alation every eighth year. In fact, the 71/2 days
by which two lunar years exceeded two solar years, amounted to thirty days,
or a full month, in eight years. By inserting, therefore, three additional
months instead of four in every period of eight years, the coincidence
between the solar and lunar year would have been exactly restored if the
latter had contained only 354 days, inasmuch as the period contains 354 x 8
+ 3 x 30 = 2922 days, corresponding with eight solar years of 3651/4 days
each. But the true time of 99 lunations is 2923.528 days, which exceeds the
above period by 1.528 days, or thirty-six hours and a few minutes. At the
end of two periods, or sixteen years, the excess is three days, and at the
end of 160 years, thirty days. It was therefore proposed to employ a period
of 160 years, in which one of the intercalary months should be omitted; but
as this period was too long to be of any practical use, it was never
generally adopted. The common practice was to make occasional corrections
as they became necessary, in order to preserve the relation between the
octennial period and the state of the heavens; but these corrections being
left to the care of incompetent persons, the calendar soon fell into great
disorder, and no certain rule was followed till a new division of the year
was proposed by Meton and Euctemon, which was immediately adopted in all
the states and dependencies of Greece.

The mean motion of the moon in longitude, from the mean equinox, during a
Julian year of 365.25 days (according to Hansen's _Tables de la Lune_,
London, 1857, pages 15, 16) is, at the present date, 13 x 360 deg. +
477644".409; that of the sun being 360 deg. + 27".685. Thus the corresponding
relative mean geocentric motion of the moon from the sun is 12 x 360 deg. +
477616".724; and the duration of the mean synodic revolution of the moon,
or lunar month, is therefore 360 deg. / (12 x 360 deg. + 477616".724) x 365.25 =
29.530588 days, or 29 days, 12 hours, 44 min. 2.8 sec.

The _Metonic Cycle_, which may be regarded as the _chef-d'oeuvre_ of
ancient astronomy, is a period of nineteen solar years, after which the new
moons again happen on the same days of the year. In nineteen solar years
there are 235 lunations, a number which, on being divided by nineteen,
gives twelve lunations for each year, with seven of a remainder, to be
distributed among the years of the period. The period of Meton, therefore,
consisted of twelve years