Ancient Jewish Computers


Passover, the Bible tells us (Exodus 34:18), is Hag Hamatzot (Holiday of The Matzot) whose time is Mo’ed HaAviv, a spring festival, that begins on the 15th day of Nisan, on the night of a full moon after the vernal equinox (“Tekufat HaShana”), following the Passover sacrifice on the 14th.

It’s an unusually precise specification. To ensure that Passover did not start before spring, the tradition in ancient Israel held that the first day of Nisan would not start until the barley is ripe, that being the test for the onset of spring.

Complying with the biblical mandate and working out a predictable (computed) calendar rather than one based on observation was quite a feat. It required adopting a lunisolar calendar, where the days are determined by the time it takes for the earth to make one complete rotation, months are determined by the lunation of the moon (the time it takes for it to make a complete revolution around the earth) and the years are determined by the time of the earth’s travel around the sun.

This whole business is complicated by the fact that the orbits of the earth and the moon are “eccentric,” to use the astronomer’s term, meaning they deviate from a perfect circle. So even though an “average” lunar month is 29.530589 days, lunations (complete turns) vary up to 15 hours from this mean, and it’s not very convenient to have months that aren’t an integral (whole number) of days. As a result, some months have 29 days, some 30, and some either 29 or 30, depending on where we are in a 19-year cycle. This cycle is known as the Metonic cycle, named for the Greek astronomer Meton of Athens.

A computer nerd, I have always been fascinated by the Jewish calendar, perfected about 1,650 years ago, and I’ve always wondered how such complex calculations could be accomplished in the pre-computer age.

I’ve recently discovered how, and it’s a fascinating story.

Famously, and for a long time, the Jewish calendar was not fixed. The new moon was declared by the Sanhedrin in Jerusalem, based on the observation of competent witnesses. Eventually, the Sanhedrin was prevented from meeting. During the presidency of Rabban Gamaliel IV (270–290) the Sanhedrin was forced to drop its name; its authoritative decisions were issued under the name of Beth HaMidrash. Shortly afterwards, persecution by an increasingly Christianized Rome prevented the Sanhedrin from being legally constituted.

In response to the increasing oppression, Hillel the Patriarch (Hillel II), in what was probably the last official act of the Sanhedrin, fixed the Jewish calendar “for all time” in 357/9 CE by publishing the detailed rules for fixing the calendar and for adding intercalary (leap) months. The modern Jewish calendar, being rule-based, does not rely on observations of the barley crop.

Shortly after Hillel’s decree, Theodosius I, Roman Emperor from 379 to 395, forbade the Sanhedrin to assemble in any form and declared smicha (ordination), the mechanism for appointing members of the Sanhedrin, illegal; despite a few attempts over the centuries, it has never been reconstituted.

We know that since the Muslim calendar is totally lunar, Ramadan can occur during any month of the solar year, and in any season. We also know that Pope Gregory had to reform the Christian calendar in 1582 by dropping 10 days and changing the rules for determining a leap year.

Hillel’s rules are known as Sod Ha-Ibbur, the secret of the pregnant (i.e., intercalary) months.

The fact that they are called sod, or secret, is testament to their complexity. Other complex determinations, such as the formula for manufacturing a costly perfume derived from persimmons in the area near the Dead Sea are also referred to as sod.

Just how accurate is the Jewish calendar? Despite the approximations necessary to provide a fixed calendar, the accumulated discrepancy from current astronomical values amounts to only one day in 14,000 years, according to H. S. Schloss in the journal Computers & Mathematics with Applications (June 2000).

That is just astonishing accuracy.

So how were the Rabbis able to solve all the complexities?

A video about the Lego Antikythera Mechanism more than hints at the answer. After watching this short video, it became clear to me in a flash how they did it.

They did use computers, just not electronic ones.

They must have figured out the calendar using mechanical gears, based on a device similar to the Antikythera mechanism (pronounced ANT-i-ki-THEER-ə), an ancient mechanical computer designed to calculate astronomical positions.

Two thousand years ago, a Greek mechanic set out to build a machine that would model the workings of the known Universe. The result was a complex clockwork mechanism that displayed the motions of the sun, moon and planets on precisely marked dials. By turning a handle, the creator could watch his tiny celestial bodies trace their paths through the sky.

The mechanic’s name is now lost, and the ship carrying the instrument sank. But his machine, dubbed the Antikythera mechanism, has been found, and it is by far the most technologically sophisticated artifact that survives from antiquity according to Jo Marchant in NatureNews (Nov. 2010).

The Antikythera mechanism was recovered in 1900–01 from the Antikythera wreck, by divers off Point Glyphadia on the Greek island of Antikythera, but decades would pass before its complexity and significance were understood.

Based on carbon dating, domestic utensils and objects from the ship, as well as coins and amphoras recovered from the wreck, the Antikythera mechanism is believed to have been built around the first century BCE, about 350 years before Hillel II fixed the Jewish calendar.

In 1974, Yale University professor Derek de Solla Price argued convincingly that the mechanism was a calendar computer. From gear settings and inscriptions on the mechanism’s faces, he concluded that the mechanism was made about 87 BCE and lost at sea only a few years afterwards.

In other words, the Rabbis had the motive (the biblical mandate and the dissolution of the Sanhedrin) and with the Antikythera Mechanism (or something like it), they had the means.

But here’s the really fascinating part. Scientists who dug into the astronomical theories encoded in this supposedly quintessentially Greek device (based on its markings) have concluded that they are not Greek at all, but Babylonian, and so the concepts (not the physical device) are much older.

This finding forced historians to rethink a crucial period in the development of astronomy.

For Jews, however, it is no surprise at all that astronomy was highly developed in Babylon, which in its heyday boasted the largest Jewish community in the world.

The famous Babylonian Talmudist, Samuel (Shmuel) bar Abba of Nehardia was a second century first-generation Babylonian Amora who also studied extensively in Israel. A renaissance man, he was not only head of the great Academy of the Yeshiva at Nehardea, but also a noted physician and an astronomer. Samuel was known as “Yarhina’ah” or “Yarhinai” (the month-maker) because of his ability to independently determine the beginning of the month from calculations.

Samuel taught calendric science to his colleagues and pupils. His astronomical studies of the revolutions of the moon enabled him to predict the beginning of the month (Rosh Chodesh) from calculations, a determination that was still then officially made in Israel by the Sanhedrin, and not in Babylon. Samuel reached his conclusion, I can assure you, from calculations, without connecting via the Internet to a sky-pointing telescope in Jerusalem. Samuel was quoted as saying, “The paths of heaven are as clear to me as the pathways of Nehardea” (B. Talmud Ber. 58b).

Not only that, but Samuel claimed to be able to obviate the necessity of celebrating Yom Tov Sheni (double holy days) in the diaspora (B. Talmud R. H. 20b; comp. Rashi ad loc.), as the certainty of his calculations, he felt, removed the doubt that formerly arose when word was spread by messenger or by bonfires. Tradition (minhag avotenu beyadenu), however, ended up trumping science (B. Talmud Beitza 4b).

“R. Zera said … now that we are well acquainted with the fixing of the new moon, why do we observe two days? — Because they sent [word] from there [Land of Israel]: Give heed to the customs of your ancestors which have come down to you; for it might happen that the government might issue a decree and it will cause confusion [in ritual].”

Though rebuffed in that attempt, Samuel computed a 60-year Hebrew calendar and subsequently sent it to R. Johanan, the head of the Palestinian academies, as proof of his knowledge (B. Hul. 95b).

While we haven’t found an astronomical instrument with Samuel’s fingerprints or DNA, it’s now clear that Talmudic statements relating to intercalating the solar and lunar calendars must be taken at face value. Jews from early Talmudic days knew how to determine the calendar based on rules, and likely used early mechanical calendars, like the Antikythera mechanism in order to do so.

David E. Y. Sarna has contributed on Jewish topics to The Washington Post, The Jerusalem Post, Tablet Magazine and Jewish Voice and Opinion. A writer and technologist, he is the author of six books, more than 120 published articles and four patents. He blogs at