3. By combining in a single system the solar and lunar years as described above. This has been done in various ways by different peoples, by adopting a cycle of years of varying length, in which the resultants of the two bases of calculation should be brought into harmony as nearly as possible. In other words, though the difference between a single solar year and a single lunar year is more than 11 days, it is possible, by taking a number of years together and reckoning them as lunar years, one or more of them being lengthened by an additional month, to make the whole period very nearly coincide with the same number of solar years. Thus the Athenians adopted for this purpose at different times groups or cycles of 8 and 19 years. In the Octaeteris or 8-year cycle there were 99 lunar months, 3 months of 30 days being added in 3 of the 8 years—a plan which falls short of accuracy by about 36 hours. Later on a cycle of 19 years was substituted for this, in which the discrepancy was greatly reduced. The Roman year in historical times was calculated on a system of this kind, though with such inaccuracy and carelessness as to lose all real relation to the revolutions both of earth and moon. But there was a tradition that before this historical calendar came into use there had been another system, which the Romans connected with the name of Romulus. This year was supposed to have consisted of 10 months, of which 4—March, May, July, October—had 31 days, and the rest 30; in all 304. But this was neither a solar nor a lunar year; for a lunar year of 10 months = 295 days 7 hours 20 minutes, while a solar year = 365-1/4. Nor can it possibly be explained as an attempt to combine the two systems. Mommsen has therefore conjectured that it was an artificial year of 10 months, used in business transactions, and in periods of mourning, truces[2], &c., to remedy the uncertainty of the primitive calculation of time; and that it never really was the basis of a state calendar. This view has of course been the subject of much criticism[3]. But no better solution has been found; the hypothesis that the year of 10 months was a real lunar year, to which an undivided period of time was added at each year’s end, to make it correspond with the solar year and the seasons, has not much to recommend it or any analogy among other peoples. It was not, then, the so-called year of Romulus which was the basis of the earliest state-calendar, but another system which the Romans themselves usually ascribed to Numa. This was originally perhaps a lunar year; at any rate the number of days in it is very nearly that of a true lunar year (354 days 8 hours 48 minutes)[4]. It consisted of 12 months, of which March, May, July, October had 31 days, and the rest 29, except February, which had 28. All the months therefore had an odd number of days, except the one which was specially devoted to purification and the cult of the dead; according to an old superstition,