In the beginning, observation by direct skywatching was the only method of identifying planetary motion. By Babylonian times a body of written records based on observation, stretching back centuries, was used for prediction. These written records provided the basis for the next step - prediction from tables.
The Greeks applied complex mathematics to the process and developed theories of how planets moved, in time and space. These theories may not have been 'correct' by our modern standards but they provided a basis for refining a mathematical model of the solar system. Ptolemy provided a mathematics which was used up till the renaissance based on a geocentric view (though some Greeks had developed a heliocentric model). Ptolemy's model was used for the creation of ephemerides, tables giving the position of the planets with fairly good accuracy, though obviously it did not deal with the outer planets, asteroids or other bodies, which had not been discovered.
In the renaissance, Copernicus developed a model of heliocentric movement, based largely on the work of Arab Astrologer/Astronomers (which in turn can be traced back to Greek times). Refinements to the the theory of planetary motion came from Galileo and more importantly Kepler. Add in Newton's mathematics and you get a very precise means of predicting planetary position without ever having to look at the sky or through a telescope.
A mathematical model based on Newton and Kepler works very well for the prediction of planetary motion but not so well when you try to predict outside the solar system. Hence Einstein's theories provide modifications for much of NASA's modelling. Bigger and better computers, bigger and better telescopes allow for refinement but essentially you can predict using a PC and a mathematical model based on Newton's theory.
In practice you don't actually do this directly, you simply install the Swiss Ephemeris (or other ephemeris) on your PC, that contains the nescessary mathematical model and all you need to do is to ask for the positions at any given time on any given date. Virtually all Astrological software comes with an ephemeris 'built in' so the process never really impinges on the Astrologer any more.
I think it's still the case though that students learning Astrology have to master the use of an ephemeris and the calculation of planetary movement during the day, using two midnight positions from a hard copy ephemeris. Strictly speaking the mathematics is an approximation in that process but the error is so small it can be ignored,