## Riemann on Riemann sums

Though Riemann sums had been considered earlier, at least in particular cases (for example, by Cauchy), the general version we consider today was introduced by Riemann, when studying problems related to trigonometric series, in his paper Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe. This was his Habilitationsschrift, from 1854, published posthumously in 1868.

Riemann’s papers (in German) have been made available by the Electronic Library of Mathematics, see here. The text in question appears in section 4, Ueber den Begriff eines bestimmten Integrals und den Umfang seiner Gültigkeit. The translation below is as in

• A source book in classical analysis. Edited by Garrett Birkhoff. With the assistance of Uta Merzbach. Harvard University Press, Cambridge, Mass., 1973. MR0469612 (57 #9395).

Also zuerst: Was hat man unter $\displaystyle \int_a^b f(x) \, dx$ zu verstehen?

Um dieses festzusetzen, nehmen wir zwischen $a$ und $b$ der Grösse nach auf einander folgend, eine Reihe von Werthen $x_1, x_2,\ldots, x_{n-1}$ an und bezeichnen der Kürze wegen $x_1 - a$ durch $\delta_1$, $x_2 - x_1$ durch $\delta_2,\ldots,$ $b - x_{n-1}$ durch $\delta_n$ und durch $\varepsilon$ einen positiven ächten Bruch.  Es wird alsdann der Werth der Summe $\displaystyle S = \delta_1 f(a + \varepsilon_1 \delta_1) + \delta_2 f(x_1 + \varepsilon_2 \delta_2) + \delta_3 f(x_2 + \varepsilon_3 \delta_3) +\cdots$ $\displaystyle +\delta_n f(x_{n-1} +\varepsilon_n \delta_n)$

von der Wahl der Intervalle $\delta$ und der Grössen $\varepsilon$ abhängen.  Hat sie nun die Eigenschaft, wie auch $\delta$ und $\varepsilon$ gewählt werden mögen, sich einer festen Grenze $A$ unendlich zu nähern, sobald sämmtliche $\delta$ unendlich klein werden, so heisst dieser Werth $\displaystyle \int_a^b f(x) \, dx$.

In Birkhoff’s book:

First of all: What is to be understood by $\displaystyle \int_a^b f(x)\,dx$?

In order to establish this, we take the sequence of values $x_1,x_2,\ldots, x_{n-1}$ lying between $a$ and $b$ and ordered by size, and, for brevity, denote $x_1 - a$ by $\delta_1$, $x_2 - x_1$ by $\delta_2,\ldots,$ $b - x_{n-1}$ by $\delta_n$, and proper positive fractions by $\varepsilon_i$. Then the value of the sum $\displaystyle S = \delta_1 f(a + \varepsilon_1 \delta_1) + \delta_2 f(x_1 + \varepsilon_2 \delta_2) + \delta_3 f(x_2 + \varepsilon_3 \delta_3) +\cdots$ $\displaystyle +\delta_n f(x_{n-1} +\varepsilon_n \delta_n)$

will depend on the choice of the intervals $\delta_i$ and the quantities $\varepsilon_i$. If it has the property that, however the $\delta_i$ and the $\varepsilon_i$ may be chosen, it tends to a fixed limit $A$ as soon as all the $\delta_i$ become infinitely small, then this value is called $\displaystyle \int_a^b f(x) \, dx$.

(Of  course, in modern presentations, we use $\Delta_i$ instead of $\delta_i$, and say that the $\delta_i$ approach $0$ rather than become infinitely small. In fact, we tend to call the collection of data $x_1,\dots,x_{n-1}$, $\varepsilon_1,\dots,\varepsilon_n$ a tagged partition of ${}[a,b]$, and call the maximum of the $x_{i+1}-x_i$ the mesh or norm of the partition.)