How did Ptolemy use mathematics to predict the positions of the Sun, Moon, and planets?

Ptolemy predicted the positions of the Sun, Moon, and planets by developing a geocentric mathematical model that combined uniform circular motions to account for observed irregularities. His primary work, the Almagest, utilized trigonometry to calculate these celestial paths.1234

Key mathematical techniques included:

• Eccentric Motion: Ptolemy placed Earth at a position offset from the center of a body's circular path. This allowed him to explain why heavenly bodies appear to move at varying speeds, sweeping out equal angles in unequal times from a terrestrial perspective.15
• Epicyclic Model: To explain planetary motion, including retrograde motion, he used an epicycle—a small circle revolving along a larger circular path called a deferent.1
• The Equant: To refine planetary predictions, he introduced the equant, a point from which a planet's epicycle appears to move at a uniform angular speed, even though its motion relative to the deferent's center is non-uniform.1
• Trigonometric Tables: Ptolemy constructed a table of chords, which functioned similarly to modern sine tables, enabling precise calculations of celestial positions based on angular measurements.4