Modelling AssumptionsThe most detailed account of the regulation of the intercalary months is given in Macrobius

Saturnalia1.13.11-14. He explains that the theoretical model was 16 years of intercalations in alternate years, the intercalations being alternately 22 days and 23 days each, followed by 8 years in which there were three 22-day intercalations. Thus there was a 24 year cycle in which there were 7 intercalations of 22 days and 4 of 23. A slightly different account is provided by Censorinus 20.6, who states that intercalary months initially occurred in alternate years in the pre-Julian calendar, until it was realised that this sytem was advancing the calendar against the solar year. The pontiff were then given the power to insert intercalations, which they were said to have abused.Macrobius' account has been used by many scholars, particularly in the nineteenth century, as the basis for their reconstructions of Republican chronology. However, the sequences of intercalation which can be recovered independently from the historical record simply do not agree with this model. While it can be shown that the density of intercalations corresponds to an average frequency of an intercalation roughly every alternate year, they were certainly less frequent than this in Caesar's pontificate, and during the period immediately after the Second Punic War. On the other hand, it is clear that there were several pairs of consecutive intercalations early in the second century. Finally, seasonal and eclipse data shows that Roman months could be up to four months out of sync with their Julian namesakes.

Additionally, recoverable statistics for intercalational lengths in other periods do not conform to the Macrobian model. It has long been known that there were 12 intercalations between A.U.C. 564 = 190 and A.U.C. 586 = 168, and that 9 of these were 23 days long. It is shown here that the month-lengths for the ordinary intercalations between A.U.C. 677 = 77 and A.U.C. 708 = 46 can mostly be recovered, and that either 11 of 12 were 23 days long or 9 out of 11 were 23 days long. It is also extremely likely that all but 4 of the intercalations between A.U.C. 586 = 168 and A.U.C. 696 = 58 were 23 days long. The demonstrable statistics for the second and first centuries simply do not support the 7:4 ratio between 22 and 23 day intercalations implied by the Macrobian model, let alone the 1:1 ratio implied by Censorinus.

Accordingly, it is held here, with A. K. Michels,

The Calendar of the Roman Republic, 169, that Macrobius' description probably reflects the theories of a later chronographer, rather than the actual behaviour of the Republican calendar in the period under review. An alternative conclusion, but one with the same effect, has been recently proposed by L. Magini,Astronomy and Calendar in Ancient Rome: The Eclipse Festivals. Magini argues that the Roman calendar was originally regulated by a highly sophisticated astronomical system derived from the Middle East. He uses Macrobius' account to reconstruct this system and to explain the supposed original significance of the Roman festivals. He proposes that this system was operated by the pontifices, who closely controlled the secret knowledge required to understand it, and that it finally broke down when the calendar was published by the augur Cn. Flavius in, traditionally, A.U.C. 450 = 304 (Livy 9.46). Whatever other merits Magini's analysis may possess, this last proposition has the virtue of completely removing it from the realm of historical verifiability.Thus, I see no justification for using Macrobius' model of intercalary regulation for chronological purposes, and in the reconstruction proposed here it has not been used at any point. The data for the second and first century appears to support Censorinus' account. Hence, so far as possible, this model is based strictly on historical data, combined only with the model of the annual calendar and the nundinal cycle.

However, the data does permit a likely reconstruction of the regulatory provisions of the

Lex Acilia, which controlled intercalation after A.U.C. 563 = 191. These may be summarised as follows:

Intercalary years were normally 378 days between regular years

- There were no more than two regular years in a row

- Pairs of consecutive intercalary years separated by at least one regular year were permitted to compensate for omitted intercalations, notably those omitted during and after the Second Punic war

- The first intercalation of such pairs was 377 days long, the second 378
While the third century data is arguably consistent with the model of Censorinus, there are some weak indications that this period too favoured 23-day intercalations over 22 day ones. As far as I can determine, there are as yet no available synchronisms which are secure or precise enough to allow the actual ratio of lengths in that period to be determined. However, there is a plausible solution for the eclipse of Ennius which implies that all intercalations were 23 days in length at this time. Accordingly, the third century is modelled on this basis, although the model also indicates the range of uncertainty.

Additionally, one other assumption has been made to help choose between alternative solutions: In long periods of uncertainty, model the distribution of intercalations as close to regular behaviour as possible. This is a simple application of Occam's Razor.

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