A History of Mathematics From Mesopotamia to Modernity

Greeks,Practical andTheoretical 67

‘al-majist ̄i’, the greatest), by which it is usually known. Like Euclid, it was a standard textbook for
over a thousand years, much commented and occasionally revised and criticized in the light of new
theories but only losing its popularity in the seventeenth century as the theories of Copernicus,
Kepler, and Newton came to form a solid alternative of a very different kind.
Ptolemy was not the first to develop the theory found in his book; the ideas on how the planets
move were supposedly first framed by Apollonius (second centurybce) and dealt with in a textbook
by Hipparchus (a little later); but their works, superseded by theAlmagest, have not survived. What
Ptolemy apparently added to Hipparchus was more accuracy and a simpler method of calculation.
His book, however, though a major primary source and very interesting (translation by Toomer
1984), is not a straightforward read, and this also is worth thinking about. Where a mathematics
textbook typically has one general subject, and starts from first principles to show you how to solve
quadratic equations, or prove Pythagoras’s theorem, or differentiate, an astronomy textbook has to
set up a large and complex apparatus of general theory, observation, and particular verifications;
and this is true of Newton’sPrincipiapart III (the explanation of the System of the World) as much
as of Ptolemy. In theAlmagest, Ptolemy first goes through the foundational assumptions: that the
Earth is spherical and fixed in the middle of the universe, that the heavens move in circles around
it, and so on. At this point, we have to set aside the fact that we ‘know’ that Ptolemy’s system is
wrong, since it is in fact an extremely good mathematical explanation of what is observed, and the
mathematics is what should concern us. So we must accept these assumptions. We then observe
that the stars havetwomotions: in a day they describe circles about the pole star, and in a year
they describe circles about a different axis, defined by the path of the sun—the ‘ecliptic’ (see Fig. 6).

North

celestial

pole North

polar

distance

Star

Earth

Autumn

equinox

Celestial

equator

Ecliptic

South

ecliptic

pole

South

celestial

pole

Spring

equinox

Degrees along

celestial equator

North

ecliptic

pole

Celestial

latitude

Celestial

longitude

Fig. 6The relation between the Earth, the poles, the equator, and the ecliptic in the geocentric (Earth-centred) model.