*“After I had discovered true intervals of the orbits by ceaseless labour over a very long time and with the help of Brahe’s observations, finally the true proportion of the orbits showed itself to me. On the 8*^{th} of March of this year 1618, if exact information about the time is desired, it appeared in my head. But I was unlucky when I inserted it into the calculation, and rejected it as false. Finally, on May 15, it came again and with a new onset conquered the darkness of my mind, whereat there followed such an excellent agreement between my seventeen years of work at the Tychonic observations and my present deliberation that I at first believed that I had dreamed and assumed the sought for in the supporting proofs. But it is entirely certain and exact that the proportion between the periodic times of any two planets is precisely one and a half times the proportion of the mean distances.”

With these words Johannes Kepler announced, in his usual overblown flowery style, his third law of planetary motion in the fifth book of his Harmonices Mundi i.e. P_{1}^{2}/P_{2}^{2}=R_{1}^{3}/R_{2}^{3}

This law is one of the most important in the entire history of astronomy. Newton was able to demonstrate that, under assumption of his three laws of motion, his inverse square law of gravity and Kepler’s third law are equivalent. As Cassini had already shown that not only the then known planets of the solar system but also the satellite systems of Jupiter and Saturn obeyed Kepler’s third law, Newton had the necessary empirical confirmation for his law of gravity.

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Kepler’s laws capture the meaning of the word “beautiful” as applied to mathematical descriptions of nature.

Well said!

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I just found your blog (from your tweet, which is also new to me) and love it..thanks for these .. I do a calendar of math related events and just linked here for the details of Kepler’s third.. Imagine I will do that frequently .. really nice stuff you do..

I found your blog because you followed my tweets, I’m now following yours! Saw your tweet about Kepler’s third law and remembered this post and so I tweeted it. Thanks for the memory 😉

I remember the delight of accidentally ‘discovering’ Kepler’s Third Law when I was at school. We had just started studying Newton’s laws of motion, when I noticed that I could combine them to get a pretty formula with t^2 on one side and r^3 on the other. The teacher was moderately impressed, informing me that I had just done a Kepler (as well as scooping his next lesson).

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