π: A Line of Thirty Cubits Going all Around
by tillerofthesoil
“’Whatever has a circumference of three handspans is one handspan in diameter’ (Mishnah). Is it so? Rabbi Yoḥanan said: The verse says: And he made the cast metal sea ten cubits from edge to edge, circular all around, and five cubits in height and a line of thirty cubits going all around (1 Kings 7:23). But surely there was its brim? Rav Papa replied: Its brim, [was merely] like the petal of a lily, as is written: its thickness was a handspan, and its rim like the design of a cup’s rim, blossom and lily, two thousand bats did it hold (ibid., 26) [the bat is a relatively large unit of liquid measure, but the precise quantity is uncertain]. But there was [still] a fraction at least? When it was reckoned it was that of the inner circumference [this is the only instance where a doubt is raised in the Talmud in connection with a mathematical statement, proving that the Rabbis were well aware of a more exact ratio between the diameter and circumference; the ratio of 1:3 was accepted by the Rabbis simply as a workable number for religious purposes, cf. BT Eruvin 76a; Sukkah 7b]” (BT Eruvin 14a).
“Rabbi Abba opened, saying, ‘He made the sea (1 Kings 7:23), and it is written: Standing upon twelve oxen: three facing north, three facing west, three facing south, and three facing east, with the sea set upon them above… (ibid., 25), and it is written: the twelve oxen underneath the sea (ibid., 44). Standing upon twelve oxen—certainly so! For this sea [Shekhinah] is arrayed by twelve in two worlds: by twelve above, appointed by chariots; by twelve below, twelve tribes” (Zohar 1:241a).
And he made the cast metal sea ten cubits from edge to edge, circular all around, and five cubits in height, (וקוה (וְקָו and a line, of thirty cubits going all around (1 Kings 7:23).
“This is essentially a metal pool, to be used by the priests to bathe hands and feet before they perform their ritual duties. The Hebrew uses the rather grand term יָם (yam), sea, perhaps to suggest a cosmic correspondence between the structure of the temple and of creation at large. The translation reads, with many Hebrew manuscripts, קָו (qav), line, for the incomprehensible Masoretic קוה (qavoh)” (Robert Alter, Ancient Israel, p. 638).
Dividing the gematriyya (numerical value) of each spelling—קָו (qav) and קוה (qavoh)—(i.e., the disparity of the text as it is קְרֵי וּכְּתִיב [qeri u-ketiv], vocalized and written) and multiplying by three yields the first five digits of the irrational number π: 111:106 x 3 = 3.1415094…. 111:106 is the third best ratio for numbers under 10,000, and it is the best that can be generated from a single letter difference, see Matityahu Hacohen Munk, “Three Geometric Problems in the Bible and the Talmud” Sinai, Vol. 51, 1962, p. 218–227, cf. Tsaban and Garber, “On the Rabbinical Approximation of π,” Historia Mathematica 25 (1998), 75–84, article no. HM972185; Adam Simon Levine, Jewish Geometery, National Havurah Committee Summer Institute 2011.
“You should know that the ratio of the circle’s diameter to its circumference is not known and it is never possible to express it precisely. This is not due to a lack in our knowledge, as the fools think; but it is in its nature that it is unknown, and there is no way [to know it], but it is known approximately. The geometers [such as Archimedes] have already written essays about this, that is, to know the ratio of the diameter to the circumference approximately, and the proofs for this. This approximation which is accepted by the educated people is the ratio of one to three and one seventh. Every circle whose diameter is one handspan, has in its circumference three and one seventh handspan approximately. As it will never be perceived but approximately, the [rabbis] took the nearest integer and said that every circle whose circumference is three fists is one fist wide, and they contented themselves with this for their needs in the religious law” (Maimonides, Perush al Mishnah Eruvin 1:5).
Maimonides uses Archimedes’ approximation for π of 22/7 which is approximately 3.14 and is an elegant and useful approximation.
יִקָּווּ הַמַּיִם מִתַּחַת הַשָּׁמַיִם אֶל מָקוֹם אֶחָד וְתֵרָאֶה הַיַּבָּשָׁה (Yiqqavu ha-mayim mitaḥat ha-shamayim el-maqom eḥad ve-tera’eh ha-yabasha), Let the waters under heaven be gathered together to one place, and let the dry land appear (Genesis 1:9)—קָו מִדָה (qav middah), measuring line (Ba’al ha-Turim, cf. Bereshit Rabbah 5:1; Zohar 1:15a, 18a; 2:199a, 233a–b, 258a; ZḤ 56d–58d [QhM], 86a; TZ 18, 37b).
The carpenter stretches out his rule; he marks it out with a קָו (qav), line; he fashions it with planes, and he marks it out with the compass, and makes it after the figure of a man, according to the beauty of a man; that it may remain in the house (Isaiah 44:13).
And קוה (קָו) הַמִּדָּה (qav ha-middah), the measuring line, shall yet go straight over to the hill Gareb, and shall then turn to Goah (Jeremiah 31:38).
Therefore thus says YHWH; I have returned to Jerusalem with compassions: My house shall be built in it, says YHWH of armies, (וקוה (וְקָו (ve-qav), and a line, shall be stretched forth upon Jerusalem (Zechariah 1:16).
For He looks to the ends of the earth, beneath all the heavens He sees, to gauge the heft of the wind, מִשְׁקָל וּמַיִם תִּכֵּן בְּמִדָּה (mishqal u-mayim tiken be-middah), and to weigh water with a measure, when He fixes a limit for rain and a way for the thunderhead (Job 28:24–26)—קָו מִדָה (qav middah), measuring line.
“Qav ha-Middah: Length and Breadth. Qav ha-Middah: Depth and Height. Qav ha-Middah: Circle and Square” (Zohar Ḥadash 57a).
“What is the difference between them [ס (samekh) and ם (mem)]? Well, when enclosed and hidden within itself, within the supernal point above, She [Binah] assumes the form of the letter ס (samekh), enclosed within and hidden, ascending above. And when She returns and sits רְבִיעָא (revi’a), crouching, over children below to suckle them, She assumes the form of the letter ם (mem) [which is] רְבִיעָא (revi’a), quadrilateral, enclosed in four directions of the world” (Zohar 2:127a, cf. BT Shabbat 104a).
“First mystery: י (yod) primordial point, standing upon nine pedestals supporting it. These are stationed in four directions of the world, just as End of Thought, final point, is stationed in four directions of the world—except that this is male and that is female. This one stands without a body; and when clothed in a garment, it stands upon nine pedestals in mystery of the letter ם (mem), not circular. Although the letter ס (samekh) is circular, assuming a circle, in the mystery of letters engraved in dazzling points above they are square, while below circular…” (Zohar 2:180a, cf. BT Eruvin 76b; Bahir §§114–116).
For some people, Pi Day is an occasion to marvel at circles, long revered as symbols of perfection, reincarnation and the cycles of nature. But it is the domestication of infinity that we really should be celebrating. Mathematically, pi is less a child of geometry than an early ancestor of calculus, the branch of mathematics, devised in the 17th century, that deals with anything that curves, moves or changes continuously.
As a ratio, pi has been around since Babylonian times, but it was the Greek geometer Archimedes, some 2,300 years ago, who first showed how to rigorously estimate the value of pi. Among mathematicians of his time, the concept of infinity was taboo; Aristotle had tried to banish it for being too paradoxical and logically treacherous. In Archimedes’s hands, however, infinity became a mathematical workhorse.