From Archimedes to Riemann

The idea of computing an area by summing infinitely many little pieces is extremely old: it travelled through more than two thousand years before receiving, with Bernhard Riemann, the rigorous definition we still learn today.

Archimedes’ method of exhaustion

In the 3rd century BC Archimedes could already “integrate” without having the symbol $\int$. With the method of exhaustion — inherited from Eudoxus — he approximated a curved figure with ever finer inscribed and circumscribed polygons, and “trapped” the area between two sequences squeezing it from above and below. In the Quadrature of the Parabola he proved in this way that the area of a parabolic segment equals exactly $\frac{4}{3}$ of the inscribed triangle, summing the geometric series

$1 + \frac{1}{4} + \frac{1}{16} + \cdots = \frac{4}{3}.$

It was an argument by double reductio ad absurdum, correct but laborious: a general algorithm was still missing (Boyer, Netz).

Indivisibles and the birth of the calculus

In the seventeenth century Bonaventura Cavalieri took up the idea again with the method of indivisibles, picturing an area as a sum of infinitely many “chords” and a volume as a sum of infinitely many sections (see Cavalieri’s principle). A few decades later Isaac Newton and Gottfried Leibniz recognised that behind tangents, velocities and areas lay a single tool — the calculus — with general rules and an efficient notation, and the fundamental theorem tied the integral to the derivative. The calculus worked beautifully, yet it rested on “infinitesimals” whose logical status was still uncertain.

Riemann’s rigour

It fell to the nineteenth century to provide solid foundations. In 1854, in his habilitation dissertation at Göttingen, Bernhard Riemann defined the integral $\int_a^b f(x)\,dx$ as the limit of the sums

$\sum_{i=1}^{N} f(\xi_i)\,\Delta x_i,$

as the largest width of the subintervals tends to zero, with no further need to speak of “infinitely small” quantities. A function is Riemann integrable when this limit exists and is the same whatever the choice of the sample points $\xi_i$ and of the partition. The sums appearing in the definition of the integral — and in the simulation with the little rectangles — are indeed called Riemann sums in his honour: with the language of the limit, they close the circle opened by Archimedes twenty centuries earlier (Boyer, Katz).

Links

Topics: Integral
Concepts: Area beneath a curve · Definite integral · Riemann sum
People: Archimedes · Bernhard Riemann · Isaac Newton · Gottfried Leibniz