Claudius Ptolemy

Born: about 85 in Egypt

Died: about 165 in Alexandria, Egypt

One of the most influential Greek astronomers and geographers of his time, Ptolemy propounded the geocentric theory in a form that prevailed for 1400 years. However, of all the ancient Greek mathematicians, it is fair to say that his work has generated more discussion and argument than any other. We shall discuss the arguments below for, depending on which are correct, they portray Ptolemy in very different lights. The arguments of some historians show that Ptolemy was a mathematician of the very top rank, arguments of others show that he was no more than a superb expositor, but far worse, some even claim that he committed a crime against his fellow scientists by betraying the  ethics and integrity of his profession.

We know very little of Ptolemy's life. He made astronomical observations from Alexandria in Egypt during the years AD 127-41. In fact the first observation which we can date exactly was made by Ptolemy on 26 March 127 while the last was made on 2 February 141. It was claimed by Theodore Meliteniotes in around 1360 that Ptolemy was born in Hermiou (which is in Upper Egypt rather than Lower Egypt where Alexandria is situated) but since this claim first appears more than one thousand years after Ptolemy lived, it must be treated as relatively unlikely to be true. In fact there is no evidence that Ptolemy was ever anywhere other than Alexandria.

His name, Claudius Ptolemy, is of course a mixture of the Greek Egyptian 'Ptolemy' and the Roman 'Claudius'. This would indicate that he was descended from a Greek family living in Egypt and that he was a citizen of Rome, which would be as a result of a Roman emperor giving that 'reward' to one of Ptolemy's ancestors.

We do know that Ptolemy used observations made by 'Theon the mathematician', and this was almost certainly  Theon of Smyrna who almost certainly was his teacher. Certainly this would make sense since  Theon of Smyrna was both an observer and a mathematician who had written on astronomical topics such as  conjunctions, eclipses,  occultations and  transits. Most of Ptolemy's early works are dedicated to Syrus who may have also been one of his teachers in Alexandria, but nothing is known of Syrus.

If these facts about Ptolemy's teachers are correct then certainly in  Theon of Smyrna he did not have a great scholar, for  Theon of Smyrna seems not to have understood in any depth the astronomical work he describes. On the other hand Alexandria had a tradition for scholarship which would mean that even if Ptolemy did not have access to the best teachers, he would have access to the libraries where he would have found the valuable reference material of which he made good use.

Ptolemy's major works have survived and we shall discuss them in this article. The most important, however, is the Almagest which is a treatise in thirteen books. We should say straight away that, although the work is now almost always known as the Almagest that was not its original name. Its original Greek title translates as The Mathematical Compilation but this title was soon replaced by another Greek title which means The Greatest Compilation. This was translated into Arabic as "al-majisti" and from this the title Almagest was given to the work when it was translated from Arabic to Latin.

The Almagest is the earliest of Ptolemy's works and gives in detail the mathematical theory of the motions of the Sun, Moon, and planets. Ptolemy made his most original contribution by presenting details for the motions of each of the planets. The Almagest was not superseded until a century after  Copernicus presented his  heliocentric theory in the De revolutionibus of 1543. Grasshoff writes in:-

Ptolemy's "Almagest" shares with  Euclid's "Elements" the glory of being the scientific text longest in use. From its conception in the second century up to the late Renaissance, this work determined astronomy as a science. During this time the "Almagest" was not only a work on astronomy; the subject was defined as what is described in the "Almagest".

Ptolemy describes himself very clearly what he is attempting to do in writing the work (see for example ):-

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 might have been, will be discussed at length to the best of our ability.

Ptolemy first of all justifies his description of the universe based on the earth-centred system described by  Aristotle. It is a view of the world based on a fixed earth around which the sphere of the fixed stars rotates every day, this carrying with it the spheres of the sun, moon, and planets. Ptolemy used geometric models to predict the positions of the sun, moon, and planets, using combinations of circular motion known as  epicycles. Having set up this model, Ptolemy then goes on to describe the mathematics which he needs in the rest of the work. In particular he introduces trigonometrical methods based on the chord function Crd (which is related to the sine function by sin a = (Crd 2a)/120).

Ptolemy devised new geometrical proofs and theorems. He obtained, using chords of a circle and an  inscribed 360-gon, the approximation

p = 3 17/120 = 3.14166

and, using 3 = chord 60,

3 = 1.73205.

He used formulas for the Crd function which are analogous to our formulas for sin(a + b), sin(a - b) and sin a/2 to create a table of the Crd function at intervals of 1/2 a degree.

This occupies the first two of the 13 books of the Almagest and then, quoting again from the introduction, we give Ptolemy's own description of how he intended to develop the rest of the mathematical astronomy in the work (see for example ):-

[After introducing the mathematical concepts] we have to go through the motions of the sun and of the moon, and the phenomena accompanying these motions; for it would be impossible to examine the theory of the stars thoroughly without first having a grasp of these matters. Our final task in this way of approach is the theory of the stars. Here too it would be appropriate to deal first with the sphere of the so-called 'fixed stars', and follow that by treating the five 'planets', as they are called.

In examining the theory of the sun, Ptolemy compares his own observations of  equinoxes with those of  Hipparchus and the earlier observations Meton in 432 BC. He confirmed the length of the  tropical year as 1/300 of a day less than 365 1/4 days, the precise value obtained by  Hipparchus. Since, as Ptolemy himself knew, the accuracy of the rest of his data depended heavily on this value, the fact that the true value is 1/128 of a day less than 365 1/4 days did produce errors in the rest of the work. We shall discuss below in more detail the accusations which have been made against Ptolemy, but this illustrates clearly the grounds for these accusations since Ptolemy had to have an error of 28 hours in his observation of the equinox to produce this error, and even given the accuracy that could be expected with ancient instruments and methods, it is essentially unbelievable that he could have made an error of this magnitude. A good discussion of this strange error is contained in the excellent article .

Based on his observations of  solstices and equinoxes, Ptolemy found the lengths of the seasons and, based on these, he proposed a simple model for the sun which was a circular motion of uniform angular velocity, but the earth was not at the centre of the circle but at a distance called the eccentricity from this centre. This theory of the sun forms the subject of Book 3 of the Almagest.

In Books 4 and 5 Ptolemy gives his theory of the moon. Here he follows  Hipparchus who had studied three different periods which one could associate with the motion of the moon. There is the time taken for the moon to return to the same longitude, the time taken for it to return to the same velocity (the anomaly) and the time taken for it to return to the same latitude. Ptolemy also discusses, as  Hipparchus had done, the  synodic month, that is the time between successive  oppositions of the sun and moon. In Book 4 Ptolemy gives  Hipparchus's epicycle model for the motion of the moon but he notes, as in fact  Hipparchus had done himself, that there are small discrepancies between the model and the observed parameters. Although noting the discrepancies,  Hipparchus seems not to have worked out a better model, but Ptolemy does this in Book 5 where the model he gives improves markedly on the one proposed by  Hipparchus. An interesting discussion of Ptolemy's theory of the moon is given in .

Having given a theory for the motion of the sun and of the moon, Ptolemy was in a position to apply these to obtain a theory of eclipses which he does in Book 6. The next two books deal with the fixed stars and in Book 7 Ptolemy uses his own observations together with those of  Hipparchus to justify his belief that the fixed stars always maintain the same positions relative to each other. He wrote (see for example ):-

If one were to match the above alignments against the diagrams forming the constellations on  Hipparchus's celestial globe, he would find that the positions of the relevant stars on the globe resulting from the observations made at the time of  Hipparchus, according to what he recorded, are very nearly the same as at present.

In these two book Ptolemy also discusses precession, the discovery of which he attributes to  Hipparchus, but his figure is somewhat in error mainly because of the error in the length of the tropical year which he used. Much of Books 7 and 8 are taken up with Ptolemy's star catalogue containing over one thousand stars.

The final five books of the Almagest discuss planetary theory. This must be Ptolemy's greatest achievement in terms of an original contribution, since there does not appear to have been any satisfactory theoretical model to explain the rather complicated motions of the five planets before the Almagest. Ptolemy combined the epicycle and  eccentric methods to give his model for the motions of the planets. The path of a planet P therefore consisted of circular motion on an epicycle, the centre C of the epicycle moving round a circle whose centre was offset from the earth. Ptolemy's really clever innovation here was to make the motion of C uniform not about the centre of the circle around which it moves, but around a point called the equant which is symmetrically placed on the opposite side of the centre from the earth.

The planetary theory which Ptolemy developed here is a masterpiece. He created a sophisticated mathematical model to fit observational data which before Ptolemy's time was scarce, and the model he produced, although complicated, represents the motions of the planets fairly well.

Toomer sums up the Almagest in as follows:-

As a didactic work the "Almagest" is a masterpiece of clarity and method, superior to any ancient scientific textbook and with few peers from any period. But it is much more than that. Far from being a mere 'systemisation' of earlier Greek astronomy, as it is sometimes described, it is in many respects an original work.

We will return to discuss some of the accusations made against Ptolemy after commenting briefly on his other works. He published the tables which are scattered throughout the Almagest separately under the title Handy Tables. These were not merely lifted from the Almagest however but Ptolemy made numerous improvements in their presentation, ease of use and he even made improvements in the basic parameters to give greater accuracy. We only know details of the Handy Tables through the commentary by  Theon of Alexandria but in  the author shows that care is required since  Theon was not fully aware of Ptolemy's procedures.

Ptolemy also did what many writers of deep scientific works have done, and still do, in writing a popular account of his results under the title Planetary Hypothesis. This work, in two books, again follows the familiar route of reducing the mathematical skills

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