Kepler's laws
Commentary 




Commentary.  
It is interesting to note that Kepler derived all three
of these laws by dead reckoning: He did not have calculus available
to assist him (it wasn't invented yet!). He was a brilliant
astronomer, but could not gain access to data which was accurate
enough by himself: He needed the help of Tycho Brahe (and their
relationship is an entertaining story by itself). He began to think
of planets as moving in ellipses when his data from the orbit of Mars
showed that it did not follow the path of a circle  which is what
everyone assumed at the time. It is interesting to note that, should Kepler have been studying another planet, he would not have noticed any discrepancy between its true and predicted (circular) orbit. Mars is the only planet at the right distance from Earth and with an eccentricity large enough so that its orbit could be distinguished from a circle with the equipment available at the time. Kepler also attempted to show a universal significance between the regular solids and the "spheres of the heavens," and failed miserably. His last years were spent in misery as his mother was convicted and jailed as a witch. Later, Newton used his new calculus and laws of motion and gravitation to show that Kepler was right. One day in 1689 he came up to his friend, Edmund Halley, and casually mentioned to him that he'd proved that, with a 1/r^{2} force law like gravity, planets orbit the sun in the shapes of conic sections. This undoubtedly took Halley aback, as Newton had just revealed to him the nature of the Universe (at least the Universe as it was known then). Halley then pressed Newton to publish his findings, but he realized that he'd forgotten the proof. After struggling to remember how he had proved the theorem, he published his work and it later appeared in full form in his Principia. Later, Halley examined the orbits of comets, and, armed with Newton's new understanding of the effects of gravity, computed the orbit of Halley's comet. He predicted that it orbits the Sun once every 76 years, and named the date of next passage. Unfortunately, he died before that year, but astronomers saw the comet come just when he said it would. For that, the comet received his name: Halley's Comet.


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