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When did it become possible to predict the appearance of comets ?

Orbits of comets seen in 1531, 1607 and 1682 calculated to be very similar by Edmund Halley

The appearance of comets became predictable after Halley, an English scientist correctly predicted the reappearance of the comet seen in 1683. Using the newly developed gravitational theory of Newton, he predicted that this comet would return in 1758. His calculations were refined later by others to include the effects of Jupiter and Saturn on the orbit. When the comet was indeed found in December 1759 as expected, it was named after Halley, although Halley was not alive at that time. It is the most famous of all comets, having been traced back in history to 240 B.C. It was a triumph of mathematical understanding over fear and superstition. Since that time it has become possible to predict the paths of comets quite accurately. So there is really no need to fear comets any more unless they are on collision course with the earth!
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