Is there such a thing as Jewish mathematics? Some people protest that such a question is ridiculous. After all, isn’t science purportedly the pursuit of objective truth? Is there anything unique about a British biologist, a Chinese physicist or a Russian software engineer? Why should Jewish math be any different? About ninety years ago, the Lithuanian Education Minister posed a similar question to Rav Yosef Shlomo Kahaneman, the Ponevezer Rav.

“Why do you Jews want separate schools and Jewish teachers even for general subjects like mathematics?” he challenged him. “Is there such a thing as Jewish mathematics?”

Rav Yosef Shlomo answered the minister’s question with another question,

“From what number do you start your calculations?”

“From zero, of course!” replied the surprised minister.

“We, however, start our calculations from One!” replied the Ponevezer Rav.

Of course, his answer was philosophical. He meant that when a Jew learns anything, it must be taught from the standpoint of belief in *Hashem*’s unity and providence and not as a mass of facts sprawling in an existential vacuum.

**SOD HA’IBUR**

Another answer to our question is that Jewish mathematics does actually exist. This is the information *Hashem *entrusted to *Klal Yisroel *in order to calculate the new moon; as the Gemara (Shabbos 75a) comments on the verse, “For this is your wisdom and your understanding before the eyes of the nations” (Devarim 4:6) –

“What is the wisdom and understanding which is before the nations? Say that this is the calculation of *tekufos *(the times of the solstices and equinoxes) and *mazalos *(when each of the twelve constellations will ‘rise’ during the year)*.*”

In the same vein, the Abarbanel (Shemos 12:1) cites the Mechilta that discusses how Moshe had diffi culty understanding *kiddush hachodesh *until *Hashem *showed him the new moon and explained, “Through this they wanted to say that Moshe and Aharon did not achieve this wonderful wisdom through their knowledge and investigation, but the *sod ha’ibur *(foundation of knowing the birth of the new month) and the roots of its wisdom are *halacha *from Sinai and based on prophecy.”

Based on this axiom, the Abarbanel then mounts an attack against the Karaites who were trying to develop their own calendar based on purely human knowledge. In Hilchos Kiddush HaChodesh (11:1, 4), the Rambam, too, mentions that Chazal’s teachings are an integral part of his calculations, writing as follows:

“Because we said in these *halachos *that *beis din *calculated with great exactitude and knew whether the moon would be seen or not be seen, we know that whoever has a correct spirit, and whose heart yearns for matters of wisdom and to understand the foundations, will yearn to know those ways with which they calculated so that a person can know whether the moon will be seen on this night or not seen…

“And do not let these ways (of calculating the new moon that we will present here) be light in your eyes because we do not need them nowadays, since these ways are distant and deep ways, and this is the *sod ha’ibur *that the great sages knew and that one does not reveal to any person except to those who are *semuchim *(have *semicha*) and understanding.”

At the end of Hilchos Kiddush HaChodesh (19:16), the Rambam implies that his detailed mathematical calculations have the status of *talmud Torah*: “Thus we have explained the calculations of all the ways that one requires to know the [time of] seeing [the new moon], and of examining witnesses, so that all should be known and understood and no approach of the ways of Torah should be missing, and they should not wander to search after it in other books. Learn from the *sefer *of *Hashem *and read. Not one of them is missing!”

In addition, the Gemara (Shabbos 75a) writes, “Whoever knows how to calculate *tekufos *and *mazalos *and does not calculate, concerning him the verse (Yeshayahu 5:12) says, ‘And they did not look at the work of *Hashem *and the work of His hands they did not see’… From where (do we know) that it is a *mitzvah *on every person to calculate *tekufos *and *mazalos*? Because it says, ‘And you shall guard and do, for this is your wisdom and understanding before the eyes of the nations.’”

Because of these considerations, the Rambam’s calculations of *sod ha’ibur *are more extensively studied with each passing year and generating a whole new genre of Torah literature. This is Jewish math!

**JEWISH ANGLE TO NUMBERS**

Chazal say that *Hashem *looked in the Torah and created the world. Thus to the Jew, mathematics and science are not cold facts but facets of Heavenly wisdom; numbers down here are rooted in spiritual sources.

This is clearly indicated in the Tanchuma (Vayechi 15) which says, “‘All these are the *shevatim *(tribes) of Yisroel, twelve.’ But later (Bereishis 17) it writes, ‘He (Yishmoel) will give birth to twelve princes,’ (not using the word *shevatim*), because these (tribes of Yisroel) are *shevatim *(controlling forces) over the order of the world – twelve hours in the day, twelve hours in the night, and so the *mazalos *and the months and the twelve stones of the *ephod. *Therefore, these are the *shevatim *of Yisroel, twelve!”

Jews are also drawn to math because of its essential function in understanding many *sugyos *of the Gemara, such as *eiruvin*, *sukkah*, *terumos*, *ma’asros *and even mixtures of milk and meat. As Rav Moshe Chaim Luzatto (the Ramchal) writes, “Without it (certain sciences), a person can never reach correct, clear knowledge, and is like a craftsman who does not have his tools… For example, geometry, fractions and astronomy are needed for the *mitzvos *of leap-years, *kilayim*, *techumin *and suchlike.”

This inherent link between Torah and math led to a flourishing literature. The first Jewish math book was the Mishnas Midos, written in the time of the *Gaonim. *The Ibn Ezra was an expert mathematician. His Sefer HaMispar was the first European book to fully explain the Arab numeric system and is used universally nowadays. Until then, Europeans relied on the *alefbeis *or clumsy Roman numerals to get by. Rav Eliyahu Mizrachi (the Re’em), author of the famous commentary on Rashi and of other *seforim*, took out time to write a mathematical treatise, Meleches HaMispar.

Rav Avraham bar Chiya HaSephardi (the Raava”ch) wrote Chibbur HaMeshicha v’HaTashboret in 4876/1116 after noticing that “most of the sages of our generation in France are not expert in land surveying… because the necessity for this is great in measuring and dividing land between heirs and partners, so that no one can measure them and divide them accurately and justly, until he utilizes this wisdom.” He was the first person to systematize Hebrew terms for scientific concepts, such as *hikish *(compare), *golem *(object), *mutzak *(pyramid), *tzir *(center) and suchlike, and the first Jewish mathematician to discuss trigonometry in Hebrew and create Jewish terms for sine (*meitar*), cosine (*meitar she’eiris*) and tangent (*tzel omeid*).

In addition, he formulated the famous method for calculating the area of a circle which anyone who studies Gemara has seen cited by Tosfos a number of times. This is where Tosfos draws a diagram of a circle, built of concentric layers like an onion, being sliced through to its center, and two more diagrams where the layers are rearranged as a triangle and then as a square.

The greatest Jewish mathematician of all was Rav Levi ben Gershom (the Ralbag, born in 5048/1288 in France), grandson of the Ramban. He pioneered mathematical innovations that passed unnoticed in his time as his Hebrew works were incomprehensible to non-Jewish scholars.

Although the Ralbag rarely discusses math in his monumental commentary on the Tanach, he devotes attention to the famous verse where the *yam shel Shlomo *(the water container Shlomo built for the *Beis HaMikdash*) is described as being ten *amos *wide and 30 *amos *round. The famous problem is that, according to the radiuscircumference ratio of *pi *(3.14), the pool’s circumference should have been not 30 *amos *but 31.4 *amos*! The Gemara answers that the verse is giving an approximate measure.

The Ralbag gives another answer, based on an earlier verse stating that the walls of the container were a hands-breadth thick. Accordingly, he explains that the verse is telling us the smaller inner circumference of Shlomo’s container.

An even more ingenious answer is mentioned by the Vilna Gaon whose famous math book, Ayil MeShulash, was published posthumously in 5594/1834. The introduction to this *sefer *states that its purpose was to present the science of mathematics to Torah scholars, so that they need not have to resort to non-Jewish sources. The Gra also invented the Kramer Theorem, named after his surname, Kramer.

**A BRILLIANT OBSERVATION**

Drawing on an anonymous source, the Gra insists that the value of *pi *is, indeed, mentioned in the verse: instead of spelling the word *kav *(circumference) with the two letters *kav *and *vav *as it is pronounced, it adds an extra *hei *to the end of the word writing it as *kavah. *Take the *gematria *(numerical value) of the word as it is spelled (111), says the Gra, and divide it by the *gematria *of how the word is pronounced (106), and you will end up with the number 1.04716981321, etc. Then multiply this number by 3 (the diameter/circumference ratio of the verse) and you will come up with 3.1415… Keep going long enough you will reach the value of pi to the 10,000th decimal point.

Because of the importance of mathematics to so many fields of Torah learning, it has generally been an adjunct of Jewish education throughout the ages. For example, Rav Akiva Eiger writes, “Torah with *derech eretz *is good – to teach them the art of writing and calculation about an hour or two a day.”

However, how much time to devote to this and at what age to begin are determined by the *gedolim *of every generation, according to circumstances because like every other aspect of Jewish life, math, too, must be subjected to vigorous inspection. As Chazal (Bava Basra 78b) comment on the verse (Bamidbar 21:27), “Therefore the rulers say, come to *Cheshbon*” – “therefore the rulers over their inclination say, come make and let us calculate the calculation of the world, the loss of a *mitzvah *compared to its rewards and the reward of a sin against its loss.”

(Partial source: Chochmas HaTashbores: HaMatematike be’Aspaklaria HaYehudis by Rachel Rosenbaum, Mossad HaRav Kook, Jerusalem, 2003)