Imagine yourself back in time thousands of years ago. You have no clocks or calendars. How can you predict when winter will arrive (or spring)? How about the time of year when aurochs migrate? When do we travel to find ripe berries? Being able to tell what time of year it was enhanced your chances of survival. How do you keep track of time?
Well the answer is obvious if you simply watch the sky. The sun rises and the sun sets. The moon waxes from new to full and then wanes back to new. Sunrise and sunset are sometimes further south (the time when it's cold and snowy and the sun low in the sky) and sunset and sunrise are sometimes further north (the time when it's warm and green and the sun high in the sky). The stars and contellations visible in the sky, and when they rise and set, changes over the course of the seasons.
Let's look at one of those cycles. One thing easy to observe are the phases of the moon and every ancient culture developed a lunar calendar. We still use them today in modern times - they're the reason why religious holidays like Yom Kippur, Passover, Easter, and Ramadan always fall on different days of the year on our civil calendar. The dates of these holidays are based on ancient lunar calendars.
The cycle of lunar phases is called the synodic month and lasts 29.53059 days (note that it's not an even number of days).
Another cycle which was obvious to all ancient cultures is the change in position of sunrise and sunset over the course of the year. For, example, here in the Hudson Valley, if you go out and observe where the Sun rises each morning, you'll see the following in December, March, June, September, and December again.
In December, the Sun is rising toward the south and never gets high in the sky. In March, the sunrise is almost due east. By June, the Sun is rising toward the north and gets high in the sky. In September, the sunrise position swings back toward rising in the east. And, finally, when December rolls around again, the Sun is rising in the south. Over and over again as each year passes with the changing seasons.
The time when the Sun rises (and sets) furthest in the south is the winter solstice. The time when it rises (and sets) furthest in the north in the summer solstice. The time when it rises (and sets) due east are the vernal (spring) and autumnal (fall) equinoxes. If you wanted, you could erect some stones to mark these directions.
Like the lunar cycle, you can measure the time for this cycle (which is a year) but it's also not an even number of days. This period of time is called the tropical year (from the Greek tropikos which means "to turn") and is 365.24163 days long.
Let's look for a relationship between these two cycles.
It turns out that nineteen tropical years:
(365.24163 days/year) * (19 years) = 6,939.59097 days
How many synodic months are in 6,939.59097 days?
(6,939.59097 days) / (29.53059 days/month) = 234.99669 months
Which rounds very closely to 235 months (there's about a 2.3 hour difference).
In other words, 235 synodic months is almost exactly 19 tropical years. This is called the Metonic Cycle (another mouthful of a term is the Enneadecaeteris which is derived from the ancient Greek word εννεαδεκαετηρις, which means "nineteen years").
This cycle was named for Meton of Athens, a Greek mathematician who lived in the 5th century BC. While Meton utilized this cycle for reforming the calendar at the time, it was know to the Babylonians in the 6th century BC who utilized it in their calendars as well.
So what do we get when we divide 235 months by 19 years?
(235 months) / (19 years) = 12.36842 months/year
That's why years still have 12 months. But, after a 12 synodic month year, you're left with 0.36842 of a month which is:
(0.36842 months) * (29.53059 days/month) = 10.88 days
So you have roughly 11 extra days between a lunar year of 12 cycles of lunar phases and the solar year from solstice to solstice. Most cultures dealt with these extra days by periodically inserting an extra month.
The traditional Jewish calendar, for example, has twelve months alternating between 29 and 30 days (averaging the 29.5 day lunar phase cycle). But, because of those 11 extra days per year, they periodically add an extra month. In years 3, 6, 8, 11, 14, 17, and 19 of the nineteen year Metonic cycle, they add a "leap month." In normal years, Adar occurs around February/March before the vernal equinox, in leap months, Adar become Adar Bet (Adar 2) and the 30 day month of Adar Aleph (Adar 1) is added to the calendar.
Even our modern civil calendar, while no longer tied to lunar phases, still has 12 months in a year reflecting its derivation from older lunar-based calendar. Months are longer than 29.5 days, however, to keep the calendar in sync with the tropical year (and therefore the seasons).
Let's look at the days in each month with respect to the average number of days in the synodic month of lunar phases:
What a surprise, the difference adds up to 11 days. It still doesn't exactly match up which is why we periodically insert that extra day in February every four years.
I love watching the sky and seeing that change in lunar phases (it was a bright full January "Wolf Moon" out there last night) and the change in the position of the Sun. I can already feel the days lengthening and the Sun moving higher in the sky as we move into February. Very soon, we'll start to see some of the earliest signs of spring in the neverending dance of the Earth, Moon, and Sun.