moon and the positions of the sun in the heavens do not recur in a full number of days, and that a solar year does not contain a full number of months; the various phases of the moon requiring a little more than 2912 days, and the positions of the sun somewhat less than 36514 days; thus making the solar year contain about 12 months and 11 days.
To illustrate: if months of 29 and 30 days be alternated, if 365 days be allotted to each solar year, and after 99 such months (equivalent to eight years), the counting of the months be made from the same date of the year, a tolerable agreement with celestial signs could, indeed, be attained, but after a number of years the date of the month would differ from that indicated by the moon, and with a greater lapse of time the accepted solar year would show a decided variation from the annual position of the sun and moon, and, in fact, from the yearly heating effects of the sun. An improvement could, indeed, be made if, in the alternation of months of 29 and 30 days, two months of 30 days each be allowed to succeed each other at intervals of 33 months; also, if each fourth solar year were to contain 36514 days instead of 365, and if, finally, once in nineteen years, the so-called golden cycle, the reckoning of the lunar months began with the same date of the sun; but a uniform and perfect coincidence of these cycles of days, months, and years would also, in this manner, be unattainable. In short, the problem was purely an arithmetical one, especially difficult so long as it was sought to make the minor divisions of the year agree with the appearances of the moon, and it is, accordingly, no matter of wonder that many fruitless attempts at improvement were made before a judicious and adequate method was discovered. The founder of Islam clearly recognized the danger which the unsuccessful attempts to construct a calendar for future time engendered for the authority of the leaders in that religion, and hence arose the fear that the unavoidable conflict of different calendars would promote the formation of religious sects. Mohammed forbade, therefore, the establishment of any connection of the lunar month with the solar year, and ordered all calculations to be made from the observation and proclamations of the new light of the moon. While the peculiar difficulties of the chronological problems had afforded, on the one hand, advance in the observation and understanding of the movements of the heavenly bodies, and, on the other, had brought calculations of time into discredit, these astronomical inquiries produced still other important results at an early date, as in Babylon, many centuries before the conquest of Alexander the Great. These results, extending beyond the limits of chronology, secured to astronomers a mighty influence over the minds of the people, notwithstanding the repeated inaccuracies occurring in the calendar. This was attained through the teachings of astrology.
The systematic and long-continued recording of the celestial phe-
