**Hipparchus** was a Greek mathematician who compiled an early example of trigonometric tables and gave methods for solving spherical triangles.

- Reasonably enough Hipparchus is often referred to as Hipparchus of Nicaea or Hipparchus of Bithynia and he is listed among the famous men of Bithynia by Strabo, the Greek geographer and historian who lived from about 64 BC to about 24 AD.
- There are coins from Nicaea which depict Hipparchus sitting looking at a globe and his image appears on coins minted under five different Roman emperors between 138 AD and 253 AD.
- This seems to firmly place Hipparchus in Nicaea and indeed Ptolemy does describe Hipparchus as observing in Bithynia, and one would naturally assume that in fact he was observing in Nicaea.
- However, of the observations which are said to have been made by Hipparchus, some were made in the north of the island of Rhodes and several (although only one is definitely due to Hipparchus himself) were made in Alexandria.
- If these are indeed as they appear we can say with certainty that Hipparchus was in Alexandria in 146 BC and in Rhodes near the end of his career in 127 BC and 126 BC.
- It is not too unusual to have few details of the life of a Greek mathematician, but with Hipparchus the position is a little unusual for, despite Hipparchus being a mathematician and astronomer of major importance, we have disappointingly few definite details of his work.
- Only one work by Hipparchus has survived, namely Commentary on Aratus and Eudoxus and this is certainly not one of his major works.
- It is however important in that it gives us the only source of Hipparchus's own writings.
- Although Ptolemy obviously had studied Hipparchus's writings thoroughly and had a deep respect for his work, his main concern was not to transmit it to posterity but to use it and, where possible, improve upon it in constructing his own astronomical system.
- Where one might hope for more information about Hipparchus would be in the commentaries on Ptolemy's Almagest Ⓣ(The major thesis: from the Arabic 'al-majisti' -- the Arabic translation of the Greek 'Mathematike Syntaxis' later translated into Latin as 'Magna Syntaxis').
- There are two in particular by the excellent commentators Theon of Alexandria and by Pappus, but unfortunately these follow Ptolemy's text fairly closely and fail to add the expected information about Hipparchus.
- Since when Ptolemy refers to results of Hipparchus he does so often in an obscure way, at least he seems to assume that the reader will have access to the original writings by Hipparchus, and it is certainly surprising that neither Theon or Pappus fills in the details.
- One can only assume that neither of them had access to the information about Hipparchus on which we would have liked them to report.
- Let us first summarise the main contribution of Hipparchus and then examine them in more detail.
- Hipparchus calculated the length of the year to within 6.5 minutes and discovered the precession of the equinoxes.
- Hipparchus's value of 46" for the annual precession is good compared with the modern value of 50.26" and much better than the figure of 36" that Ptolemy was to obtain nearly 300 years later.
- We believe that Hipparchus's star catalogue contained about 850 stars, probably not listed in a systematic coordinate system but using various different ways to designate the position of a star.
- This last point shows that in any detailed discussion of the achievements of Hipparchus we have to delve more deeply than just assuming that everything in the Ptolemy's Almagest Ⓣ(The major thesis: from the Arabic 'al-majisti' -- the Arabic translation of the Greek 'Mathematike Syntaxis' later translated into Latin as 'Magna Syntaxis') which he does not claim as his own must be due to Hipparchus.
- So major shifts have taken place in our understanding of Hipparchus, first it was assumed that his discoveries were all set out by Ptolemy, then once it was realised that this was not so there was a feeling that it would be impossible to ever have detailed knowledge of his achievements, but now we are in a third stage where it is realised that it is possible to gain a good knowledge of his work but only with much effort and research.
- Let us begin our detailed description of Hipparchus's achievements by looking at the only work which has survived.
- Hipparchus's Commentary on Aratus and Eudoxus was written in three books as a commentary on three different writings.
- Thirdly there was commentary on Aratus by Attalus of Rhodes, written shortly before the time of Hipparchus.
- It is certainly unfortunate that of all of the writings of Hipparchus this was the one to survive since the three books on which Hipparchus was writing a commentary contained no mathematical astronomy.
- As a result of this Hipparchus chose to write at the same qualitative level in the first book and also for much of the second of his three book.
- However towards the end of the second book, continuing through the whole of the third book, Hipparchus gives his own account of the rising and setting of the constellations.
- Towards the end of Book 3 Hipparchus gives a list of bright stars always visible for the purpose of enabling the time at night to be accurately determined.
- As we noted above Hipparchus does not use a single consistent coordinate system to denote stellar positions, rather using a mixture of different coordinates.
- This would tend to confirm that this work by Hipparchus was done near the end of his career.
- Hipparchus's "Commentary" contains his own observations of the stellar positions, great in number but inaccurate in operation, despite all his ability for accurate observations.
- Within an interval of 10 years everything can happen, particularly in the case of a man like Hipparchus.
- Those views which consider Hipparchus's astronomical activities at his two different epochs as similar are completely unfounded.
- Perhaps the discovery for which Hipparchus is most famous is the discovery of precession which is due to the slow change in direction of the axis of rotation of the earth.
- This work came from Hipparchus's attempts to calculate the length of the year with a high degree of accuracy.
- Of course the data needed by Hipparchus to calculate the length of these two different years was not something that he could find over a few years of observations.
- Hipparchus also made a careful study of the motion of the moon.
- In calculating the distance of the moon, Hipparchus not only made excellent use of both mathematical techniques and observational techniques but he also gave a range of values within which be calculated that the true distance must lie.
- Although Hipparchus's treatise On sizes and distances has not survived details given by Ptolemy, Pappus, and others allow us to reconstruct his methods and results.
- Hipparchus's calculations led him to a value for the distance to the moon of between 59 and 67 earth radii which is quite remarkable (the correct distance is 60 earth radii).
- Hipparchus appears to know that 67 earth radii for the distance of the moon comes from this upper limit of solar parallax, while the lower value of 59 earth radii corresponds to the sun being at infinity.
- Hipparchus not only gave observational data for the moon which enabled him to compute accurately the various periods, but he developed a theoretical model of the motion of the moon based on epicycles.
- Hipparchus was also able to give an epicycle model for the motion of the sun (which is easier), but he did not attempt to give an epicycle model for the motion of the planets.
- Finally let us examine the contributions which Hipparchus made to trigonometry.
- The documentary evidence comes from Ptolemy and Theon of Alexandria who explicitly says that Hipparchus wrote a work on chords in 12 books.
- He then goes on to show that the table can be computed from some basic formulae which would be known to Hipparchus, one of which is the supplementary angle theorem, essentially Pythagoras's theorem, and the half-angle theorem.
- The only trace of Hipparchus's tables that survives is in Indian tables which are thought to have been based on that of that of Hipparchus.
- it seems highly probable that Hipparchus was the first to construct a table of chords and thus provide a general solution for trigonometrical problems.
- A corollary of this is that, before Hipparchus, astronomical tables based on Greek geometrical methods did not exist.
- If this is so, Hipparchus was not only the founder of trigonometry but also the man who transformed Greek astronomy from a purely theoretical into a practical predictive science.

Born 190 BC, Nicaea (now Iznik), Bithynia (now Turkey). Died 120 BC, probably Rhodes, Greece.

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Analysis, Ancient Greek, Astronomy, Geography, Geometry, Origin Turkey, Physics

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive