Pi

Ohalot (12:6) | Yisrael Bankier | 15 years ago

In a number of Mishnayot discuss the relationship between the circumference of a circle and its diameter is mentioned. In mathematics, this relationship is defined by an irrational constant (3.14159…). The Greek letter Pi (π) has been popularised due to its adoption to represent that value.

Having a mathematic and engineering background, the adoption of three in the Mishnah and Gemara had been at first surprising. Matters are further complicated as the Gemara (Eiruvin 14a) brings a pasuk from Melachim (I 7:23) in support of this position:

And [Shlomo HaMelech] made the pool of cast metal ten amot from rim to rim circular all around and five amot was its height and a line of thirty amot would encircle it all around.

The Rambam (Eiruvin 1:5) notes that Pi is irrational and cannot be known. He adds that in his time ²²/7 (3.1429…) was the well accepted approximation. The Rambam explains that since the number is irrational the Chachamim took a large figure, the nearest integer, as a satisfactory approximation.

In this manner, the Rosh (Teshovot 2:19) answered the Rashba’s question of why it was necessary to bring a pasuk in support of a matter than can be solved scientifically. He explains that the source was required to teach that an approximation of three is sufficient for legal matters. In fact the Aruch HaShulchan (Orach Chaim 363:22) rules that this is indeed a divine decree that three is used for all legal questions.

Conversely, the Tashbetz (1:165) explains that the Chachamim did have accurate approximations and used them for practical applications. Nonetheless they followed the principle that “One should always teach his student in a brief manner” (Pesachim 3b) and the approximation of three achieved this end.

A person shared with me an insight in the name of the Vilna Gaon on this issue. The source brought for the relationship being three is from the Yam Shel Shlomo; a ‘sea’ that was one of the utensils constructed by Shlomo HaMelech in the Beit Ha’Mikdash. It is described in two places in Tanach: in Melachim (1 7:23) and Divrei Ha’Yamim (2 4:2). In Melachim however there appears to be a ‘spelling mistake’ - the word for line (קָו) is written קוה. In other words the line that is taken and multiplied by three, as stated in the pasuk, is distorted. Taking the numerical value (gematria) of קוה (111) and dividing by the numerical value of קו (106) and then multiply by three as stated, the result is 3.141509... far more accurate than ²²/7 that I grew up with!⁴¹

To be honest I do not remember the conclusion and I have since heard different endings from the above exercise. Nonetheless, it appears to support everything stated above. The Navi is conveying that the value 3 is suitable for all intents and purpose – whether for teaching alone or even practical applications. Nonetheless we find, albeit encoded, a far more accurate value that aligns so perfectly with the meaning of the text. It appears to be recognition that three, or any other such value adopted for practical purpose, is just an approximation.⁴²

⁴¹ Note that engineers use approximation to four or five decimal place.

⁴² As an aside the Guinness record for reciting the digits of Pi from memory is 67,890 taking him 24 hours and 4 minutes. Baruch sh’natan Torah l’amo Yisrael!

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