Moréri, Le grand dictionnaire historique (Paris, 1759; this edition only). G. Wertheim wrote on Logistica in Bibliotheca mathematica, 2 (1901), 213–219. On Opera geometrica and De quadratura, see Moritz Cantor, Vorlesungen über Geschichte der Mathematik, II (Leipzig, 1913), 561–563, but with the emendations by G. Eneström, in Bibliotheca mathematica, 12 (1912), 253.

J. J. Verdonk

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