Gerard of Brussels

(fl. first half of the thirteenth century)

geometry

Gerard played a minor but not unimportant role in the development of kinematics and the measure of geometrical figures. His career remains obscure except for his having written treatise entitled Liber de motu, which remains in six manuscripts. Four of these date from the thirteenth century. Written sometime between 1187 and 1260, the Liber de motu quotes the translation of Archimedes’ De quadratura circuli (“On the Measurement of the Circle”) by Gerard of Cremona; the translation was completed before the translator died in 1187. On the other hand, the Liber de motu is mentioned in the Biblionomia of Richard of Fournival, who lived in 1260

Gerard seems to have known the Liber philotegni de trangulis of Jordanus de Nemore, whose exact dating is as difficult to determine as that of Gerard but who can be placed, with some confidence, in the early decades of the thirteenth century. The similarity in their names suggests that Gerard may be identified with the unknown mathematician Gernardus who wrote an arithmetical tract Algorithmus demonstratus. That Gerard is referred to as magister in the title of Liber de motu and his apparent knowledge of Jordanus suggest a university milieu for this work—perhaps the University of Paris.

The Liner de motu contains thirteen propositions, in three books. In these propositions the varying curvilinear velocities of the points and parts of geometrical figures in rotation are reduced to uniform rectilinear velocities of translation. The four propositions of the first book relate to lines in rotation, the five of the second to areas in rotation, and the four of the third to solids in rotation, Gerard’s proofs are particularly noteworthy for their ingenious use of an Archimedean-type reductio demonstration, in which the comparison of figures is accomplished by the comparison of their line elements. In this latter technique Gerard assumed that if the ratio of the elements of two figures taken in pairs is the same, then the ratio of the totalities of the elements of the figures is the same. Such a technique resembles the procedure followed in Archimedes’ Method, which Gerard could not have read. The proposition most influential on later authors was the first: “Any part as large as you wish of a radius describing a circle … is moved equally as its middle point.”¹ This is similar in a formal way to the rule for uniform acceleration, which appears to have originated with William of Heytesbury at Merton College, Oxford, in the 1330’s². This rule asserted that a body which is uniformly acclerated traverses the same space in the same time as a body which moves with a uniform velocity equal to the velocity that is the mean between the initial and final velocities of the accelerating body. Gerard’s proposition concerns itself with movements that uniformly vary over some part or all of the linear magnitude rotating, while Heytesbury’s rule through some period of time. The “middle velocity” is used in both rules to convert the movements to uniformity.

Gerard’s influence on Thomas Bradwardine, the founder of the Merton school of kinematics, is evident, for Bradwardine knew and quoted Gerad’s tract. Furthermore, Nicole Oresme’s De configurationibus qualitatum, written in the 1350’s, shows some possible dependency on the De motu³

NOTES

1.Liber de motu M. Clagett, ed., p. 112.

2. For a discussion of the Merton rule, see M. Clagett, Science of Mechanics, ch. 5.

3. M. Clagett, Nicole Oresme and the Medieval Geometry of Qualities and Motions (Madison, Wis., 1968), p. 466.

BIBLIOGRAPHY

The Liber de motu has been edited by M. Clagett, “The Liber de motu of Gerard of Brusels,” in Osiris, 12 (1956), 73-175. See M. Clagett, Science of Mechanics in the Middle Ages (Madison Wis., 1959; repr. 1961), ch. 3; G. Sarton Introduction to the History of Science II (Baltimore, 1931), 629; and V. Zubov, “Ob ’Arkhimedovsky traditsii’ v srednie veka (Traktat Gerarda Bryusseleskogo ’Odvizhenii’)” (“The Archimedean Tradition in the Middle Ages [Gerhard of Brussels’ Treatise on Motion]”) in Istorikomatematicheskie issledovaniya,16 (1965), 235-272.

Marshall Clagett