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Differential Analyzers
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Gathered below are a collection of references to a selection of material relating to Differential Analyzers. This is a work in progress. If you have references not included here, I would be pleased to hear from you.
  



Differential Analyzer Bibliography Table
Adler, M., "Meccano Torque Amplifier", 2001 Online description of a demonstration torque amplifier using standard Meccano parts.
Amble, O., "On a principle of Connexion for Bush Integrators," Journal of Scientific Instruments, 23, 284-287, 1946. A survey of regenerative connections of one or two integrators, by which it is possible to obtain functions such as the logarithm, square root, or any rational power.
Anon., "Sir William Thomson's Harmonic Analyser," Engineering, 30, 561, 1880. A description of a seven integrator version of William Thomson's harmonic analyser (see Thomson (1878)) constructed by R. W. Munro for the Meteorological Office. Contains a detailed engraving.
Anon., "With the Editor. Meccano Aids Scientific Research," Meccano Magazine, XIX, 6, 441, 1934. A one page introduction to a following article Anon. (1934B) discussing the Manchester Meccano differential analyzer and Bush's prototype at MIT.
Anon., "Machine Solves Mathematical Problems. A Wonderful Meccano Mechanism," Meccano Magazine, XIX, 6, 442-444, 1934. A good article, with many excellent pictures of both the Manchester Meccano differential analyzer and Bush's prototype at MIT. A short extract from this is reproduced in Wright (1978).
Anon., "Servo-Mechanism Demonstrated Today by Professor Harold L. Hazen," The MIT Tech, LIV, 9, 1, 1934. Describes a servo mechanism designed to follow a curve on the differential analyzer input table.  Can be found online here: P1, P4.
Anon., "A Diligent Machine," The Manchester Guardian, 25 Jan 1934, p 8. A curious short column commenting on the report Anon (1934E) later in the same issue.
Anon., "A Calculating Machine Working by Curves for Manchester University," The Manchester Guardian, 25 Jan 1934, pp 9-10. More than a full column devoted to the full scale analyzer being built for Manchester University, describing in layman's terms the kinds of applications it would be used for.
Anon. "Differential Analyser at Manchester University," Engineering, 140, 3268, 88-92, 1935. A description of the Metropolitan-Vickers machine at Manchester. Includes good pictures including the digital revolution counters and time interval camera.
Anon., "Differential Analyser for the University of Manchester," Nature, 135, 535, 1935. A brief column reporting on the opening ceremony for the Manchester Metropolitan-Vickers machine.
Anon., "The Differential Analyser," The Engineer, 160, 4149, July 19 & 26, 1935. An excellent two part article describing the mechanical principles and construction of the differential analyzer.
Anon. "The Differential Analyser in Electrical Engineering," Nature, 143, 36, 1939. A review of the two papers Hartree (1938C) and Hartree (1938D).
Anon., "The Electro-Mechanical Brain - MIT.'s Differential Analyzer Advances Science by Freeing it from Pick-and-Shovel Work of Mathematics," LIFE, Jan 14, 1946, pp 73-76. A brief article with outstanding pictures describing the Rockefeller differential analyzer at MIT.
Anon., Annual Report of the University of Malaya 1949-50, pp 44-45. Contains a brief mention of Prof. J. C. Cooke's Meccano differential analyzer; specifically that it then had three integrators, with a fourth planned.
Anon., "A Meccano Calculating Machine. Solving Complex Mathematical Equations," Meccano Magazine, XXXVI, 1, 11, 1951. A one page article describing the Meccano differential analyzer built by Prof. J. C. Cooke at the University of Malaya, Singapore.
Anon., "Mechanical Differential Analyser with SKF Bearings," The SKF Ball Bearing Journal,  #3,  67-72, 1953. Description of the Chalmers University DA, with excellent pictures. This machine is unusual in having the interconnect oriented vertically with integrators hanging below.  The integrator design is also unconventional.
Anon., "Differential Analyser," The GMM Series of Modern Supermodels No. 4, London: The Chief Meccanoman, 1967. The first set of model building instructions published for a Meccano model of Bush's differential analyzer.
Anon., "Pioneer Computer Goes To Washington," USC University Bulletin, 26, 13, Jan 23, 1978.
 Reports on the dismantling of the UCLA differential analyzer.  It was sent to the Smithsonian, where it remains in storage.
Anon. "Among the Model Builders," Meccano Magazine, 58, 3, 54-55, 1973. Reports on the rediscovery of the Cambridge Meccano differential analyzer in New Zealand at MOTAT. Includes a picture reproduced from the New Zealand Herald (Anon (1973B)). The text is somewhat confused as to the provenance of this machine.
Anon., "Toy Used to Build 'Brain Box' in 1930s," New Zealand Herald, 2 June, 1973. A brief report on the rediscovery of the Cambridge Meccano differential analyzer and its installation at MOTAT. Includes a good picture with Dr. Whale, which is reproduced in Anon (1973).
Anon., "Computer Display," Museum News, MOTAT, Sep. 1973. A brief report on the then new computer display at MOTAT, featuring the Cambridge Meccano differential analyzer.
Anon., "Among the Model Builders," Meccano Magazine, 59, 2,  36, 1974. A follow up article reproducing a letter from a Mr. Barcroft who was a laboratory assistant working under the direction of A Porter operating a Meccano differential analyzer at the Air Defence and Research Establishment, Malvern in 1942.  It is unclear just which machine he refers to.
Anon. "New Computer Display,", Museum News, MOTAT, Sep. 1981. A brief report on the recently updated computer display at MOTAT.  The article states that at this time the Meccano differential analyzer "is still capable of performing calculus."
Anon., "That Which was Lost has been Found," New Zealand Federation of Meccano Modellers Magazine, 17, 3, 1993. The editorial reproduces a picture from the New Zealand Herald of the Cambridge Meccano differential analyzer after it was rediscovered at MOTAT. There are brief notes plus a reproduction of another article from The Dominion of June 23, 1993.
Anon., "Meccano Differential Analyser No.2," New Zealand Federation of Meccano Modellers Magazine, 25, 1, 2001. Although no author is identified, these notes are edited from Tee (1993). See also Irwin (2001).
Ashurst, F. G., Pioneers of Computing. London: Frederick Muller, 1983. An excellent collection of short biographies. Chapter 7 is devoted to Vannevar Bush and the differential analyzers.
Asprey, W. (Editor), Computing Before Computers. Ames, Iowa: Iowa State University Press, 1990. Chapter five "Analog Computing Devices" contributed by Alan G. Bromley provides a good general survey of analog computation devices, from early planimiters, Kelvin's harmonic analyzer, differential analyzers, to modern electronic analog techniques. The full text is available online here.
Barton, J. C., Campbell, D. A., and Read, R. C., "An Analog Method for Studying Multiple Scattering," Proceedings of the Physical Society, LXX, 8A, 605-614, 1957. This paper describes a simulation of multiple scattering conducted using a source of random motion "coupled to a differential analyser, built in Meccano, which is similar to one built by Hartree and Porter (1935)."
Beard, R. E. "The Differential Analyser" Royal College of Science Journal, 12, 127-138, 1942. The text of a lecture delivered before the Society on 24 February 1942, describing the basic principles of the differential analyzer and including a picture of Beard's own machine.
Beard, R. E., "The Construction of a Small Scale Differential Analyser and its Application to the Calculation of Actuarial Functions," Journal of the Institute of Actuaries. LXXI (part II), 193-227, 1942. A brief description of the machine, with a more extensive discussion on its application to actuarial work. A summary of the ensuing discussion after the paper was presented is also included.
Berends, T., "Historic Machine found by MOTAT," New Zealand Herald, 29 June, 1993. A report on the rediscovery of the Meccano differential analyzer at MOTAT.  Pictured with the machine are Dr. H. Whale and R. Dearing, director of the museum, who states that it will be restored and displayed  in a prime position in the museum.
Berry, T. M., "Polarized-Light Servo System," AIEE Transactions, 63, 4, 195-198, 1944. Description of the photo electric follower system used on the General Electric differential analyzer (see Kuehni (1944)).
Blackett, P. M. S. and Williams, F. C., "An Automatic Curve Follower for the Differential Analyser," Proceedings of the Cambridge Philosophical Society, 35, 494-505, 1939. This design uses a photoelectric slope detector, a mechanical arrangement to generate the tangent of an angle, plus an integrator to follow the slope of the curve more smoothly than with the simple servo arrangement of Hazen (1936).
Boerdijk, Ir. A. H., "Constructive Use of Friction in Torque Amplifiers and Constant Torque Devices," Constructor Quarterly, 4, 24-25, June 1989. Describes the principles of the torque amplifier and related torque limiting devices.
Bowles, M. D., "U.S. Technological Enthusiasm and British Technological Skepticism in the Age of the Analog Brain," IEEE Annals of the History of Computing, 18, 4, 5-15, 1996.

 This article is a comparative analysis of the British and U.S. differential analyzers from 1930 to 1945. The author examines the development of the Bush and Hartree analyzers in the context of the U.S. engineering community and the British scientific community. Includes many interesting details.
Bückner, H., The Differential Analyser, Gõttingen, undated.
 This monograph develops a mathematical theory of set-ups with several free inputs both from a topological and an analytical point of view using the theory of Pfaffian systems.  It was developed independently of Shannon (1941).  It is very remote from the practicalities of actual machine operation.
Bush, V., "The Differential Analyzer. A New Machine for Solving Differential Equations," Journal of the Franklin Institute, 212, 447-488, 1931. Bush's original paper giving a detailed account of the first differential analyzer built at MIT in 1930.
Bush, V., and Caldwell, S. H., "A New Type of Differential Analyzer," Journal of the Franklin Institute, 240, 255, 1945. A lengthy paper describing the very large scale second generation machine at MIT (the Rockefeller DA). This system made extensive use of shaft angle encoders and servo motors, allowing the mechanical integrators to be interconnected electrically through a matrix of telephone switching relays rather than though mechanical shafting, and programmed from punched paper tape.
Bush, V., Pieces of the Action, New York: William Morrow and Company, 1970. Bush's autobiography.  Contains only brief mention of the differential analyzer work.
Cairns, W. J., Crank, J., and 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. UK National Archives reference DEFE 15/751  C20779. Describes improvements made to the Cambridge model to enhance reliability and usability: stronger output arms in the torque amplifiers, lighter integrator discs, clutches in the lead screw drives, and a double input table similar to Hartree's for time lag problems. Applications described include problems in heat flow, explosive detonations, and transmission line simulations.
Campbell, S. M. "Beatrice Helen Worsley: Canada's Female Computer Pioneer," IEEE Annals of the History of Computing, 25, 4, 51-62, 2003. A short biography of Beatrice Worsley. Includes a brief mention of the Meccano differential analyzer she built over a 6 week period in the summer of 1948.
Cook, A. C., and Maginniss, F., J., "More Differential Analyzer Applications." General Electric Review, 52, 8, 14-20, 1949. This paper is a follow up to Maginniss (1945), reporting on eight more engineering applications of the General Electric analyzer. The front cover of this issue has an excellent picture of the analyzer.
Cook, A. C., "Special Devices Aid Differential Analyzer Solution of Complex Problems," Transactions of AIEE, 69, 1365-1370, 1950.
 Describes a range of special devices added to the GE differential analyzer. These incluse a curve follower, vector summation device, sinusoid generator, multiplier, and others.
Cossons, N. (ed.), Making of the Modern World, London: John Murray, 1992.
 An illustrated volume highlighting 100 key inventions from the collections of the London Science Museum. Includes a beautiful picture of Hartree's full scale machine.
Cresswell, J., MOTAT: Museum of Transport and Technology of New Zealand (Inc.), Auckland, New Zealand: Hamlyn, 1976. Almost too late, a small group of enthusiasts joined together in an attempt to preserve the remaining relics of New Zealand's transport and engineering history. The result of their enterprise is the subject of this book. The Meccano differential analyzer is discussed on p102.
Croarken, M., Early Scientific Computing in Britain, Oxford: Oxford Science Publications, 1990. Chapter 5 is devoted to a discussion of the Manchester and Cambridge differential analyzers
Croarken, M., "The Emergence of Computing Science Research and Teaching at Cambridge, 1936-1949," IEEE Annals of the History of Computing, 14, 4, 10-15, 1992.
 This article describes the motivation behind the creation of the laboratory.  It covers the period during which both the model and full scale differential analyzers were installed and operated.
Croarken, M., "Computing in Britain During World War II," IEE History of Technology Summer Meeting 6th July 2002, London, 2002. Includes a discussion of the use of the Manchester and Cambridge differential analyzers for military calculations. Includes a very extensive set of references.
Crank, J., The Differential Analyser, London: Longmans, 1947. An excellent introduction by the person in charge of operation of the full scale differential analyzer in the Mathematical Laboratory at the University of Cambridge. Includes many pictures and diagrams and a fairly detailed account of the construction of the Meccano machines.
Cundy, H. M., and Rollett, A. P,. Mathematical Models, Oxford: Oxford University Press, 1961. Brief description of the principles of Bush's differential analyzer.  Mentions the possibility of Meccano construction.
Dalton, J., "Continuing the Saga of the Differential Analyser," Meccanoman's Newsmag, #68, 1994. Reports on an interview with Maurice Wilkes on the early history of the Cambridge Meccano differential analyzer.
Darwin, C. G., "Douglas Rayner Hartree 1897-1958," Biographical Memoirs of Fellows of the Royal Society, 4, 103-116, 1958. An excellent short biography of Hartree. Includes a bibliography listing his published works.
Eames, C., and Eames, R. A., Computer Perspective, Cambridge, MA: Harvard University Press, 1973. This book is based on an exhibition conceived and assembled for IBM, displaying aspects of the intellectual and socio-economic environments in the sixty years leading up to the modern computer.  Page 119, titled "Meccano and Quantum Mechanics", is devoted to the Manchester Meccano differential analyzer.
Eyres, N. R., "Meccano in the Classroom," Mathematical Gazette, 54, 389, 282-283, Oct 1970.
 A short article describing a two integrator Meccano differential analyzer with no torque amplifiers, used for educational purposes.
Fail, R., "Electro-mechanical Servo for Differential Analyser," Meccanoman's Journal, #12, 310,1968. A rudimentary electro-mechanical servo design to replace a torque amplifier.
Fail, R., "Mini Differential Analyzer," Midlands Meccano Guild Gazette, No. 16, 4-9, April 1993. A small demonstration model from standard Meccano parts with full construction details. Two integrators and output table. Uses electro-mechanical servos instead of torque amplifiers.
Fischer, C. F., "Reminiscences at the end of the Century," Molecular Physics, 98, 1043-1050, 2000 A collection of auto-biographical notes including an interesting discussion of the author's work as a research student with Hartree around the time he was building the first Meccano differential analyzer. Available online here.
Fischer, C. F., Douglas Rayner Hartree - His Life in Science and Computing, Singapore: World Scientific, 2003. This scientific biography of Douglas R. Hartree not only describes important events in his life but also outlines his contributions to a number of fields.  Hartree was very interested in the process of computation. When he learned of a differential analyzer for solving differential equations, he first built a model using Meccano.
Ford, H. C., "Mechanical Movement" United States Patents 1,317,915, 1,317,916, 1919. The disk/ball/cylinder integrator used by Hannibal Ford in military fire control systems. I am not aware of this type of integrator ever being used in a differential analyzer, but it has the advantage of being able to transmit substantial torque without requiring a torque amplifier. The second patent has a derivative capable of even greater loading.
Gray, E., "The Torque Amplifier," New Zealand Federation of Meccano Modellers Magazine, Oct 1992. An unusual application of the torque amplifier to amplify the outputs of servo motors in a motor chassis.
Hartree, D. R., F.R.S., and Porter, A., "The Construction and Operation of a Model Differential Analyser," Memoirs and Proceedings of the Manchester Literary & Philosophical Society, 79, 51-74, 1935. A detailed  account of the construction and operation of the Meccano differential analyzer at Manchester University including a number of photographs of the machine.
Hartree, D. R., F.R.S., and Ingham, J., "Note on the Application of the Differential Analyser to the Calculation of Train Running times," Memoirs and Proceedings of the Manchester Literary & Philosophical Society, 83, 1-15, 1938. An interesting paper on a relatively simple second order equation. Although the work was done on the full scale machine, reference is made to the fact this would be a suitable problem for the Meccano differential analyzer described in Hartree (1935).
Hartree, D. R., "The Mechanical Integration of Differential Equations," Mathematical Gazette, 22, 342-364, 1938. A comprehensive account of the construction and application of the Manchester differential analyzers with a good close up photograph of the integrators of the full scale machine.
Hartree, D. R, and Nuttall, A. K., "The Differential Analyser and its Application in Electrical Engineering," Journal of the Institution of Electrical Engineers, 83, 643-647, 1938. A fairly detailed account of the Metropolitan-Vickers machine with good pictures, including of the special input table. A somewhat briefer section discusses actual applications.
Hartree, D. R., and Porter, A., "The Application of the Differential Analyzer to transients on a Distortionless Transmission Line," Journal of the Institution of Electrical Engineers, 83, 648-656, 1938. Examines the behavior of transients on a finite distortionless transmission line. This problem required use of a special input table that can feed back a solution after a fixed delay. Application to lightening arresters is discussed.
Hartree, D. R., "The Thirty-fourth Kelvin Lecture: Mechanical Integration in Electrical Problems," Journal of the Institution of Electrical Engineers, 90, 435-442, 1943. Lecture delivered before The Institution on 29, April, 1943. After a brief description of the differential analyzer (including mention of Kelvin's contribution) details solutions to several problems in electrical engineering obtained using the machine.
Hartree, D. R., "Differential Analyser," Ministry of Supply Permanent Records of Research and Development No. 17-502, 1946/9. A detailed account of the use of the Manchester differential analyzer during the war.  Probably the largest collection of applications gathered in a single document.  Section 1.4 documents other differential analyzers in the UK, including Meccano models.
Hartree, D. R., F.R.S., Calculating Instruments and Machines, Urbana: University of Illinois Press, 1949. An extensive discussion of differential analyzers with many photographs including the Meccano differential analyzer described in Hartree (1935). Includes a chapter on its application to partial differential equations. There is a suggestion for the solution of a set of nonlinear simultaneous equations for a problem in spherical, sound waves remarkable in that it uses no integrators, just an intricate interconnection if input tables, output tables and multiplier/divider units. Extensive references are provided.
Hartree, D. R., "The Bush Differential Analyser and its Applications," Nature, 146, 3697, 319-323, 1940. A general description of the differential analyzer. Despite the title, the pictures are actually of the Metropolitan-Vickers machine at Manchester.
Hartree, J., and Tee, G., "Toy Story," New Scientist, 4 Mar 2000. Letters to the editor discussing the differential analyzer, in response to the question from a reader as to whether any significant inventions or principles owe their discovery to the use of Meccano. Available online here.
Hazen, H. L., Jaeger, J. J., and Brown, G. S., "An Automatic Curve Follower," Review of Scientific Instruments, 7, 353-357, 1936. Describes an automatic curve follower using a photocell, servo motor, and torque amplifier, which can follow the black/white boundary of a curve on an input table.  See also Blackett (1939).
Heffron, W. G., "Operation and Application of the Differential Analyzer," Product Engineering, 23, 4, 164-170, 1952. General description of the operation of a differential analyzer with particular reference to the General Electric 14 integrator machine.
Hey, T., The Quantum Universe, Cambridge University Press, 1987. A brief reference to Hartree's atomic structure calculations, including a picture of Hartree and Porter with the Meccano differential analyzer.
Hogle, H., "Torque Amplifier," Canadian MeccaNotes, 6, 19, June 1997. An unusual torque amplifier design in Meccano parts. Probably not sensitive enough for use in a differential analyzer.
Holst, P. A., "Svein Rosseland and the Oslo Analyzer," IEEE Annals of the History of Computing, 18, 4, 16-26, 1996. At one time the Oslo analyzer was the world's largest; technically advanced, highly accurate, and used by theoretical physicists from around the world.  A discussion of the machine, and the man who created it.
Irwin, W., "Differential Analyser No. 2," New Zealand Federation of Meccano Modellers Magazine, 25, 2, 2001. Letter to the editor identifying the source of the article in the preceding issue (Anon (2001)) and reporting on more recent efforts to restore the Meccano differential analyzer at MOTAT.
Irwin, W., "Differential Analyser Myths," New Zealand Federation of Meccano Modellers Magazine, 25, 3, 2001. A short piece exposing a number of myths circulating about Meccano differential analyzers.
Irwin, W., "The Differential Analyzer Explained," New Zealand Federation of Meccano Modellers Magazine, 26, 3, 2002. Describes the principles of operation of a differential analyzer.  Illustrated with a picture of an earlier Meccano machine by the current author and the Science Museum exhibit of an integrator from the Meccano differential analyzer described in Hartree (1935).  Available online here.
Irwin, W., "Meccano Differential Analyser . . . and New Zealand's First Computer," The International Meccanoman,  #46, IX.2005. An account of the differential analyzer exhibits at the 2005 NZFMM Easter Convention in Auckland, New Zealand, which included the restored section of  the Cambridge Meccano machine and the author's version of  the model in Fail (1993) .
Irwin, W., "Differential Analyzer - Adding Unit," New Zealand Federation of Meccano Modellers Magazine, 31, 6, 8-9, 2007. Details of the adding unit used in the original Cambridge Meccano differential analyzer together with a modern reconstruction.
Irwin, W., "Propagation of an Urban Legend, a Differential Analyser Myth," New Zealand Federation of Meccano Modellers Magazine, 31, 6, 14-15, 2007. Debunks the myth that the Cambridge Meccano differential analyzer was used by Barnes Wallis for the design of the "bouncing bomb".
Jackson, A. S., Analog Computation, New York: McGraw-Hill, 1960.
 Although primarily about electronic analog machines, this book includes a concise section on mechanical differential analyzers (pp. 573-578).
Janssen, E., and Lebell, D., "Applications of the Mechanical Differential Analyzer to Electrical Engineering," Electrical Engineering, 70, 432-435, 1951. Applications for the differential analyzer to magnetic amplifiers, pulse transformers and electron accelerators.
King, D., "Historic Computer Lost from MOTAT," New Zealand Herald, section 4, page 5, April 20, 1993. A half page report on the loss of the Meccano differential analyzer from MOTAT. Gives a history of the machine and a picture of it in the museum's former computer display.
King, D., "Heritage Rusts to Bits in Rain," New Zealand Herald, section 3, page 5, April 27, 1993. Another report on the events surrounding the loss of the Meccano differential analyzer from MOTAT.
Kuehni, H. P., and Peterson, H. A., "A New Differential Analyzer," AIEE Transactions, 63, 5, 221-228 (discussion 429-431), 1944. Detailed technical description of the General Electric 14 integrator differential analyzer which used a Polaroid optical follower system on the integrators.  The follower system is described in Berry (1944).
Kryloff, A., "Sur un intégrateur des équations différentielles ordinaires," Bulletin de l'Académie Impériale des Sciences de St.-Pétersbourg, Ser. V, T.XX, 1, Jan. 1904. In this little known work, Kryloff describes a machine based on the work of Thompson (Thompson (1876A)), but using radically different integrators.  It is unclear if the machine was successful as the paper was written while it was still under construction. This paper is in French.
Lennard-Jones, J. E., Wilkes, M. V., and Bratt, J. B., "The Design of a Small Differential Analyser,"  Proceedings of the Cambridge Philosophical Society, 35, 485, 1939. A detailed description of the construction and testing of the Meccano differential analyzer at Cambridge University.  In an early test using four integrators, the wave equation of the hydrogen atom was solved giving the value of the ground state energy correct to one part in 500.
Lowe, I., "Ancient Computer Down and Out," New Scientist, 138, 1873, p.50, 15 May 1993. (May not appear in all editions.) A follow up to articles in the New Zealand Herald (King (1993A) and King (1993B)) reporting on the loss of the Cambridge Meccano differential analyzer from MOTAT.
Macauley, T., "Operating the Meccano Differential Analyser,", unpublished, MOTAT, Auckland, New Zealand, 1978. A brief operating manual for the Meccano differential analyzer at MOTAT which indicates that it was in operation in 1978 for demonstrations.  One of the 5 integrators was not functional.
Maginniss, F.J., "Differential Analyzer Applications,", General Electric Review, 48, 5, 54-59, 1945. A discussion of eight applications of the differential analyzer to engineering problems and the techniques used to handle them.
Marsh, P., "The Meccano Set Computer," New Scientist, 80, 1134, (supplement 28-29), 1978. A popular article on the history of the Manchester differential analyzers.
Massey, H. S. W., Wylie, J., Buckingham, R. A., and Sullivan, R., "A Small Scale Differential Analyser - Its Construction and Operation," Proceedings of the Royal Irish Academy, 45A, 1, 1-21, 1938. A four integrator machine.  All the spur gears used in this machine are of Meccano manufacture.  Helical gears and sprockets are from Bond's. Everything else is of custom design. The paper contains some excellent pictures.
Michel, J. G. L., "Extensions in Differential Analyzer Technique," Journal of Scientific Instruments, 25, 10, 357-361, 1948. In this paper, a constructive technique is developed for obtaining the results arrived at by Amble (1946) from analytical considerations. The technique is extended to include the integral of a quotient, and the inversion of functions. Mentions that one of these techniques was applied on the Cambridge Meccano differential analyzer.
Michel, J. G. L., "Errors of Friction Wheel Integrators," Journal of Scientific Instruments, 32, 2, 43-44, 1955. Analyzes the error inherent in a wheel and disk integrator whenever there is sliding motion of the wheel. This error is approximately proportional to the torque which must be applied to the wheel.
Mindell, D. A., Between Human and Machine - Feedback, Control, and Computing Before Cybernetics, Baltimore: Johns Hopkins University Press, 2002. Mindell shows how the modern sciences of systems emerged from disparate engineering cultures and how they converged during W.W.II. Chapter 5, Analog Computing at MIT, is devoted to the work of Vannevar Bush's lab.
Myers, D. M., and Blunden, W. R., "The C.S.I.R.O. Differential Analyser," Proceedings of Conference on Automatic Computing Machines, Sydney Australia, 1951. A description of the construction of a 10 integrator differential analyzer at the University of Sydney. This machine uses electrical interconnections between the units using a system called "M-type" transmission.  Foe a more detailed account, see Myers (1952).
Myers, D. M., and Blunden, W. R., "The C.S.I.R.O. Differential Analyser," Journal of the Institution of Engineers, Australia, 24, 195-204, Oct-Nov, 1952. A more comprehensive description of the C.S.I.R.O. analyzer than given in Myers (1951).  It includes a section on applications to which the instrument has been applied.
Nieman, C. W., "Bethlehem Torque Amplifier," American Machinist, 66, 21, 895-897, 1927. Nieman's original description of the torque amplifier, with application to automobile power steering.  The caption of one figure contains the phrase "adaptable to computing machines" though there is no other mention in the text, and this article considerably predates Bush's application.  See also US patents 1751645, 1751647, and 1751652.
O'Neill, R., "Meccano 'Dam Busters' computer stars at MOTAT," Computerworld, 16 July, 2007.  A review of the new computing exhibit at MOTAT which features the recently restored  Cambridge Meccano differential analyzer.
Owens, L., "Vannevar Bush and the Differential Analyzer: The Text and Context of an Early Computer," Technology and Culture, 27, 1, 1986, 63-95.  Reprinted in Nyce, J. M. From Memex To Hypertext, Academic Press, 1991. Discusses the  background to Bush's machines, from the early product integraphs through the mechanical analyzer to the mighty Rockefeller electro-mechanical analyzer.
Partridge, A., "Torque Amplifier," Midlands Meccano Guild Gazette, No 7, 5, September 1982. A simple servo based design using a motor and differential to operate electrical contacts.
Partridge, A., "Torque Amplifiers," Constructor Quarterly, 19, 40-42, March 1993. Constructional details of two torque amplifier designs using Meccano parts.
Paynter, H. M., "The Differential Analyzer as an Active Mathematical Instrument," IEEE Control Systems Magazine, 9, 7, 3-8, 1989. This presentation features the essential role played by amplification and control in the successful development of the differential analyzer. Mentions the Meccano machines, but incorrectly states that their torque amplifiers were made entirely from Meccano parts.
Peierls, R., Bird of Passage, Princeton University Press, 1985. The highly readable autobiography of Rudolf Peierls.  Contains a brief mention of Hartree and the Meccano differential analyzer (p104).  Peierls was the supervisor of A. M. Wood when he was building a Meccano machine at Birmingham (see Wood (1942)) and on p137 he confirms this machine was never completed.
Porter, A., "An Approximate Determination of the Atomic Wave Functions of the Chromium Atom," Memoirs and Proceedings of the Manchester Literary & Philosophical Society, 79, 75-81, 1934. Presents the results of approximate calculations of the wave functions of the chromium atom, carried out on the Meccano differential analyzer described in Hartree (1935).
Porter, A., Differential Analyser Log Book, MS474 London: Science Museum Library. Hand written day to day notes recorded from July to December 1935 as the full scale analyzer at Manchester was being commissioned. These notes make fascinating reading. It would appear that getting reliable operation from the machine was very challenging.
Porter, A., The Differential Analyser and Some Applications, University of Manchester PhD Thesis, 1936. Porter's PhD Thesis in which he describes both the model and full size machines, including modifications to handle time-lag problems.  He covers in detail numerous applications.
Porter, A., Introduction to Servomechanisms, London: Mehuen & Co., 1950. This is a general treatment of the theory of servomechanisms. However, the first chapter uses as examples both the mechanical torque amplifier of the original Bush differential analyzer, and the electromechanical servo system used on the later Rockefeller differential analyzer.
Porter, A., "Building the Manchester Differential Analyzers: A Personal Reflection," IEEE Annals of the History of Computing, 25, 2, 86-92, 2003. A delightful memoir by one of the pioneers.  Written in his 93rd year.  This memoir is extracted from chapter 4 of Porter (2004).
Porter, A., So Many Hills to Climb: My Journey from Cumbria to North Carolina, Silver Spring: The Beckham Publications Group, 2004. Arthur Porter describes a remarkable life with eloquent sensitivity and charming candor. Chapter 4 covers his time at Manchester University and the building of the differential analyzers. This chapter was in large part reprinted in Porter (2003).
Robinson, T. B., "The Meccano Set Computers," IEEE Control Systems Magazine, 25, 4, 74-83, 2005. A history of the Meccano based small scale differential analyzers.
Robinson, T. B., "A Reconstruction of the Differential Analyzer in Meccano," IEEE Control Systems Magazine, 25, 4, 84-89, 2005. Description of the author's modern reconstruction in Meccano of a differential analyzer based on Bush's prototype.
Rose, H. E., "The Mechanical Differential Analyser: Its Principles, Development, and Applications," Proceedings of the Institute of Mechanical Engineers, 159, 1948, 46-54 and 62-80.
 This is a review paper which contributes nothing original. It is followed by a record of the discussion which ensued when it was presented, in which A. Porter leads a blistering attack for both its lack of originality and for the arbitrary change in notation which Rose adopted. It is further followed by miscellaneous communications relating to mechanical and electrical analyzers.
Rosseland, von S., "Mechanische Integration von Differentialgleichungen," Die Naturwissenschaften, 27, 44, 729-735, 1939. A description of the 12 integrator full scale machine at the Institut für Theoretische Astrophysik, Oslo, Norway. The paper is in German.
Shannon, C. E., "Mathematical Theory of the Differential Analyzer," Journal of Mathematics and Physics, XX, 4, 1941.  Reprinted in Claude E. Shannon, Collected Papers (ed. Wyner A. D. and Sloane, N. J. A.) Wiley-IEEE Press, 1993. A heavy duty mathematical analysis of the class of problems that can be addressed by the differential analyzer, assuming only that the machine has an unlimited number of integrators and adders. An interesting paper, but somewhat remote from the practicalities of actual machine operation.
Small, J. S., The Analogue Alternative: The Electric Analogue Computer in Britain and the USA, 1930-1975 (Studies in the History of Science, Technology and Medicine), Routledge, 2001. While primarily concerned with electronic analog machines, chapter 2 provides a good introduction, including Hartree's development of differential analyzers at Manchester. Includes a picture of the Meccano differential analyzer.  Extensive references.
Smillie, K., "People, Languages, and Computers: A Short Memoir," IEEE Annals of the History of Computing, 26, 2, 62-74, 2004. A biographical memoir which briefly mentions work by Jim Howland to extend Beatrice Worsley's Meccano differential analyzer in 1951. Essentially the same material can be found online here and here.
Smillie, K., "A J Simulator for a Meccano Differential Analyzer," Vector, 23, 3 2008.
 This paper describes a five-integrator differential analyzer simulator implemented in J and gives two examples of its use.
Smillie, K., "Simulation of Meccano Differential Analyzers", 2008.
 A more extended version of Smillie (2008a) published on line.  The simulator code is available for download via this page.
Smith, P., "Who sez it cannot be done!," Meccano Engineer, #11, March 1996. A short article with a picture of a one integrator Meccano DA constructed by T. Brooker, in which the torque amplifier is constructed entirely from standard Meccano parts.  Probably the first person to have done it.
Soroka, W. W., Analog Methods in Computation and Simulation, New York: Mc Graw-Hill. A comprehensive survey a vast array of both mechanical and electrical methods current at the time of writing. Includes much material on the differential analyzer with extensive references.
Spackman, L., "A Meccano Differential Analyser," Meccanoman's Newsmag, #67, 1993. Discussion of the history of the Cambridge Meccano differential analyzer in New Zealand.
Strong, C. L., "The Amateur Scientist: A plan for an analogue computer that can be built for about $50," Scientific American, June 1968. A design for a home built two integrator differential analyzer with an example application. Construction of the machine would be greatly simplified by the use of Meccano!
Schultes, D., "On the Development and Use of Differential Analyzers," April 2004. A short essay presenting the development of the differential analyzer. In addition to discussing the mechanical machines it includes a section on the later electronic machines and a short comparison with present day technology.
Tee, G. J., "Meccano Differential Analyser No. 2," unpublished notes, 1993. Notes on the complex history of the Cambridge Meccano differential analyzer, shipped to New Zealand in 1950, almost dismantled in the 1960's, displayed at MOTAT in the 70's then lost in 1993 and finally found again in damaged condition.
Thomson, J., "An Integrating Machine having a new Kinematic Principle," Proceedings of the Royal Society, 24, 262, 1876. Description of the ball and disk integrator in which a ball rests under gravity between a horizontal cylinder and an inclined integrator disk thus avoiding the sliding necessary in a wheel and disk integrator. Inspired by the work of Prof. James Clark Maxwell who applied a similar approach to an improved planimeter design. Reprinted in Thomson (1879).
Thomson, Sir W., "An Instrument for Calculating the Integral of the Product of two Given Functions," Proceedings of the Royal Society, 24, 266, 1876. Application of the ball and disk integrator described in Thomson (1876A) to the evaluation of the integral of a product of functions. Reprinted in Thomson (1879).
Thomson, Sir W.,  "Mechanical Integration of Linear Differential Equations of the Second Order with Variable Coefficients,"  Proceedings of the Royal Society, 24, 269, 1876. Sir William Thomson first suggested that the integrators developed by his brother could be interconnected to produce solutions of differential equations. The idea was not practical at the time because of the lack of torque amplifiers. Reprinted in Thomson (1879).
Thomson, Sir W.,  "Mechanical Integration of the General Linear Differential Equation of any Order with Variable Coefficients," Proceedings of the Royal Society, 24, 271, 1876. A generalization of the scheme proposed in Thomson (1876C) to equations of arbitrary order. Reprinted in Thomson (1879).
Thomson, Sir W.,  "Harmonic Analyzer," Proceedings of the Royal Society, 27, 371, 1878. A description of a harmonic analyzer using up to eleven of the ball and  disk integrators described in Thomson (1876A) and the principles detailed in Thomson (1876B) where one of the functions in the product is a sine or cosine. This machine was used to extract the coefficients of a Fourier series for the purposes of tide prediction. Reprinted with an addition dated April, 1879 in Thomson (1879).
Thomson, Sir W., and Tait, P. G., Treatise on Natural Philosophy,  Vol. 1, Cambridge: Cambridge University Press, 1879. Although a textbook on physics, it contains an Appendix (curiously called Appendix B' even though it's the only one) which contains reprints of several papers on integrating devices previously published in the Proceedings of the Royal Society. See Thomson (1876A), Thomson (1876B), Thomson (1876C), Thomson (1876D), and Thomson (1878),
Travis, I., "Differential Analyzer Eliminates Brain Fag," Machine Design, 7, 7, 15-18, 1935. Description of the ten integrator differential analyzer built at the Moore School of Engineering, University of Pennsylvania. This machine included two polar input tables.
Wildes, K. L. and Lindgren N. A., A Century of Electrical Engineering and Computer Science at MIT, 1882-1982, Cambridge: MIT Press, 1985.
 Chapter 4 provides a non-technical historical account of four generations of analog technology developed under Bush's guidance, from the early integraphs through to the mighty Rockefeller DA.
Wilkes, M. V., Memoirs of a Computer Pioneer, Cambridge, MA: The MIT Press, 1985. Describes his encounter with the Meccano differential analyzer at Cambridge and how he came to take responsibility for it. He also describes the addition of the fifth integrator by Miss E Monroe in 1937.
Williams, M. R., "UTEC and Ferut: The University of Toronto's Computation Centre," Annals of the History of Computing, 16, 2, 4-12, 1994. Contains a brief reference to Beatrice (Trixie) Worsley's Meccano differential analyzer at the Unversity of Toronto. There is a picture of the machine in an early stage of construction..
Williams, M. R., A History of Computing Technology, 2nd Edition, Wiley-IEEE Computer Society Press, 1997 Broad survey of computing history. Chapter 5 covers analog machines in general with a section on the differential analyzers. Mentions the Hartree Meccano differential analyzer and has an interesting reference to a Meccano machine built in Toronto by Beatrice (Trixie) Worsley.
Winston, B., Media Technology and Society: A History: From the Telegraph to the Internet, Routledge, 1998 Contains a brief reference to Hartree's construction of a differential analyser in Meccano after visiting MIT to see Bush's machine.
Wood, A. M., The Design and Construction of a Small Scale Differential Analyser and its Application to the Solution of a Differential Equation, University of Birmingham MSc. Thesis, 1942. The design of a small scale machine (6 integrators) built largely of Meccano parts is described.  Torque amplifiers and integrator carriages are custom built. Because of war time material shortages, only two integrators were completed at the time the thesis was written.  A second section describes the solution of a equation performed using the Meccano differential analyzer at Cambridge University. Peierls (1985) confirms it was never completed.
Worsley, B. H., "Construction of a Model Differential Analyzer,"  Worsley Archives, box 3, folder 10, Queen's University Archives, Ontario, 1948. A memo to Dr. B. A. Griffith, Toronto Computation Center, dated 10 September, 1948 describing the construction of a three integrator Meccano differential analyzer.
Worsley, B. H., "Differential Analyzer," Worsley Archives, box 3, folder 10, Queen's University Archives, Ontario, undated. A set of course notes from the University of Toronto Department of Physics, for a 4th year practical course using Worsley's model differential analyzer.  Undated, but must be 1949 or later because of a reference to Hartree (1949).
Wilson, A. H., "The Binding Energies of the Hydrogen Isotopes" Proceedings of the Cambridge Philosophical Society, 34, 365-374, 1938. Although fundamentally a paper about nuclear physics, it is interesting because this is the problem which resulted in the addition of a fifth integrator to the Cambridge Meccano differential analyzer.
Wright, G., The Meccano Super Models (The Hornby Companion Series, Vol. 2), London: New Cavendish, 1978. Reproduced on p.31 is a small extract from Anon (1934B).
Copyright (c) 2010 Tim Robinson