# Calculation and Computation - Premodern, Early Modern, Non-western

### leibniz calculating machine tables

The difference in the meaning of "calculation" and "computation" that are found in Steinmetz and Bush seems to have been built on the precapitalist use of terms formed by the roots *calcul*-and *comput*-, respectively. Historiographically, the late medieval period offers an example when one compares the interest shown in quantifying the heterogeneity of motion by the Merton College theorists known as the "Oxford calculatores" to the practice shown by the ecclesiastical community toward homogenizing the standardization of time through a technique known as *computus.* During the late medieval and early modern period, the attack by the algorists upon the abacists—that is, the promoters of computing who relied upon previous and private memorization of tables of numerical relationships versus the defenders of the ancient tradition who placed emphasis on live calculations performed in public by moving pebbles (that is, calculi) along designated lines—compares favorably to the conflict between the digital (programming) and the analog (living labor) of the recent decades.

The physical embodiment of complex numerical relationships through interconnected mechanical parts that were concealed by a case, hence masking the motion of the gears, displayed only the input and output numbers. Some celebrated early modern examples were Blaise Pascal's (1623–1662) adding machine (1642) and Leibniz's adding and multiplication machine (1685). Earlier in the same century (1623), the German mathematician and linguist Wilhelm Schickard (1592–1635) had mechanized the set of numbered sliding rods that John Napier (1550–1617) had devised in 1617 to simplify astronomical calculations. Both Pascal and Leibniz sought to profit selling their machines to merchants and natural philosophers. Galileo Galilei (1564–1642) tried the same with his improved computing dividers. Many of the early modern natural philosophers were heavily involved in calculating innovations. Additional contributions include "calculation by analysis" to the coordinates of René Descartes (1596–1650), by differentiation and integration in the calculus tradition that prefigures in Simon Stevin (1548–1620) and materializes in Newton and Leibniz, and by the analysis of multiplication and division into addition and subtraction through the logarithms that Napier introduced in 1614.

As with the method of algorists, the speed in calculation by logarithms assumed the availability of relevant tables. The transformation of these tables into scales inscribed in, first, circles sharing a fixed center and, soon after, scales that slid beside each other while sharing a fixed framework, found its ultimate presentation in the logarithmic slide rule, configured by William Oughtred (1574–1660) as early as 1621. The interactive proliferation of both tables of logarithms and logarithmic slide rules determined the history of calculation from the early modern period until the very recent decades. The dilemmas of computation in the recent decades have prefigured in the construction and worldwide use of tens (if not hundreds) of millions of slide rules, linear, circular, cylindrical, or hybrid, wooden, metal, or plastic, handmade or mass produced, cheap or expensive, with accessories such as cursors and magnifying glasses to increase the accuracy without increasing the size. The recent debates over special versus general purpose software and over competing software operating systems was long rehearsed in debates over choice of general or special purpose slide rule scales and scale system standards. Moreover, as scales of all sorts slid beside each other or within each other, the logarithmic slide rule turned out to be only one of innumerable versions of slide rules. Material culture scholars see no end to collection of calculation wheels devised to compute phenomena ranging from a menstrual cycle to a baseball season.

The coevolution of logarithmic slide rules and tables of logarithms is no different from the codevelopment of Leibniz's "calculus" and his "calculating machine." Taken together they point to the paired constitution of the scientific and the technical since earlier modernity. If Leibniz sought to sell his machine to merchants, Charles Babbage (1792–1871), a Cambridge mathematics professor, was interested in a calculating engine organized internally according to the symbolic efficiency and rationality of the nascent industrial order emerging within the first half of the nineteenth century in Great Britain. Partially funded by the government, Babbage failed for economic reasons to have his "difference engine" constructed because he was forced to depend on one of the most skillful workers of the period, Joseph Clement (1779–1844), in the workshop of whom the engine to calculate as if powered by "steam" was to be constructed. While the construction of the difference engine encountered the problem of skilled labor, Babbage realized that the use of the engine itself would be limited to special purposes, which would not eliminate calculation's dependence on skilled labor. In 1833, he started drafting sketches of a second engine, the "analytical engine," which he sought to be independent of labor skill. He kept making modifications to the plans until his death in 1871 without ever managing to advance beyond programmatic descriptions of such an engine.

In Babbage's *The Exposition of 1851; or, Views of Industry, the Science and the Government of England,* published in London in 1851 following his review of the panorama of industrial capitalism staged at the Crystal Palace World Fair, he repeatedly touched on the ideal of an engine that would render laborers mere "attendants," unable to influence the production of calculation. Although Babbage's calculating engines matched his promotion of a version of continental analysis that he thought to be more appropriate to a calculation to sustain further industrial growth, he became known for his emphasis on the mechanization rather than the organization of work. The organization of work has been the pursuit of the division-of-labor scheme that the French engineer Gaspard Clair Prony (1755–1839) devised in 1792 to have a new set of logarithmic and trigonometric tables produced as a monument to the new French Republic. Six eminent mathematicians who selected the appropriate equations formed the peak of his pyramid, a layer of ten mathematicians below them advanced the analysis to the point where everything was converted to simple arithmetic, and a group of one hundred humans, recruited even from hairdressers, performed simple operations on a part of a problem before passing it to the next person.

The expansive reproduction of this scheme by setting smaller or larger groups during the nineteenth and the twentieth century, after taking advantage of sources available both within and outside the Western society, resulted in the formation of an army of "computers." By the introduction of commercial calculating machines such as Thomas de Colmar's (1785–1870) Arithmometer, exhibited at the 1855 Paris Exposition, references to "human computers" are found from as near as a British male scientist's environment of female friends and as far as male Indians working in British engineering initiatives in their country. The relative continuity between Leibniz's calculating machine and the Arithmometer—the most widely available calculating machine based on improvements on Leibniz's machine by 1871—is an index of the relative continuity between merchant and industrial capitalism. Leibniz had promoted his machine to a society marked by the dynamic appearance of merchants by stating that "it is unworthy of excellent men to lose hours like slaves in the labor of calculation, which could safely be relegated to anyone else if the machine were used" (Leibniz, quoted in David Eugene Smith, *A Source Book in Mathematics* [1929], pp. 180–181). Babbage's "attendants" were the industrial version of Leibniz's slaves. Babbage wanted his calculating engines used for the production of general and special purpose tables, including tables for the navigational pursuits of the British empire. Batteries of human computers—along with other implements such as calculating machines and slide rules—produced numerous tables and charts for an assortment of military and civilian purposes (scientific, engineering, and commercial).

The lack of a synthesis of scholarly studies limits scholars' ability to compare the modern European experience with calculation and computation to those of non-European societies. What is known about the Inca knotted strings known as *quipu*
and the Chinese knotted cords suggests that societies paradigmatic of civilization in other continents relied heavily on what is now identified as calculation and computation. Ongoing historical interpretation of archaeological findings from ancient Mesopotamia point to the few who knew how to connect computations routinely produced by the many to a calculating coefficient that could make the abstract concrete. Projecting Western conceptual demarcations to non-Western societies may prove problematic, especially considering the parallel dead ends from having projected late modern Western demarcations to ancient and early modern histories. Interpretation, for example, of the Hellenistic Antikythera mechanism as "analog," and accordingly, a technically inferior computer, has blocked historians from taking into account the digitalization introduced by the complicated geared structure underneath the disk representing analogically the universe. As a result, the search for how the technical accuracy of the artifact matched with social interests of the period has been replaced by the assumption of limits in the accuracy due to inferiority from belonging to an essentialist inaccurate technical genre. Similarly, a historiography aiming at interpreting the analog motion of the pebbles on ancient and early modern abacuses has been blocked by the late modern emphasis on the resting pebbles—that is, on the perception of pebbles as digits. The tradition of the Chinese and the Western abacus, therefore, has been flatly situated under the "digital," thereby making it impossible to acknowledge differences in the employment of abacus analogies between and within traditions.

## User Comments

over 2 years ago

Specifications

Can multiply, devide, add and substract. Mechanical device made of copper and steel. Carriage is performed with a stepped wheel, which mechanism is still in use today.

Chronology

Contrary to Pascal, Leibniz (1646-1716) successfully introduced a calculator onto the market. It is designed in 1673 but it takes until 1694 to complete. The calculator can add, subtract, multiply, and divide. Wheels are placed at right angles which could be displaced by a special stepping mechanism.

The speed of calculation for multiplication or division was acceptable. But like the Pascaline, this calculator required that the operator using the device had to understand how to turn the wheels and know the way of performing calculations with the calculator.(2)