Who first calculated the orbits of comets based on historical records dating back to antiquity?

Sir Edmund Halley

Explanation:

Sir Edmund Halley was the first person who presented the calculation for 24 cometary orbits in the year 1705. One of these he predicted to the proximity of Earth in the Year 1758. That comet indeed returned as predicted by him, the comet is called Halley's comet. Halley's comet can be seen in the atmosphere every 76 years. Last Halley's comet was seen in the year 1986.