**Ptolemy** was the most influential of Greek astronomers and geographers of his time. He propounded the geocentric theory of the solar system that prevailed for 1400 years.

- 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.
- 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.
- 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'.
- 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.
- 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 he did not have a great scholar, for Theon 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 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.
- Ptolemy first of all justifies his description of the universe based on the earth-centred system described by Aristotle.
- 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.
- Ptolemy devised new geometrical proofs and theorems.
- 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.
- 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.
- 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.
- In Books 4 and 5 Ptolemy gives his theory of the moon.
- 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.
- 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.
- 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.
- 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.
- Ptolemy's really clever innovation here was to make the motion of CCC 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.
- We will return to discuss some of the accusations made against Ptolemy after commenting briefly on his other works.
- 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.
- 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.
- Ptolemy does this rather cleverly by replacing the abstract geometrical theories by mechanical ones.
- Ptolemy also wrote a work 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.
- 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.
- Another work on Optics is in five books and in it Ptolemy studies colour, reflection, refraction, and mirrors of various shapes.
- 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.
- 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, 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.
- 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.
- Ptolemy had to consider those values as 'observed' which could be confirmed by theoretical predictions.

Born about AD 85, Egypt. Died about AD 165, Alexandria, Egypt.

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African, Analysis, Ancient Arab, Ancient Babylonian, Ancient Greek, Ancient Indian, Applied Maths, Astronomy, Geography, Geometry, Origin Egypt, Number Theory, Physics, Special Numbers And Numerals

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**Oâ€™Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive