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were inserted at even intervals as far as possible, in the third fifth, and eighth year." But, as Geminus proceeds to point out, the length assumed for the lunar month (29½ days) was incorrect. He puts it at 29½ days + $1/33$ of a day. Ninety-nine months on this computation would make up 2923½ days, but eight solar years make up 2922 days; i.e. in a cycle of eight years, the lunar time is a day and a half in excess of the solar time. The discrepancy can be remedied by adding three days to the solar time in sixteen years, but this remedy makes the solar time three days too long in sixteen years, or 30 days too long in 160 years. The system was therefore corrected by leaving out one intercalary month in 160 years.

Even this is incorrect. The lunar month, even as fixed at 29½ and $1/33$ days was found to be 25" too short (according to the calculation of Hipparchus). In 160 years, 25" a month amounts to a deficiency of 13 hours, and as the month of 30 days, omitted once in 160 years, is 11 hours too long for a lunar month, the deficiency (11 + 13 hours) was made up by adding one day in 160 years.

Whether these corrections were ever carried out is doubtful; they were probably superseded by a better system, before sufficient time had elapsed to put into practice the corrections which had been made in theory. In 432, Meton invented a different cycle, consisting of 19 years. In 19 years there are 6940 days and 235 months, including 7 intercalated months, and each year averages 365$5/19$ days. The 235 months were divided into 110 full and 125 hollow. The regular alternation of full and hollow was discontinued, every 64th day being omitted, in order to bring down 7050 days (235 months X 30 days) to the required number of 6940.

This period was faulty in regard to the solar year, which was $1/76$ of a day too long. It was therefore corrected by Calippus, who instituted a cycle of 76 years=four Metonic cycles, and omitted a day in each cycle. This period was regarded by Geminus as the most accurate.

The cycle of Meton was invented in 01. 87.1.=432, but