We’ve been predicting eclipses for over 2000 years. Here’s how.

With records stretching back to about 700 BC, Mesopotamians were able to determine the length of a Saros Cycle—the interval between when the Moon, Earth, and Sun line up for an eclipse. A cycle happens once every 18 years, 10 days (11 days on leap years), and eight hours, tracing a shadow on the Earth. That extra eight hours means that the position of the eclipse shifts over time as the Earth rotates.

Read more here
http://www.popsci.com/people-have-been-able-to-predict-eclipses-for-really-long-time-heres-how