Grégoire de Saint-Vincent
Grégoire de Saint-Vincent (22 March 1584 Bruges – 5 June 1667 Ghent) was a Flemish Jesuit and mathematician. He is remembered for his work on quadrature of the hyperbola.
Grégoire gave the "clearest early account of the summation of geometric series." He also resolved Zeno's paradox by showing that the time intervals involved formed a geometric progression and thus had a finite sum.
The contribution of Opus Geometricum was in
For example, the "ungula is formed by cutting a right circular cylinder by means of an oblique plane through a diameter of the circular base." And also the "’double ungula formed from cylinders with axes at right angles." Ungula was changed to "onglet" in French by Blaise Pascal when he wrote Traité des trilignes rectangles et leurs onglets.
Grégoire wrote his manuscript in the 1620s but it waited until 1647 before publication. Then it "attracted a great deal of attention...because of the systematic approach to volumetric integration developed under the name ductus plani in planum. "The construction of solids by means of two plane surfaces standing in the same ground line" is the method ductus in planum and is developed in Book VII of Opus Geometricum
In the matter of quadrature of the hyperbola, "Grégoire does everything save give explicit recognition to the relation between the area of the hyperbolic segment and the logarithm."
Saint-Vincent found that the area under a rectangular hyperbola (i.e. a curve given by xy = k) is the same over [a,b] as over [c,d] when
This observation led to the natural logarithm. 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 This functional property characterizes logarithms, and it was 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.
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