In 1878 William Thomson, the later Lord Kelvin, published a concept for a mechanical device for the solution of systems of linear equations. This device was certainly a byproduct of his work on tide predicting machines where he computed weighted sums of sine functions using pulley systems.
- W. Thomson, On a Machine for the Solution of Simultaneous Linear Equations, Proc. R. Soc. Lond. 28 (1878), 111-113.
- W. Thomson, P. G. Tait, Treatise on Natural Philosophy, Cambridge University Press 1988, 482-487.
From 1934 to 1936 John Wilbur developed the MIT Simultaneous Calculator. He completed a prototype for the solution of systems with two equations on October 27, 1934. The final device could solve a system of nine homogeneous equations in ten unknowns. I have collected some links to newspaper, magazine and journal articles on the web.
- Calculating Machine for the Solution of Simultaneous Equations invented here, The MIT Tech LIV (40), October 30, 1934.
- Calculating Machine for Simultaneous Equations, Nature 134 (1934), p.877.
- A New Mechanical Brain, The Michigan Technic, January 1935, p.66.
- New Machine Solves Algebraic Problems, The Pittsburgh Press, December 7, 1936, p.33.
- Machine Solves Mathematical Problems in Few Seconds, The Times Record, Troy N.Y., December 15, 1936, p.18.
- Solves Algebraic Equations, The Rocky Mount Herald, January 15, 1937, p.1.
- Solves Algebraic Equations, The Montgomery Press 49 (9), January 15, 1937.
- Robot Solves Problems, Popular Science, February 1937, p.37.
- Robot Mathematician Solves Nine Simultaneous Equations, Science News Letter for February 13, 1937, p.101.
- The Mechanical Solution of Simultaneous Equations, Journal of the Franklin Institute 222 (6), December 1936, 715-724.
The most prominent user was Wassily W. Leontief who later won a Nobel prize in 1973 for his Input-Output analysis.
The Simultaneous Calculator is briefly described in the fantastic book A Computer perspective by the office of Charles and Ray Eames, and was on display in the underlying IBM exhibition in 1973.
Unfortunately, the device seems to be lost at some later point of time.
I recommend to read the essay W. Cauer and his Mathematical Device by H. Petzold in the book Exposing Electronics about the Simultaneous Calculator and other very interesting, but more or less unsuccessful devices for the solution of systems of linear equations in the pre-digital era.
In 1944 Sasaki Tatsujiro, Shiga Makoto, Miita Junichi built a calculator that closely resembled the Simultaneous Calculator. It is on permanent display at the National Museum of Nature and Science, Tokyo.
- M. Maejima, Short History and Calculation of the Wilbur Machine (Japanese), Bull. Natn. Sci. Mus. Tokyo, Ser. E, 24 (2001), 25-29.
- E. Wada, Analogue calculator for nine simultaneous equations, IPSJ Magazine 50 (2009), 914–918.
- F. W. Sinden, Mechanisms for Linear Programs, Operation Research 7 (1959), 728-739.
- J. Stringer, K. B. Haley, The Application of Linear Programming to a Large-Scale Transportation Problem, Proc of First International Conference on Operational Research, Oxford, 1957.
The first one deals with mechanisms where the coefficients are 0 or 1 and shows nicely how duality in linear programming is reflected in dual mechanisms. The second one describes at the end a concrete device to solve the transportation problem.