7 years in every 19 as years of 13 months each. This
thirteenth month is called an intercalary month, and in the present
Jewish calendar it is the month Adar which is reduplicated under the
name of Ve-Adar. But, though from the necessity of the case, this
intercalation, from time to time, of a thirteenth month must have been
made regularly from the first institution of the feast of unleavened
bread, we find no allusion, direct or indirect, in the Hebrew Scriptures
to any such custom.

Amongst the Babylonians a year and a month were termed "full" when they
contained 13 months and 30 days respectively, and "normal" or
"incomplete" when they contained but 12 months or 29 days. The
succession of full and normal years recurred in the same order, at
intervals of nineteen years. For 19 years contain 6939 days 14-1/2
hours; and 235 months, 6939 days 16-1/2 hours; the two therefore
differing only by about a couple of hours. The discovery of this cycle
is attributed to Meton, about 433 B.C., and it is therefore known as the
Metonic cycle. It supplies the "Golden Numbers" of the introduction to
the Book of Common Prayer.

There are two kinds of solar years, with which we may have to do in a
luni-solar calendar--the tropical or equinoctial year, and the sidereal
year. The tropical year is the interval from one season till the return
of that season again--spring to spring, summer to summer, autumn to
autumn, or winter to winter. It is defined as the time included between
two successive passages of the sun through the vernal equinox, hence it
is also called the equinoctial year. Its length is found to be 365 days,
5 hours, 49 minutes, and some ancient astronomers derived its length as
closely as 365 days, 6 hours, by observing the dates when the sun set at
exactly the opposite part of the horizon to that where it rose.

The sidereal year is the time occupied by the sun in apparently
completing the circuit of the heavens from a given star to the same star
again. The length of the sidereal year is 365 days, 6 hours, 9 minutes.
In some cases the ancients took the sidereal year from the "heliacal"
risings or settings of stars, that is from the interval between the time
when a bright star was first seen in the morning just before the sun
rose, until it was first so seen again; or last seen just after the sun
set in the evening, until it was last so seen again.

But to connect the spring new moon with the day when the sun has
returned