The geometric mean is an ancient concept, but hyperbolic angle was developed in this configuration by Gregoire de Saint-Vincent. He was attempting to perform quadrature with respect to the rectangular hyperbola "y" = 1/"x". That challenge was a standing open problem since Archimedes performed the quadrature of the parabola. The curve passes through (1,1) where it is opposite the origin (mathematics) in a unit square. The other points on the curve can be viewed as rectangles having the same area as this square. Such a rectangle may be obtained by applying a squeeze mapping to the square. Another way to view these mappings is via hyperbolic sectors. Starting from (1,1) the hyperbolic sector of unit area ends at (e, 1/e), where e is 2.71828…, according to the development of Leonhard Euler in "Introduction to the Analysis of the Infinite" (1748).