If we inquire into the basis of the duodecimal system and consider how it can have happened that, in addition to ten, twelve should have been so early and universally singled out from the equal series of numbers, we shall probably be able to find no other source to which it can be referred than a comparison of the solar and lunar periods. Still more than the double hand of ten fingers did the solar cycle of nearly twelve lunar periods first suggest to man the profound conception of an unit composed of equal units, and thereby originate the idea of a system of numbers, the first step towards mathematical thought. The consistent duodecimal development of this idea appears to have belonged to the Italian nation, and to have preceded the first contact with the Greeks.
But when at length the Hellenic trader had opened up the route to the west coast of Italy, the measures of surface remained unaffected, but the measures of length, of weight, and above all of capacity - in other words those definite standards without which barter and traffic are impossible - experienced the effects of the new international intercourse. The oldest Roman foot has disappeared; that which we know, and which was in use at a very early period among the Romans, was borrowed from Greece, and was, in addition to its new Roman subdivision into twelfths, divided after the Greek fashion into four hand-breadths (palmus) and sixteen finger-breadths (digitus). Further, the Roman weights were brought into a fixed proportional relation to the Attic system, which prevailed throughout Sicily but not in Cumae - another significant proof that the Latin traffic was chiefly directed to the island; four Roman pounds were assumed as equal to three Attic minae, or rather the Roman pound was assumed as equal to one and a half of the Sicilian litrae or half-minae[7]. But the most singular and chequered aspect is presented by the Roman measures of capacity, as regards both their names and their proportions. Their names have come from the Greek terms either by corruption (amphora, modius after medimnos, congius from choeus, hemina, cyathus) or by translation (acetabulum from ozubaphon); while conversely zesteis is a corruption of sextarius. All the measures are not identical, but those in most common use are so; among liquid measures the congius or chus, the sextarius, and the cyathus, the two last also for dry goods; the Roman amphora was equalized in water-weight to the Attic talent, and at the same time stood to the Greek metretes in the fixed ratio of 3:2, and to the Greek medimnos of 2:1. To one who can decipher the significance of such records, these names and numerical proportions fully reveal the activity and importance of the intercourse between the Sicilians and the Latins. The Greek numeral signs were not adopted; but the Roman probably availed himself of the Greek alphabet, when it reached him, to form ciphers for 50 and 1000, perhaps also for 100, out of the signs for the three aspirated letters which he had no use for. In Etruria the sign for 100 at least appears to have been obtained in a similar way. Afterwards, as usually happens, the systems of notation among the two neighbouring nations became assimilated by the adoption in substance of the Roman system in Etruria.
In like manner the Roman calendar - and probably that of the Italians generally - began with an independent development of its own, but subsequently came under the influence of the Greeks. In the division of time the returns of sunrise and sunset, and of the new and full moon, most directly arrest the attention of man; and accordingly the day and the month, determined not by cyclic calculation but by direct observation, were long the exclusive measures of time. Down to a late age sunrise and sunset were proclaimed in the Roman market-place by the public crier, and in like manner it may be presumed that in earlier times, at each of the four phases of the moon, the number of days that would elapse from that phase until the next was proclaimed by the priests. The mode of reckoning therefore in Latium - and the like mode, it may be presumed, was in use not merely among the Sabellians, but also among the Etruscans - was by days, which, as already mentioned, were counted not forward from the phase that had last occurred, but backward from that which was next expected; by lunar weeks, which varied in length between 7 and 8 days, the average length being 7 3/8; and by lunar months which in like manner were sometimes of 29, sometimes of 30 days, the average duration of the synodical month being 29 days 12 hours 44 minutes. For some time the day continued to be among the Italians the smallest, and the month the largest, division of time. It was not until afterwards that they began to distribute day and night respectively into four portions, and it was much later still when they began to employ the division into hours; which explains why even stocks otherwise closely related differed in their mode of fixing the commencement of day, the Romans placing it at midnight, the Sabellians and the Etruscans at noon. No calendar of the year had, at least when the Greeks separated from the Italians, as yet been organized, for the names for the year and its divisions in the two languages have been formed quite independently of each other. Nevertheless the Italians appear to have already in the pre-Hellenic period advanced, if not to the arrangement of a fixed calendar, at any rate to the institution of two larger units of time. The simplifying of the reckoning according to lunar months by the application of the decimal system, which was usual among the Romans, and the designation of a term of ten months as a "ring" (annus) or complete year, bear in them all the traces of a high antiquity. Later, but still at a period very early and undoubtedly previous to the operation of Greek influences, the duodecimal system (as we have already stated) was developed in Italy, and, as it derived its very origin from the observation of the fact that the solar period was equal to twelve lunar periods, it was certainly applied in the first instance to the reckoning of time. This view accords with the fact that the individual names of the months - which can only have originated after the month was viewed as part of a solar year - particularly those of March and of May, were similar among the different branches of the Italian stock, while there was no similarity between the Italian names and the Greek. It is not improbable therefore that the problem of laying down a practical calendar which should correspond at once to the moon and the sun - a problem which may be compared in some sense to the quadrature of the circle, and the solution of which was only recognized as impossible and abandoned after the lapse of many centuries - had already employed the minds of men in Italy before the epoch at which their contact with the Greeks began; these purely national attempts to solve it, however, have passed into oblivion.
What we know of the oldest calendar of Rome and of some other Latin cities - as to the Sabellian and Etruscan measurement of time we have no traditional information -is decidedly based on the oldest Greek arrangement of the year, which was intended to answer both to the phases of the moon and to the seasons of the solar year, constructed on the assumption of a lunar period of 29 1/2 days and a solar period of 12 1/2 lunar months or 368 3/4 days, and on the regular alternation of a full month or month of thirty days with a hollow month or month of twenty-nine days and of a year of twelve with a year of thirteen months, but at the same time maintained in some sort of harmony with the actual celestial phenomena by arbitrary curtailments and intercalations. It is possible that this Greek arrangement of the year in the first instance came into use among the Latins without undergoing any alteration; but the oldest form of the Roman year which can be historically recognized varied from its model, not indeed in the cyclical result nor yet in the alternation of years of twelve with years of thirteen months, but materially in the designation and in the measuring off of the individual months. The Roman year began with the beginning of spring; the first month in it and the only one which bears the name of a god, was named from Mars (Martius), the three following from sprouting (aprilis) growing (maius), and thriving (iunius), the fifth onward to the tenth from their ordinal numbers (quinctilis, sextilis, september, october, november, december), the eleventh from commencing (ianuarius)[8], with 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" (mercedoniusat 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 follow them but very imperfectly, for it is scarcely likely that the calendar would be handled with greater skill than was manifested in its original arrangement.
7. I. XIII. Etrusco-Attic and Latino-Sicilian Commerce
8. I. XII. Nature of the Roman Gods
9. From the same cause all the festival-days are odd, as well those recurring every month (