Grégoire de Saint-Vincent
- 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 were 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 theorem 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."
- In 1647, Gregoire de Saint-Vincent published his book, Opus geometricum quadraturae circuli et sectionum coni (Geometric work of squaring the circle and conic sections), vol. 2 (Antwerp, (Belgium): Johannes and Jakob Meursius, 1647). In Book 6, part 4, page 586, Proposition CIX, he proves that if the abscissas of points are in geometric proportion, then the areas between a hyperbola and the abscissas are in arithmetic proportion. This finding allowed Saint-Vincent's former student, Alphonse Antonio de Sarasa, to prove that the area between a hyperbola and the abscissa of a point is proportional to the abscissa's logarithm, thus uniting the algebra of logarithms with the geometry of hyperbolas.
See also: Enrique A. González-Velasco, Journey through Mathematics: Creative Episodes in Its History (New York, New York: Springer, 2011), page 118.
- W. W. Rouse Ball (1912) A Short Account of the History of Mathematics, 5th edition, p 308, Macmillan Publishers
- Gregoire de Saint-Vincent (1647) Opus geometricum quadraturae circuli et sectionum coni, 2 volumes, Antwerp.
- Margaret E. Baron (1969) The Origins of the Infinitesimal Calculus, Pergamon Press, Oxford et al., see pp. 135 – 47.
- C.H. Edwards, Jr. (1979) The Historical Development of the Calculus, pp. 154–8, Springer-Verlag, ISBN 0-387-90436-0 .
- David Eugene Smith (1923) History of Mathematics, Ginn & Co., v.1, p. 425.
- Hans Wussing (2008) 6000 Jahre Mathematik: eine kulturgeschichtliche Zeitreise, S. 433, Springer, ISBN 9783540771920 .
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- Gregory Saint Vincent, and his polar coordinates from Jesuit History, Tradition and Spirituality by Joseph F. MacDonnell.
- O'Connor, John J.; Robertson, Edmund F., "Gregorius Saint-Vincent", MacTutor History of Mathematics archive, University of St Andrews.