The definition of a “month” is not nearly as simple as it seems. After all, April, the month in which Pesach falls out this year, is not the same as Nisan, the month in which Pesach occurs every year. Nisan is a chodesh, a lunar month, while April is an ordinary month and not connected to the phases of the moon. Therein lies the difference between the Torah’s concept of a chodesh and the secular definition of a month: a chodesh is closely (although not exactly) associated with the phases of the moon.
In a baraisa cited in Maseches Rosh HaShanah 25a Rabban Gamliel provides the time duration of what would be referred to scientifically as the “mean synodic month”: “Thus have I received a tradition from the house of my father’s father: the rebirth of the moon is not less than 29½ days two-thirds of an hour and 73 chalakim.” This is the time that elapses from molad to molad i.e. the time between lunar conjunctions when the moon Earth and sun are aligned so that the moon is totally invisible. An hour is comprised of 1 080 chalakim (Rambam Hilchos Kiddush HaChodesh 6:2).
Thus Rabban Gamliel’s tradition indicates a period of exactly 29 days 12 hours and 793 chalakim. This exact value written in a different mathematical format known as sexagesimal (meaning mathematics based on 60 as opposed to the decimal based on ten with which we are familiar) appears in the Almagest authored by the Greek astronomer Ptolemy in the year 150 CE. The duration of the mean synodic month according to both Rabban Gamliel and the Almagest then is 29.5305941358 days.
Earth-based measurement of the synodic month duration is extremely difficult. It was only with the advent of space-based measurements that a highly precise result was obtained at 29.5305888531 days. The value provided by Rabban Gamliel as well as l’havdil Ptolemy is only 0.456 seconds more than the value obtained by NASA.
We can understand how Rabban Gamliel possessed a value so strikingly close to the measurement obtained by modern scientists but how did a Greek astronomer come up with a number anywhere near the one obtained by today’s sophisticated measurements?
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