A Mechanical Calculator for Arithmetic Sequences (1844–1852): Part 1, Historical Context and Structure

Prior to 1900, almost every mechanical calculating machine was aimed at facilitating a standard arithmetic operation, and it tried to do so in a general way. For instance, the first machines were adding machines, and they could add any two numbers, depending on the size of the machine. In some cases, the machines were tailored for specific needs, such as adding nondecimal monetary units. Most of the early multiplying machines were in fact adding machines, but the multiplicand could be stored, shifted, and reused in a new addition, although usually not automatically. Some machines were equipped with automatic shifts, and toward the end of the 19th century, some multiplication machines were based on stored tables, allowing for shortcuts. All these machines were aimed at general multiplication.

Specialized numerical calculating machines were much more rare, especially in the 19th century. The main type of such a machine is the difference engine, such as those of Babbage, Scheutz, Wiberg, Grant, and others. These machines also perform additions, but the additions concern several numbers simultaneously, or nearly simultaneously, for instance, to compute the interpolated values of a table.¹ Such machines are not general adding machines. For the 20th century, we can cite several examples of special-purpose mechanical calculating machines, such as the machine built by Christel Hamann for the interpolations used in constructing Bauschinger and Peters’s table of eight-place logarithms (1910–1911)² and the one used by Thompson for his Logarithmetica Britannica (1952).³

These machines were not the only special-purpose mechanical calculators, however. There have been special calculators for arithmetic purposes (such as the Carissan congruence machine and Lehmer sieve) as well as analog calculators, such as Torres’ algebraic machine (1893). There were undoubtedly others. This article gives, for the first time, a complete description of the machine developed around 1850 by Jean-Baptiste Schwilgué (1776–1856) for the mechanical computation of arithmetic sequences, of particular use in setting gear-cutting machines. The first part in a two-part series, this article describes the machine’s historical context and structure. The second part will present the details behind its basic operations.

A Calculator of Sequences (1852)

The machine shown in Figures 1 and 2 was completed by Jean-Baptiste Schwilgué, most certainly in 1852.⁴ Schwilgué (1776–1856) is above all known for building the third astronomical clock (1838–1843) of the Strasbourg cathedral, but he was in fact a clockmaker and engineer. He began as a clockmaker’s apprentice. He later worked his way to weights and measures controller and to a professor of mathematics. In the 1820s, he worked on improving scales and moved to Strasbourg in 1827. He was particularly interested in tower clocks and he constructed about 500 of them, many still in existence. In the 1840s, he patented a number of mechanisms, including a small adding machine, which turns out to be the currently oldest known key-driven adding machine.⁵

Figure 1.

Schwilgué’s machine in its custom-made box, probably in 1920. The weight and the string are now missing. (Musée des arts et métiers, inv 19151-0001)

Clocks are made of gears, or teethed wheels. Cutting the teeth of a wheel requires the clockmaker to turn the wheel by a certain angle, cut a tooth, turn it again, cut a new tooth, and so on. The angular position of the wheel depends on the number of teeth to cut and on the particular tooth at work (see Figure 3). For the astronomical clock, Schwilgué had to use a number of wheels with a nonstandard number of teeth, sometimes large prime numbers, and he constructed a wheel-cutting machine in which the angular position could be varied with a high accuracy. The machine used for the astronomical clock unfortunately no longer exists, but another one, constructed around 1842 (that is, during the time Schwilgué constructed his clock), does still exist, although it has been modified in places, including in the measurement of the divisions. With the latter machine, Schwilgué could...