How did early astronomers calculate accurate solar system positions?

In the pre-computer age, say 50 years ago and back, how did astronomers calculate accurate positions of the Sun, Moon, and planets for predicting an eclipse or a transit of Venus?

Venus Ring of Light

This "ring of light" effect is one of the many phenomena observed as the planet crosses the outer limb of the Sun's disk. Unlike the black drop, few observers question the validity of this effect.
Courtesy Lorenzo Comolli

They did it by hand, with the help of numerical tables. These weren’t the trigonometric and logarithmic tables you may remember from school. They were cleverly designed sets of tables, arranged so that all you had to do was look up a bunch of intermediate quantities and combine them to get the answer.

For example, Simon Newcomb’s Tables of Venus, published by the US Naval Observatory in 1895, required 27 separate lookups and yielded a single, highly accurate position after about 20 minutes of pencil-and-paper arithmetic. Similar works existed for each of the other planets and the Sun. The Moon posed by far the greatest difficulty. Yale University astronomer Ernest W. Brown (1866 –1938) devoted a large portion of his professional career to what was called the “lunar theory.” With the aid of his three-volume Tables of the Motion of the Moon (1919), an experienced astronomer could obtain a single accurate position of the Moon in about six hours!

You can find these works in any major observatory library. With slight modifications, they remained the gospel on planetary positions right up through the 1950s.

— Roger W. Sinnott