The light of the moon that reaches us is the reflection of sunlight, and is not produced by the moon itself. From the beginning of the Hebrew month, when the new moon first becomes visible, until the 15th of the month, the portion of the moon that faces us and is illuminated by the sun increases in size, and so the moon appears to grow larger. From the 15th of the month onwards, the illuminated portion of the moon decreases in size, and the moon seems to grow smaller, until it disappears altogether from our view. It is hidden for 24 hours, and then reappears. Remarkably, the precise duration of this cycle was known to the ancient Jewish sages, based on a tradition originating from Sinai.

Rabban Gamliel and the 29.53-Day Cycle

The Talmud (Rosh Hashanah 25a) cites the following comment of Rabban Gamliel:

‘So have I received from the house of the father of my father:
The renewal of the moon is not ever less than 29 and a half days
(12 hours), and 2/3 of an hour, and 73 portions of an hour.”

Rabban Gamliel here presents his tradition as to the duration of the lunar cycle. The expression “not ever less than” might seem to imply that the cycle could be longer, but it is clear from the Gemara that Rabban Gamliel used this phrase to correct the mistake made by those who wanted to declare the new month long before its proper time. Rabban Gamliel asserted that these people were wrong, for the average cycle is 29 and a half days, and it could never be significantly shorter than this period, because there is only a slight difference between the average lunar cycle and the actual lunar cycle of any particular month.[1]

To fully understand Rabban Gamliel’s intent, we must first try to understand the terms “2/3 of an hour” and “73 portions ofan hour.”

The Rambam writes (Hilchot Kiddush Hahodesh6:3):

“An hour is divided into 1080 ‘portions.’ The Sages divided an hour into this number of portions because this number has a half, quarter, eighth, third, sixth, ninth, fifth, and tenth; these names [i.e., fractions] have many portions, as well.”

The Rambam explains that the sages divided the hour into 1080 portions in order to simplify calculations. Thus, when Rabban Gamliel speaks of 2/3 of an hour, he refers to 720 portions
(1080 x 0.66). Andwhen we add 73 portions, we arrive at
793 portions.

Converting this calculation to our decimal system, we arrive at 0.734259 (793 ÷ 1080) hours, or 0.030594 (0.734259 ÷ 24) days.

Now we add on the 29.5 days, and the result is 29.530594
(29.5 + 0.030594)days, which is the average duration of the lunar cycle according to Rabban Gamliel’s calculation.

Today, scientists have calculated the precise duration of the average lunar cycle, based on measurements made from satellites and using atomic clocks of extreme precision. Amazingly, the result is almost exactly the same as the conclusion reached by our Sages millennia ago. According to modern science, the average duration of the synodic month (the period between successive new moons) is 29.530588.[2]
The difference between the results of the scientists today and the words of the Sages is miniscule, equal to 0.5184 seconds.

This knowledge was in the possession of the Sages thousands of years ago, when the means of measuring such matters did not exist. Indeed, Rabban Gamliel stated that he received this information “from the house of the father of my father.”It was not derived from astronomical calculations, but rather was transmitted as a tradition for generations in the family. Incidentally, Rabban Gamliel,as we know from other sources, was a descendent of the royal house of King David.

Rav Hai Gaon (939-1038) thus explains, “The calculation of the ‘secret of the intercalation of the month’ has been in the possession of the people of Israel from the days of Moses.”

The Ibn Ezra (1089-1164) similarly writes in his book Ha’ibur: “The cycle is a tradition in the possession of the family of David, the man of Gd, that in 19 years of lunar years, each month is
29 days, 12 hours and 793 portions, as Rabban Gamliel cited.”

The difference between the lunar cycle identified by our Sages and that determined by modern scientists is not significant in any practical sense, and has no effect at all on the calendar fixed by the sages two millennia ago. Nevertheless, we should ask, if this information has a divine source, then why is it not exactly equivalent to the result of scientific measurement (assuming that the scientific data is indeed exact)?

A possible answer is that the deviation originates from Rabban Gamliel’s method of dividing hours into 1080 portions. In this division, the number 793 gives the maximum precision and the closest result to the current scientific findings. Had he said 792 or 794, the discrepancy would have been greater.

But why, in fact, did Rabban Gamliel use this system of dividing the hour into1080 portions, rather than a system that allows for greater precision?

The answer emerges from the aforementioned comment of the Rambam that this system of division is the simplest to use because it is easily divisible by other numbers. Since this slight discrepancy has no significant effect on the declaration of new months, even over the course of thousands of years, Rabban Gamliel preferred using this system, even though he may have had a more precise figure as a tradition from his family.[3]

Divine Calculations

Claudius Ptolemaeus (Ptolemy, 85-165) of Alexandria, the Greek mathematician, geographer, astronomer and astrologist who is widely considered the most prominent astronomer in the ancient world, expressed his astonishment when he heard of the Jewish sages’ calculation of the lunar cycle. He acknowledged that the lunar cycle could not be calculated using the primitive technology of his day, and said that this proves that the Jews had prophecy.

Ptolemy’s astonishment was described many centuries later by Rabbi Don Yitzchak Abarbanel (1437 – 1508), who wrote:

“The wisdom that was in the possession of the Jewish sages in this matter caused the Greek Ptolemaeus to praise and extol the one who invented the 19-year cycle with the work of intercalation of the month that the Jews possessed, and he wrote that this demonstrates that they had prophecy.”

Who could know better than that ancient astronomer that it was impossible with the very limited means available to astronomers of his time, and certainly of earlier times, to achieve such a precise result without prophecy?

Some have argued that the sages determined the precise duration of the lunar cycle simply through empirical observation, by systematically tracking the new moon over a long period of time, counting the days between each new moon. Perhaps, this argument contends, the sages appointed someone to track the new moon and mark down how many days passed from one new moon to the next, over the course of many years. Before his death, the observer would pass on this information to somebody else, who would continue the observations from the point where his predecessor left off. These observations could have continued for a long enough period of time,for example, 200 years, for an accurate calculation of the average lunar cycle to be made. Of course, the longer the period of observation was, the greater the level of accuracy.

Although this method of measuring the length of the lunar cycle is theoretically possible, a thorough examination of the facts makes it difficult to imagine that such a system was ever implemented. It seems clear that this was not the method used by the Sages when they calculated the lunar cycle’s duration, for a number of reasons:

After 200 years of empirical observation, it would be impossible to reach the precise figure of 29.530594. The number would either be slightly smaller (29.530416) or slightly larger (29.530833). (Incidentally, both these numbers would be slightly different from the calculation of modern science –29.530588.) Therefore, it is evident that the source of their figure was not from empirical observation.


If such observations were made over the course of hundreds of years, they would have assuredly been documented. It is hard to imagine that such a vital undertaking could have taken place over such a long period without any record.


It does seem plausible that ancient civilizations could have implemented such a lengthy process of consistent, monthly monitoring, over hundreds of years, without missing even one month, especially considering they had no technology with which to view the moon under cloudy conditions. Moreover, it is hardly likely that the findings could have been preserved from one generation to the next without the transmission being interrupted by war, exile and the like.


It is not possible that the sages of Israel, who afforded great importance to the declaration of the new month at the proper time, would base their calculation on the observations of anonymous men, passed from generation to generation, without the possibility of verifying the information.

We must also bear in mind that in ancient times, there were noavailable means of calculating with the accuracy of six decimal points – neither with a sand hourglass, a sundial, or a
water-powered clock. The Sages of Israel were equipped with wisdom from a divine source – the tradition that originates from Moshe, whoreceived this information at Mount Sinai.