“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 8th 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. P12/P22=R13/R23
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.