A History of Mathematics From Mesopotamia to Modernity

Greeks,Practical andTheoretical 69

Planet (or sun)

Deferent

Epicycle

Earth

Fig. 8The epicycle model. The planet moves clockwise round a small circle (epicycle) whose centre moves anticlockwise round a

large circle (deferent) centred on the Earth. This explains why some planets (e.g. Mars) sometimes move backwards in relation to the

stars.

explanation is not so much to say what ‘really’ happens as to give an accurate description from

which you can make predictions. Again, the physics, as we would call it, is not considered. In the

first model (‘epicyclic’), the sun moves at constant speed round a small circle or ‘epicycle’; and the

centre of the epicycle itself moves at constant speed round the ecliptic, with the Earth as centre (see

Fig. 8).

The second (‘eccentric’) supposes that the sun is travelling at uniform speed around a circle, but

that the Earth is not the centre of the circle. It follows that the sun appears to be travelling more

slowly when it is further from the Earth than when it is nearer (see Exercise 8). In Appendix B, I give

the detailed working out by Ptolemy, from his observations, of where the centre of the orbit is in

relation to the Earth. This is a detailed piece of Greeknumericalmathematics, based on the geometric

tradition. Is it practical? Very much so, in that it makes possible (the beginnings of ) the calculation

of where the sun will be. But it is supported by a formidable theoretical apparatus of results about

chords in a circle, angles, and so on. Considered as a whole—and this is the justification for devoting

so much time to his work—Ptolemy’sAlmagestgives more of an impression of the range and variety

of Greek mathematics than any other text which we have.

Exercise 6.Show thatCrd( 60 ◦)= 60 , andCrd( 36 ◦)= 60 ((

√

5 − 1 )/ 2 ).

Exercise 7.How would you findCrd(θ/ 2 )givenCrd(θ)?

Exercise 8.Explain the variation in the sun’s apparent speed, on the eccentric hypothesis.

Research problem.Find the two reasons why the length of the day varies (a) as we would understand it,

(b) as Ptolemy would have put it. (This is called the ‘equation of time’.)

5. On the uncultured Romans

With the Greeks geometry was regarded with the utmost respect and consequently none were held in greater honour

than mathematicians, but we Romans have delimited the size of this art to the practical purposes of measuring and

calculating. (Cicero,Tusculan Disputations, tr. Serafina Cuomo, in Cuomo 2001, p. 192)

The above quotation heads Serafina Cuomo’s chapter on the Romans; and her recent book makes
the first serious attempt to investigate and indeed question a view of their mathematics accepted