Grégoire de Saint-Vincent

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Grégoire de Saint-Vincent (March 22, 1584 Bruges – June 5, 1667 Ghent), a Jesuit, was a mathematician who discovered that the area under a rectangular hyperbola (i.e. a curve given by xy = c) is the same over [a,b] as over [c,d] when a/b = c/d. This discovery was fundamental in for the developments of the theory of Logarithms and an eventual recognition of the natural logarithm (whose name and series representation where discovered by Nicholas Mercator, but was only later recognized as a log of base e). The stated property allows one to define a function A(x) which is the area under said curve from 1 to x, which has the property that A(xy) = A(x)+A(y). Since this functional property characterizes logarithms, it has become mathematical fashion to call such a function A(x) a logarithm. In particular when we choose the rectangular hyperbola xy = 1, one recovers the natural logarithm.

To a large extent, recognition of de Saint-Vincent's achievement in quadrature of the hyperbola is due to his student and co-worker Alphonse Antonio de Sarasa, with Marin Mersenne acting as catalyst. A modern approach to his theorm uses squeeze mapping in linear algebra.

Although a circle-squarer he is known for the numerous theorems which he discovered in his search for the impossible; Jean-Étienne Montucla ingeniously remarks that "no one ever squared the circle with so much ability or (except for his principal object) with so much success." He wrote two books on the subject, one published in 1647 and the other in 1668, which cover some two or three thousand closely printed pages; the fallacy in the quadrature was pointed out by Christiaan Huygens. In the former work he used Bonaventura Cavalieri's method of the indivisibles. An earlier work entitled Theoremata Mathematica, published in 1624, contains a clear account of the method of exhaustions, which is applied to several quadratures, notably that of the hyperbola.

[edit] References

Gregoire de Saint-Vincent (1647) Opus geometricum quadraturae circuli et sectionum coni, 2 volumes, Antwerp.
David Eugene Smith (1923) History of Mathematics, Ginn & Co., v.1, p.425.
Margaret E. Baron (1969) The Origins of the Infinitesimal Calculus, Pergamon Press, Oxford et.al., see pp. 135 - 47.