It's March 14! Pi Day! To celebrate, let's eat some pie and learn about how this mysterious and magical number representing the ratio of a circle's circumference to its diameter appears in Jewish tradition! Our exploration of the Jewish encounter with π begins with Solomon, who while constructing the Jerusalem Temple inadvertently records the first approximation of π in Jewish tradition.
(כג) וַיַּ֥עַשׂ אֶת־הַיָּ֖ם מוּצָ֑ק עֶ֣שֶׂר בָּ֠אַמָּ֠ה מִשְּׂפָת֨וֹ עַד־שְׂפָת֜וֹ עָגֹ֣ל ׀ סָבִ֗יב וְחָמֵ֤שׁ בָּֽאַמָּה֙ קוֹמָת֔וֹ (וקוה) [וְקָו֙] שְׁלֹשִׁ֣ים בָּאַמָּ֔ה יָסֹ֥ב אֹת֖וֹ סָבִֽיב׃
(23) Then he made the tank of cast metal, 10 cubits across from brim to brim, completely round; it was 5 cubits high, and it measured 30 cubits in circumference.
While Solomon's temple was pretty good, there's an issue here: π ≠ 3. While you could chalk this up to the fact that Solomon lived over 3000 years ago, the Babylonians and Egyptians both knew that π was greater than three long prior with Babylonian mathematicians defining the number at 3.125 and Egyptians at 3.160. While this may not have troubled Solomon, it troubled later Jewish thinkers. By the Talmudic period, we see rabbis debating this known inconsistency within the biblical text.
אָמַר רַבִּי יוֹחָנָן: סוּכָּה הָעֲשׂוּיָה כְּכִבְשָׁן, אִם יֵשׁ בְּהֶקֵּיפָהּ כְּדֵי לֵישֵׁב בָּהּ עֶשְׂרִים וְאַרְבָּעָה בְּנֵי אָדָם — כְּשֵׁרָה, וְאִם לָאו — פְּסוּלָה. כְּמַאן — כְּרַבִּי, דְּאָמַר: כׇּל סוּכָּה שֶׁאֵין בָּהּ אַרְבַּע אַמּוֹת עַל אַרְבַּע אַמּוֹת פְּסוּלָה. מִכְּדִי, גַּבְרָא בְּאַמְּתָא יָתֵיב, כֹּל שֶׁיֵּשׁ בְּהֶקֵּיפוֹ שְׁלֹשָׁה טְפָחִים יֵשׁ בּוֹ רוֹחַב טֶפַח, בִּתְרֵיסַר סַגִּי? ... מִכְּדֵי כַּמָּה מְרוּבָּע יוֹתֵר עַל הָעִיגּוּל — רְבִיעַ, בְּשִׁיתְּסַר סַגִּי? הָנֵי מִילֵּי בְּעִיגּוּל דְּנָפֵיק מִגּוֹ רִיבּוּעָא, אֲבָל רִיבּוּעָא דְּנָפֵיק מִגּוֹ עִגּוּלָא — בָּעֵינַן טְפֵי, מִשּׁוּם מוּרְשָׁא דְקַרְנָתָא. מִכְּדֵי כׇּל אַמְּתָא בְּרִיבּוּעָא אַמְּתָא וּתְרֵי חוּמְשֵׁי בַּאֲלַכְסוֹנָא, בְּשִׁיבְסַר נְכֵי חוּמְשָׁא סַגִּיא? לָא דָּק. אֵימוֹר דְּאָמְרִינַן לָא דָּק פּוּרְתָּא, טוּבָא מִי אָמְרִינַן לָא דָּק?
§ Rabbi Yoḥanan said: With regard to a sukka that is shaped like a furnace and is completely round, if its circumference has sufficient space for twenty-four people to sit in it, it is fit, and if not, it is unfit. The Gemara asks: In accordance with whose opinion did Rabbi Yoḥanan rule that the sukka must be so expansive? The Gemara answers: It is undoubtedly in accordance with the opinion of Rabbi Yehuda HaNasi, who said: Any sukka that does not have an area of four cubits by four cubits is unfit... The Gemara asks further: Now, by how much is the perimeter of a square inscribing a circle greater than the circumference of that circle? It is greater by one-quarter of the perimeter of the square. If that is the case, a circle with a circumference of sixteen cubits is sufficient. Why, then, does Rabbi Yoḥanan require a circumference of twenty-four cubits? The Gemara answers: This statement with regard to the ratio of the perimeter of a square to the circumference of a circle applies to a circle inscribed in a square, but in the case of a square circumscribed by a circle, the circle requires a greater circumference due to the projection of the corners of the square. In order to ensure that a square whose sides are four cubits each fits neatly into a circle, the circumference of the circle must be greater than sixteen cubits. The Gemara calculates precisely how much greater the circumference must be in order to circumscribe the four-by-four-cubit square. Now, in every square whose sides each measure one cubit its diagonal measures one and two-fifths cubits, and in a circle that circumscribes a square, the diagonal of the square is the diameter of the circle. In this case, the circumscribed square measures four by four cubits; therefore, the diagonal of the square, which is the diameter of the circle, measure five and three-fifths cubits. Since the Gemara calculates the circumference of the circle as three times its diameter, a circular sukka with a circumference of seventeen cubits minus one-fifth of a cubit should be sufficient. The Gemara answers: Rabbi Yoḥanan was not precise and rounded the dimensions of the circular sukka to a number higher than the absolute minimum. The Gemara wonders: Say that we say that the Sage was not precise when the difference between the number cited and the precise number is slight; however, when the difference is great, do we say the Sage was not precise? After all, Rabbi Yoḥanan stated that the minimum measure is twenty-four cubits, a difference of more than seven cubits.
While the discussions in the Talmud continue, it is clear that Chazal were wrestling with the question of the error between π and the approximations used for practical halachic purposes to represent it. This is particularly clear in the Rambam who intuits a deep truth about π, albeit without proof.
יש לך לדעת כי יחוס אלכסון העגולה אל המסבב אותה בלי ידוע וא"א לדבר בו לעולם באמת וחסרון זו ההשגה אינה מאתנו כמחשבת הכת הנקראת גהלי"ה אבל הוא בטבעי זה הדבר בלי ידוע ואין במציאותו שיושג אבל (ידוע) [יודע] זה בקרוב וכבר חברו חכמי התשבורת לזה חבורים לידע יחוס האלכסון אל המסבב בקרוב ודרך המופת בזה הקרוב אשר עליו סומכין חכמי החכמות הלמודיות הוא יחוס האחד לשלשה ושביעית וכל עגולה שיהיה באלכסון שלה אמה יהיה בהיקפה ג' אמות ושביעית בקרוב ולפי שזה לא יושג לעולם אלא בקרוב לקחו הם בחשבון הגדול ואמרו כל שיש בהיקפו ג"ט יש בו רחב טפח וסמכו ע"ז במה שהוצרכו אליו מן המדידה בתורה:
The ratio of the diameter to the circumference of a circle is not known and will never be known precisely. This is not due to a lack on our part (as some fools think), but this number [π] cannot be known because of its nature, and it is not in our ability to ever know it precisely. But it may be approximated ...to three and one-seventh. So any circle with a diameter of one has a circumference of approximately three and one-seventh. But because this ratio is not precise and is only an approximation, they [the rabbis of the Mishnah and Talmud] used a more general value and said that any circle with a circumference of three has a diameter of one, and they used this value in all their Torah calculations.
Thus, we see that for the Rambam, the text from 1 Kings is not being referenced to say that π has a value equal to 3, but rather that this text is proof that we are halachically sanctioned to use this approximation for practice. The Vilna Gaon devised an ingenious reading of this text that enabled him to extract a better approximation of π from the 1 Kings text using כתיב קרי.
He notes that the written text of 1 Kings states קוה but the Masoretic pointing indicates a reading of קו. Employing gematria, one sees that קוה has a value of 111 and קו is 106. Taking the ratio of these two values 111/106=1.047 and multiplying it by the given value of π≈3 we find that π≈333/106=3.14151! This gives a secret approximation hidden within the Tanach!
Furthermore, this is the second-best approximation for π with a denominator less than 30,000! The approximation π≈333/106 corresponds to [3; 7,15] on π's continued fraction representation. The next approximation [3; 7, 15, 1, 292] is 10399/333102.
(Maybe Solomon was smart?)
Before wrapping up this journey into Jewish mathematical genius, let's take a look at medieval Catalonia. Into this melting pot of Christian, Muslim, and Jewish culture lived the philosophical and mathematical polymath Rabbi Abraham bar Hiyya ha-Nasi (RAHN). In his treatise חיבור המשיחה והתשבורת ("Treatise on Measurement and Calculation"), Rahn synthesized Classical and Islamic mathematics, bringing ideas such as the quadratic equation to Europe for the first time. However, beyond transferring these traditions, the Rahn developed a proof for the area of a circle more intuitive than previous ones. This proof, presented below, was ultimately amended into the Talmud by the Tosafists.
Talmud - How much larger is [the area of] a square than a circle - one fourth
Tosafot - This cannot be proven from the fact that a square of 3x3 has a perimeter of 12 surrounding it, and a circle with a diameter of 3 has a circumference of (approximately) 9 surrounding it as [according to the Talmud's calculation] a circle with a circumference of 3 has a diameter of 1. (Our hypothesis is that any shape's area is equal to its perimeter or circumference. If this is so, then the Talmud's conjecture is seemingly proved, as the square's perimeter is 1/4 larger than the circle's circumference and they have the same distance across (3).]
[Why is this not a proof -] As proof cannot be brought from the perimeter/circumference [of a shape] to its area. [For example, according to the hypothesis that the perimeter/circumference of a shape is equal to its area,] you would think that a circle with a diameter of 4 would have an area of only as much as a square of 3x3, since their perimeter/circumference is equal (the square is 3 per side, thus its perimeter is 3x4=12, and the circle's circumference is 12 as we multiply its diameter, 4, by 3. However, [in truth,] when you divide a square of 3x3 into a grid of 3 strips horizontally and 3 strips vertically, you will only find 9 square cubits (see diagram). But a circle with a diameter of 4 must have 12 strips of one square cubit each. For a square of 4x4, when it is divided into 4 strips of a cubit's width along its height, and also along its width, will result in 16 strips of one square cubit. And if [as the Talmud conjectures] a square is only one-fourth larger than a circle, it comes out that you are saying that a circle [with a corresponding diameter of 4] has an area of 12 square cubits. (Thus we find that a circle and square can have the same perimeter/circumference, 12, and still have different areas - the square has 9 but the circle has 12.) Rather, [we must say] that there is no proof from the circumference/perimeter [to the area] at all.
Sources:
Belaga. “On The Rabbinical Exegesis of an Enhanced Biblical Value of π.” 2011: https://philarchive.org/archive/BELOTR
Epstein, Sheldon, and Murray Hochberg. “A Talmudic Approach to the Area of a Circle.” Mathematics Magazine 50, no. 4 (1977): 210–210. https://doi.org/10.2307/2690221.