# Mathematical Treasures: Early Calculating Machines

Author(s):
Frank J. Swetz (The Pennsylvania State University)

### Roman Hand Abacus

Figure 1. Reconstruction of a Roman "hand abacus" after a bronze original held by the Bibliothèque Nationale de France, in Paris. This 1977 reconstruction resides in the Römisch-Germanisches Zentralmuseum or Romano-Germanic Central Museum (known as the RGZ Museum), an archaeological and historical museum and research center in Mainz, Germany. (Source: Wikimedia Commons, courtesy of photographer Mike Cowlishaw, CC BY-SA 3.0, May 2004)

The facsimile shown above of a Roman hand abacus - or calculator (plural calculatores), also a term for the specialist (or specialists) who used it - is of an original that dates to the first century. This is a bi-quinary device with the beads in the uppermost columns denoting 5's and those in the aligned bottom columns denoting 1's. From right to left, beginning with the third column, the columns denote increasing powers of ten, from units (100 ) to millions (106). In the third, fourth, and fifth columns from the right, one can see the familiar Roman numerals I, X, and C, for 1, 10, and 100, respectively. Originally these columns would have contained five beads each, one in the upper slot and four in the lower slot. As it is, the third through ninth columns of the abacus could display a number only as high as 2,436,177.

The first two righthand columns are employed to denote fractional parts of a transaction. The second of these columns originally had one bead in its upper slot and five beads in its lower slot so that it could represent from 0 to 11 twelfths of the Roman as, the basic unit of the monetary system. One-twelfth of an as was an uncia, or ounce. The bead in the upper slot was worth six ounces and those in the lower slot one ounce each. The slots in the righthand column were, from top to bottom, for one-half, one-fourth, and one-third ounce and originally had, respectively, one, one, and two beads each.

Reference: Georges Ifrah, The Universal History of Numbers, Harvill Press, 1998, pp. 209-211

### The Pascaline

Figure 2. This calculator, designed by Blaise Pascal (1623-1662) and dubbed the "Pascaline," resides in the Musée des arts et métiers in Paris, France. (Source: Wikimedia Commons, courtesy of photographer David Monniaux, June 2005, © 2005 David Monniaux)

Blaise Pascal’s calculator or Pascaline was introduced in 1645. This is a gear driven device with each counter (gear set) able to record digits from 1 to 9, at which point a “carry-over” linkage transfers a unit to the next counter (10's carry). This calculator was efficient for its time.

### The Stepped Reckoner

Figure 3. This replica of the Stepped Reckoner (or Staffelwalze) machine designed by Gottfried Wilhelm Leibniz (1646-1716) is on display at the Technische Sammlungen Museum (also known as the Deutsches Museum), in Dresden, Germany. (Source: Wikimedia Commons, courtesy of photographer Kolossos, CC BY-SA 3.0, July 2006)

Gottfried Wilhelm Leibniz’ Stepped Reckoner was conceived along the lines of the Pascaline, although it was more advanced as it could also perform multiplication by repeated additions. Leibniz actually saw the binary number system as most efficient for calculations, but instead used decimal-based counters in his machine. Introduced in 1671, the calculator proved too expensive for popular adoption.

### The Difference Engine

Figure 4. The London Science Museum's construction of Charles Babbage's Difference Engine (Source: Wikimedia Commons, courtesy of photographer Geni, CC BY-SA, August 2008)

The Babbage Difference Engine was an analog computer conceived by the British mathematician Charles Babbage (1791-1871). Under contract to the British Navy, Babbage proposed and designed a gear-driven engine that employed a method of “finite differences” to generate and explore function values. The machine required a thousand sets of gears to produce 50-digit results. At the time, it was felt that such a machine could not be built and the project faltered. However, in the twentieth century, the Science Museum, London, following Babbage’s original plans and employing the technical skills available in the late nineteenth century, constructed and successfully operated the Difference Engine.

Figure 5. Patent drawing of Babbage computing unit (1853).

Index of Mathematical Treasures

Index of Mathematical Objects

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasures: Early Calculating Machines," Convergence (October 2015)