The following is from Historical Eclipses and Earth’s Rotation, F. Richard Stephenson, pages 93-97
More celestial observations are preserved from Babylon than from any other contemporary civilisation. Yet until about a century ago, when large numbers of clay tablets devoted to astronomy began to be unearthed at the site of Babylon, little was known about the achievements of the skywatchers of this once great city. What could be established was mainly based on ancient Greek texts and the Old Testament. Both the Prophet Isaiah (e.g. 47:13) and the ancient Greek writer Strabo (Geography, XVI, 1.6) stress the Babylonian preoccupation with astrology. ... The ancient Greek historian Diodorus Siculus (Library of History, II, 9) implies that the lofty ziggurat - built during the reign of Nebuchadrezzar II (604-563 BC) was used as an observatory. ...
Among writers of the ancient Greek and Roman world whose works are still extant, only the great Alexandrian astronomer Claudius Ptolemy (c. AD 150) hints at the true scale on which celestial observation was practised at Babylon. In his Mathematike Syntaxis (Mathematical Systematic Treatise) - which later became known as the Almagest - Ptolemy specifically mentions sets of Babylonian eclipse observations to which he had access. ...
It is regrettable that Ptolemy actually cites no more than ten Babylonian records of lunar eclipses from the apparently large number available to him. Furthermore, no observations of solar eclipses from Babylon are preserved in the Almagest, although there are a few references to other celestial phenomena reported from this site, such as conjunctions of planets with stars. These various observations range in date from 721 to 229 BC, lunar eclipses covering the period from 721 to 382 BC.
The discovery of vast numbers of astronomical cuneiform texts at the site of Babylon during the 1870s and 1880s was eventually to revolutionise knowledge of Babylonian astronomy. These texts, which are in the form of inscribed clay tablets, range in date from about 730 BC to AD 75. Many of the eclipse records which they contain have proved so important in studies of the Earth's past rotation. ... In particular, the earlier observations must have formed the ultimate source of the material used by Hipparchus and Ptolemy. Despite the existence of this huge archive, the ten Babylonian lunar eclipse records cited in the Almagest form an interesting set of data and deserve to be analysed in their own right. ... It should be stressed that [the Assyrian cuneiform tablets of the eighth and seventh centuries BC] are mainly in the form of astrological texts and - unlike the Babylonian material - contain little quantitative information. 4.2 The Babylonian lunar eclipse records in the Almagest The very earliest astronomical records quoted in the Almagest are exclusively of Babylonian origin. ... Ptolemy evidently did not have access to material much older than 721 BC - the earliest Babylonian eclipse record which he discusses. He states (III, 7) that beginning with the reign of King Nabonassar (correct name: Nabu-nasir) of Babylon (747-733 BC) 'the ancient observations are, on the whole, preserved down to our own time'. This is apparently why Ptolemy chose the era of Nabonassar (747 BC) for numbering years. ... It is a fact that Babylonian history and chronology are extremely weak in the two centuries immediately prior to Nabonassar's reign. Even the names of several kings who ruled during this interval are unknown (cf. Oates, 1979, p.201). However, beginning with Nabonassar, Babylonian chronology is securely established.
It seems very likely that Ptolemy did not compile the list of Babylonian observations himself. The evidence points instead to Ptolemy's great predecessor Hipparchus of Rhodes (c. 150 BC). As noted above, Ptolemy specifically mentions a series of eclipse observations which had been 'brought over from Babylon' and investigated by Hipparchus. Toomer (1988) is of the opinion that the entire Babylonian record available to Ptolemy was compiled by Hipparchus, further suggesting that Hipparchus 'arranged them in a form suitable for use by Greek astronomers'. It is clear from reading the accounts of individual eclipses quoted by Ptolemy that he did not receive them at first hand. Sadly, apart from the material preserved in the Almagest, all trace of Hipparchus' compilation has long been lost.
No ancient manuscripts of the Almagest now exist; the earliest copies date from the ninth century AD (Toomer, 1984, pp. 2-4). However, as Toomer points out, there is in general very close accord between the text of individual manuscripts.
For the very earliest eclipse observations (721 and 720 BC) which he cites, Ptolemy numbers the year from the accession of Mardokempad (Marduk-apla-iddin), who was ruler of Babylon at the time. However, all later years are counted from the era of Nabonassar. Although the eclipse dates would originally be expressed in terms of the Babylonian luni-solar calendar, Ptolemy invariably specifies the month in terms of the Egyptian calendar. Numbering days from a fixed epoch was considerably simplified on this latter system. Each Egyptian year contained 12 equal months of 30 days followed by 5 epagomenal ('additional') days. This fixed year of 365 days was not adjusted to the solar year by intercalation so that the first day of the year gradually retrograded through the seasons, making a complete circuit in 1460 years (the Sothic cycle). ...
In quoting days of the month for lunar eclipses, Ptolemy was in the habit of using double dates - e.g. Thoth 18/19. This is because although the civil day began at sunset in Babylon, according to Egyptian convention it commenced at the following sunrise (Toomer, 1984, p. 12). ...
Comparison with the many lunar records found on the astronomical tablets recovered from Babylon makes it clear that the eclipse times quoted by Ptolemy are not in original form (i.e. using time-degrees) but have been modified to correspond to the Greek method (equinoctial or seasonal hours). Presumably Hipparchus was responsible for these reductions. It is a pity that the original measurements are not preserved; it is likely that some loss of accuracy would occur when the times were reduced to the Greek system. However, no attempt at restoration is possible. Only in a single case (523 BC) is there a parallel inscription on an extant cuneiform tablet, and even this is problematical.