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Ptolemy is often taught as the man who got the solar system wrong. That is true and also unfair. His book, the Almagest, was the standard astronomy text for about 1,400 years because its predictions worked, and it contains mathematics you study today. This post explains what was in it, how his model of planetary motion worked, and what went wrong, with two worked examples from the book's own trigonometry.
Who was Ptolemy?
According to the MacTutor biography of Ptolemy from the University of St Andrews, Claudius Ptolemy lived from about AD 85 to about AD 165. We know little of his life, but his dated observations were made in Alexandria between 26 March 127 and 2 February 141. His name mixes a Greek-Egyptian name with a Roman one.
MacTutor notes that his work "has generated more discussion and argument than any other" ancient mathematician's. Some historians rank him among the greatest; others accuse him of adjusting data.
What is the Almagest?
It is a treatise in 13 books. Its Greek title meant "The Mathematical Compilation"; later "The Greatest Compilation", which became the Arabic al-majisti and then Almagest in Latin. MacTutor outlines the contents:
Books 1 and 2: the Earth-centred framework and the mathematics, including a table of chords at half-degree intervals.
Book 3: the Sun's motion, with the Earth slightly off the centre of the Sun's circle.
Books 4 to 6: the Moon, building on Hipparchus, then a theory of eclipses.
Books 7 and 8: the fixed stars, precession and a star catalogue of over a thousand stars.
Books 9 to 13: the five known planets, Ptolemy's most original work.
MacTutor quotes the historian Toomer calling it "a masterpiece of clarity and method, superior to any ancient scientific textbook".
How did epicycles and the equant work?
Planets sometimes appear to stop and move backwards against the stars for a few weeks, called retrograde motion. A single circle around the Earth cannot produce that. Ptolemy combined two circles. Each planet moves on a small circle, the epicycle, whose centre moves around a larger circle. That larger circle is centred slightly away from the Earth.
His key innovation, MacTutor explains, was the equant: a point placed opposite the Earth, on the other side of the circle's centre. The epicycle's centre moves at a uniform angular speed as seen from the equant, not from the centre. This let the model fit the observed speeding up and slowing down of planets fairly well.
Worked example 1: Ptolemy's chords and your sines
Ptolemy used a circle of radius 60 and the chord function Crd. MacTutor gives the link to the sine you know: sin a = Crd(2a) ÷ 120.
The chord of 60° in a circle is equal to the radius, because it forms an equilateral triangle with two radii. So Crd 60° = 60.
sin 30° = Crd 60° ÷ 120 = 60 ÷ 120 = 0.5. Correct.
The chord of 90° is the hypotenuse of a right isosceles triangle with legs 60, so Crd 90° = 60√2 ≈ 84.85.
sin 45° = 84.85 ÷ 120 ≈ 0.7071. Correct again.
MacTutor also reports Ptolemy's value of π as 3 17/120. As a decimal that is 3 + 0.14167 = 3.14167, which differs from the true value 3.14159 by less than 0.0001.
Worked example 2: Ptolemy's theorem gives Pythagoras
MacTutor notes Ptolemy used a theorem about four points on a circle to derive his chord addition formulas, the ancestors of sin(a + b). Today it is called Ptolemy's theorem: for a quadrilateral ABCD inscribed in a circle, AC × BD = AB × CD + AD × BC.
Apply it to a rectangle with sides p and q and diagonal d. Every rectangle fits in a circle.
The diagonals are both d, so AC × BD = d².
Opposite sides are equal, so AB × CD = p × p and AD × BC = q × q.
Therefore d² = p² + q².
That is the Pythagorean theorem as a special case. Our post on the history of the Pythagorean theorem follows that rule further back.
Where did the Almagest go wrong?
The obvious error is the fixed central Earth. A subtler one concerns the year. MacTutor reports that Ptolemy confirmed Hipparchus' tropical year of 365¼ days minus 1/300 of a day, while the true value is 365¼ minus 1/128.
1/128 − 1/300 = (300 − 128) ÷ 38,400 = 172 ÷ 38,400 ≈ 0.00448 day per year.
That is about 0.00448 × 1,440 ≈ 6.5 minutes too long each year.
Because so much depended on that value, MacTutor says it caused errors elsewhere, and it is one reason some historians accuse him of fitting observations to expected results. The 2022 study of Hipparchus' recovered catalogue, by Gysembergh, Williams and Zingg, also found that Ptolemy's star catalogue was not based solely on Hipparchus' data, and that Hipparchus' was more accurate.
Common mistakes: saying Ptolemy invented the geocentric idea (it was older, from Aristotle's physics), and assuming epicycles were added at random. They were a deliberate mathematical tool.
What can a student take from Ptolemy?
Write for your reader. Ptolemy set out his plan and taught the maths before using it. Organise your notes the same way.
Judge a model by its predictions, and keep asking whether it is true. These are different questions.
Check your foundations. One wrong constant, the length of the year, spread errors through his work.
For the Sun-centred alternative, read our posts on Copernicus and Kepler's laws. If you cannot picture how two circles produce a backward loop, the Fermi Simulation Lab has interactive simulations for ideas easier to see than read.
Next step: compute sin 60° from Ptolemy's chord of 120°, using the fact that it equals the side of an equilateral triangle inscribed in the circle.
