Babbage Designs a Mechanical Calculator Summary

  • Last updated on November 10, 2022

Charles Babbage designed an early form of the mechanical calculator that could produce accurate mathematical tables. In 1833, he completed a portion of a machine that could automatically compute and print table values. He later perfected this work and produced a primitive device that anticipated the electronic computers of the late twentieth century.

Summary of Event

Charles Babbage wanted to master numbers mechanically. His mathematical work and research required the use of the many mathematical tables of his era, but these tables were riddled with errors, and so he set out to produce tables that were more accurate. As early as 1812, Babbage began to think about the possibilities for a newly designed mechanical calculator, and in 1819, he started to work seriously on the project. Babbage, Charles Mathematics;calculating machines Inventions;calculating machines Calculating machines Clement, Joseph [kw]Babbage Designs a Mechanical Calculator (1819-1833) [kw]Designs a Mechanical Calculator, Babbage (1819-1833) [kw]Mechanical Calculator, Babbage Designs a (1819-1833) [kw]Calculator, Babbage Designs a Mechanical (1819-1833) Babbage, Charles Mathematics;calculating machines Inventions;calculating machines Calculating machines Clement, Joseph [g]Great Britain;1819-1833: Babbage Designs a Mechanical Calculator[0980] [c]Mathematics;1819-1833: Babbage Designs a Mechanical Calculator[0980] [c]Inventions;1819-1833: Babbage Designs a Mechanical Calculator[0980] [c]Engineering;1819-1833: Babbage Designs a Mechanical Calculator[0980]

Large mathematical tables on multiplication, logarithms, and other functions were used heavily at the beginning of the nineteenth century. Tables reduced the tedious work of complex calculations and provided ready references. Teams of calculators—the term for people performing simple arithmetic tasks—worked over long periods to produce results, which were then gathered and given to a typesetter, who would set the figures and print the tables. This procedure left much room for error, not only in computation but also in copying and typesetting.

Mechanical calculators had been performing additions and multiplication by means of continual additions, but the calculators were hand cranked, and their accuracy and practical use had been restricted by a lack of precision engineering. Babbage, facing the same mechanical and mathematical problems, decided to use the method of “differences” and to have his machine not only perform the computations but also set the type to avoid the human error of transferring the figures from machine to writing to print.





Table makers had used a “tool” called the method of differences when constructing tables of polynomial function values. This tool could eliminate the difficult operations of multiplication and division by replacing them with simple additions. For a polynomial function such as f(x) = 3x + 1 (see table), note the differences between each adjacent value of f(x) when evaluated for successive values of x. Thus, one finds the constant differences. In complex functions, it may be necessary to calculate the differences of differences (or second differences) before finding a constant difference. To evaluate a function for many values of x in a table, it is simpler to do so by adding the constant difference to the difference above, then adding that difference to the one above it, and so on, until reaching the function value.

In 1822, Babbage completed a small working model with spare parts in his basement. The model calculated six-digit numbers and could evaluate functions with a constant second difference. Babbage called his machine the “difference engine.” It could store a series of numbers and perform addition with those numbers. The stored numbers stood for the polynomial value, its first difference, second difference, and so on. The machine added each difference to the next higher one until it reached the polynomial value. In using the method of differences for his machine, Babbage needed only to mechanically replicate addition, not multiplication.

Charles Babbage.

(Library of Congress)

While Babbage had managed to build the working model on his own, a full-size engine required the development of more precision engineering, which would be costly. Using his model, he asked the Royal Society for help with petitioning the government for funds to build a full-scale machine. In 1823, he received £1,500 and agreed to put up £3,000 to £5,000 of his own funds. He hired the machinist Joseph Clement to help develop new techniques and depended on him for all the practical mechanical work. Babbage designed the parts for the difference engine, and he and Clement designed and created tools to make those parts. The team followed a pattern of design, toolmaking, and redesign, a process that created great advances in the toolmaking trade. Clement trained numerous machinists in his workshop and disseminated new methods throughout the industry in Great Britain.

Babbage had expected the government to refund his portion of the research and development costs upon completion. However, that part of the bargain was forgotten by 1827. Through pressure from some of Babbage’s friends, the government eventually gave him some additional funds. A pattern of work stoppages began, however, when Babbage’s personal funds ran out. He had to let go of trained personnel. When he did receive additional funds, he rehired and often had to retrain the same workers.

Babbage’s work also was delayed when he and Clement broke off their association. Clement had established a successful machinist shop and had many customers besides Babbage. Babbage had expected Clement to move his shop, and the engine, to a location near Babbage’s home, but Clement declined. The ownership of the tools that they had designed for the construction of the machine belonged to Clement, and he took those tools and the designs with him.

During the work stoppage after Clement’s departure, Babbage conceived an idea for an “analytical engine.” His design included the prescient use of punched cards to control or program the machine. He adapted the programming (control mechanism) on Jacquard looms to mechanical mathematical calculation. Technically, Babbage’s system was identical to that of Jacquard—punched cards to control the action of small, narrow, circular metal rods that in turn governed the actions required to weave or calculate (by controlling positions of cogwheels). Babbage designed this engine with five basic parts—the store, mill, control, input, and output—which remained the basic units found in electronic computers of one century later.

Babbage tried to persuade the government that it would take less time and money to construct the new design than to finish the old one. They debated his proposal for more than eight years, and eventually the government told Babbage no more money would be forthcoming.

The government had paid a total of £17,000 while Babbage had contributed £20,000. For this financial outlay, the government received a portion of a larger machine in 1833 that could calculate tables for functions with two or three orders of difference; what Babbage delivered was a crude form of the more modern calculator. A full machine, if built, would have been 10 feet high, 10 feet wide, and 5 feet deep, and would have seven vertical steel axles, each of which would have eight brass wheels. The axles each represented one of six orders of differences, with the seventh representing the value of the function. Each of the wheels would have the digits 0 to 9 engraved around the circumference, and the wheel’s position would designate which number it represented.

Babbage had designed his machine to be as error-free as possible. The machine would include mechanisms to handle problems such as wheels getting out of place. The printing capability of this machine set it apart from earlier machines. The mechanism created a stereotyped plate for the printing presses and removed human typesetting errors.


Charles Babbage wrote up plans for an improved difference engine in 1849 and attempted to get the government to construct it; the government refused. In his design work for a second difference engine (Difference Engine No. 2) and for the analytical engine, Babbage was far ahead of his time. His designs included many of the elements of the computers built one hundred years later. Yet, the development of the machines meant frustration, financial loss, and ultimate failure on a personal level.

The main impact of Babbage’s difference engine was perhaps indirect. In the process of designing his mechanical calculator, his demands and his work with Joseph Clement set new, higher standards of precision for the mechanical engineering field. Standards of toolmaking advanced greatly through the work of Clement and his apprentices, which included the toolmaking industry pioneer Joseph Whitworth. Furthermore, Babbage’s struggle with the British government between 1823 and 1848 over funds set an example for future relations between scientific researchers and government finance.

Further Reading
  • citation-type="booksimple"

    xlink:type="simple">Ashworth, William J. “Charles Babbage, John Herschel, and the Industrial Mind.” Isis 87 (1996): 629-653. Analyzes Babbage’s mathematical thought processes.
  • citation-type="booksimple"

    xlink:type="simple">Babbage, Charles. Passages from the Life of a Philosopher. Edited by Martin Campbell-Kelly. Piscataway, N.J.: IEEE Press, 1994. Babbage’s personal reminiscences regarding the design and development of the mechanical calculators and other projects.
  • citation-type="booksimple"

    xlink:type="simple">Babbage, Henry P., ed. Babbage’s Calculating Engines: A Collection of Papers Relating to Them—Their History, and Construction. Los Angeles: Tomash, 1982. Facsimile reprint of 1889 edition published in London, with a new introduction by Allan G. Bromley. Contains thirty-three items, most of them assembled by Charles Babbage before his death. Additions and editing provided by his youngest son.
  • citation-type="booksimple"

    xlink:type="simple">Collier, Bruce. The Little Engines That Could’ve: The Calculating Machines of Charles Babbage. New York: Garland, 1990. A good account of Babbage’s struggles to develop his calculator.
  • citation-type="booksimple"

    xlink:type="simple">Hyman, Anthony. Charles Babbage: Pioneer of the Computer. Princeton, N.J.: Princeton University Press, 1982. A standard biography with a rich account of Babbage’s life and times.
  • citation-type="booksimple"

    xlink:type="simple">Lindgren, Michael. Glory and Failure: The Difference Engines of Johann Muller, Charles Babbage and Georg and Edvard Scheutz. Cambridge, Mass.: MIT Press, 1990. Provides a detailed exploration of similar work being done in the field during Babbage’s time.
  • citation-type="booksimple"

    xlink:type="simple">Swade, Doron. The Difference Engine: Charles Babbage and the Quest to Build the First Computer. New York: Viking Press, 2001. A short and readable account of Babbage’s work on the difference engine and on the attempt to build his design in the late twentieth century.

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