A History of Mathematics From Mesopotamia to Modernity

66 A History ofMathematics

dates of both are in dispute), Diophantus invented an ‘algebraic’ notation which might have made

such questions easier; but although his works were studied and commented, they had no apparent

influence on the solving of everyday puzzles for more than a thousand years.^6

Exercise 5.(a) Show that, in a right-angled triangle with short sides a,b and other notation as above,

r=^12 (a+b−c), (b) explain what Heron is doing in terms of this formula.

4. Astronomy, and Ptolemy in particular

Mathematicians sometimes have difficulty in thinking about astronomy as a part of their subject;

however, historically it is essential. Unified as a subject of study with mathematics from very early

times, it has often provided work for mathematicians and motivation for much of their enquiries;

the trigonometric functions (sin, cos, and so on) and the study of geometry on the sphere owe more

to the study of the heavens, at least in their beginnings, than to the geography of the Earth.

There are considerable problems, touched on lightly in the last chapter, in setting up a

mathematical astronomy. One could list some of them:

• find the length of the day


• find the length of the year


• before doing either of these, find a reliable way of measuring time


• find the path of the sun in the heavens, at a given place, at a given time of year.

It is worth taking time to think about these problems, and what instruments, observations, and/or

calculations you would need to answer them. All of them are non-trivial, and all need to be

understood with some finesse before we can begin to answer the subtler questions about the paths

of stars, planets, and so on, let alone eclipses of the sun and moon. Necessarily, a great deal must

already have been worked out for Plato (if it was his idea, as tradition has it) to pose the problem of

accounting geometrically for the motions of the various heavenly bodies. The restriction classically

put on these was that they must be ‘composed of uniform circular motions’—we shall see later what

this implied. The restriction had a practical advantage (circular motion is easy to work with) and

a philosophical one (it corresponds to some ideal of perfection). The explanation, as I mentioned

in the previous chapter, had to be descriptive, giving you the possibility of predicting where a body

would be at a given time. This, incidentally, made it essential for astrology, which has almost always

been a major concern for astronomers and by extension for mathematicians in general.^7 On the

other hand, there was no call for a physical explanation of what force might make the heavenly

bodies move; this theory, which is not easy to reconcile with the descriptive one, was given by

Aristotle, and does not really concern us.

The major textbook of astronomy which has survived is the work of Ptolemy, who worked in

Egypt in the second centuryce. While he named it theMathematical Syntaxis(‘treatise’), from the

time of the first Arabic translations it came to acquire the name ‘Almagest’ (=Arabicized Greek

6. Specifically, until the Renaissance (chapter 6). Hypatia among the Greeks and Qusta ibn L ̄uqa among the Arabs are known as
   students and commentators of Diophantus, and half of his work only survives in ibn L ̄uqa’s translation.


7. Whether or not one believes in the influence of the planets’ positions at a given time, the actual calculations which determine
   them are often quite hard mathematics. Ptolemy, in hisHandy Tables, simplified the work so that the practising astrologer could look
   up the answer without reading his heavily theoreticalAlmagest.