Claudius Ptolemy was a famed Greek scholar whose work in astronomy and geometry helped form the basis of trigonometry. Ptolemy's earliest and most famous work was the *Almagest*, which focused on astronomy. In it, he developed geometrical proofs, which included the first notable value for *π *since **Archimedes**' time, to explain the motion of the sun, the moon, and the planets. Although his geocentric theory of astronomy mistakenly identified the Earth instead of the sun as the center of the solar system, this work served as the authoritative text on astronomy for over 1,400 years. Ptolemy also wrote works on music, optics, and astrology. His treatise, *Geography*, survived for centuries as a primary reference source and included the first practical use of parallels and longitudes.

Ptolemy and his life remain a mystery outside of his scientific written works. Estimates of the era in which he lived are based on his recorded astronomical observations. The name Ptolemy was common in Egypt, leading to the belief that he was a native of that country. His first name, Claudius, suggests that he was recognized as a Roman citizen, probably due to a legacy handed down through his Greek ancestors. The exact date and cause of Ptolemy's dead are unknown.

Ptolemy probably lived in Alexandria, Egypt, where he made most of his astronomical observations. The city's famous library provided him with access to the largest collection of accumulated knowledge during his time. From his introduction to the *Almagest*, Ptolemy appears to adhere to the Aristotelian school of philosophy, but he was also influenced by other philosophies, such as stoicism. To his credit, Ptolemy acknowledged that he based much of his work on that of previous scholars, especially the astronomical theories of **Hipparchus**.

### Advances the Use of Trigonometry through Astronomy

Like Hipparchus before him, Ptolemy's interest in astronomy and measurements involving the "heavenly" bodies led to his important contributions in geometry and trigonometry. Although Hipparchus is credited as the founder of trigonometry, Ptolemy's earliest known work, the *Almagest*, comprehensively systemized the sum of Greek knowledge in both trigonometry and astronomy.

Ptolemy's Greek title for the *Almagest *was *Syntaxis*, translated as "mathematical compilation." It was quickly recognized as a ground breaking work and came to be called "the great compilation" by the Greeks, probably to distinguish it from earlier and more elementary works on astronomy. It eventually became known as the *Almagest*through early Arabic translations and subsequent Latin translations as the "almagesti" or "almagestum."

Consisting of 13 books, the *Almagest *includes original theories and proofs, including a method of calculating the chords of a circle. These chords were used by the Greeks in spherical geometry and required trigonometry to make various calculations for solar, lunar, and planetary positions and eclipses of the sun and moon. Ptolemy's model of astronomy used combinations of circles, known as epicycles, or small circular orbits centered on a larger circle's circumference. Although his geocentric theory that the heavenly bodies rotated around a central, stationary Earth was eventually discredited, the theorems and system that he advanced in this work provided an admirable and ready means to construct tables representing the movements of the sun, moon, and planets. Its geometrical representations of these movements represent a testimonial to both Ptolemy and the great mathematical thinkers he borrowed from. The *Almagest *also included a catalogue of stars based largely on the catalogue developed by Hipparchus.

Ptolemy also wrote several other works on astronomy. In his *Handy Tables*, he improved upon and systematically organized the Table of Chords from the *Almagest *into one volume for practical use. Ptolemy also wrote two treatises that demonstrated the Greeks had mastered more than basic "classical" geometry. In the *Analema* and *Planisphaerium*, Ptolemy uses both trigonometry and other mathematical techniques to discuss the projection of points on the celestial sphere and stereographic projection. These works include methods for constructing sundials and the basis for the astrolabe, an astronomical instrument used during the Middle Ages. In the *Planetary Hypotheses*, he carries on his astronomical work from the *Almagest* and includes new theories of planetary latitudes and for determining the size of planets and their distances from the Earth. Ptolemy also wrote other astronomical works, which are either lost or only partially survive.

### Establishes Scientific Foundation for Geography

Ptolemy's second great scientific work was *Geography*, an early attempt to map the world as it was known in his time. Although much of it is inaccurate, the book placed the accumulated Greek and Roman knowledge of the Earth's geography on a solid scientific foundation. Ptolemy's inaccuracies were inevitable, considering that modern surveying techniques were unknown and Ptolemy himself knew little of the world outside of the Roman empire. As a result, Ptolemy had to rely on information from travelers, including soldiers in the Roman army.

Much of *Geography *is based on the work of Marinus of Tyre, as well as Hipparchus and Strabo. Ptolemy's important contribution in the work was his introduction of the first systematic use of latitudes and longitudes. Not only did the book contain a number of maps, Ptolemy included directions followed by geographers until the Renaissance for creating maps. Unfortunately, Ptolemy miscalculated the Earth's circumference and many of his estimates of latitude were incorrect. Nevertheless, *Geography *became a profoundly influential work whose popularity led to the creation of opulent manuscripts for the wealthy.

Like many of the learned men of his day, Ptolemy displayed a wide ranging interest in the sciences. His treatise, *Optics*, uses mathematical equations to develop theories of light and refraction (especially as it pertains to the planets and stars) and also discusses vision and mirrors. His work *Harmonica* is only partially intact and represents a treatise on musical theory, including the mathematical intervals of notes.

In the time of Ptolemy, astrology was also considered a science, and Ptolemy wrote a four-book work called *Tetrabibilios*. Always the educator, Ptolemy viewed this work as the logical follow up to the *Almagest *and created it as a textbook for casting horoscopes. To provide this work with a scientific basis, he attempted to relate his efforts in astronomy to astrology.

While modern day science has proven many of Ptolemy's works to be inaccurate, he deserves his place in history as a great intellect. On the basis of the longtime worldwide interest in his works, Ptolemy had a profound impact on the thoughts and philosophies of many generations. A compiler of knowledge and teacher, Ptolemy exhibited a clarity of style and ingenuity, which made both the *Almagest *and *Geography *standard textbooks well into the Renaissance period and beyond. As a result, Ptolemy, who was the last great astronomer of the Alexandrian school, was long recognized as the authority on "all things in heaven and on Earth."

- Gow, James.
*A Short History of Greek Mathematics*. New York: Chelsea Publishing Company, 1968, pp. 202-301. - Heath, Thomas L.
*The Manual of Greek Mathematics*. New York: Dover Publications, 1963. pp. 402-14. - Magill, Frank N., editor.
*The Great Scientists.*Danbury, CT: Grolier Educational Corporation, 1989, pp. 32-37. - Toomer, G. J. "Claudius Ptolemy," in
*Dictionary of Scientific Biography*. Volume XI. Edited by Charles Coulston Gillispie. New York: Charles Scribner's Sons, 1975, pp. 186-206.