Grégoire de Saint-Vincent

Ductus plani in planum

The contribution of Opus Geometricum was in

making extensive use of spatial imagery to create a multitude of solids, the volumes of which reduce to a single construction depending on the ductus of a rectilinear figure, in the absence of [algebraic notation 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."

The manuscript also claimed to solve the ancient problem of squaring the circle, for which it was criticized by others, including Vincent Léotaud in his 1654 work Examen circuli quadraturae.

Quadrature of the hyperbola

Saint-Vincent found that the area under a rectangular hyperbola (i.e. a curve given by ${\displaystyle xy=k}$) is the same over ${\displaystyle [a,b]}$ as over ${\displaystyle [c,d]}$ when

${\displaystyle \frac{a}{b}=\frac{c}{d}.}$

This observation led to the hyperbolic logarithm. The stated property allows one to define a function ${\displaystyle A(x)}$ which is the area under said curve from ${\displaystyle 1}$ to ${\displaystyle x}$, which has the property that ${\displaystyle A(xy)=A(x)+A(y).}$ This functional property characterizes logarithms, and it was mathematical fashion to call such a function ${\displaystyle A(x)}$ a logarithm. In particular when we choose the rectangular hyperbola ${\displaystyle xy=1}$, 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.

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 the Asymptote of an Hyperbola, in a Geometrical Progression, and the Spaces that the Perpendiculars, thereon erected, made in the Hyperbola, were equal one to the other.”

A historian of the calculus noted the assimilation of natural logarithm as an area function at that time:

As a consequence of the work of Gregory St. Vincent and de Sarasa, it seems to have been generally known in the 1660s that the area of a segment under the hyperbola ${\displaystyle y=\frac{1}{x}}$ is proportional to the logarithm of the ratio of the ordinates at the ends of the segment.