The Greeks adopted this calendar, and, in consequence, considered the days of their month as representing the changes of the moon: the last day of the month was called evn kai véa, “the old and new," as belonging to both the waning and the reappearing moon': and their festivals and sacrifices, as determined by the calendar, were conceived to be necessarily connected with the same periods of the cycles of the sun and moon. "The laws and the oracles," says Geminus, "which directed that they should in sacrifices observe three things, months, days, years, were so understood." With this persuasion, a correct system of intercalation became a religious duty. The above rule of alternate months of 29 and 30 days, supposes the length of the months 29 days and a half, which is not exactly the length of a lunar month. Accordingly the Months and the Moon were soon at variance. Aristophanes, in "The Clouds," makes the Moon complain of the disorder when the calendar was deranged. Οὐκ ἄγειν τὰς ἡμέρας Οὐδὲν ὀρθῶς, ἀλλ' άνω τε καὶ κάτω κυδοιδοπαν 18 Aratus says of the moon, in a passage quoted by Geminus, p. 33: Αιει δ ̓ ἄλλοθεν ἄλλα παρακλίνουσα μετωπὰ Ειρῃ, ἀποστάιη μήνος περιτέλλεται πως. As still her shifting visage changing turns By her we count the monthly round of morns. Ἡνίκ ̓ ἂν ψευσθῶσι δείπνου κἀπίωσιν οἴκαδε Τῆς ἑορτῆς μὴ τυχόντες κατὰ λόγον τῶν ἡμερῶν. CHORUS OF CLOUDS. Nubes, 615-19. The Moon by us to you her greeting sends, The correction of this inaccuracy, however, was not pursued separately, but was combined with another object, the securing a correspondence between the lunar and solar years, the main purpose of all early cycles. Sect. 5.-Invention of Lunisolar Years. THERE are 12 complete lunations in a year; which according to the above rule, would make 354 days, leaving 121 days of difference between such a lunar year and a solar year. It is said, that at an early period, this was attempted to be corrected by interpolating a month of 30 days every alternate year; and Herodotus 20 relates a conversation of Solon, implying a still ruder mode of intercalation. This 19 This passage is supposed by the commentators to be intended as a satire upon those who had introduced the cycle of Meton (spoken of in Sect. 5), which had been done at Athens a few years before "The Clouds" was acted. 20 B. i. c. 15. can hardly be considered as an improvement in the Greek calendar already described. The first cycle which produced any near correspondence of the reckoning of the moon and the sun, was the Octaëteris, or period of 8 years: 8 years of 354 days, together with 3 months of 30 days each, making up (in 99 lunations,) 2922 days; which is exactly the amount of 8 years of 365 days each. Hence this period would answer its purpose so far as the above lengths of the lunar and solar cycles are exact; and it might assume various forms, according to the manner in which the three intercalary months were distributed. The customary method was to add a thirteenth month at the end of the third, fifth, and eighth year of the cycle. This period is ascribed to various persons and times; probably different persons proposed different forms of it. Dodwell places its introduction in the 59th Olympiad, or in the 6th century, B. C.: but Ideler thinks the astronomical knowledge of the Greeks of that age was too limited to allow of such a discovery. This cycle, however, was imperfect. The duration of 99 lunations is something more than 2922 days; it is more nearly 29231; hence in 16 years there was a deficiency of 3 days, with regard to the motions of the moon. This cycle of 16 years (Heccodecaeteris), with 3 interpolated days at the end, was used, it is said, to bring the calculation right with regard to the moon; but in this way the origin of the year was displaced with regard to the sun. After 10 revolutions of this cycle, or 160 years, the interpolated days would amount to 30, and hence the end of the lunar year would be a month in advance of the end of the solar. By terminating the lunar year at the end of the preceding month, the two years would again be brought into agreement: and we have thus a cycle of 160 years11. This cycle of 160 years, however, was calculated from the cyle of 16 years; and was probably never used in civil reckoning; which the others, or at least that of 8 years, appear to have been. The cycles of 16 and 160 years, were corrections of the cycle of 8 years; and were readily suggested, when the length of the solar and lunar periods became known with accuracy. But a much more exact cycle, independent of these, was discovered and introduced by Meton22, 432 years B. C. This cycle consisted of 19 years, and is so correct and convenient, that it is in use among ourselves to this day. The time occupied by 19 years, and by 235 lunations, is very nearly the same; (the former time is less than 6940 days by 9 hours, the latter, by 71⁄2 hours.) Hence, if the 19 years be divided into 235 months, so as to agree with the changes of the moon, at the end of that period the same succession may begin again with great exactness. 21 Geminus. Ideler. 22 Ideler, Hist. Unters. p. 208. In order that 235 months, of 30 and 29 days, may make up 6940 days, we must have 125 of the former, which were called full months, and 110 of the latter, which were termed hollow. An artifice was used in order to distribute 110 hollow months among 6940 days. It will be found that there is a hollow month for each 63 days nearly. Hence if we reckon 30 days to every month, but at every 63d day leap over a day in the reckoning, we shall, in the 19 years, omit 110 days; and this accordingly was done. Thus the 3d day of the 3d month, the 6th day of the 5th month, the 9th day of the 7th, must be omitted, so as to make these months 'hollow.' Of the 19 years, seven must consist of 13 months; and it does not appear to be known according to what order these seven years were selected. Some say they were the 3d, 6th, 8th, 11th, 14th, 17th, and 19th; others, the 3d, 5th, 8th, 11th, 13th, 16th, and 19th. The near coincidence of the solar and lunar periods in this cycle of 19 years, was undoubtedly a considerable discovery at the time when it was first accomplished. It is not easy to trace the way in which such a discovery was made at that time; for we do not even know the manner in which men then recorded the agreement or difference between the calendar day and the celestial phenomenon which ought to correspond to it. It is most probable, that the length of the month was obtained with some exactness, by the observation |