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I've just watched this replica of the Antikythera mechanism. I've heard also about Babagge's analytical machine and the Curta calculator. I got curious: What did they use to build computers made of gears? We have nowadays some treatises to make logic gates with transistors, how did they manage to build logic gates or to do whatever was needed for building their computers?

I'm not sure if this question really fits this forum, but it was the nearest one I could think about.

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This is a very broad question, with a different answer for each kind of device. The computers before modern times did NOT use logic gates.

For example the January 2004 Scientific American article "The Curious History of the First Pocket Calculator" covers how Curta calculators work, in addition to the history.

The Antikythera mechanism used gear reductions (gears with differing teeth counts) to implement the ratios of times of various periodic events to predict positions. Given the [known even then] frequencies, it is straightforward to compute the required gear ratios. http://www.cs.virginia.edu/~robins/Decoding_an_Ancient_Computer.pdf

In addition (as mentioned in a comment above) there were many many different flavors of analog computers, that a quick google search should be able to find explanations for many of them.

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as in other answer, this is a broad question, but there is some general study of mechanical computers and how their conceptual elements led to modern computers. note that ref to the Antikythera mechanism and some other physical systems as a "computer" is a modern projection and more of an analogy/ metaphor. one book that comes close to answering your question is Pebbles to computers: the Thread by Blohm/ Beer. it takes an achaeological pov on the question and looks at the construction of very old computer-like devices. note that the Antikythera mechanism while very old (~2Millenia) and (re)discovered/ excavated about ~1Century ago was not well understood until more recently through scientific research.

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