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Before the Computer: Mechanical Calculation

Abstract

Long before electricity, people built machines to do arithmetic. Over two thousand years they moved from beads on rods to brass gears that carried tens automatically, from logarithm tables that turned multiplication into addition to a loom that “programmed” patterns with punched cards. None of these devices was a computer (most could not even be reprogrammed) yet together they assembled the ideas the computer would later need: mechanized calculation, the stored instruction, and the conviction that thought could be delegated to a machine. This is the deep prehistory that Charles Babbage inherited.

The Antikythera Mechanism

In 1901 sponge divers pulled a corroded lump of bronze from a Roman-era shipwreck off the Greek island of Antikythera. It turned out to be the most sophisticated mechanism known from antiquity: the Antikythera mechanism, dated to roughly the 2nd–1st century BCE, a hand-cranked device of at least 30 interlocking bronze gears that modeled the motions of the Sun, Moon, and planets and predicted eclipses.

Its front and back carried scales for the calendar and for solar and lunar eclipses, which modern imaging has made legible; the philologist Albert Rehm photographed the fragments in 1905 or 1906, and the Bavarian State Library put his photographs online in 2020. It is the world’s oldest known analog computer, and a historical dead end. Nothing of comparable mechanical complexity reappears in the surviving record for over a thousand years. The knowledge to build it was lost, which is the recurring lesson of this prehistory: invention without transmission leads nowhere.

The Abacus: Calculation as a Tool, Not a Machine

For most of recorded history the dominant calculating aid was the abacus, beads or counters on rods or grooves, used across Mesopotamia, China (the suanpan), Japan (the soroban), and Rome. It does not calculate by itself; it is a memory aid that lets a trained human hold and manipulate numbers in a positional system faster than pen and paper.

The Inca administration stored decimal numbers in knotted cords, the quipus, and calculated on a counting board, the yupana. The abacus held its own against machines well into the twentieth century: in a contest in Tokyo on 11 November 1946, the Japanese clerk Kiyoshi Matsuzaki with a soroban beat the American soldier Thomas Wood with an electric calculator in four of five rounds.

That distinction matters. The abacus mechanizes bookkeeping of digits, not the operation of arithmetic. The leap that defines this whole era is moving the operation itself (the carry, the multiply) into the device.

Logarithms and the Slide Rule: Turning Multiplication into Addition

The first such leap was mathematical, not mechanical. In 1614 the Scottish laird John Napier published Mirifici Logarithmorum Canonis Descriptio, introducing logarithms, which convert multiplication into addition and division into subtraction. Napier also devised Napier’s bones (1617), numbered rods that simplified multiplication, and a second device, the promptuary; he died on 4 April 1617 near Edinburgh. The Swiss instrument maker Jost Bürgi (1552–1632) invented logarithms independently.

Building on Napier’s logarithms and Edmund Gunter’s logarithmic scale, the English clergyman William Oughtred placed two such scales side by side around 1622 and created the slide rule: slide one scale against the other and you read off products and quotients directly. The circular version was invented twice between 1621 and 1630, by Oughtred and by Richard Delamain, and Oughtred devised the two-piece linear rule in the 1630s. In the nineteenth century calculating cylinders with scales metres long extended the principle, and Maurice d’Ocagne’s nomography turned formulas into charts read with a ruler. The slide rule was the working engineer’s calculator for the next 350 years (it sized bridges, designed aircraft, and rode to the Moon on Apollo) until the pocket electronic calculator killed it in the 1970s.

The Geared Calculators: Pascal and Leibniz

The first calculating machine was German. On 20 September 1623 Wilhelm Schickard (born in Herrenberg on 22 April 1592), professor of Hebrew and mathematics at Tübingen, wrote to Johannes Kepler about a machine that combined a six-digit adding mechanism with carry and six cylinders bearing the multiplication table after the manner of Napier’s rods. The first copy was lost; a second, built for Kepler, burned in a workshop fire in February 1624. Schickard never published it, and only after his letters were rediscovered did the Tübingen philosopher Bruno von Freytag-Löringhoff build a working reconstruction, copies of which stand in several museums.

The mechanical breakthrough that Europe knew about came in France. In 1642, the nineteen-year-old Blaise Pascal, trying to ease his tax-official father’s arithmetic, built the Pascaline, a box of geared wheels that performed addition and subtraction with automatic carry: when a wheel passed nine, a mechanism advanced the next wheel. Mechanizing the carry was the conceptual core; it made the machine, not the operator, responsible for the arithmetic.

Why the Pascaline Didn’t Catch On

Pascal built around 20 machines, but they were fragile, expensive, hard to manufacture to tolerance, and could not reliably multiply. They were marvels and status objects more than working tools, a pattern that haunts mechanical computing: the idea outruns the manufacturing precision of its age.

The German polymath Gottfried Wilhelm Leibniz went further. Around 1672–1694 he designed the Stepped Reckoner, which could add, subtract, multiply, and divide. Its key invention, the Leibniz wheel (stepped drum), became the standard multiplying mechanism for two centuries. Leibniz also championed binary arithmetic (the base-2 system at the heart of every modern computer): his manuscript De progressione dyadica of 1679 even sketched a binary calculating machine in which marbles stood for ones; Ludolf von Mackensen built a working version in 1972, and the Arithmeum in Bonn another. In England, Samuel Morland’s adding instrument of 1666 had no carry; the first English machine that deserved the name was designed in 1775 by Charles Stanhope, who also built a small logic device, the “Demonstrator,” for testing syllogisms.

The Arithmometer and the Jacquard Loom

Two early-19th-century French inventions turned mechanical calculation from curiosity into industry.

Before it, the Hessian engineer Johann Helfrich Müller had demonstrated a four-function calculator in Göttingen on 24 June 1784 and used it in 1788 to compute tables for foresters, and Jacob Leupold’s Theatrum Arithmetico-Geometricum (1727) had surveyed calculating devices. Charles Xavier Thomas de Colmar, head of a Paris insurance company, patented the Arithmometer on 18 November 1820, using Leibniz’s stepped drum, and built it in series from the 1850s; in 1852 he presented ornate examples to King Ferdinand II of Portugal and Pope Pius IX. It was the first mass-produced, commercially successful mechanical calculator, reliable enough to sit on the desks of insurers and engineers for decades. Calculation had finally become a product.

The more consequential machine wove cloth. Joseph-Marie Jacquard’s (born in Lyon on 7 July 1752) loom (introduced in 1804; Napoleon watched it demonstrated in Lyon in 1805) used a chain of punched cards to control which threads lifted on each pass, automating the weaving of complex patterns. The card was the program: change the cards, change the cloth. This was the first widespread use of an exchangeable, machine-readable set of instructions, and it is the direct ancestor of the punched cards that ran Hollerith’s tabulators and, a century and a half later, mainframe computers.

The Calculator Industry

The United States became the home of the adding machine. Caroline Winter of Piqua, Ohio, received a patent for an adding machine on 12 April 1859, the first woman known to have invented a calculating device. Frank Baldwin patented a pinwheel calculator in 1875 and built ten, later working with Jay Monroe. Dorr E. Felt built his first key-driven adder from a grocer’s wooden box in November 1884, founded Felt & Tarrant in 1889, and had built some 6,500 Comptometers by 1903, while William Burroughs developed a printing adder. Europe excelled at multiplication: Léon Bollée’s direct multiplier won a gold medal at the Paris World’s Fair of 1889, and the Millionär, designed by Otto Steiger, patented in Germany in 1892 and built by Hans W. Egli near Zurich from 1899 to 1935 (4,655 by the usual count, 5,099 according to a later study by Gerald Saudan, at up to 2,772 francs in 1914), multiplied with a mechanical multiplication-table body in one turn per digit. The adding machine reached the stage on 19 March 1923, when Elmer Rice’s play The Adding Machine opened at New York’s Garrick Theatre: its bookkeeper Mr. Zero is fired after 25 years and replaced by an adding machine, kills his boss, and ends up operating a giant adding machine in heaven. Even Otto Lilienthal patented a disc device for the multiplication table, on 18 September 1888. The pinwheel machine that spread furthest came from St. Petersburg, where the Swedish engineer Willgodt Odhner built his first in 1873, patented it in several countries in 1878 and 1879, and began series production of an improved model in 1890. In 1892 he sold his German branch to Grimme, Natalis & Co. of Braunschweig, which sold the design as the Brunsviga; after the revolution Odhner’s works were nationalised and moved to Moscow, where from the late 1920s it was sold as the Felix, named after the secret-police chief Felix Dzerzhinsky, who had overseen the move; production, later in Kursk and Penza, ran until 1978, with about 300,000 made in 1969 alone.

Some machines did only one job. In 1926 the Berkeley number theorist Derrick Lehmer built a sieve from bicycle chains to find factors of numbers, followed in 1932 by a geared sieve that tested about 5,000 combinations a second and stopped when light passing through a line of aligned holes reached a photocell; the Computer History Museum holds three of his sieves. Lehmer later moved to electronic machines. As director of the Institute for Numerical Analysis in Los Angeles he directed the run in which Raphael Robinson’s program on the SWAC found the Mersenne prime 2521 − 1 on 30 January 1952, the first Mersenne prime found by an electronic computer.

German makers specialised in compactness and new mechanisms. Christel Hamann’s Gauss, prize-winner at the Paris World’s Fair of 1900 and built in series from 1905, was the first small calculating machine, 12.5 cm across, and his proportional lever of 1903 went into the Mercedes-Euklid calculators of Zella-Mehlis; Carl Kübler’s Addiator company, founded in Berlin on 13 February 1920, sold pocket slide adders worldwide. The endpoint of mechanical miniaturisation was the Curta of the Viennese engineer Curt Herzstark, who had filed his first patent in 1938 and worked out the design while a prisoner in Buchenwald; the Contina company in Liechtenstein showed it in Basel in May 1949, sold it from that November, and built about 140,000 by 1971. A German rival, the Alpina from Kaufbeuren, appeared in 1961 and vanished the same year. Electronics ended the line. Hamann’s Berlin successor firm brought out the electric Model 600 in 1968, a four-function machine with a printer and no transistors, at a time when electronic desk calculators were already on sale; its overlapping gear mechanisms jammed at the slightest input error or machining tolerance, its designer Günter Hornauer took his own life the same year, and when the works closed in 1970, 2,500 of the 5,000 machines built went from the warehouse to the scrapyard. The Japanese Calcu-pen of 1975 put a four-function calculator into a ballpoint pen.

Before electronic computers, mechanical calculators were also run by teams. From 1 February 1938 the Mathematical Tables Project, a Works Progress Administration scheme in a Manhattan office building, employed 125 out-of-work New Yorkers under Arnold Lowan and Gertrude Blanch to compute tables of functions, organised by operation; it ran until 1949.

The office that bought these machines left its own records. The Berlin statistician Paul Mayet visited the Paris World’s Fair of 1900 (14 April to 12 November, some 48 million visitors) and wrote a report for the Imperial Statistical Office on which of the calculators on show it could use. His office already owned thirteen Burroughs adders, six Burkhardt stepped-drum machines and one Brunsviga; new to him was the Arithmachine, a small chain adder from Chicago by the Vienna-born Henry Goldman, later sold as the Arithstyle and Contostyle. The jury in the analogue class gave a gold medal to the Serbian mathematician Mihailo Petrović for a calculating device that worked with liquids. A quarter-century later the first history of the trade appeared: Die Rechenmaschinen und ihre Entwicklungsgeschichte (April 1925), by “Ernst Martin”, the pen name of the office-machine journalist Johannes Meyer (1885 to 1949), who had worked for Underwood and wrote from the Franconian town of Pappenheim. It sorted the machines into stepped-drum and pinwheel calculators, printing adders with full or ten-key keyboards, and small adders, and ran chronologically from the Pascaline (Martin did not yet know of Schickard); an enlarged edition followed in 1936 and an English translation of the first edition in 1992.

The Bridge to Babbage

These threads, mechanized arithmetic from Pascal and Leibniz, mass production from Thomas, and the programmable punched card from Jacquard, converged in the mind of Charles Babbage. Explicitly inspired by the Jacquard loom, he designed the Analytical Engine (from 1837): a general-purpose mechanical computer with a separate “store” (memory) and “mill” (processor), controlled by punched cards. Ada Lovelace saw that such a machine could manipulate symbols as well as numbers, the conceptual birth of programming. Babbage’s machine was never completed, but it is the hinge where mechanical calculation becomes the idea of the computer. A generation later, the Spanish engineer Leonardo Torres Quevedo showed that the engine Babbage could not finish in brass could be built in electromagnets and relays, and even constructed a chess automaton, the first machine to decide a game on its own.

⚠️ Dead End: The Limits of Brass and Gears

Mechanical calculation hit a hard ceiling, and it is worth naming why. Cost and organisation could not keep pace with design ambition: Babbage’s engines failed for lack of money and through his constant redesigning more than for lack of machining precision, as the Science Museum’s working reconstruction of his Difference Engine No. 2 showed (see Charles Babbage). Speed and wear: gear trains are slow, and friction and backlash accumulate errors and break teeth. Scale: a truly general machine needed thousands of precise parts, making it ruinously expensive and unreliable.

The whole approach was a dead end not because the ideas were wrong (they were almost exactly right) but because the medium was wrong. Arithmetic mechanized in brass could never be fast, cheap, or reliable enough. The ideas survived; the gears did not. They had to wait for a frictionless medium with no moving parts: the electron. Relays, then vacuum tubes, then transistors would finally give Pascal’s carry and Jacquard’s program a body that could keep up with them.

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