## Ptolemy and the Almagest

Ptolemy (Latinized as Claudius Ptolemaeus) was one of the great scholars of antiquity, and mathematical astronomy was dominated by his ideas for nearly 1500 years following his death. Little is known of his life, but he taught in Alexandria and quoted the results of his observations made between AD 127 and 141. He was responsible for a number of great works, each of which place him among the most important ancient authors. The earliest of these is his masterpiece of mathematical astronomy, the Almagest, and others include the Tetrabiblos (on astrology) and the Geography (on mathematical geography). These works exercised a colossal influence over mankind for the next 2000 years. Ptolemy's very wide range of interests is indicated by his works on other subjects, e.g. music, optics, and logic. His Harmony, in which he described musical consonances and their relationship to an underlying universal harmony, was later an inspiration to Kepler.

The name Almagest - which is the name by which his astronomical treatise is usually known - is a corruption by medieval Latin translators of the Arabic

15 Ptolemy Almagest, Book IX, 2. Translation from Toomer (1984).

All quotations from the Almagest are taken from the translation by Toomer (1984).

17 See Martens (2000), Chapter 6.

word for 'the greatest', and was given to the book long after it was written. The original Greek title translates as Mathematical Synthesis or Mathematical Collection and the work, which survives in the original Greek, is sometimes referred to as the Syntaxis. Most of our knowledge of Greek astronomy is derived from this work, which was originally written around AD150, translated into Latin in the twelfth century, and first printed in the sixteenth century. The huge success of the Almagest resulted in the loss of most of the work of Ptolemy's predecessors, notably that of Hipparchus.

Ptolemy was, as far as we know, the first person to show how to convert observational data into numerical values for the parameters so as to make existing planetary models fit the observations. Unlike mathematical astronomers before him, who used unspecified observational data, Ptolemy used specific dated observations - indeed, many of the observations Ptolemy used, which span over 800 years, are only preserved through his work. The rigour with which Ptolemy developed his theories in the Almagest set a very high standard for future astronomical works.

The model of the Solar System as set out in the Almagest will be described in detail below. The various geometrical models he used are quite complicated and so, before describing these, we will begin with a brief description of the overall structure. In Ptolemy's universe, the spherical Earth is situated at the centre of the heaven, which is itself spherical. The Sun, Moon and planets all orbit the Earth and as to their order, Ptolemy wrote:

... we see that almost all the foremost astronomers agree that all the spheres are closer to the earth than that of the fixed stars, and farther from the earth than that of the moon, and that those of the three [outer planets] are farther from the earth than those of the other [two] and the sun, Saturn's being greatest, Jupiter's the next in order towards the earth, and Mars' below that. But concerning the spheres of Venus and Mercury, we see that they are placed below the sun's by the most ancient astronomers, but by some of their successors these too are placed above [the sun's], for the reason that the sun has never been obscured by them [Venus and Mercury] either. To us, however, such a criterion seems to have an element of uncertainty, since it is possible that some planets might indeed be below the sun, but nevertheless not always be in one of the planes through the sun and our viewpoint, but in another [plane], and hence might not be seen passing in front of it, just as in the case of the moon, when it passes below [the sun] at conjunction, no obscuration results in most cases.18

Thus, Ptolemy decided to side with those astronomers who put Mercury and Venus nearer than the Sun as this naturally separates those planets which can be seen at any longitude with respect to the Sun and those which always remain

18 Ptolemy Almagest, Book IX, 1.

close to the Sun. Ptolemy's authority was such that the order of the planets that he proposed (Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn) was accepted by virtually all subsequent astronomers until the sixteenth century.

In the Almagest, the motion of each celestial body is considered in turn and, if all the models are put together, we get the world system sketched in Figure 3.5. The diagram illustrates the epicyclic nature of the planetary models, but not the many other elements used by Ptolemy to reflect more accurately the observational evidence. An important thing to note is the fact that, for Mercury and Venus, the line joining the Earth to the centre of their epicycles is the same as that joining the Earth to the Sun, whereas for the outer planets, which may appear anywhere with respect to the Sun, the radius connecting the planet to the centre of its epicycle is parallel to the Earth-Sun line. Thus, it is evident that the Sun does not simply orbit the Earth like the other planets, but has a much more significant role. The motion of the Sun also plays a role in the lunar theory.

As the Sun has an important function in the theories of all the heavenly bodies, it is logical to begin with a solar theory, and this is what Ptolemy does. He goes on then to consider the Moon and, finally, the planets. For each celestial body, Ptolemy describes the type of phenomena that must be accounted for, goes on to propose a geometrical model suitable for the purpose, shows how to use observations to derive the numerical values of the various geometrical parameters and, finally, produces tables to enable others to determine the position of the body on a given date. Ptolemy obtained data easily for the Sun, Moon, and all the planets from Venus to Saturn, but his over-reliance on the poor available data for Mercury - much the most difficult of the then-known planets to observe - led him to introduce a complicated geometrical device in order accurately to reproduce erroneous data.

The Almagest is a complete exposition of Greek mathematical astronomy divided into thirteen books. As Ptolemy himself says:

We shall try to note down everything which we think we have discovered up to the present time; we shall do this as concisely as possible and in a manner which can be followed by those who have already made some progress in the field. For the sake of completeness in our treatment we shall set out everything useful for the theory of the heavens in the proper order, but to avoid undue length we shall merely recount what has been adequately established by the ancients. However, those topics which have not been dealt with [by our predecessors] at all, or not as usefully as they

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