THE LUNAR CYCLE.
The Lunar cycle, or the cycle of the moon, is a period of nineteen years,
after which the new and full moons fall on the same days of the year as
they did nineteen years before. This cycle was invented by Meton, a
celebrated astronomer of Athens, and may be regarded as the masterpiece of
ancient astronomy. In nineteen solar years there are 235 lunations, a
number which, on being divided by nineteen, gives twelve lunations, with
seven of a remainder, to be distributed among the years of the period.
The period of Meton, therefore, consisted of twelve years containing
twelve months each, and seven years containing thirteen mouths 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 30 days, and 110 deficient
months of 29 days each. The number of days in the period was, therefore,
6940; for (125 x 30) + (110 x 29) = 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 7050 days, from which 110 days are to be
deducted; for (235 x 30) = 7050; 7050 - 110 = 6940, as before. 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 (365d, 5h, 48m, 49.62s.) x 19 = 6939d, 14h,
27m, 42.78s.; hence, the period, which is exactly 6940 days, exceeds
nineteen annual revolutions of the earth by a little more than nine and a
half hours. On the other hand, the exact time of the synodic revolution of
the moon is 29d, 12h, 44m, 2.87s.; 235 lunations, therefore, contain 235 x
(29d, 12h, 44m, 2.87s.) = (6939d, 16h, 31m, 14.45s.), so that the period
exceeds 235 lunations by nearly seven and a half hours.
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, in order to make a correction of the Metonic cycle, 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 to the moon,
consequently, amounted to only six hours, or to one day in 304 years.
This period exceeds seventy-six true solar years by 14h, 9m, 8.88s., but
coincides exactly with seventy-six Julian years; and in the time of
Calippus the length of the solar year was almost universally supposed to
be exactly 365-1/4 days.