Ptolemy's Almagest is the only surviving comprehensive ancient treatise on astronomy. Babylonian astronomers had developed arithmetical techniques for calculating astronomical phenomena; Greek astronomers such as Hipparchus had produced geometric models for calculating celestial motions. Ptolemy, however, claimed to have derived his geometrical models from selected astronomical observations by his predecessors spanning more than 800 years, though astronomers have for centuries suspected that his models' parameters were adopted independently of observations. Ptolemy presented his astronomical models in convenient tables, which could be used to compute the future or past position of the planets. The Almagest also contains a star catalogue, which is a version of a catalogue created by Hipparchus. Its list of forty-eight constellations is ancestral to the modern system of constellations, but unlike the modern system they did not cover the whole sky (only the sky Hipparchus could see). Across Europe, the Middle East and North Africa in the Medieval period, it was the authoritative text on astronomy, with its author becoming an almost mythical figure, called Ptolemy, King of Alexandria. The Almagest was preserved, like most of extant Classical Greek science, in Arabic manuscripts (hence its familiar name). Because of its reputation, it was widely sought and was translated twice into Latin in the 12th century, once in Sicily and again in Spain. Ptolemy's model, like those of his predecessors, was geocentric and was almost universally accepted until the appearance of simpler heliocentric models during the scientific revolution.  His Planetary Hypotheses went beyond the mathematical model of the Almagest to present a physical realization of the universe as a set of nested spheres, in which he used the epicycles of his planetary model to compute the dimensions of the universe. He estimated the Sun was at an average distance of 1,210 Earth radii, while the radius of the sphere of the fixed stars was 20,000 times the radius of the Earth.  Ptolemy presented a useful tool for astronomical calculations in his Handy Tables, which tabulated all the data needed to compute the positions of the Sun, Moon and planets, the rising and setting of the stars, and eclipses of the Sun and Moon. Ptolemy's Handy Tables provided the model for later astronomical tables or zijes. In the Phaseis (Risings of the Fixed Stars), Ptolemy gave a parapegma, a star calendar or almanac, based on the hands and disappearances of stars over the course of the solar year.

Answer this question "what geometrical models did he use?" by extracting the answer from the text above.
Ptolemy presented a useful tool for astronomical calculations in his Handy Tables, which tabulated all the data needed to compute the positions of the Sun, Moon and planets,