Any study of the Seventy Weeks Prophecy would have to show how one would count toward the coming of Jesus. We would need a day or date to count from (a beginning) and a day or date to count to (an end). Furthermore, since the Seventy Weeks Prophecy points to several markers along the way, we would also need a calendar that shows exactly what God is doing as the along at those times. A calendar measures time. It also gives a sense of ‘controlling’ time which is another way of saying one has control of one’s life. But more than this, calendars were often produced for the purpose of worship. For example the calendar we use today is the Gregorian calendar, named after Pope Gregory and was calculated over a concern that we were celebrating Easter out of its season.
Today the Hebrew calendar is lunar/solar. It is determined entirely by calculation of the heavenly bodies. This was not done in ancient times. The months were determined by observing the occurrences of the new moons. It was determined, however, that three new moons occur roughly in every 59 days. Therefore, it could be predicted that one month would be 30 days long and the month following would be 29.
This would be satisfactory, if it were not for the fact that the Hebrew calendar was also a seasonal calendar. That is, the religious activities depended upon their occurring during the spring and the fall respectively. With a normal lunar year of 354 days, there would occur an error of at least 11 days every solar year, which is what our Gregorian calendar is. This error would continue to compound until the months would not occur in their seasons. Therefore, in order that the religious festivals would not occur outside their respective seasons, an intercalary month was added about every two to three years. However, this too was done by observation. The first month, Abib had to occur during the time of the spring equinox, because nothing could be harvested from the fields until a portion of ripe fruits were offered to God. If they were not ripe, how could they be offered? If they were not offered, how could the regular harvest be sanctified? Therefore, a periodic 13th lunar month was needed.
Sometime after the destruction of Jerusalem, about the year 359 AD, Rabbi Hillel II fixed these intercalary months to 7 within a 19-year time cycle (3, 6, 8, 11, 14, 17, 19). The 19-year time cycle was probably not in use during the time of Christ. However, if all one did was observe the heavens one would know that an intercalary month was sometimes needed after two years or three years, so a reasonable facsimile was in practice. This would keep the Hebrew calendar in sequence with the planting and harvest seasons and also keep lunar and solar reckoning within reasonable agreement.
How Were the Calculations Made?
As it is today, the Hebrew calendar is very complicated and far from the simple tool it had to have been in ancient times. There had always been some calculation used with observation, but it was kept simple so the average person in Israel could tell when one month ended and another began. This was especially important when the new moon landed on a Sabbath or was itself a Sabbath as was the case of the 7th new moon. The nation had to prepare for the Sabbath. How could one prepare, if no one knew when the month began, if, indeed, the priests in Jerusalem had to announce its arrival by their observations?
In the case of the 7th new moon, the Feast of Trumpets, it didn’t do any good if the day was declared a Sabbath at 5 or 6 AM. The day began at sunset the previous evening. Furthermore, it doesn’t make any sense to declare the month an intercalary month (13th) month on the 14th of the month, because the lamb had to be set aside on the 10th of the month and continually inspected for blemishes. Leavening agents also had to be removed from the dwellings and probably took more than one day. Finally, the priesthood served in courses. The first course began its service for the ecclesiastical year on the Sabbath immediately preceding the first day of the first month (Abib or Nisan). “Observing” the seasons entirely by observation of the heavens would not be logical. There must be a better way.
It is evident from Scripture that some predictability was observed when using the calendar. For example, David and Jonathan knew that the following day was the day of the new moon (1Samuel 20:5, 18, 24). How could they have understood this, if new moons were not declared until the first sighting of the moon’s faint crescent? Therefore, it seems very likely that new moons were observed, but only to keep the festivals in their respective seasons. The months were predictable so that someone like David or Jonathan knew ahead of time when the new month would occur. So it is likely that each month was fixed with either 30 or 29 days, knowing the distance between three new moons is two full months or 59 days. So the months probably alternated 30 in the first and 29 in the second etc. without varying from year to year. The even months would always be 29 days and the odd would always be 30 days, except for the occasional 13th month. In this case the month could be either 30 or 29 days according to how many days the previous intercalary month had. If the previous leap year contained 30 days in the 13th month, then the next would have 29 days. So it would vary with each successive intercalary month just as the regular months did throughout the year. The calendar needed to be kept simple, since they were not written out for each family to own. People needed to know how to calculate the new month on their own. Simplicity was necessary.
The mean or average lunar month is 29.530594 days. Seven intercalary months or 206.714158 days were added to each 19-year time cycle. Rabbi Hillel II’s calculations put seven intercalary months into every 19-year time cycles. As I said above, it would not have been unusual for the ancient Jews to know through observation, that from time to time an intercalary month was needed two years from the previous intercalary year instead of the usual three years.
A 19-year time cycle having seven 30-day months = 210 days or a difference of +3.285842 days more than the lunations in a 19-year cycle. After two 19-year time cycles the calendar would be off the regular lunation cycle by nearly 7 days. In order to keep the calendar from getting out of sequence, one would have to subtract days from the regular months, making its calculation complicated and not user friendly. The average person would not know when the next month began. Therefore, this method was probably not used.
A 19-year time cycle having four 30-day months and three 29-day months = 207 days or a difference of +0.285842 days in a 19-year time cycle.
A 19-year time cycle having four 29-day months and three 30-day months = 206 days or a difference of – 0.714158 days in a 19-year time cycle
In exactly two 19-year time cycles of seven 30-day months and seven 29-day months the solar calendar would be behind the lunations by less than a half a day (+0.428316). In such a case, every 100 years (cir. 5 nineteen-year cycles) the Jews could subtract one day from the end of the last intercalary month having 30 days causing it to have only 29 days. In so doing, there would be no interference with how the holy festivals fell during the year, and the lunar and solar years would be kept in sync.
In conclusion, I submit that the ancient Hebrew calendar in use at the time of Christ was one dependent upon the seasons and needed a periodic additional month to keep the religious festivals in their seasons. This month varied in its number of days just like the succeeding months during any given year. The intercalary, 13th month, therefore, had 30 days one leap year and 29 days in the next intercalary year.
 There were intercalary months added, however, even if a 19-year time cycle was not kept. Third century apologist, Julius Africanus said: “…the Greeks and the Jews insert three intercalary months every 8 years.” Ante-Nicene Fathers, Vol. 6: Julius Africanus; The Extant Fragments of Five Books; Chapter 16: “On the Seventy Weeks of Daniel” – Paragraph 3.
 This has been suggested by some scholars, including Isaac Newton in his: “The Chronology of Ancient Kingdoms Amended”
 “The observation of the autumnal equinox, i.e., ‘the going out of the year’ (Ex. 23:16), and of the spring or vernal equinox, called ‘the return of the year’ (1Ki. 20:26; 2Ch. 36:10), was important for controlling the calendar and consequently the festivals. Thus the year began with the new moon nearest the vernal equinox when the sun was in Aries (cf. Josephus, Ant. 3.201 (Ant. III.x.5)), and the Passover on the fourteenth day of Nisan coincided with the first full moon (Ex 12:2-6)” (The Illustrated Bible Dictionary, J. D. Douglas, N. Hillyer, editors, IVP, 1980; Calendar, Vol 2, p 223).
 A solar year = 365.2425 days and 19 solar years = 6939.6075 days.
A lunar year = 29.530594 (one lunation) X 12 = 354.367128 days + an intercalary (13th) month of 29.530594 days 7 times in a 19-year time cycle. Therefore, 19 lunar years = 6939.68959 days or 235 lunations (new moons).
This is a difference of 0.08209 days, or the lunar cycle is greater than the solar cycle by 1hour, 58 minutes and 12.6 seconds in the 19-year cycle.
 Adding a 13th month to the Hebrew calendar every three years or so would technically necessitate making the 13th month 30 days for every intercalary year. However, the ancient Hebrews were observing seasons and not parts of days. They knew that two new moons occurred every 59 days, making one 30 days long and the other 29. There is no reason to assume they did otherwise with the intercalary month. One year the 13th month would be 30 days and the next intercalary year it would be 29. It is only logical to conclude that the ancient Jews did this. Every three years the lunation would have an error of one tenth of a day (approximately 2hrs. and 24 min.). They would not have been able to observe this in the heavens. This required sophisticated calculation.