We should first discuss the perplexing question of the period in which Cleomedes lived. The only certainty here is that On the Circular Motions of the Celestial Bodies discusses the work of Posidonius at length and so is clearly written after the middle of the first century BC. In fact On the Circular Motions of the Celestial Bodies ends with the words (see for example ):-
The preceding teachings are not the author's own opinion but collected from older or more recent summaries; much of it is taken from Posidonius.It is hard to estimate from these words how long after Posidonius the author, Cleomedes, is writing.
Heath  favours a date in the middle of the first century BC. He also points out:-
As [Cleomedes] seems to know nothing of the works of Ptolemy, he can hardly... have lived later than the beginning of the second century AD.Neugebauer, however, disagrees with these conclusions of Heath and proposes that Cleomedes wrote his text around 370 AD. His argument is based on a comment by Cleomedes in the text where he remarks than there are two bright stars (Aldebaran and Antares) such that the rising of one and the setting of the other take place at the same time. These stars Cleomedes claims lie at 15° of their sign. Using Ptolemy's positions for the stars at the time the Almagest Ⓣ was written and Ptolemy's value of 1° per 100 years for precession, Neugebauer gets his date of 371 AD for Cleomedes writings, to which Neugebauer estimates a maximum error of 50 years on either side.
Astronomically Neugebauer's calculations are of course perfectly correct. However they are suspect for a number of other reasons. Firstly the data in On the Circular Motions of the Celestial Bodies does not seem to be due to Cleomedes but to a variety of sources. Of course accepting this argument would make Cleomedes dates later still. Secondly the data in Cleomedes is of widely differing degrees of accuracy. Some is very good, while other data has errors of 20%. Thirdly the actual astronomical event of Aldebaran setting and Antares rising at the same instant can never be observed.
Heath's comment that Cleomedes knows nothing of the works of Ptolemy is also less certain than it might at first appear. Cleomedes is writing an elementary textbook and it is certainly not always the case that one mentions recent research in a low level textbook. For example many elementary textbooks on applied mathematics still use Newton's gravitational methods (and for good reason) 100 years after Einstein gave an improved theory of gravitation.
In  Dicks suggests that the most likely date for Cleomedes is the first century AD and we have taken that as the best available compromise from what is known. That is not to say that Neugebauer's date is impossible. In fact there are other features of the text which would tend to support the fourth century AD as a date, despite the lack of references to Ptolemy. Not least of these is the fact that this was a period when many second rate textbooks of this nature were written and the style is not unlike that of other fourth century AD texts, some of which give the same astronomical data as Cleomedes.
On the Circular Motions of the Celestial Bodies is a work in two volumes. We commented above that it was important for a number of reasons. The most important certainly is that it gives us the best indication that we have of the contents and the style of a text by Posidonius. As Heath comments :-
... the very long first chapter of Book II (nearly half of the Book) ... seems for the most part to be copied bodily from Posidonius.But this is not Cleomedes' main aim in writing the text. It is written to attack the Epicureans who believed among other odd beliefs, that the sun was as large as it looked, namely one foot across. Cleomedes spends much time in his text showing that this is false, but it does seem as if he is going to extremes when he compares Epicureans unfavourably with rats, reptiles and worms. Cleomedes' own philosophical views show that he is a Stoic.
Since On the Circular Motions of the Celestial Bodies is compiled from a number of sources, there is a fair variation in the quality, and in many places the book fails to be consistent where the various sources disagree. Whenever a piece of text is thought to be due to Cleomedes himself, there is much evidence that his understanding of the topic was very limited and, but for the quality of his sources, one feels that Cleomedes would not fare any better than the Epicureans for naiveté.
Lunar eclipses are described well in the text, and the conical shape of the earth's shadow shows an interesting depth of understanding (at least of Cleomedes' source). He also correctly explains the reports of lunar eclipses seen when both the sun and moon are above the horizon as being due to refraction.
Neugebauer gives samples of the sense and nonsense which are mixed in Cleomedes :-
Cleomedes states that no fixed star has an apparent diameter less than one finger (a rather absurd exaggeration) while the apparent diameter of Venus should be two fingers, i.e. 1/6 of the lunar or solar diameter. Of some interest is the remark that the absolute size of fixed stars may reach, or even surpass, the sun ... it is [also] said that the earth, seen from the sun, would appear at best a very small star.One further interest in Cleomedes' work is that it is in On the Circular Motions of the Celestial Bodies that we learn of Eratosthenes method of measuring the circumference of the earth. This is one of the best known of the achievements of early mathematical astronomy and we are indebted to Cleomedes for relating the method. Of course not everyone believes that the story of Eratosthenes' measurement is authentic but, despite this, it is widely accepted.
Finally we should remark that Neugebauer suggests from a study of certain astronomical data given in On the Circular Motions of the Celestial Bodies that Cleomedes lived in the Hellespont on the Black Sea, suggesting the city of Lysimachia. Neugebauer admits that the city of Lysimachia was destroyed in 144 BC which seems at odds with his own date of 370 AD for Cleomedes but he is able to show that despite the disaster of 144 BC records of the city certainly extend up to the fourth century AD. The weakness of Neugebauer's argument must surely be that almost all of Cleomedes' text and data is taken from the works of others so Neugebauer's arguments seem only to give strong evidence for one of Cleomedes' sources having written in Lysimachia.
Article by: J J O'Connor and E F Robertson