How did Claudius Ptolemy contribute to mathematics?

Claudius Ptolemy made significant contributions to mathematics, primarily through his application of quantitative analysis to astronomy and geography. His work, particularly the Almagest, represents a pinnacle of Greco-Roman science.1

His key mathematical achievements include:

• Trigonometry: Ptolemy compiled the earliest surviving table of a trigonometric function. In the Almagest, he constructed a table of chords for angles ranging from 0° to 180°, which functioned essentially as a table of sines. By utilizing the Babylonian sexagesimal system (base 60), he provided a precise mathematical framework for calculating celestial positions.12
• Cartography: In his Guide to Geography, Ptolemy developed sophisticated mathematical techniques for mapmaking. He devised two distinct map projections to represent the spherical Earth on a flat surface, using circular arcs to map latitude and longitude.3
• Optics: He established the mathematical principle that an object and its mirror image must form equal angles relative to a mirror.1

Ptolemy’s work was driven by the conviction that the physical world could be explained through rigorous mathematical terms.3