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 needed by a reader.
Ptolemy does this rather cleverly by replacing the abstract geometrical
theories by mechanical ones. Ptolemy also wrote a work on astrology. It may
seem strange to the modern reader that someone who wrote such excellent
scientific books should write on astrology. However, Ptolemy sees it rather
differently for he claims that the Almagest allows one to find the positions of
the heavenly bodies, while his astrology book he sees as a companion work
describing the effects of the heavenly bodies on people's lives.

In a book entitled Analemma he discussed methods of
finding the angles need to construct a sundial which involves the projection of
points on the celestial sphere. In
Planisphaerium he is concerned with
stereographic projection of the celestial sphere onto a plane. This is
discussed in where it is stated:-

In the stereographic projection treated by Ptolemy in
the "Planisphaerium" the celestial sphere is mapped onto the plane of
the equator by projection from the south pole. Ptolemy does not prove the
important property that circles on the sphere become circles on the plane.

Ptolemy's major work Geography, in eight books,
attempts to map the known world giving coordinates of the major places in terms
of latitude and longitude. It is not surprising that the maps given by Ptolemy
were quite inaccurate in many places for he could not be expected to do more
than use the available data and this was of very poor quality for anything
outside the Roman Empire, and even parts of the Roman Empire are severely
distorted. In Ptolemy is described as:-

... a man working [on map-construct without the
support of a developed theory but within a mathematical tradition and guided by
his sense of what is appropriate to the problem.

Another work on Optics is in five books and in it
Ptolemy studies colour, reflection,
refraction, and mirrors of various shapes. Toomer comments in:-

The establishment of theory by experiment, frequently
by constructing special apparatus, is the most striking feature of Ptolemy's
"Optics". Whether the subject matter is largely derived or original,
"The Optics" is an impressive example of the development of a
mathematical science with due regard to physical data, and is worthy of the
author of the "Almagest".

An English translation, attempting to remove the
inaccuracies introduced in the poor Arabic translation which is our only source
of the Optics is given in .

The first to make accusations against Ptolemy was
Tycho Brahe. He discovered that there
was a systematic error of one degree in the longitudes of the stars in the star
catalogue, and he claimed that, despite Ptolemy saying that it represented his
own observations, it was merely a conversion of a catalogue due to Hipparchus corrected for precession to
Ptolemy's date. There is of course definite problems comparing two star
catalogues, one of which we have a copy of while the other is lost.

After comments by
Laplace and Lalande, the next to attack Ptolemy vigorously was Delambre. He suggested that perhaps the
errors came from Hipparchus and that
Ptolemy might have done nothing more serious than to have failed to
correct Hipparchus's data for the time
between the equinoxes and solstices. However
Delambre then goes on to say (see):-

One could explain everything in a less favourable but
all the simpler manner by denying Ptolemy the observation of the stars and
equinoxes, and by claiming that he assimilated everything from Hipparchus, using the minimal value of the
latter for the precession motion.

However, Ptolemy was not without his supporters by any
means and further analysis led to a belief that the accusations made against
Ptolemy by Delambre were false. Boll
writing in 1894 says:-

To all appearances, one will have to credit Ptolemy
with giving an essentially richer picture of the Greek firmament after his
eminent predecessors.

Vogt showed clearly in his important paper that by considering Hipparchus's Commentary on Aratus and
Eudoxus and making the reasonable assumption that the data given there agreed
with Hipparchus's star catalogue, then
Ptolemy's star catalogue cannot have been produced from the positions of the
stars as given by Hipparchus, except
for a small number of stars where Ptolemy does appear to have taken the data from Hipparchus. Vogt writes:-

This allows us to consider the fixed star catalogue as
of his own making, just as Ptolemy himself vigorously states.

The most recent accusations of forgery made against
Ptolemy came from Newton in . He begins this book by stating clearly his
views:-

This is the story of a scientific crime. ... I mean a
crime committed by a scientist against fellow scientists and scholars, a
betrayal of the ethics and integrity of his profession that has forever
deprived mankind of fundamental information about an important area of astronomy
and history.

Towards the end Newton, having claimed to prove every
observation claimed by Ptolemy in the Almagest was fabricated, writes :-

[Ptolemy] developed certain astronomical theories and
discovered that they were not consistent with observation. Instead of
abandoning the theories, he deliberately fabricated observations from the
theories so that he could claim that the observations prove the validity of his
theories. In every scientific or scholarly setting known, this practice is
called fraud, and it is a crime against science and scholarship.

Although the evidence produced by Brahe,
Delambre, Newton and others certainly do show that Ptolemy's errors are
not random, this last quote from is, I
[EFR] believe, a crime against Ptolemy (to use Newton's own words). The book
[8] is written to study validity of these accusations and it is a work which I
strongly believe gives the correct interpretation. Grasshoff writes:-

... one has to assume that a substantial proportion of
the Ptolemaic star catalogue is grounded on those Hipparchan observations
which Hipparchus already used for the
compilation of the second part of his "Commentary on Aratus".
Although it cannot be ruled out that coordinates resulting from genuine
Ptolemaic observations are included in the catalogue, they could not amount to
more than half the catalogue.

... the assimilation of Hipparchan observations can no
longer be discussed under the aspect of plagiarism. Ptolemy, whose intention
was to develop a comprehensive theory of celestial phenomena, had no access to
the methods of data evaluation using arithmetical means with which modern
astronomers can derive from a set of varying measurement results, the one
representative value needed to test a hypothesis. For methodological reason,
then, Ptolemy was forced to choose from a set of measurements the one value
corresponding best to what he had to consider as the most reliable data. When
an intuitive selection among the data was no longer possible ... Ptolemy had to
consider those values as 'observed' which could be confirmed by theoretical
predictions.

As a final comment we quote the epigram which is
accepted by many scholars to have been written by Ptolemy himself, and it
appears in Book 1 of the Almagest, following the list of contents (see for
example [11]):-

Well do I know that I am mortal, a creature of one
day.

But if my mind follows the winding paths of the stars

Then my feet no longer rest on earth, but standing by

Zeus himself I take my fill of ambrosia, the divine
dish.