Gregoire was born in Bruges8 September 1584. After reading philosophy in Douai, he entered the Society of Jesus21 October 1605. His talent was recognized by Christopher Clavius in Rome. Gregoire was sent to Louvain in 1612, and was ordained a priest 23 March 1613. Gregoire began teaching in association with François d'Aguilon in Antwerp from 1617 to 20. Moving to Louvain in 1621, he taught mathematics there until 1625. That year he became obsessed with squaring the circle and requested permission from Mutio Vitelleschi to publish his method. But Vitelleschi deferred to Christoph Grienberger, the mathematician in Rome. On 9 September 1625, Gregoire set out for Rome to confer with Grienberger, but without avail. He returned to the Netherlands in 1627, and the following year was sent to Prague to serve in the house of Emperor Ferdinand II. After an attack of apoplexy, he was assisted there by Theodorus Moretus. When the Swedes raided Prague in 1631, Gregoire left and some of his manuscripts were lost in the mayhem. Others were returned to him in 1641 through Rodericus de Arriaga. From 1632 Gregoire resided with The Society in Ghent and served as a mathematics teacher.
Ductus plani in planum
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 methodductus 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."
Quadrature of the hyperbola
Saint-Vincent found that the area under a rectangular hyperbola is the same over as over when This observation led to the natural logarithm. The stated property allows one to define a function A 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 a logarithm. In particular when we choose the rectangular hyperbola xy = 1, one recovers the natural logarithm. A student and co-worker of Saint-Vincent, A. A. de Sarasa noted that this area property of the hyperbola represented a logarithm, a means of reducing multiplication to addition. A modern approach to his theorem uses squeeze mapping in linear algebra. In 1651 Christiaan Huygens published his Theoremata de Quadratura Hyperboles, Ellipsis, et Circuli which referred to the work of Saint-Vincent. The quadrature of the hyperbola was also addressed by James Gregory in 1668 in True Quadrature of Circles and Hyperbolas While Gregory acknowledged Saint-Vincent's quadrature, he devised a convergent sequence of inscribed and circumscribed areas of a general conic section for his quadrature. The term natural logarithm was introduced that year by Nicholas Mercator in his Logarithmo-technia. A historian of the calculus noted the assimilation of natural logarithm as an area function at that time:
