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In this answer to a question of mine on the stackexchange physics site, I learned about the tautochrone problem. Apparently it was solved by Huygens in 1659, which is before Newton's work on mechanics.

  1. What did he base his assumptions on? Were​ there phenomenological laws, analogous to Kepler's laws for planetary motion?
  2. What were his assumptions and how did he solve the problem?

Answers just addressing one of these points are very welcome.

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    $\begingroup$ Interesting question. I don't read Latin and haven't Huygens' Horologium Oscillatorium, but my guess would be along the following lines. The acceleration of an object moving down an incline $\theta$ is proportional to $\sin\theta$, and the proof of this fact doesn't require Newton's laws. Galileo established it empirically and also linked it to a body of theoretical facts. I think this is all that's really needed in terms of physics in order to attack this problem. $\endgroup$ – Ben Crowell Jun 11 '17 at 3:28
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    $\begingroup$ See Joella Yoder, Unrolling Time: Christiaan Huygens and the mathematization of nature, Cambridge UP (1988) $\endgroup$ – Mauro ALLEGRANZA Jun 11 '17 at 6:56
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    $\begingroup$ This PDF titled "Huygens Discovers the Isochrone" gives Huygens result as Proposition XI (page marked 174). You will need to read through to the pages 180/181 to see how it relates to the tautochrone. See math.nmsu.edu/~history/mm-3-2-huygens.pdf $\endgroup$ – Nick Jun 11 '17 at 16:45
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Everything he had to know about gravity and motion to state and solve the Tautochrone problem was well-known since Galileo at least. This problem assumes the uniform gravity (everything happens near the Earth surface), so Newton's law of gravity has nothing to do with it. Newton's law is only relevant for motions at great distances from Earth, like planet and satellite motions.

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  • $\begingroup$ When referring to Newton's work I was thinking more of the equations of motion and calculus. I understand that assuming a uniform downward force doesn't require anything deep, but I think the difficulty comes after that. Do you know which assumptions about the motion he made? And then the most interesting part: how did he get to the solution from these? Thanks. $\endgroup$ – doetoe Jun 11 '17 at 12:20
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    $\begingroup$ 1. Neither calculus not equations of motion were invented by Newton out of nowhere. All of these gradually developed before and after Newton. 2. If you want to see Huygens' argument exactly, why don't you just consult his book? There is an English translation. 3. Many similar problems were solved in Europe before Newton's calculus became available. $\endgroup$ – Alexandre Eremenko Jun 11 '17 at 12:47

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