Eruvin 14 – The Value of Pi

In a discussion of various shapes and strengths that beam over alley might take, the Mishnah (page 13b) makes this statement:

[IF IT WAS] ROUND IT IS LOOKED UPON AS THOUGH IT WERE SQUARE.

This is a legal fiction to allow approximations. A straw beam is thought of as metal, a curved beam is thought of as straight. All well and good. But then, the Mishnah enters into dangerous territory:

WHATSOEVER HAS A CIRCUMFERENCE OF THREE HANDBREADTHS IS ONE HANDBREADTH IN DIAMETER

Thereby establishing a 3:1 proportion of circumference to diameter. But we know that that ratio is actually described by pi to 3.14159 (etc.) This was known in the rabbi’s world to at least a few digits. And in a unique question – the only time in the entire Talmud that this occurs – the rabbis ask:

Whence are these calculations deduced?

All this stems from a well-known mathematical problem in the Bible. In a description of items created by King Solomon there is this statement:

And he made a molten sea, ten cubits from one brim to the other; it was round all around, and its height was five cubits; and a line of thirty cubits measured the circle around it. (I Kings 7:23)

Again the ratio of 3:1 is explicitly stated – and impossible! A precisely 30 cubit line could not encompass a circle whose diameter was precisely 10 cubits. The line would have to be 31.415 (etc.) long!

This has caused grief for generations for those who view the bible as inerrant. For Jews, of course, there is a big difference between Torah (the 5 books of Moses) and Kituvin (the scriptural writings, including Kings). These measurements are seen as human, not divine. None-the-less, it has been a point discussed in rabbinic literature and no less than Maimonides is thought to be the first to declare pi an irrational number.

The internet is rife with theories and explanations. Some good detailed discussions are in Abarim Publications and Rationalist Judaism. (For more on numbers in the bible, see Jewish Virtual Library.)

Some have resorted to numerology to find hidden codes of pi. Others even declared that the value calculated humans is simply wrong.

The rabbis in our discussion on this page are not quite so apoplectic. Maybe the point, particularly in our Mishnah, is to give a handy approximation. Or maybe not everything has been considered:

But surely there was [the thickness of] its brim?

R. Papa replied: Of its brim, it is written in Scripture [that it was as thin as] the flower of a lily; for it is written: And it was a handbreadth thick, and the brim thereof was wrought like the brim of a cup, like the flower of a lily; it held two thousand baths.

But there was [still] a fraction at least? — When [the measurement of the circumference] was computed it was that of the inner circumference.

So the 3:1 ratio could have been approximate, or it could have been the difference between the inner and outer measurements! In any case it makes for some fascinating reading. For Jews this is not threatening.