At the science museum in London they have this very cute little gadget used by mapmakers 150 years ago: an axle with a rubber ring around it, and the ring pressing against a cone. The whole lot is attached to a metal stylus; you trace around an area on a map with the stylus and a little reader tells you the area of what you've traced around. I always found that ingenious. The exhibit in London then goes on to show how you can use the same idea to integrate and hence solve differential equations, and finishes with a monster machine that can solve ordinary 4th order ODEs using basically the same trick; you set the coefficients with dials and then the machine draws a graph of the output. I'm afraid I know neither the name of the cute gadget nor the machine :-( but it strikes me as being appropriate for a "math museum"...
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