Gottfried Wilhelm Leibniz

Abstract

Gottfried Wilhelm Leibniz (1646–1716) designed the first calculating machine that could multiply on its own, worked out binary arithmetic in 1679 and published it in 1703, and sketched machines that were built only centuries later: a binary calculator that ran on marbles, an equation-solving linkage, a cipher machine and a binary pocket watch. Only one of his calculators survives, in Hanover, and it never quite worked. This article covers his computing work; his calculus, philosophy and diplomacy are left aside except where they touched the machines.

Leipzig, Jena and Paris

Leibniz was born in Leipzig on 1 July 1646 (21 June by the Julian calendar still used in Saxony), the son of the university professor Friedrich Leibnütz; he wrote his name as Leibniz from 1671. In 1663 he spent a semester at Jena with the mathematician Erhard Weigel, a popular teacher who lectured in German rather than Latin, filled his house with inventions and in 1672–73 promoted a number system with base four. Leibniz went on to serve the Elector of Mainz, Johann Philipp von Schönborn.

In March 1671 he read the preface to Pascal’s Pensées, which mentioned Pascal’s adding machine, and began to think about a machine that would also multiply. A year later his employer sent him to Paris, where he arrived on 31 March 1672 to put a plan to Louis XIV for a French conquest of Egypt. The plan went nowhere; Leibniz stayed four years, met Christiaan Huygens and the scientists of the new Académie des sciences, and turned his idea into hardware.

The Machine That Multiplied

Pascal’s machine could multiply only by repeated addition done by hand. Leibniz wanted the shift and the repeated additions done by the machine. His first sketch connected gears by a chain; drawings from Paris in 1672, probably by a clockmaker working with him, show wheels with retractable teeth (a pinwheel) and then a cylinder whose teeth grew in length along its axis, the stepped drum (Staffelwalze). Turning the drum once advanced a neighbouring wheel by zero to nine teeth, depending on where the wheel sat along it.

On 1 February 1673 (22 January in England’s Julian calendar) Leibniz showed a wooden model to the Royal Society in London; two brass models followed in Paris. In 1676 he entered the service of the dukes of Brunswick-Lüneburg in Hanover, where he lived for the rest of his life. An “older machine,” built from 1679 mainly by a French mechanic, worked by 1694 and was shown to Elector Ernst August in 1695; a “younger machine,” built by clockmakers from about 1690 until Leibniz’s death, is the only one that survives. It handled eight-digit inputs and a longer result, but its carry mechanism passed a carry only across two places, so that 99 + 1 worked and 999 + 1 needed help from the operator. Leibniz published a description, the Brevis descriptio machinae arithmeticae, in 1710; Jacob Leupold reprinted it in German in 1727.

The lost machines

In 1701 Leibniz handed both machines to his former secretary Rudolf Christian Wagner, a mathematics professor at Helmstedt, who spent about ten years with local clockmakers trying to improve them. A letter of 12 August 1712 in Leibniz’s papers, dictated by a craftsman named Johannes Volckmahr Blumenfeld, proposes a smaller version with half the digits. Apart from the younger machine, now in the Leibniz exhibition of Leibniz University Hanover, every one of these devices has disappeared.

The stepped drum outlived its inventor. It sat at the core of Charles Xavier Thomas’s Arithmometer of 1820 and of most mechanical calculators for the next century (see Before the Computer: Mechanical Calculation). The Dresden computer pioneer Nikolaus Joachim Lehmann reconstructed the surviving machine, and working replicas stand in several museums.

Binary Arithmetic

Leibniz did not invent the binary system: the Spanish bishop Juan Caramuel y Lobkowitz described it in print in his Mathesis biceps of 1670. Leibniz developed it. On 15 March 1679 he wrote De progressione dyadica, a three-page Latin manuscript on calculating with 0 and 1, which ends with a machine. A box with holes that can be opened (1) or closed (0) releases marbles into channels; moved from column to column as multiplication requires, it adds shifted copies of one number, and the channels carry the result. He published the arithmetic, not the machine, in his Explication de l’arithmétique binaire for the Paris Academy in 1703.

He looked for ways to make binary popular. In 1697 he designed a medal showing binary numbers for the Duke of Brunswick-Wolfenbüttel, and in a note found in his papers by the philosopher Lloyd Strickland, who edited Leibniz on Binary (2022) with Harry Lewis, he proposed a pocket watch whose dial marked the hours in binary with raised and flat dots, so that the time could be read by touch at night.

The marble machine was built nearly three centuries later. Ludolf von Mackensen made a working version at the Deutsches Museum in Munich in 1972 (another account says 1971), a copy followed in Kassel in 1985, the precision mechanic Gerhard Weber built a new model in 2004, and the Arithmeum in Bonn has another. Independently, Miles Libbey of London patented a marble-driven binary adder for teaching in 1961.

Machines for Equations and Ciphers

In December 1674 Leibniz described in two manuscripts a Constructor, a mechanical linkage that solved algebraic equations such as ax + bx² + cx³ = d: the user set the coefficients and a trial value of x, and the linkage displayed the left-hand side as a length to compare with d. He never published or built it. The mathematician Johann Andreas von Segner invented a similar device independently, and the English clergyman John Rowning built one in 1768.

In the late 1670s he conceived a machina deciphratoria, a cipher machine with a keyboard and a rotating drum carrying several scrambled alphabets that advanced after a set number of letters, so that the cipher alphabet kept changing. He described it in 1688 for Emperor Leopold I, who was occupied with other matters. The philosopher Nicholas Rescher rediscovered the design around 2010, and a working model was built.

Logarithms interested him too. In letters of May and June 1712 he asked the mine surveyor Bernhard Ripking in the Harz to make him a logarithmic calculating line, and rejected Ripking’s counter-proposal of a circular slide rule; an appendix to the 1726 German edition of his Theodicy describes a brass cylinder with two sliding bands carrying 10,000 marks, in effect a slide rule for addition and subtraction.

Logic

Leibniz wanted reasoning itself to become calculation. In his New Essays on Human Understanding, a dialogue answering John Locke’s dismissal of syllogisms, he defended formal logic and hinted that it was incomplete; the book appeared only in 1765. His work on a logical calculus stayed among his papers, partly published by Johann Eduard Erdmann in 1840 and properly only in Louis Couturat’s La logique de Leibniz (1901) and Couturat’s edition of the manuscripts (1903), at the time when Frege, Russell and Whitehead were founding modern logic (see George Boole and Boolean Logic). He also took part in the old argument over logical proofs of God’s existence, criticising Descartes’ version in a text he sent to Antoine Arnauld in 1686.

Leibniz died in Hanover on 14 November 1716, leaving some 200,000 sheets of papers. The tercentenary of his death made 2016 a “Leibniz Year” in Germany, and the Heinz Nixdorf MuseumsForum in Paderborn shows a working replica of his calculator in its gallery of pioneers.