# How could Huygens have solved the tautochrone problem before Newton's theory of gravity and equations of motion?

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?

• 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.
– user466
Jun 11, 2017 at 3:28
• See Joella Yoder, Unrolling Time: Christiaan Huygens and the mathematization of nature, Cambridge UP (1988) Jun 11, 2017 at 6:56
• 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
– nwr
Jun 11, 2017 at 16:45
• The link to the PDF excerpt "Huygens discovers the isochrone" given in the comment by nwr no longer works. Working link as of november 24, 2022: web.nmsu.edu/~davidp/history/mm-3-2-huygens.pdf Nov 24, 2022 at 17:00