Mechanisms by Chebyshev

I built a replica of Chebyshev's calculating machine, using parts from the Actobotix robot building system. Exceedingly simple mechanism, which ingeniously removes the need for any carry mechanism.

The only problem is that the digits showing the results are not in a nice, straight line. The second digit rotates 10x slower than the first digit, the third digit rotates 10x slower than the second digit, etc.... after a while, it becomes difficult to sus out what digits the resulting number actually has.

After dorking around with it for a few months, it struck me, that I could make the problem go away if I used a signed-digit positional number system, viz, instead of having the digits 0,1,2,3,4,5,6,7,8, and 9, I used digits -5, -4, -3, -2, -1, 0, 1, 2, 3, 4. Signed-digit number systems take a bit of getting used to, but the advantage is that they represent negative numbers as naturally as they do positive numbers. No need to use complement arithmetic. Cf. Knuth’s favorite base, balanced ternary—just for base 10, not base 3.

The digits still are unaligned, but it was clear and unambiguous what the result was. What's more, you could annotate the dials such that you could read off the 10's complement result if you wanted to---which meant that for positive numbers, you didn't even have to learn signed-digit arithmetic, you just got the result you'd be expecting to get.

In other words, Chebychev came within INCHES of realizing that he had invented the world's first--perhaps still the only for base 10–calculating machine which could natively handle negative numbers, with no need for complement arithmetic.

I consider this my small revenge on Chebychev....when I was taking advanced probability theory, every time I saw the name "Chebychev" I groaned...because I knew I was in for weeks of trying to understand an absolutely impenetrable theorem, proving an unbelievable formula...which was infinitely useful :-(

Still, it's a shame he didn't make that jump--his calculating machine has none of the finicky components a carry mechanism does, so its parts are cheap to manufacture and its dead-simple to assemble. The 1870's could have been like the 1970's, when pocket calculators became affordable enough for everybody to buy them.