68 A History ofMathematics
ABOrr
φφFig. 7The chord of an angle. Ifθ=angle AOB= 2 φ, then the chord Crd(θ)=AB= 2 rsinφin our terms, where normallyr=60.Again, we would say that one is ‘caused’ by the daily rotation of the Earth and the other by its
annual rotation around the sun; but that is not how it looks to the naive observer.
Having got this far, Ptolemy needs to calculate the angle between the two axes, and this leads
him into trigonometry or the theory of calculations about angles. The function which he uses is the
‘chord’ of the angle, which modern translators usually write Crdθ; it is the chord in the sector of
the circle whose angle isθ, so Crdθ= 2 rsin(θ/ 2 ), whereris the radius. The angle is written in our
modern units (degrees, minutes, and so on), while the chord is written in units of length, in a circle
whose radius is 60, using (Babylonian type) sexagesimal fractions, as for the angle (see Fig. 7). This
again is interesting; we saw Archimedes calculating with Egyptian type fractions for the circle, and
this may have been usual, but for setting results out formally in a table—which is what Ptolemy
did—sexagesimals are clearly better. They point forwards towards modern decimal notation, as well
as backwards to Babylon, and they were to be used in astronomy continuously until recently.
The table of chords (in intervals of half a degree) is worked out fairly quickly using some basic
Euclidean geometry, with some results of Ptolemy’s own. Square roots are often extracted, as they
need to be, with no indication of how; but we might suppose that something like the method Heron
uses for finding√
720 (see Appendix A) is being used.
Next, (again this is typical of the variety of topics in the book) Ptolemy describes an instrument
for measuring the angle between the two circles, the ecliptic and the equator. He derives the angle
from his measurement, using the table of chords, as 23;51,20◦. We therefore think of the sun as
moving around this circle at a uniform speed through the 12 signs of the zodiac (each of which
takes up roughly 30◦), in one solar year of around 365^14 days. We can then (one would think) find
where the sun will be at any time on any day of the year; and in particular, which sign it will be in.
The problem is that, as seen from the Earth, the sun does not move through the signs at uniform
speed. The 90◦from spring equinox to summer solstice, for example, takes two days longer than the
90 ◦from summer solstice to autumn equinox. The sun is (or appears) slower in travelling through
Taurus (May) than through Leo (August).
This ‘anomaly’, one among many astronomical anomalies which have to be explained, could
be dealt with by supposing that the sun actually had a variable speed, and finding how it varied;
this was Kepler’s idea, arrived at around 1600, and it is what one would say today. But, both for
ease in calculation and because of the general (Platonic) theory, Greek astronomy did not work in
that way. Instead, Ptolemy puts forward two models of how the sun moves. They are, as he says,
equivalent (you may try to work out why), but that is not really important, since the aim of an