Sanhedrin 011

14:
However, if we multiply the time of one lunation (29:12:44:3.33) by twelve months we find that the sum is short of the solar year by more than 11 days. If no allowance is made for this discrepancy the seasons will recede into the lunar year by approximately 11 days every lunar year of 354 days. (This is, indeed, true of the Moslem calendar, which is an unmodified lunar calendar.) For the reasons already indicated in the previous section, the Torah does not permit our Jewish calendar to go unmodified, in order to ensure that Pesaĥ always falls in the spring. However, we cannot just add on 11 days to every year, since the calendar is a lunar calendar, and the months must be determined by the sighting of the new moon, as explained in our last shiur. Some time during the 8th century BCE the Babylonians discovered a curiosity that was also "rediscovered" by the Greek astronomer, Meton, around 432 BCE. The discovery was that 19 solar years are almost exactly equal to 235 lunations (Jewish months). 235 lunations is 19 times 12 with another seven left over. It was now but a simple step to create a situation that seven of every nineteen years should have thirteen months instead of twelve, and the seasons would more or less fall into place – even according to the lunar calendar. This, basically, is the modified lunar calendar still in use by Jews.

15:
235 months of 29 days, 12 hours, 44 minutes and 3.33 seconds results in 6939 days, 16 hours, 33 minutes and 3.33 seconds. The Earth makes nineteen complete revolutions on the sun's axis in 6939 days, 14 hours, 34 minutes and 48 seconds. This is a discrepancy over each nineteen year cycle of only about 0.0769 of a day. According to the fixed calendar established by the President of the Sanhedrin, Hillel II, in 358 CE, the following years of each Metonic cycle have thirteen months (or, in rabbinic parlance, are 'pregnant'): 3rd, 6th, 8th, 11th, 14th, 17th and 19th. (If you want to know whether any given year is 'pregnant' or not, divide the currant Jewish year by 19; the remainder will indicate the present year in the Metonic cycle.)

16:
However, before the institution of the permanent calendar intercalation (adding an extra month into the year) was announced by a decision of the Sanhedrin, a decision which was based on several factors – agricultural, meteorological and practical. Once a decision had been made it was announced by special messengers sent to the communities of the Diaspora. The Gemara [Sanhedrin 11b] records the following:

Cheryl Birkner Mack writes:

One thing remains unclear to me: how could witnesses see the moon on the 29th or 30th day of the month? I thought the moon was/is not visible at this time.

I respond:

The actual point of the Molad, lasts a very short time. Within minutes thereafter the thinnest crescent of the new moon would be visible. By astronomic calculations (Rémy Landau, please note!) the sages knew in what part of the sky and at what elevation the new crescent would be visible, thus they were able to check the accuracy and veracity of the witnesses.

Actually, on this last point the sages were caught on the horns of a dilemma. In order to encourage people to take the trouble to journey to Jerusalem to give evidence much fuss was made of the pair of witnesses upon whose evidence the new month was sanctified. Furthermore, all the witnesses were interviewed (even after there was no need for any more of them) so that they would not be disappointed and consider their journey a waste of time, thus refraining from making the effort next time. However, the fact that a lot of fuss was made of the witnesses must have been very tempting to glory-seekers, and those on the look out for a super dinner at the people's expense! These problems are reflected in the following Mishnah [Rosh ha-Shanah 2:8]