and integral calculus] systematic geometric transformation fulfilled an essential role. 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." Quadrature of the hyperbola Saint-Vincent found that the area under a rectangular hyperbola (i.e. a curve given by ) is the same over as over when This observation led to the hyperbolic logarithm. The stated property allows one to define a function which is the area under said curve from to , which has the property that This functional property characterizes logarithms, and it was mathematical fashion to call such a function a logarithm. In particular when we choose the rectangular hyperbola , 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. An approach to Vincent−Sarasa theorem may be seen with hyperbolic sectors and the area-invariance of squeeze mapping. 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. Saint-Vincent was lauded as Magnan and "Learned" in 1688: “It was the great Work of the Learned Vincent or Magnan, to prove that distances reckoned in