The quadrature of the hyperbola and the logarithm

One of the most elegant results of the seventeenth century springs directly from the rectangular hyperbola $xy = k$, that is, from the graph of the inverse proportionality $y = \tfrac{k}{x}$. The problem that fascinated mathematicians of the time was quadrature: computing the area of the region bounded by the curve, the $x$-axis and two vertical lines.

In 1647, in his monumental Opus geometricum, the Flemish Jesuit Grégoire de Saint-Vincent discovered a remarkable property of this area. Write $A(a,b)$ for the area under the hyperbola $y = \tfrac{1}{x}$ between the abscissas $a$ and $b$. Saint-Vincent observed that multiplying both endpoints by the same factor leaves the area unchanged: $A(a,b) = A(\lambda a, \lambda b).$ The geometric reason is that stretching the abscissas by a factor $\lambda$ compresses the ordinates by the same factor, keeping the area constant. From this the key property follows at once: $A(1, xy) = A(1, x) + A(1, y).$ In other words, if the abscissas grow in geometric progression, the corresponding areas grow in arithmetic progression. But this is exactly the behaviour that, a few decades earlier, John Napier had made the foundation of his logarithms.

It was Saint-Vincent’s pupil, Alfonso Antonio de Sarasa, who made the link explicit in 1649: the area under the hyperbola is a logarithm. In modern language, $\int_a^b \frac{dx}{x} = \ln b - \ln a = \ln\frac{b}{a}.$ The simplest curve after the straight line thus encloses the logarithm function, and the base that emerges naturally is the number $e$: this is why we speak of the natural logarithm.

The story closes in 1668, when Nicholas Mercator, in his Logarithmotechnia, derived from this same quadrature the celebrated series expansion $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots,$ one of the first examples of a power series before the systematic advent of the infinitesimal calculus of Newton and Leibniz.

Links

Topics: Homographic hyperbola
Concepts: Rectangular hyperbola · Inverse proportionality · Logarithm · Area
People: Grégoire de Saint-Vincent · Alfonso Antonio de Sarasa · Nicholas Mercator · John Napier (Nepero)