Claudius Ptolemy (85 - 165)

Ptolemy, born in Egypt, is one of those historical figures about whom little is really known, and that which is suspected can be used to portray him in all manner of ways - as a genius, and as nothing more than an orator.

What is known about the man at the centre of this enigma is that he made astronomical calculations in the Egyptian city of Alexandria from 127AD-141AD.

Luckily we do have copies of much of his work. The most famous is the Mathematical Syntaxis (widely called the Almagest) which proposed an ingenious model for the universe.

The model came to be called the Ptolemic system. This model had the natural - but wrong - idea of the earth as the centre of the universe, with the moon, sun and other known planets orbiting it in a series of ever more complicated curves to take account of known patterns of sightings and observed positions.

He created a system of planetary movement based on epicycles - this is the curve traced out by a point on the circumference of a circle, whose centre moves on the circumference of another circle.

The circle that represented the orbit of the planet about its centre was called an “epicycle”, and the circle on which this centre moved was called the “deferent”.

However, to accord with observation even this was not enough.

Ptolemy adapted his model by making it more sophisticated - piling epicycles on other epicycles, and in another refinement some of the centres involved were offset.

Ptolemy’s system involved at least 80 epicycles to explain the motions of the Sun, the Moon, and the five planets known in his time - Mercury, Venus, Sun, Mars, Jupiter and Saturn. Ptolemy’s book - The Almagest - was also an encyclopaedia of astronomy.

It predicted eclipses; it had a star catalogue containing 48 constellations, using the names we still use today; and even a breakdown on the length of months.

Ptolemy’s mathematics

What Ptolemy did which set his work apart from other ancient astronomers was to create a mathematical model which absorbed and accounted for the things that people observed in the heavens.

But behind the model lay some equally sophisticated mathematics and the rest of his book is devoted to giving a good account of this. His model, at its heart, was based on circles so he constructed a table giving the lengths of chords in a circle. In particular he introduced a trigonometric function which he called ‘the chord function’ and denoted it by Crd. This is related to our modern sine function, by the formula:

Crd(a) = 120.\sin({a \over 2})

He developed and used formulas for his Crd function which are analogous to the modern addition and subtraction trigonometric formulae:

\sin(a + b) = \sin a.\cos b + \sin b.\cos a;

\sin(a - b) = \sin a.\cos b - \sin b.\cos a.

These were of practical use and enabled him to create a table of the Crd function at intervals of half a degree. He obtained, using his tables on a 360-gon (that is, a regular polygon with 360 sides) inscribed in a circle, the approximation

\pi \approx {317 \over 120} = 3.14166

and, since Crd(60\deg)=\sqrt{3}

he also obtained from his tables, the approximation,

\sqrt{3} \approx 1.73205

Let us conclude with Ptolemy’s own words from the introduction to his book,

“Well do I know that I am mortal, a creature of one day. But if my mind follows the winding paths of the stars then my feet no longer rest on earth, but standing by Zeus himself I take my fill of ambrosia, the divine dish.”