**Callippus of Cyzicus**are guesses but he is known to have been working with Aristotle in Athens starting in 330 BC.

We know that Callippus was a student in the School of Eudoxus. We also know that he made his astronomical observations on the shores of the Hellespont, which can be deduced from the observations themselves. Simplicius writes in his commentary on *De caelo* by Aristotle (see for example [1]):-

Callippus made accurate determinations of the lengths of the seasons and constructed a 76 year cycle comprising 940 months to harmonise the solar and lunar years which was adopted in 330 BC and used by all later astronomers. This calendar of Callippus is examined in detail by van der Waerden in [6]. Ptolemy gave us an accurate date for the beginning of this cycle in 330 BC in theCallippus of Cyzicus, having studied with Polemarchus, Eudoxus's pupil, following him to Athens dwelt with Aristotle, correcting and completing, with Aristotle's help, the discoveries of Eudoxus.

*Almagest*saying that year 50 of the first cycle coincided with the 44

^{th}year following the death of Alexander.

The Callippic period is based on the Metonic period devised by Meton (born about 460 BC). Meton's observations were made in Athens in 432 BC but he gave a length for the year which was ^{1}/_{76} of a day too long. The relation between Callippus's period and that of Meton are explained in [2] as follows:-

Callippus introduced a system of 34 spheres to explain the motions of the heavenly bodies. The Sun, Moon, Mercury, Venus and Mars each had five spheres while Jupiter and Saturn had four and the stars had one. This addition of six spheres over the system proposed by Eudoxus increased the accuracy of the theory while preserving the belief that the heavenly bodies had to possess motion based on the circle since that was the 'perfect' path. Heath writes [4]:-Callippus of Cyzicus(c.370-300BC)was perhaps the foremost astronomer of his day. He formed what has been called the Callippic period, essentially a cycle of four Metonic periods. It was more accurate than the original Metonic cycle and made use of the fact that365.25days is a more precise value for the tropical year than365days. The Callippic period consisted of4×235, or940lunar months, but its distribution of hollow and full months was different from Meton's. Instead of having totals of440hollow and500full months, Callippus adopted441hollow and499full, thus reducing the length of four Metonic cycles by one day. The total days involved therefore became(441×29)+(499×30), or27,759and27,759÷(19×4)gives365.25days exactly. Thus the Callippic cycle fitted940lunar months precisely to76tropical years of365.25days.

Other contributions of Callippus to mathematical astronomy included his observation of the inequality in the lengths of the seasons. He accounted for this in his model by making the velocity of the Sun vary through the year and this was achieved with the two extra spheres described above.Callipus tried to make the system of concentric spheres suit the phenomena more exactly by adding other spheres; he left the number of spheres at four in the case of Jupiter and Saturn, but added one each to the other planets and two each in the case of the sun and the moon ... . This would substitute for the hippopede[see the Eudoxus article]a still more complicated elongated figure ...

The Callippic period contributed to the accuracy of later astronomical theories. Kieffer writes in [1]:-

Although the system of concentric spheres gave way to epicycles and eccentrics, Callippus's period became the standard for correlating observations accurately over many centuries, and thus contributed to the accuracy of later astronomical theories.

**Article by:** *J J O'Connor* and *E F Robertson*