Differential analyser

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This article is about analogue differential analysers. For the digital implementation, see Digital Differential Analyzer.

The differential analyser is a mechanical analogue computer designed to solve differential equations by integration, using wheel-and-disc mechanisms to perform the integration.[1] It was one of the first advanced computing devices to be used operationally.[2] The original machines could not add, but then it was noticed that if you turn the two wheels of a rear differential, the drive shaft will compute the average of the left and right wheels. A simple gear ratio of 1:2 then enables you to multiply by two, so addition (and subtraction) are achieved. Multiplication is just a special case of integration, namely integrating a constant function.[3]


Kay McNulty, Alyse Snyder, and Sis Stump operate the differential analyser in the basement of the Moore School of Electrical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania, c. 1942–1945.
A differential analyser at the NACA Lewis Flight Propulsion Laboratory, 1951

Research on solutions for differential equations using mechanical devices, discounting planimeters, started at least as early as 1836, when the French physicist Gaspard-Gustave Coriolis designed a mechanical device to integrate differential equations of the first order.[4]

The first description of a device which could integrate differential equations of any order was published in 1876 by James Thomson, who was born in Belfast in 1822, but lived in Scotland from the age of 10.[5] Though Thomson called his device an "integrating machine", it is his description of the device, together with the additional publication in 1876 of two further descriptions by his younger brother, Lord Kelvin, which represents the invention of the differential analyser.[6]

One of the earliest practical uses of Thomson's concepts was a tide-predicting machine built by Kelvin starting in 1872-3. On Lord Kelvin's advice, Thomson's integrating machine was later incorporated into a fire-control system for naval gunnery being developed by Arthur Pollen, resulting in an electrically driven, mechanical analogue computer, which was completed by about 1912.[7] Italian mathematician Ernesto Pascal also developed integraphs for the mechanical integration of differential equations and published details in 1914.[8]

However, the first widely practical general purpose differential analyser was constructed by Harold Locke Hazen and Vannevar Bush at MIT, 1928–1931, comprising six mechanical integrators.[9][10][11] In the same year, Bush described this machine in a journal article as a "continuous integraph".[12] When he published a further article on the device in 1931, he called it a "differential analyzer".[13] In this article, Bush stated that "[the] present device incorporates the same basic idea of interconnection of integrating units as did [Lord Kelvin's]. In detail, however, there is little resemblance to the earlier model." According to his 1970 autobiography, Bush was "unaware of Kelvin’s work until after the first differential analyzer was operational."[14] Claude Shannon was hired as a research assistant in 1936 to run the differential analyzer in Bush's lab.[15]

Douglas Hartree of Manchester University brought Bush's design to England, where he constructed his first "proof of concept" model with his student, Arthur Porter, during 1934: as a result of this, the university acquired a full scale machine incorporating four mechanical integrators in March 1935, which was built by Metropolitan-Vickers, and was, according to Hartree, "[the] first machine of its kind in operation outside the United States".[16] During the next five years three more were added, at Cambridge University, Queen's University Belfast, and the Royal Aircraft Establishment in Farnborough.[17] One of the integrators from this proof of concept is on display in the History of Computing section of the Science Museum (London) alongside a complete Manchester machine.

In Norway, the locally built Oslo Analyser was finished during 1938, based on the same principles as the MIT machine. This machine had 12 integrators, and was the largest analyser built for a period of four years.[18]

In the United States, further differential analysers were built at the Ballistic Research Laboratory in Maryland and in the basement of the Moore School of Electrical Engineering at the University of Pennsylvania during the early 1940s.[19] The latter was used extensively in the computation of artillery firing tables prior to the invention of the ENIAC, which, in many ways, was modelled on the differential analyser.[20] Also in the early 1940s, with Samuel H. Caldwell, one of the initial contributors during the early 1930s, Bush attempted an electrical, rather than mechanical, variation, but the digital computer built elsewhere had much greater promise and the project ceased.[21] In 1947, UCLA installed a differential analyser built for them by General Electric at a cost of $125,000.[22] By 1950, this machine had been joined by three more.[23]

Early computer-and-plotter dating to 1944, solving complex equations again 70 years after.[24]

At Osaka Imperial University (present-day Osaka University) around 1944, a complete differential analyser machine was developed (illustrated) to calculate the movement of an object and other problems with mechanical components, and then draws graphs on paper with a pen. It was later transferred to the Tokyo University of Science and has been displayed at the school’s Museum of Science in Shinjuku Ward. Restored in 2014 is one of only two still operational differential analyzers produced before the end of World War II.[24]

In Canada, a differential analyser was constructed at the University of Toronto in 1948 by Beatrice Helen Worsley, but it appears to have had little or no use.[25]

A differential analyser may have been used in the development of the bouncing bomb, used to attack German hydroelectric dams during World War II.[26] Differential analysers have also been used in the calculation of soil erosion by river control authorities.[27]

The differential analyser was eventually rendered obsolete by electronic analogue computers and, later, digital computers.

Use of Meccano[edit]

MOTAT's Meccano differential analyser in use at the Cambridge University Mathematics Laboratory, c. 1937. The person on the right is Dr Maurice Wilkes, who was in charge of it at the time

The model differential analyser built at Manchester University in 1934 by Douglas Hartree and Arthur Porter made extensive use of Meccano parts: this meant that the machine was cheaper to build, and it proved "accurate enough for the solution of many scientific problems".[28] A similar machine built by J.B. Bratt at Cambridge University in 1935 is now in the Museum of Transport and Technology (MOTAT) collection in Auckland, New Zealand.[28] A memorandum written for the British military's Armament Research Department in 1944 describes how this machine had been modified during World War II for improved reliability and enhanced capability, and identifies its wartime applications as including research on the flow of heat, explosive detonations, and simulations of transmission lines.[29]

In 1948, this machine was bought by Professor Harry Whale of Auckland, for 100 pounds sterling, and he then took it to Auckland for use at the Seagrave Radio Research Centre.[30]

It is estimated that "about 15 Meccano model Differential Analysers were built for serious work by scientists and researchers around the world".[31] More recently, building differential analysers with Meccano parts has become a popular project among serious Meccano hobbyists. An example is the differential analyser built at Marshall University, which is now used for educational purposes, in that a student not only solves a differential equation but also becomes the "calculator" by operating the machine, and so develops a better understanding of what a differential equation is.[32]

See also[edit]


  1. ^ Irwin, William (July 2009). "The Differential Analyser Explained". Auckland Meccano Guild. Retrieved 2010-07-21. 
  2. ^ "Invention of the modern computer". Encyclopædia Britannica. www.britannica.com. Retrieved 2010-07-26. 
  3. ^ John von Neumann, The Computer and the Brain, Part 1, p.3, Yale University Press, The Silliman Memorial Lectures Series, 1958
  4. ^ Coriolis, Gaspard-Gustave (1836). "Note sur un moyen de tracer des courbes données par des équations différentielles". Journal de Mathématiques Pures et Appliquées. series I 1 (in French): 5–9. 
  5. ^ Thomson, James (1876). "An Integrating Machine having a new Kinematic Principle". Proceedings of the Royal Society. 24 (164–170): 262–5. doi:10.1098/rspl.1875.0033.  Reprinted in Thomson, James (1912). Joseph Larmor & James Thomson, ed. Collected Papers in Physics and Engineering by James Thomson. Cambridge University Press. pp. xvii, 452–7. ISBN 0-404-06422-1. 
  6. ^ Hartree, D.R. (September 1940). "The Bush Differential Analyser and its Implications". Nature. 146 (3697): 319. doi:10.1038/146319a0. . Lord Kelvin's descriptions: Thomson, William (1876). "Mechanical Integration of Linear Differential Equations of the Second Order with Variable Coefficients". Proceedings of the Royal Society. 24 (164–170): 269–71. doi:10.1098/rspl.1875.0035.  Thomson, William (1876). "Mechanical Integration of the general Linear Differential Equation of any Order with Variable Coefficients". Proceedings of the Royal Society. 24 (164–170): 271–5. doi:10.1098/rspl.1875.0036. 
  7. ^ Pollen, Anthony (1980). The Great Gunnery Scandal – The Mystery of Jutland. Collins. p. 23. ISBN 0-00-216298-9. 
  8. ^ Pascal, Ernesto (1914). Miei Integrafi per Equazioni Differenziali (in Italian). Naples: B. Pellerano.  See also Integraph.
  9. ^ Karl L. Wildes and Nilo A. Lindgren, A Century of Electrical Engineering and Computer Science at MIT, 1882-1982 (Cambridge, Massachusetts: MIT Press, 1985), pages 90-92.
  10. ^ Robinson, Tim (June 2005). "The Meccano Set Computers A history of differential analyzers made from children's toys". IEEE Control Systems Magazine. 25 (3): 74–83. doi:10.1109/MCS.2005.1432602. . Hartree, D.R. (September 1940), op. cit.
  11. ^ Bush's differential analyser used mechanical integrators. The output of each integrator was intended to drive other parts of the machine; however, the output was too feeble to do so. Hazen recognized that a "torque amplifier", which had been invented in 1925 by Henry W. Nieman and which was intended to allow workers to control heavy machinery, could be used to provide the necessary power. See: Stuart Bennett, A History of Control Engineering 1930-1955 (London, England: Peter Peregrinus Ltd., 1993), page 103. See also Nieman's U.S. patents: (1) "Servo mechanism", U.S. patent no. 1,751,645 (filed: 28 January 1925; issued: 25 March 1930) ; (2) "Servo mechanism", U.S. patent no. 1,751,647 (filed: 8 January 1926; issued: 25 March 1930); (3) "Synchronous amplifying control mechanism", U.S. patent no. 1,751,652 (filed: 8 January 1926; issued: 25 March 1930).
  12. ^ Bush, V.; Gage, F.D.; Stewart, H.R. (January 1927). "A continuous integraph". Journal of the Franklin Institute. 203 (1): 63–84. doi:10.1016/S0016-0032(27)90097-0. .
  13. ^ Bush, V. (October 1931). "The differential analyzer. A new machine for solving differential equations". Journal of the Franklin Institute. 212 (4): 447–488. doi:10.1016/S0016-0032(31)90616-9. .
  14. ^ Robinson, Tim (June 2005), op. cit., citing Bush, Vannevar (1970). "Pieces of the Action". New York NY: Morrow. .
  15. ^ Gleick, James (2011). The Information: A History, a Theory, a Flood (ebook). Patheon. p. 342/1102. ISBN 978-0-00-742311-8. 
  16. ^ Robinson, Tim (June 2005), op. cit., Hartree, D.R. (September 1940), op. cit. Hartree and Porter wrote about the model in their paper "The Construction and Operation of a Model Differential Analyser". Memoirs and Proceedings of the Manchester Literary & Philosophical Society. 79: 51–74. 1935. .
  17. ^ Robinson, Tim (2005-12-07). "Other Differential Analyzers". Tim Robinson's Meccano Computing Machinery web site. Retrieved 2010-07-24.  Includes summaries of "Meccano Differential Analyzers" and "Full Scale Differential Analyzers".
  18. ^ Holst, P.A. (Oct–Dec 1996). "Svein Rosseland and the Oslo analyzer". IEEE Annals of the History of Computing. 18 (4): 16–26. doi:10.1109/85.539912. 
  19. ^ Randell, Brian (ed.), The Origins of Digital Computers Selected Papers (3rd edition, 1982), Berlin, Heidelberg, New York: Springer-Verlag. p. 297. Google Books. Retrieved 25 July 2010.
  20. ^ Bunch, B. & Hellemans, A., The History of Science and Technology: A Browser's Guide to the Great Discoveries, Inventions, and the People who Made Them, from the Dawn of Time to Today (2004), New York: Houghton Mifflin, p. 535. Google Books. Retrieved 25 July 2010.
  21. ^ Randell, Brian (Oct 1982). "From Analytical Engine to Electronic Digital Computer: The Contributions of Ludgate, Torres, and Bush" (PDF). IEEE Annals of the History of Computing. IEEE Computer Society. 4 (4): 327–41. doi:10.1109/MAHC.1982.10042. Retrieved 2010-07-25. 
  22. ^ "UCLA's Bush Analyzer Retires to Smithsonian" (Google News). Computerworld. 1978-01-09. Retrieved 2010-07-22. 
  23. ^ "The Thinking Machine". UCLA Engineering. Archived from the original on 2010-07-10. Retrieved 2010-07-22. 
  24. ^ a b HISATOSHI KABATA (2014), "Early computer dating to 1944 solving complex equations again after long 'reboot'", The Asahi Shimbun/Technology.
  25. ^ Campbell, Scott M. (2003, Oct.-Dec.). "Beatrice Helen Worsley: Canada's Female Computer Pioneer" (PDF). IEEE Annals of the History of Computing. IEEE Computer Society. 25 (4): 53–4. doi:10.1109/MAHC.2003.1253890. Retrieved 2010-07-24. [Worsley's] research was suggested by Samuel H. Caldwell, of MIT’s electrical engineering department, who had helped Vannevar Bush design recent analyzers. … Over six weeks during summer 1948, Worsley constructed a differential analyzer using Meccano…, based on Douglas Hartree and Arthur Porter’s 1935 article. Constructed from about CAD$75 worth of Meccano, the analyzer was minimally modified from the original design but offered slight improvements to the electrical power distribution system, the design of the torque amplifiers, and the output pen support. Unfortunately, there is no information regarding what use, if any, the analyzer was put to or why Worsley built it  Check date values in: |date= (help) For more on Beatrice Worsley, see UTEC.
  26. ^ Irwin, William (2009-07). Op. cit. "It is rumoured that a differential analyser was used in the development of the "bouncing bomb" by Barnes Wallis for the "Dam Busters" attack on the Ruhr valley hydroelectric dams in WW2. This was first mentioned in MOTAT [New Zealand] literature in 1973. However after extensive enquiries and literature searches over the last few years, no evidence can be found that the [differential analyser held by MOTAT, nor any other differential analyser, was used for this purpose. Considering the secrecy surrounding war time activities at the time it could still be possible, but most people from that era are now deceased. Two remaining personalities still alive from that era were consulted, namely Arthur Porter and Maurice Wilkes, but neither could substantiate the rumour."
  27. ^ Hally, Mike (2005), Electronic Brains: Stories from the Dawn of the Computer Age, Granta, p. xx, ISBN 9781862076631 .
  28. ^ a b (Hartree & Porter 1934–1935),"Differential Analyser". Auckland Meccano Guild. Retrieved 2010-07-21. 
  29. ^ Cairns, W. J., Crank, J., & Lloyd, E. C. Some Improvements in the Construction of a Small Scale Differential Analyser and a Review of Recent Applications, Armament Research Department Theoretical Research Memo. No. 27/44, 1944 (see Robinson, Tim (2008-06-07). "Bibliography". Tim Robinson's Meccano Computing Machinery web site. Retrieved 2010-07-26. ). The memorandum is now in The National Archives, UK: "Piece reference DEFE 15/751". The National Archives. Retrieved 2010-07-26.  For the "Armament Research Department", see Fort Halstead, and cf. the entry for 1944 in "MoD History of Innovation" (pdf). Ploughshare Innovations Ltd. Retrieved 2010-07-26. 
  30. ^ Robinson, Tim (June 2005), op. cit.: Robinson states that "[Prof. Whale] had earlier used the Meccano model in his Ph.D. studies". "Differential Analyser". NZMuseums. Retrieved 2010-07-21.  "Meccano 'Dam Busters' computer stars at MOTAT". computerworld.co.nz. 2007-07-16. Retrieved 2007-07-17.  (NB: Computerworld (NZ)'s references to "Dam Busters" and "Benmore Hydro Dam" are unsubstantiated) "2.2.Analogue computation" (php). Computing History Displays: Computer History Time Line - Precursors of Computers. University of Auckland Department of Computer Science. Retrieved 2010-07-25. 
  31. ^ Irwin, William (2009-07). Op. cit. "It is estimated by Garry Tee of Auckland University that about 15 Meccano model Differential Analysers were built for serious work by scientists and researchers around the world." For Garry Tee, see "Computing History Displays: The Displays" (php). University of Auckland. Retrieved 2010-07-22. 
  32. ^ "MU Differential Analyzer Grand Opening". Marshall University. Archived from the original on 2010-06-06. Retrieved 2010-07-22. 


  • Thomson, James (1876). "An Integrating Machine having a new Kinematic Principle". Proceedings of the Royal Society. 24 (164–170): 262–5. doi:10.1098/rspl.1875.0033. 
  • Thomson, William (1876). "Mechanical Integration of Linear Differential Equations of the Second Order with Variable Coefficients". Proceedings of the Royal Society. 24 (164–170): 269–71. doi:10.1098/rspl.1875.0035. 
  • Thomson, William (1876). "Mechanical Integration of the general Linear Differential Equation of any Order with Variable Coefficients". Proceedings of the Royal Society. 24 (164–170): 271–5. doi:10.1098/rspl.1875.0036. 
  • Bush, Vannevar (1936). "Instrumental analysis". Bulletin of the American Mathematical Society. 42 (10): 649–69. doi:10.1090/S0002-9904-1936-06390-1. 
  • Hartree, D. R.; Porter, Porter (1934–1935), "The construction and operation of a model differential analyser", Memoirs and Proceedings of the Manchester Literary and Philosophical Society, 79: 51–73, reprinted as a pamphlet July 1935 
  • Worsley, Beatrice Helen (1947). A mathematical survey of computing devices with an appendix on an error analysis of differential analyzers (Master's Thesis, MIT).
  • Crank, J. (1947). The Differential Analyser, London: Longmans, Green (this is the only book that describes how to set up and operate a mechanical differential analyser).
  • MacNee, A.B. (1948). An electronic differential analyzer (RLE, Technical Report 90, MIT. Note that this paper describes a very early electronic analogue computer, not a mechanical differential analyser: it is included because the author clearly felt that the only way to introduce such an innovation was to describe it as an "electronic differential analyser").

External links[edit]