**The Hebrew Calendar: A Marvel of Ancient Astronomy and Math**

The biggest marvel is how Iron Age Jews managed to adjust the lunar calendar to the solar one.

https://www.haaretz.com/jewish/2014-09-25/ty-article/.premium/the-secrets-of-the-hebrew-calendar/0000017f-db3f-df62-a9ff-dfff86dd0000

The calculation of the calendar was transmitted to the sages in an unbroken chain going back to Moses. … According to the ancient calculations, the exact time between one new moon and the next is 29 days, 12 hours, and 793 chalakim ‘parts of an hour’ (the hour is divided into 1080 parts). In other words, one lunar month has 29.53059 days. It is interesting to note that according to NASA (National Aeronautics and Space Administration); the time between one new moon and the next is 29.530588 days. Of course, NASA has at its disposal the most advanced and sophisticated telescopes and computers. Nevertheless, the difference between NASA’s figures and that used by Hillel II, which originated more than 3000 years ago, is .000002 or two millionths of a day, calculated for the period of one month.

Avraham Yaakov Finkel, The Essence of the Holy Days: Insights from the Jewish Sages 1993, p. 141

https://www.congregationsofgod.org/accuracy-of-jewish-calendar

**Accuracy of the Hebrew Calendar**

https://www.sciencedirect.com/science/article/pii/S0898122100001024

The Hebrew calendar must be in harmony with both the lunar and solar cycles. The fact that 19 mean solar years contain almost precisely 235 lunar months (the Metonic cycle) makes such a lunar/solar calendar possible. The accuracy of this calendar is examined. It is shown that despite the approximations necessary to provide a fixed calendar, the Hebrew calendar's mean lunar month duration discrepancy from current astronomical values (a small positive number) amounts to only one day in 14,000 years. More remarkable yet is that this error was less in the past, since there has been a decrease (very small) in the orbital period of the moon.

Let us mention several concepts from astronomy which we will need. First, there are a number of concepts of a year, but the one used for calendars is the tropical year, which is defined as the time between vernal equinoxes (late March) which we know today to be equal to 365.242199 mean solar days. A mean solar day is the average of two passages of the sun across the meridian, which we know to be 24 hours, 3 minutes, 56.55 seconds or 24.0657083 hours. Furthermore, a sidereal day (with respect to stars) is equal to 23 hours, 56 minutes, 4.10 seconds of mean solar time or 23.9344721 hours. We also have the concept of a fictitious sun. Since the sun's motion through the sky varies with the seasons, we use a fictitious sun which moves across the sky at a uniform rate. The solar day is longer than the sidereal day because of the earth's motion about the sun. That is, it has to turn more than a whole revolution to bring the sun back to the meridian. The secular calendar year is 365.2425 days, which is 365.25 - 3/400 year (from the leap year every four years except for three out of every four 100 years). The lunar month is the time to complete a cycle of lunar phases which we know to be 29.530588 days. This is also termed a synodic lunar month. Twelve synodic lunar months equals 354.36706 days, almost eleven days shorter than the tropical year. The Moslem calendar is 11 days short of the tropical year. Hence, it cycles through the full 365 days of the tropical year every 32 1/2 years. Therefore, Moslem holidays are not tied to the seasons.

**ANALYSIS**

The present Hebrew calendar year is 365 days, 5 hours, 55 minutes, and 25 25/57 seconds (25.4385965 seconds), which is 365.2468221615 days. The secular calendar year minus the tropical astronomical year equals .000301 days or one day in 3,322 years. The present Hebrew solar year minus the tropical year equals .004623 days which is 6.65712 minutes or 6 minutes and 39.4272 seconds. The Hebrew solar year is too long by one day in 228 years, that is .004623(228) equals 1.054044 days. However, as David Bleich points out, the lunations, i.e., the length of the monthly lunar cycles are established with much greater precision. The synodic lunar month is 29 days, 12 hours, 44 minutes, 2.841 seconds and the Hebrew lunar month is 29 days, 12 hours, 44 minutes, and 3 1/3 seconds. Since the lunar month is the basis of the Hebrew calendar and this discrepancy amounts to only one day in 14,000 years and since it is a crude simplification of that calendar possessed by the Sanhedrin, we can only marvel at the knowledge and skill of these sages. Since there is an infinitesimal decrease in the orbital period of the moon, this small discrepancy will grow; however, at a very small rate. However, more important, this indicates that the Hebrew calendar was even more accurate when the approximations necessary to provide a fixed calendar were made. According to David Bleich, Dr. Hugo Mandelbaum has stated that an exact lunar-solar cycle is not possible. In other words, a fixed calendar to have repeatability in lunar months relative to the solar year would have an extremely long cycle. He says it would be 689,272 years long. That, of course depends on (1) the precision of our knowledge, and (2) the degree of repeatability you desire. To the accuracy of four decimal places, a 689,272 year cycle is adequate. Hence, the solar year and the lunar month are, in fact, incommensurable.

**REFLECTIONS**

A number of issues and/or questions come to mind. What measurements were made, or even possible at the time? What technical aids were available? How were measurements made and how were they recorded? How was knowledge transmitted over generations to build the necessary databases? Recent discoveries of ancient convex and concave lenses in the Middle East are intriguing. No one has found how they were utilized other than possibly as magnifying glasses (for the convex lenses) and/or in the manufacturing of jewelry. There has been speculation that fine astronomical works were stored in the libraries of Alexandria, which were subsequently lost in destructive fires, etc., over many years. Clearly, measurements were taken and recorded over long periods of time. Perhaps some day scrolls, a la Dead Sea Scrolls, will be found that yield information on how such refined calendars, which preceded the fixed calendar, were achieved.