reference presumably to the renewal of agricultural operations that followed midwinter and the season of rest, the twelfth, and in an ordinary year the last, from cleansing (-februarius-).  To this series recurring in regular succession there was added in the intercalary year a nameless “labour-month” (-mercedonius-) at the close of the year, viz. after February.  And, as the Roman calendar was independent as respected the names of the months which were probably taken from the old national ones, it was also independent as regarded their duration.  Instead of the four years of the Greek cycle, each composed of six months of 30 and six of 29 days and an intercalary month inserted every second year alternately of 29 and 30 days (354 + 384 + 354 + 383 = 1475 days), the Roman calendar substituted four years, each containing four months—­the first, third, fifth, and eighth—­of 31 days and seven of 29 days, with a February of 28 days during three years and of 29 in the fourth, and an intercalary month of 27 days inserted every second year (355 + 383 + 355 + 382 = 1475 days).  In like manner this calendar departed from the original division of the month into four weeks, sometimes of 7, sometimes of 8 days; it made the eight-day-week run on through the years without regard to the other relations of the calendar, as our Sundays do, and placed the weekly market on the day with which it began (-noundinae-).  Along with this it once for all fixed the first quarter in the months of 31 days on the seventh, in those of 29 on the fifth day, and the full moon in the former on the fifteenth, in the latter on the thirteenth day.  As the course of the months was thus permanently arranged, it was henceforth necessary to proclaim only the number of days lying between the new moon and the first quarter; thence the day of the newmoon received the name of “proclamation-day” (-kalendae-).  The first day of the second section of the month, uniformly of 8 days, was—­in conformity with the Roman custom of reckoning, which included the -terminus ad quem- —­designated as “nine-day” (-nonae-).  The day of the full moon retained the old name of -idus- (perhaps “dividing-day").  The motive lying at the bottom of this strange remodelling of the calendar seems chiefly to have been a belief in the salutary virtue of odd numbers;(9) and while in general it is based on the oldest form of the Greek year, its variations from that form distinctly exhibit the influence of the doctrines of Pythagoras, which were then paramount in Lower Italy, and which especially turned upon a mystic view of numbers.  But the consequence was that this Roman calendar, clearly as it bears traces of the desire that it should harmonize with the course both of sun and moon, in reality by no means so corresponded with the lunar course as did at least on the whole its Greek model, while, like the oldest Greek cycle, it could only follow the solar seasons by means of frequent arbitrary excisions, and did in all probability