Could King Solomon Calculate Pi?

In honor of Pi Day: From the Bible to the Rambam, a Jewish history of this irrational number.

Elon Gilad
Elon Gilad
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The letter pi, and its number, is represented in this mosaic outside the maths building at the Technische Universität Berlin.Credit: Wikimedia Commons
Elon Gilad
Elon Gilad

March 14th is Pi Day, a celebration of the ratio of the circumference of a circle to its diameter, not of round pastries with fruit fillings, though those are definitely encouraged.

Pi is irrational. That doesn’t mean that it acts erratically and inexplicitly, but that it cannot be expressed as a ratio of two numbers. Usually pi is shortened to roughly 3.14, but the endless stream of numbers occurring after the decimal point has no particular order.

The present record for calculating digits of pi after the decimal point is 2.7 trillion, which is useless, but gets nerds all excited. For most practical matters 3.14 is more than adequate. But that was still out of the reach of the ancient Hebrews, who had only the crudest mathematical skills.

King Solomon, depicted on right handing down his famous judgment: Did he know the value of pi?Credit: Wikimedia Commons

Biblical Hebrew didn’t have words for fractions with the exception of half, third and quarter. It did however approximate pi in 1 Kings 7:23, when describing that Solomon constructed in the Great Temple of Jerusalem “a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.” (1 Kings 7:23).

In the Mishnah (Ohalot 12:6) we are told that a square surrounding a circle has a circumference a quarter longer than it. Based on my high-school geometry and algebra, that comes out to an estimated pi of 3.2. Not too shabby, though elsewhere in the Mishnah, 3 is given (Eruvin 1:5), including in a rabbinic discussion on the round sukkahs (Sukkah 7b).

At this time, the Egyptians, Greeks and Persians all have far better estimations but we caught up eventually.

Maimonides notices the irrationality

A somewhat better estimation is given in Mishnat ha-Middot, a Hebrew text of unknown date and author that was written sometime in the first millennium: "3 and a seventh," which gives 3.142857.

At this point Jewish egos may be feeling injured but don’t worry - we’ve got the great Jewish sage Maimonides.

In a commentary on the Mishnah passage giving the value of 3, Maimonides gives what may be the earliest account of pi’s irrational nature. “The ratio of the diameter of the circle and its circumference are unknown and can never be discussed with accuracy," he wrote in the 12th century. "This is not a lack of knowledge on our part, as the idiots think, but rather is that by its nature this thing is unknown, and by virtue of its reality cannot be known, though it can be estimated.”

A moment of gematriya and prescience

But wait. Could there be some mysterious biblical code giving a remarkably accurate estimation of pi, after all? Maybe, according to gematriya - an Assyro-Babylonian numerology system that Jews adopted and adapted, which assigns numerical value to a word or phrase.

It has been suggested that "molten sea" biblical passage actually encodes a way to get pretty close to pi.

The word for "diameter" in the text is kaveh, which is a strange spelling of the Hebrew word for line kav – it ends with the letter hei. Intriguingly, readers of this biblical passage are instructed in the margins (by their teachers) not to read that last letter, hei.

Now, the gematriya value of kaveh is 111, while kav's numerical value is 106. Divide kaveh by kav and you get 1.04716981.

Now: take that number and multiply it by 3 and you’ve got 3.14150943 - off by only 0.0000832.

Coincidence? Probably.