containing twelve months each, and seven years
containing thirteen months each; and these last formed the third, fifth,
eighth, eleventh, thirteenth, sixteenth, and nineteenth years of the cycle.
As it had now been discovered that the exact length of the lunation is a
little more than twenty-nine and a half days, it became necessary to
abandon the alternate succession of full and deficient months; and, in
order to preserve a more accurate correspondence between the civil month
and the lunation, Meton divided the cycle into 125 full months of thirty
days, and 110 deficient months of twenty-nine days each. The number of days
in the period was therefore 6940. In order to distribute the deficient
months through the period in the most equable manner, the whole period may
be regarded as consisting of 235 full months of thirty days, or of 7050
days, from which 110 days are to be deducted. This gives one day to be
suppressed in sixty-four; so that if we suppose the months to contain each
thirty days, and then omit every sixty-fourth day in reckoning from the
beginning of the period, those months in which the omission takes place
will, of course, be the deficient months.

The number of days in the period being known, it is easy to ascertain its
accuracy both in respect of the solar and lunar motions. The exact length
of nineteen solar years is 19 x 365.2422 = 6939.6018 days, or 6939 days 14
hours 26.592 minutes; hence the period, which is exactly 6940 days, exceeds
nineteen revolutions of the sun by nine and a half hours nearly. On the
other hand, the exact time of a synodic revolution of the moon is 29.530588
days; 235 lunations, therefore, contain 235 x 29.530588 = 6939.68818 days,
or 6939 days 16 hours 31 minutes, so that the period exceeds 235 lunations
by only seven and a half hours.

After the Metonic cycle had been in use about a century, a correction was
proposed by Calippus. At the end of four cycles, or seventy-six years, the
accumulation of the seven and a half hours of difference between the cycle
and 235 lunations amounts to thirty hours, or one whole day and six hours.
Calippus, therefore, proposed to quadruple the period of Meton, and deduct
one day at the end of that time by changing one of the full months into a
deficient month. The period of Calippus, therefore, consisted of three
Metonic cycles of 6940 days each, and a period of 6939 days; and its error
in respect of the moon, consequently, amounted only to si