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Chinese suanpan abacus

Calculation is an integral part of how societies function and has been used since ancient times to regulate trade and fix dimensions of land and buildings. Theoretical developments in mathematics, along with the growing complexity of calculations, inspired the design of calculating machines during the Early Modern period. These analogue devices, along with technologies developed for factory automation and advances in electronics engineering, gave rise to the first digital computers.

"[I]t is unworthy of excellent men to lose hours like slaves in the labor of calculation, which could safely be relegated to anyone else if machines were used." Gottfried Wilhelm von Leibniz, 1671.(1)

Early numeracy

Plate from Euclid's "Elements of Geometry"
Image 2: Plate 1, pg. 18, Euclid's Elements of Geometry, 1733 edition. Image © the Whipple Library.

Employed by the ancient Egyptians, Greeks, and Mesopotamians, the earliest calculating devices were systems of writing that used shorthand to denote specific and often large quantities. These written forms differed between cultures but usually involved groups of lines representing single units, with modified characters for intervals of five or ten.

Counting sticks, knots, and tally sticks - with values denoted by specific notches - were common forms of counting and numerical record-keeping throughout the world. These systems, along with the use of Roman numerals, persisted through the Renaissance, as many were hesitant to adopt the Hindu-Arabic numerals used today out of concern for accuracy and the potential for forgery.

The abacus is perhaps the most well known pre-modern calculating device, and is often associated with the wire-and-bead devices that originated in the Middle East. While its true origins remain debatable, the word abacus would have referred to an ancient practice of moving pebbles ('calculi') along lines written in sand.

A common abacus today is the Japanese 'soroban', which has one 'heavenly' bead per wire representing 5, and four 'earthly' beads representing 1 each. This is a simplification of the Chinese 'Suanpan', in which more beads per wire can accommodate other decimal systems such as duodecimal (i.e. base 12, rather than base 10) (Image 1, above).

Pure mathematics has its own history alongside that of counting. The origins of geometry, for example, stretch back to Ancient Greece, and Euclid's Elements, first compiled around 300 BCE, would become, in various forms, the standard mathematical textbook for nearly two millennia (Image 2).

Read more: how models were used to aid the teaching of Euclid's Elements.

The Early Modern period

The Early Modern period in Europe (roughly from the 15th to the 18th centuries) saw the development of many new calculating tools, as well as the revival and adaptation of several from classical and Middle Eastern culture. For instance the astrolabe, invented in antiquity and developed during the Islamic Golden Age (roughly the 7th to the 15th centuries), was a multi-purpose tool used to measure the heights of buildings and record planetary motions.

Slides, cranks, and dials

Most importantly, after 1400 CE new tools and techniques were developed for commerce, exploration, and natural philosophy, often serving multiple purposes. From the 17th century, the slide rule, for instance, became the most commonly used calculating device for nearly three hundred years. Beginning as a 'line of numbers' arranged on wood, paper, or brass, rulers attached to one another were used to align points along different scales to perform arithmetic and convert units.

A set of rods for use in multiplication
Image 3 Mr Hélies' rolling rods, c.1830s (Wh.1893)

The Early Modern period was also the dawn of the age of clockwork and automation, which inspired the design of calculating machines. Such devices came together gradually, and were easier to design than to build.
Scottish mathematician John Napier, who discovered the method of logarithms, first devised a set of rods for use in multiplication around 1614. A version of the rods in a box (Image 3) provided the template for a gear-based 'carry' mechanism to store values, enabling the first mechanical calculating devices. Blaise Pascal completed a number of such machines by the mid-17th century, and was followed by Samuel Morland and Gottfried Wilhelm von Leibniz.

The reduction of arithmetic to repeated mechanical manoeuvres influenced Johann Helfrich von Müller to conceive of a 'difference engine' that could handle more complex calculations. Müller's design, published in 1786, was intended to calculate tables of logarithms, replacing human 'computers' (that is, people employed to manually compute such tables) with an error-free machine.

Forty years later, the English polymath Charles Babbage designed and attempted to construct a similar machine, capable of not only calculating but also printing tables. Babbage was not able to complete his machine in his lifetime, but a fragment re-constructed by his son Henry in the 1870s proved that the concept could work.

Read more: a fragment of Babbage's difference engine

Into the digital

Texas Instruments TI-2500 Datamath calculator
Image 4: TI-2500 Datamath, c.1972 (Wh.4529.006A1)

The designs of Leibniz, Müller, and Babbage, which automated calculation with gears using 'registers' to store information as it was mechanically read, laid the foundation for the digital computers we have today.

Modern computers were first developed to solve mathematical problems. In the 1930s, German engineer Konrad Zuse built his third automatic mechanical calculator, the Z3, which carried out instructions read in by a program.

During World War II in the United States, John Mauchly and J. Presper Eckert built the Electronic Numerical Integrator and Computer (ENIAC), the fastest machine to date, to calculate firing tables for the military. At Bletchley Park, British codebreakers and engineers produced the world's first programmable electronic digital computer, Colosus, to aid in the cracking of German ciphers.

The first electronic computers with stored programs were also developed in the UK: the 'Baby' computer at Manchester and the Electronic Delay Storage Automatic Calculator (EDSAC) at Cambridge, which was used by many in the scientific community during the 1950s. Early computers were massive and expensive, so their applications had to be well defined and justified, with entire departments within universities and businesses devoted to them.

It may come as a surprise today, but when pocket electronic calculators were first introduced, manufacturers had to justify their expense to individual consumers by convincing him or her that they were in fact faster and more accurate than the ubiquitous slide rule.

Popular devices, such as the Texas Instruments Datamath series (Image 4), cost $150 upon their introduction, which was expensive for a device that only performed simple arithmetic. As they caught on, however, pocket calculators drove advances in microprocessor technology, making computer chips faster and less expensive. Just as importantly, pocket calculators helped show people how computers could fit easily into their daily routines.


  1. Quoted in: D. E. Smith, A source book in mathematics (New York, 1929), pp. 180-181.

Mikey McGovern

Mikey McGovern, 'A brief history of calculating devices', Explore Whipple Collections, Whipple Museum of the History of Science, University of Cambridge.

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