Claudius Ptolemy

Who Tabulated the Heavens · Astronomer & Geographer · c. 100 – c. 170

The Alexandrian astronomer whose Almagest governed the subject for fourteen centuries, and whose table of chords was the instrument every later computer of angles inherited.

Who was Ptolemy?

Almost nothing is known of Ptolemy's life beyond that he worked at Alexandria in the middle of the second century and made observations there between about 127 and 141. What survives is the work. The Mathematical Syntaxis, later called by its Arabic name the Almagest, is thirteen books setting out the motions of the sun, moon and planets in a system centred on a motionless Earth. It is not a book of speculation. It is a book you compute with, and to compute with angles Ptolemy needed a table, so Book I contains one: for every arc in a circle, at half-degree steps, the length of the straight line joining its ends. That line is the chord. His circle has a diameter of one hundred and twenty parts, and the entries are given in sixtieths and sixtieths of sixtieths. The construction is careful and he shows all of it. He starts from the chords that can be constructed exactly — the sides of the inscribed hexagon, square, pentagon and decagon — and uses a theorem about a four-sided figure inscribed in a circle to reach the chord of a sum and of a difference, together with a rule for halving an arc. The chord of one degree cannot be constructed by these means, so he traps it between two values he can construct and takes the result to the accuracy he needs. A chord is a length in a stated circle, not a bare ratio. The half-chord — what became the sine — is an Indian innovation of several centuries later, and Ptolemy does not have it. He also wrote the Geography, listing some eight thousand places by position and setting out how to draw a round world on a flat sheet; the Analemma, which solves problems on the sphere by dropping perpendiculars onto three mutually perpendicular planes and reading the answer off a flat figure; the Optics, on vision and refraction; a book on musical ratios; and the Tetrabiblos on astrology, which he regarded as a natural science. Later astronomers, from the tenth century onwards, have accused him of reporting observations that fit his theory more closely than his instruments could have allowed, and the charge is still argued.

Major achievements

• Wrote the Almagest, the standard work on astronomy for some fourteen centuries
• Compiled the table of chords at half-degree steps, the first usable table for computing with angles
• Proved the theorem on the inscribed four-sided figure that made the table constructible
• Trapped the chord of one degree between constructible bounds, since it cannot be built exactly
• Wrote the Geography, listing thousands of places by position with projections for mapping them

Life in brief

• c. 100 — Born: Somewhere in Roman Egypt; the details are unknown.
• 127–141 — Observations at Alexandria: The recorded observations that underpin the Almagest.
• c. 150 — The Almagest: Thirteen books, including the table of chords at half-degree steps.
• c. 150s — The Geography: Some eight thousand places listed by position, with map projections.
• c. 170 — Died: At Alexandria; the date is not securely known.

Explore the life of Ptolemy — five chapters

1. A Life Without a Record — What we know, and how little it is
2. The Table of Chords — A length in a stated circle, not a bare ratio
3. The System — Wrong at the centre and right about the appearances
4. Flattening the Sphere — The Geography, the Analemma, and the shadow of a leaning line
5. Fourteen Centuries — Greek to Arabic to Latin, and then out

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