Book 3 of the CPAE. The risings of the southern constellations. The settings of the southern constellations. The risings of the zodiacal constellations. The settings of the zodiacal constellations. The 24 hourly circles.

Within the context of Greek mathematics, as developed by figures such as Euclid, Aristarchus, Archimedes, Eratosthenes and Apollonius, the contribution of Hipparchus is not nearly so well established. Circumstantial evidence suggests that he was the principal developer of trigonometry, in both its plane and spherical forms, but no ancient source credits him with this directly. It is likely that he was driven to devise trigonometric techniques in order to solve certain specific astronomical problems involving oblique ascensions that had already been discussed in general geometrical terms by Autolycus and Euclid. In pursuit of this, he devised a chord table which related the lengths of lines to subtended angles. Using this, he was able to provide exact solutions to specific astronomical problems concerning the risings and settings of stars, as well solar and lunar theory. As suggested by slender evidence in Plutarch, Hipparchus may also have worked on algebra and combinatorics. From evidence in other writers also, it appears that he took an interest in some aspects of what today is called physics.

Hipparchus is often mentioned by ancient writers, but these reports vary greatly in quality and quantity. The two principal sources are Ptolemy and Strabo, who contribute most of what is known about the general scope of his astronomical and geographical work, respectively. By comparison, other writers tend to be less expert in their reporting or are very brief. These include Cicero, Geminus, Cleomedes, Pliny the Elder, Theon of Smyrna, Vettius Valens, Galen, Sextus Empiricus, Pappus, Firmicus, Theon of Alexandria, as well as several anonymous texts. However, they are still valuable in building up an overall picture of Hipparchus and his contribution to astronomy and mathematics. Many of these ancient sources are discussed here.

Hipparchus realised that for calculation to be effective and useful it must proceed from good observational practice. Without accuracy in the latter, the former, however sophisticated, cannot provide usable answers. Accordingly, he would have made many observations of astronomical facts and phenomena for himself, and for this he would have needed instrumentation. Very little is known for sure about what tools he used. The Antikythera Mechanism, which is roughly contemporary with Hipparchus, demonstrates the engineering skills that were available in his time, and on the reasonable assumption that such skills did not advance much by the time of Ptolemy, the latter’s description of his own instruments provides a basis for discussion. Also important in this context are notions of accuracy, precision and units of measurement. Evidence is brought to bear on these issues in relation to Hipparchus, and how they should be interpreted in the context of his works, using information gleaned from both him and from Ptolemy.

An overview of what is known about the life and times of Hipparchus. Ancient testimony shows that he was born in Nicaea in Asia Minor, and worked latterly in Rhodes, but other details of his life can only be inferred from considerations of the intellectual milieu of his time, together with an examination of his astronomical observations, as detailed by Ptolemy. From these a possible timeline of his life is sketched out.

A detailed introduction to the notional star catalogue of Hipparchus, using such information as may be extracted from the CPAE and other ancient sources. Subjects discussed are the coordinate information provided by Hipparchus, the rationale for the inclusion of certain stars, and what may regarded as a suitable astronomical epoch for the catalogue. This is determined to be the winter of 139/138 BCE. Also discussed are the various difficulties of interpretation of the star descriptions given by Hipparchus and their possible resolutions. The whole discussion is set within the context of a modern, highly accurate precession model.