The ancient people would use the Saros cycle to compute eclipses. Google for that. I don't know how accurate the ancients were. I'm guessing that they knew what dates the eclipses would occur on, and approximately what part of the world. But to get modern-day predictions (i.e. Shanghi will be 122 km north of centerline, and will receive 5:18 seconds of totality beginning at 11:42 AM), you'll need a computer.
I wrote an eclipse-predictor program for DOS using algorithms from a book. It was quite accurate. It would agree with the official predictions within minutes and a few kilometers. I didn't understand the math I borrowed. The code was thousands of lines long.